Effect of surface waviness andaspect ratio on heat transfer

inside a wavy enclosureProdip Kumar Das

Department of Mechanical Engineering University of AlbertaEdmonton Alberta Canada

Shohel Mahmud and Syeda Humaira TasnimDepartment of Mechanical Engineering University of Waterloo

Waterloo Ontario Canada

AKM Sadrul IslamDepartment of Mechanical Engineering

Bangladesh University of Engineering and Technology (BUET)Dhaka Bangladesh

Keywords Convection Finite volume methods Heat transfer Surface texture

Abstract A numerical simulation has been carried out to investigate the buoyancy induced macrowand heat transfer characteristics inside a wavy walled enclosure The enclosure consists of twoparallel wavy and two straight walls The top and the bottom walls are wavy and kept isothermalTwo straight-vertical sidewalls are considered adiabatic Governing equations are discretized usingthe control volume based regnite-volume method with collocated variable arrangement Simulationwas carried out for a range of surface waviness ratios l = 0 00-0 25 aspect ratios A = 025-05and Rayleigh numbers Ra = 100-107 for a macruid having Prandtl number equal to 10 Results arepresented in the form of local and global Nusselt number distributions streamlines and isothermallines for different values of surface waviness and aspect ratios For a special case of l = 0 andA= 10 the average Nusselt number distribution is compared with available reference The resultssuggest that natural convection heat transfer is changed considerably when surface wavinesschanges and also depends on the aspect ratio of the domain In addition to the heat transfer resultsthe heat transfer irreversibility in terms of Bejan number (Be) was measured For a set of selectedvalues of the parameters (l A and Ra) a contour of the Bejan number is presented at the end ofthis paper

The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at

httpwwwemeraldinsightcomresearchregister httpwwwemeraldinsightcom0961-5539htm

NomenclatureA = aspect ratio (= d L)Be = Bejan number (see equation (24))Br = Brinkman number (= Ec poundPr)g = gravity vector m s 2 2

Ec = Eckert numberGr = Grashof number (= r 2gbDTd 3 m 2)

h = heat transfer coefregcient W m 2 2 K2 1

k = thermal conductivity W m 2 1 K2 1

L = horizontal length of the wave mNu = Nusselt numbern = unit normal vectorPr = Prandtl number (= n k)

We would like to thank Patrick Kirchen and Shahnawaz Hossain Molla (Department ofMechanical Engineering University of Alberta Canada) for different kinds of support they gaveduring this research

Effect of surfacewaviness and

aspect ratio

1097

Received January 2003Revised May 2003

Accepted June 2003

International Journal of NumericalMethods for Heat amp Fluid Flow

Vol 13 No 8 2003pp 1097-1122

q MCB UP Limited0961-5539

DOI 10110809615530310501975

IntroductionFree or forced convection heat transfer inside geometries of regular shape(rectangular channel circular pipe etc) has many signiregcant engineeringapplications for example double-wall thermal insulation underground cablesystems solar-collectors electric machinery cooling systems ofmicro-electronic devices natural circulation in the atmosphere the moltencore of the earth etc A considerable number of published articles are availablethat deal with macrow characteristics heat transfer macrow and heat transferinstability transition to turbulence design aspects etc Curious readers aresuggested to read the paper by Yang (1987) for a comprehensive reference Onthe other hand studies dealing with convection problems inside more complexgeometries have been rather limited The phrase complex geometry coversdifferent types of geometric conreggurations namely L-shaped cavities(Mahmud 2002) trapezoidal cavities (Peric 1993) arc shaped enclosures(Chen and Cheng 2002) wavy cavities (Mahmud et al 2002c) triangularcavities (Poulikakos and Bejan 1983) etc Flow and heat transfer analysesinside such geometry is a comparatively difregcult task either experimentally oranalytically (or numerically) Therefore less effort has been given to analyzemacrow and thermal characteristics inside complex shaped geometry for the pastseveral decades The primary focus is to analyze the macrow and thermalcharacteristics inside a special class of complex shaped geometry ie ageometry with wavy walls that follows a well-deregned mathematical function(sine or cosine) Enclosures with wavy walls have always been less attractive toresearchers due to their complex macruid dynamic behavior as well as theirgeometrical complexity Sobey (1980a) published numerical results with anexperimental veriregcation (Sobey 1980b) of macrow patterns inside a furrowedchannel A furrowed channel is a special type of channel with wavy wall that

P = dimensionless pressureRa = Rayleigh number (= Gr poundPr)T = temperature KU = dimensionless velocity component in

the x-directionV = dimensionless velocity component in

the y-directionx = horizontal coordinateX = dimensionless horizontal coordinatey = vertical coordinateY = dimensionless vertical coordinate

Greek symbolsa = amplitude of the top and the bottom

walls mb = thermal expansion coefregcient K 2 1

d = inter wall spacing between the top andbottom walls m

l = surface waviness ratio (= a L)k = thermal diffusivity m2 s2 1

n = kinematic viscosity m2 s2 1

r = density of the macruid kg m 2 3

F = irreversibility distribution ratioO = dimensionless temperature

differenceH = dimensionless temperature

Subscriptav = average valueyen = at ambient conditionC = average value based on cold wallH = average value based on hot wallL = local valueW = value at the wall0 = reference valueD = difference between the two values

HFF138

1098

follows the cosine function Later Sobey extended his work by calculatingoscillatory macrow pattern inside the same geometry (Sobey 1982) Furtherextension is done by Sobey (1983) by calculating the macrow separationcharacteristics inside the wavy walled channel The macruid dynamics of theseparation phenomenon inside the wavy walled channels is still incompletethus motivating some researchers to extend the work of Sobey (1983) forexample Leneweit and Auerbach (1999) and Mahmud et al (2002ab) Theforegoing discussions that have dealt with problems of wavy walled channelsare very much restricted to macruid macrow only A limited number of forcedconvection heat transfer results are reported by Russ and Beer (1997ab) andWang and Vanka (1995) whereas considerable attention has been devotedtowards the analyses of laminar steady heat transfer and macruid macrow insidecomplex shaped domains over the past two decades However there is limitedliterature dealing with natural convection inside wavy wall domainFabrication of devices with wavy walls depend on the parameters likeamplitude wavelength phase angle inter-wall spacing etc Each of theseparameters signiregcantly affects the hydrodynamic and thermal behavior of themacruid inside it

In our earlier work (Mahmud et al 2002c) we have shown the effect ofsurface waviness on natural convection heat transfer and macruid macrow inside avertical wavy walled enclosure for a range of wave ratio (0 00 l 0 4) andaspect ratio (1 0 A 2 0) We reported a decrease in average heat transferwith the increase of surface waviness For the horizontal in-phase wavy walledenclosure (Das and Mahmud 2003) hydrodynamic and thermal behaviors ofthe macruid inside the enclosure have been investigated for a regxed aspect ratio Inthat investigation we explored the change of true periodic nature of the localNusselt number distribution with the increase of amplitude-wavelength ratiobut we have not seen any signiregcant inmacruence on the average heat transfer ratedue to the surface waviness One of our interesting regndings was the increase ofaverage heat transfer with the increase of surface waviness from zero to acertain value Recently Adjlout et al (2002) discussed natural convection in aninclined cavity with hot wavy wall and cold macrat wall They showed thedecrease in average heat transfer with the surface waviness when comparedwith macrat wall cavity However in their report they showed result for the grid of42 pound42 with an error level of 164 percent whereas heat transfer increased byonly a small amount when compared to their error level

Several research works have been conducted for convection heat transferinside enclosure or channel with wavy walls Among them Kumar (2000)presented the parametric results of macrow and thermal regelds inside a verticalwavy enclosure with porous media Kumar (2000) concluded that thesurface temperature was very sensitive to drifts in the surface undulationsphase of the wavy surface and number of the wave Hadjadj and Kyal(1999) numerically examined the effect of sinusoidal protuberances on heat

Effect of surfacewaviness and

aspect ratio

1099

transfer and macruid macrow inside an annular space using a non-orthogonalcoordinate transformation In their study it was reported that both thelocal and average heat transfer increase with the increase of protuberanceamplitude and Rayleigh number and decreasing Prandtl number Yao(1983) presented the near wall characteristics of macrow and thermal regeld of avertical wavy wall Saidi et al (1987) presented numerical and experimentalresults of macrow over and heat transfer from a sinusoidal cavity Theyreported that the total heat exchange between the wavy wall of the cavityand the macrowing macruid was reduced by the presence of vortex The vortexplays the role of a thermal screen which creates a large region of uniformtemperature in the bottom of the cavity Asako and Faghri (1987) andMahmud et al (2001) gave a regnite-volume prediction of heat transfer andmacruid macrow characteristics inside a wavy walled duct and tube respectivelyLage and Bejan (1990) documented heat transfer results near a periodically(timewise and spatial) stretching wall Hossain and Rees (1999) Moulic andYao (1989) and Rees and Pop (1995) presented similar solutions for naturalconvection macrow near wavy surface at different boundary conditions Aydinet al (1999) Elsherbiny (1996) Hamady et al (1989) Ozoe et al (1975) andSundstrom and Kimura (1996) presented results of heat transfercharacteristics inside the rectangular enclosures at different aspect ratiosand orientations without surface waviness

In this study we have presented numerical results based on theregnite-volume analysis for laminar natural convection inside a cavity whichconsists of two wavy surfaces with different amplitude-wavelength ratiosand aspect ratios Rate of heat transfer in terms of local and global Nusseltnumbers are calculated for different Rayleigh numbers Flow and thermalregelds are analyzed by parametric presentation of streamlines and isothermallines

Mathematical modelingIn this section we present the mathematical formulation of the problemconsidered in terms of the continuity momentum and energy equations andthe pertinent boundary conditions for a wavy domain containing two wavyand two macrat surfaces We begin by considering two-dimensional laminarnatural convection of an incompressible and Newtonian macruid in a cavityas shown in Figure 1 Distance between the two straight walls is L twowavy walls is d and a is the amplitude of the wavy walls Modeling themacrow as ordfBoussinesq-incompressibleordm to take into account the couplingbetween the energy and momentum equations we regard the density r asconstant everywhere except in the buoyancy term of momentum equations(equation (3)) Correspondingly the non-dimensional equations governingthe conservation of mass momentum and energy in the cavity of Figure 1 areas follows

HFF138

1100

shy U

shy X+

shy V

shy Y= 0 (1)

shy U

shy t+ U

shy U

shy X+ V

shy U

shy Y= 2

shy P

shy X+

Pr

Ra

rshy 2U

shy X 2+

shy 2U

shy Y 2

sup3 acute (2)

shy V

shy t+ U

shy V

shy X+ V

shy V

shy Y= 2

shy P

shy Y+ H +

Pr

Ra

rshy 2V

shy X 2+

shy 2V

shy Y 2

sup3 acute (3)

shy H

shy t+ U

shy H

shy X+ V

shy H

shy Y=

1PrRa

p shy 2H

shy X 2+

shy 2H

shy Y 2

sup3 acute(4)

Equations (1)-(4) are put into their dimensionless forms by scaling differentlengths with average inter wall spacing between the top and the bottom walls(d) velocity components with reference velocity (V0) which is equal to(gbDTd)12 pressure with rV 2

0 time with dV0 The dimensionless temperaturecan be deregned as H = (T 2 T yen ) DT where DT is equal to (TH2 TC)

Boundary and initial conditionsFigure 1 shows the computational grid structure and the geometry underconsideration in the present investigation along with the coordinate frameworkand different boundary conditions The surface shape of the wavy wallsfollows the proregle given in equations (5) and (6) The hot and cold walltemperatures are TH and TC respectively Initial macruid temperature inside thecavity is taken as T yen The initial temperature of the adiabatic walls is set equalto the initial macruid temperature (T yen ) inside the cavity The gravity accelerationg is acted downwards No slip boundary condition is applied for velocitycomponents at each wall For the present investigation boundary conditionsare summarized by the following equations

Figure 1Schematic representation

of the computationalgeometry under

consideration withboundary conditions

Effect of surfacewaviness and

aspect ratio

1101

0 X 1 and

0 Y l[1 2 cos (2pX ) ] U = V = 0 H = 1(5)

0 X 1 and

A Y A + l[1 2 cos (2pX )] U = V = 0 H = 0(6)

X = 0 and 0 Y A U = V = 0shy H

shy n= 0 (7)

X = 1 and 0 Y A U = V = 0shy H

shy n= 0 (8)

where X (= XA) and Y (= YA) are modireged dimensionless horizontal andvertical lengths

Numerical methodologyTo conduct a numerical simulation for the thermomacruid dynamics regelds weused the technique similar to that Hortmann et al (1990) based on theregnite-volume method as described in Ferziger and Peric (1996) Theregnite-volume method starts from the conservation equation in integral form asZ

S

rf v Ccedil n dS =

Z

S

Ggradf Ccedil n dS +

Z

V

qf dV (9)

where f is any variable G is the diffusivity for the quantity f and qf is thesource or sink of f

The solution domain is subdivided into a regnite number of contiguousquadrilateral control volumes (CV) The CVs are deregned by coordinates of theirvertices which are assumed to be connected by straight lines This simple formis possible due to the fact that the equations contain no curvature terms andonly the projections of the CV faces onto the Cartesian coordinates surfaces arerequired in the course of discretization as demonstrated below All thedependent variables solved for and all macruid properties are stored in the CVcenter (collocated arrangement) A suitable spatial distribution of dependentvariables is assumed and the conservation equations (1)-(4) are applied to eachCV leading to a system of non-linear algebraic equations The main steps ofdiscretization procedure to calculate convection and diffusion macruxes and sourceterms are outlined below

The mass macrux through the cell face ordfeordm (Figure 2) is evaluated as

Ccedilme =

Z

Se

rV dS lt (rV)eS1e = re USx1 + VSy

1

iexcl cent (10)

HFF138

1102

where S1e is the surface vector representing the area of the cell face (x = const)and Sx

1e and Sy1e denotes its Cartesian components These are given in terms of

the CV vertex coordinates as follows

Sx1e = (yn 2 ys) and S y

1e = 2 (xn 2 xs) (11)

The mean cell face velocity components Ue and Ve are obtained byinterpolating neighborsrsquo nodal values in a way which ensures the stability ofgrid scheme The convection macrux of any variable f can be expressed as

FCe =

Z

Se

rfV dS lt (rfV )eS1e = Ccedilmefe (12)

The diffusion macrux of f is calculated as

FDe = 2

Z

Se

Gf grad f dS lt 2 (Gf grad f)eS1e (13)

By expressing the gradient of f at the cell face center ordfeordm which is taken here torepresent the mean value over the whole cell face through the derivatives inx and Z directions (Figure 2) and by discretizing these derivatives with CDSthe following expression results

FDe lt Gje

S1e Ccedil PE[(fE 2 fP )(S1e Ccedil S1e) + (fn 2 fs)e(S1e Ccedil S2e)] (14)

where PE is the vector representing the distance from P to E directed towardsE S2e is the surface vector orthogonal to PE and directed outwards positive Z

Figure 2A typical 2D-control

volume and the notationfor a Cartesian grid

Effect of surfacewaviness and

aspect ratio

1103

coordinate (Figure 2) representing the area in the surface Z = 0 bounded byP and E Its x and y components are

Sx2e = 2 ( yE 2 yP ) and Sy

2e = (xE 2 xP ) (15)

The volumetric source term is integrated by simply multiplying the speciregcsource at the control volume center P (which is assumed to represent the meanvalue over the whole cell) with the cell volume ie

Qqf =

Z

V

qf dV lt (qf)PDV (16)

The pressure term in the momentum equations are treated as body force andmay be regarded as pressure sources for the Cartesian velocity componentsThey are evaluated as

Q pui

= 2Z

S

pi i dS = 2Z

V

(grad pi i) dV

lt 2 [( pe 2 pw)S1P + ( pn 2 ps)S2p] Ccedil ii (17)

where the surface vectors S1p and S2p represent the area of the CV crosssection at x = 0 and Z = 0 respectively Since CVrsquos are bounded by straightlines these two vectors can be expressed through the CV surface vectors egS1P = (S1e + S1w) 2 Terms in the momentum equations not featuring inequation (9) are discretized using the same approach and added to the sourceterm After summing up all cell face macruxes and sources the discretizedtransport equation reduces to the following algebraic equation

APfP +nb

XAnbfnb = Qf (18)

where the coefregcients Anb contains the convective and diffusive macruxcontributions and Qf represents the source term The system of equations issolved by using Stonersquos SIP solver

Discretized momentum equations lead to an algebraic equation system forvelocity components U and V where pressure temperature and macruid propertiesare taken from the previous iteration except the regrst iteration where initialconditions are applied These linear equation systems are solved iteratively(inner iteration) to obtain an improved estimate of velocity The improvedvelocity regeld is then used to estimate new mass macruxes which satisfy thecontinuity equation Pressure-correction equation is then solved using the samelinear equation solver and to the same tolerance Energy equation is then

HFF138

1104

solved in the same manner to obtain better estimate of new solution Then theabove procedure is repeated for a new time step For time marching weselected Three Time Level Method which is a fully implicit scheme of secondorder accurate Iteration is continued until the difference between the twoconsecutive regeld values of variables is less than or equal to 10 2 5

AccuracyIn the present investigation regve combinations (20 pound10 40 pound20 80 pound40160 pound80 and 320 pound160) of control volumes are used to test the effect of gridsize on the accuracy of the predicted results Figure 3 shows the distribution ofaverage Nusselt numbers of the hot wall as a function of grid sizes for fourdifferent Rayleigh numbers It is clear from the reggure that at lower Rayleighnumber average Nusselt number is almost independent of grid sizes At higherRayleigh numbers the two higher grid sizes (160 pound 80 and 320 pound160) givealmost the same result It is well known that the high mesh reregnement alwaysprovides better result in the regnite-volume method The main disadvantage intaking higher mesh number is the increase in calculation time which can bereduced by using a higher speed of Pentium processor Our goal was to getbetter results Thus throughout this study the results are presented for 320 pound160 CVsrsquo for better accuracy Predicted results of average Nusselt numbers fora square cavity (A = 1 0 l = 0) with the same boundary conditions arecompared with the results given by Bejan (1984) and Ozisik (1985)Comparisons are shown in Figure 4 where average Nusselt number isplotted as a function of Rayleigh number Predicted results show a very goodagreement with the result of Ozisic at lower Rayleigh number but differ fromBejanrsquos prediction At higher Rayleigh number present prediction shows betteragreement with Bejan compared to Ozisicrsquos results

Figure 3Variation of averageNusselt number as a

function of number of CVfor A = 0 25 and

l = 0 05

Effect of surfacewaviness and

aspect ratio

1105

Results and discussionsFlow and thermal regeldThe macrow pattern inside the wavy enclosure and the temperature proregle arepresented in terms of streamlines and isothermal lines in Figure 5(a)-(d) forA = 0 25 l = 0 10 at four selected Rayleigh numbers At low Rayleighnumber when convection current inside the cavity is comparatively weak macruidstream near the hot wall tends to move towards the centerline or crest of thecavity were two streams from the opposite direction mix and rise upwardsTwo counter-rotating vortices with small cores are observed on the macrow regeldwere bi-cellular macrow pattern divides the cavity into two symmetric parts withrespect to the centerline (X = 0 5) of the cavity Due to the uniformtemperature gradient at low Rayleigh number (= 103) circulation inside thecavity is very weak Convection is less prominent at this Rayleigh number andheat transfer is mainly dominated by conduction The basic difference betweenthe isotherms in a wavy cavity with other geometry is their shape Isothermallines are nearly parallel to each other and follow the geometry of the wavysurfaces A further increase of Rayleigh numbers increases the circulationstrength inside the cavity Ra = 104 is also characterized by the bi-cellular macrowpattern but thermal regeld is completely different compared to Ra = 103 Hereuniform temperature proregle is changed and three high gradient spot isobserved due to rapid circulation of macruid inside the cavity Isotherms turn up(convective distortion) towards the cold wall due to the strong inmacruence of theconvection current At Ra = 105 multi-cellular macrow appears with four vorticesDue to comparatively high buoyant effect another vertical stream of the macruidappears near the adiabatic walls A periodic swirling of the isothermal linesappear at Ra = 105 due to the appearance of multi-cellular macrow Two highgradient spots appear near the bottom wall where isothermal lines areconcentrated Heat transfer rate is higher in magnitude at that spots

Figure 4Validation of averageNusselt number forA = 0 25 and l = 0 0with other referencework

HFF138

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Further increase of Rayleigh number increases the strength of the vorticeswhere four-cell multi-cellular macrow pattern turns into two cell bi-cellular patternThis bi-cellular macrow pattern is qualitatively different from the macrow pattern atlow Rayleigh number Periodic swirling nature of isothermal lines disappearsdue to the occurrence of reverse transition in macrow regeld Hot spots almostoccupy the bottom wall of the cavity

Transition of cell modeThe problem of the onset of hydrodynamic and thermal instabilities in thehorizontal layers of macruid heated from bellow is well suited to illustrate themany facets mathematical and physical of the general theory ofhydrodynamic stability Specially many aspects of the stability (orinstability) of the macruid-layer trapped between the two rigid walls (bottom hotand top cold) have been treated by many researchers over the last 100 years (forexample see Chandrasekhar 1961) The transition of cell mode of the macruidinside two rigid walls (in Rayleigh-Benard convection) is also a well establishedproblem However a mathematical analysis to predict the transition parameter

Figure 5Streamlines (top)

and isothermal lines(bottom) for A = 0 25

and l = 0 10

Effect of surfacewaviness and

aspect ratio

1107

(for example the critical Rayleigh number) and the physical mechanism is anextremely challenging task in the case of wavy walled enclosure under theRayleigh-Benard convection like situation due to the presence of two additionalparameters (A and l) Instead of performing a linear stability analysis wemodireged our numerical algorithm for post processing (a programme written inFORTRAN77 and connected to the software Tecplot 8) to predict the transitionbetween unicellular macrow to bicellular or multicellular macrow Initially forconstant l and A the macrow solver is executed for a range of Rayleigh numberskeeping a considerably large gap between the two consecutive Rayleighnumbers (50-100 depending on the value l) Once we get a rough picture of thetransition of cell mode we re-execute the macrow solver for a small range ofRayleigh numbers keeping a comparatively small gap between the twoconsecutive Rayleigh numbers (1 5) For example when l = 0 01 and A =0 25 the magnitude of the starting Rayleigh number for the macrow solver is1700 Note that the regrst critical Rayleigh number for the Rayleigh-Benardconvection with rigid walls is 1708 (Chandrasekhar 1961) Figure 6(a)-(h)shows the cell pattern for l = 0 01 and A = 0 25 for eight selected values ofRayleigh number A symmetric bicellular macrow pattern characterizes the macrowreglled inside the cavity until the value of Rayleigh number reaches 1905 Twoadditional cells appear near the adiabatic walls (Figure 6(d)) at which the

Figure 6Variation of cell patterninside a cavity withl = 0 01 and A = 0 25for different Rayleighnumber

HFF138

1108

Rayleigh number changes the bicellular macrow into a four-cell multicellular macrowThese cells grow in size with the increasing Rayleigh number but growth rateof the cells becomes slower when Ra approaches 104 A second transition ofthe cell mode is observed near Ra = 106 The overall cell modes and theirtransitions are shown in Figure 7 Note that a non-uniform scale is selected forRayleigh numbers in the abscissa of the Figure 7 for convenience Howeverthis scenario is completely different for l = 0 125 as shown in Figures 8 and 9The symmetric bicellular macrow changes into an asymmetric tricellular macrowaround Ra = 13 995 This tricellular macrow pattern remains unchanged for asmall interval of Rayleigh number (Ra = 13 995 plusmn 14 877) Then a secondtransition makes it to a four-cell multicellular macrow pattern A reverse transitionoccurs around Ra = 106 The overall cell modes and their transitions areshown in Figure 9 for this case

Heat transferHeat transfer is calculated in terms of local (NuL) and average (Nuav) Nusseltnumbers using the following equations

NuL =dk

hL =dk

k

DT

dT

dn

sup3 acute

w

raquo frac14=

dH

dn

sup3 acute

w

(19)

Nuav =1

S

Z S

0

NuL ds (20a)

S =

Z 1

0

1 +dY

dX

sup3 acute2s

dX =EllipticE 2p

2 l2

pplusmn sup2

p (20b)

Figure 7Transition of the cell

mode as a functionof Rayleigh number forl = 0 01 and A = 0 25

Effect of surfacewaviness and

aspect ratio

1109

where the special function ordfEllipticEordm is the elliptic integral of second kind(Abramowitz and Stegun 1965) Local Nusselt number distribution ispresented in Figures 10 and 11 Average Nusselt number distribution ispresented in Figures 12-14 Detail discussions on heat transfer is presented inthe following two paragraphs

Local Nusselt number distribution along the hot wavy wall is shown inFigure 10 for a constant Rayleigh number and four selected surface waviness

Figure 8Variation of cell patterninside a cavity withl = 0 125 and A = 0 25for different Rayleighnumber

HFF138

1110

(l = 0 0 005 0075 01) for an aspect ratio A = 0 25 At Ra = 104 heattransfer rate shows true periodic nature with respect to the axial distance alongthe length of the bottom wall for l = 0 0 where three peaks represent not onlythe three spots of highest temperature gradient but also characterize threepairs of cell generated due to the macruid motion inside the cavity However forl = 0 05 0075 01 local Nusselt number distribution changed signiregcantlydue to low temperature gradient at the midpoint of the cavity In that case heat

Figure 9Transition of the cell

mode as a functionof Rayleigh number for

l = 0 125 and A = 0 25

Figure 10Change of the local

Nusselt number alongthe length of the bottom

wave

Effect of surfacewaviness and

aspect ratio

1111

transfer rate increases up to the X = 0 15 and falls along the wavy wallgradually up to the midpoint (X = 0 5) of the cavity This nature is similarfor other surface waviness with a slight variation in magnitudes Completelydifferent picture is observed at Ra = 105 For l = 0 0 three peaks againrepresent the three spots of highest temperature gradient with a highertemperature gradient at the mid point of the wave

Figure 12Nusselt number as afunction of Rayleighnumber for differentsurface waviness fortwo regxed aspect ratio(a) A = 0 25 and(b) A = 0 50

Figure 11Change of the localNusselt number alongthe length of the bottomwave

HFF138

1112

Figure 11 depicts the extent to which the local Nusselt number distribution isaffected by changes in the aspect ratio We observed that at Ra = 104 twopeaks move towards the center of the cavity with the increase of aspect ratioSimilar pattern was observed at Ra = 106 but the magnitude of the two peak ishigher It is also noticed that heat transfer rate is faster near the straight walland decrease rapidly with the increase of X Whatever be the values of aspectratio heat transfer rate is always minimum at the mid point of the bottom wall

Local Nusselt numbers are integrated to calculate the average value of theNusselt number according to equation (20) Average Nusselt number is plottedas a function of Rayleigh number in Figure 12 for two regxed aspect ratiosA = 0 25 and 050 for different surface waviness At lower Rayleigh number

Figure 13Nusselt number as afunction of Rayleighnumber for different

aspect ratio for two regxedsurface waviness(a) l = 0 05 and

(b) l = 0 10

Figure 14Nusselt number as a

function of surfacewaviness for regxedRayleigh number

(Ra = 5 pound104)Two lines correspond

to two different aspectratio of the enclosure asindicated in the legend

Effect of surfacewaviness and

aspect ratio

1113

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

HFF138

1114

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

follows the cosine function Later Sobey extended his work by calculatingoscillatory macrow pattern inside the same geometry (Sobey 1982) Furtherextension is done by Sobey (1983) by calculating the macrow separationcharacteristics inside the wavy walled channel The macruid dynamics of theseparation phenomenon inside the wavy walled channels is still incompletethus motivating some researchers to extend the work of Sobey (1983) forexample Leneweit and Auerbach (1999) and Mahmud et al (2002ab) Theforegoing discussions that have dealt with problems of wavy walled channelsare very much restricted to macruid macrow only A limited number of forcedconvection heat transfer results are reported by Russ and Beer (1997ab) andWang and Vanka (1995) whereas considerable attention has been devotedtowards the analyses of laminar steady heat transfer and macruid macrow insidecomplex shaped domains over the past two decades However there is limitedliterature dealing with natural convection inside wavy wall domainFabrication of devices with wavy walls depend on the parameters likeamplitude wavelength phase angle inter-wall spacing etc Each of theseparameters signiregcantly affects the hydrodynamic and thermal behavior of themacruid inside it

In our earlier work (Mahmud et al 2002c) we have shown the effect ofsurface waviness on natural convection heat transfer and macruid macrow inside avertical wavy walled enclosure for a range of wave ratio (0 00 l 0 4) andaspect ratio (1 0 A 2 0) We reported a decrease in average heat transferwith the increase of surface waviness For the horizontal in-phase wavy walledenclosure (Das and Mahmud 2003) hydrodynamic and thermal behaviors ofthe macruid inside the enclosure have been investigated for a regxed aspect ratio Inthat investigation we explored the change of true periodic nature of the localNusselt number distribution with the increase of amplitude-wavelength ratiobut we have not seen any signiregcant inmacruence on the average heat transfer ratedue to the surface waviness One of our interesting regndings was the increase ofaverage heat transfer with the increase of surface waviness from zero to acertain value Recently Adjlout et al (2002) discussed natural convection in aninclined cavity with hot wavy wall and cold macrat wall They showed thedecrease in average heat transfer with the surface waviness when comparedwith macrat wall cavity However in their report they showed result for the grid of42 pound42 with an error level of 164 percent whereas heat transfer increased byonly a small amount when compared to their error level

Several research works have been conducted for convection heat transferinside enclosure or channel with wavy walls Among them Kumar (2000)presented the parametric results of macrow and thermal regelds inside a verticalwavy enclosure with porous media Kumar (2000) concluded that thesurface temperature was very sensitive to drifts in the surface undulationsphase of the wavy surface and number of the wave Hadjadj and Kyal(1999) numerically examined the effect of sinusoidal protuberances on heat

Effect of surfacewaviness and

aspect ratio

1099

transfer and macruid macrow inside an annular space using a non-orthogonalcoordinate transformation In their study it was reported that both thelocal and average heat transfer increase with the increase of protuberanceamplitude and Rayleigh number and decreasing Prandtl number Yao(1983) presented the near wall characteristics of macrow and thermal regeld of avertical wavy wall Saidi et al (1987) presented numerical and experimentalresults of macrow over and heat transfer from a sinusoidal cavity Theyreported that the total heat exchange between the wavy wall of the cavityand the macrowing macruid was reduced by the presence of vortex The vortexplays the role of a thermal screen which creates a large region of uniformtemperature in the bottom of the cavity Asako and Faghri (1987) andMahmud et al (2001) gave a regnite-volume prediction of heat transfer andmacruid macrow characteristics inside a wavy walled duct and tube respectivelyLage and Bejan (1990) documented heat transfer results near a periodically(timewise and spatial) stretching wall Hossain and Rees (1999) Moulic andYao (1989) and Rees and Pop (1995) presented similar solutions for naturalconvection macrow near wavy surface at different boundary conditions Aydinet al (1999) Elsherbiny (1996) Hamady et al (1989) Ozoe et al (1975) andSundstrom and Kimura (1996) presented results of heat transfercharacteristics inside the rectangular enclosures at different aspect ratiosand orientations without surface waviness

In this study we have presented numerical results based on theregnite-volume analysis for laminar natural convection inside a cavity whichconsists of two wavy surfaces with different amplitude-wavelength ratiosand aspect ratios Rate of heat transfer in terms of local and global Nusseltnumbers are calculated for different Rayleigh numbers Flow and thermalregelds are analyzed by parametric presentation of streamlines and isothermallines

Mathematical modelingIn this section we present the mathematical formulation of the problemconsidered in terms of the continuity momentum and energy equations andthe pertinent boundary conditions for a wavy domain containing two wavyand two macrat surfaces We begin by considering two-dimensional laminarnatural convection of an incompressible and Newtonian macruid in a cavityas shown in Figure 1 Distance between the two straight walls is L twowavy walls is d and a is the amplitude of the wavy walls Modeling themacrow as ordfBoussinesq-incompressibleordm to take into account the couplingbetween the energy and momentum equations we regard the density r asconstant everywhere except in the buoyancy term of momentum equations(equation (3)) Correspondingly the non-dimensional equations governingthe conservation of mass momentum and energy in the cavity of Figure 1 areas follows

HFF138

1100

shy U

shy X+

shy V

shy Y= 0 (1)

shy U

shy t+ U

shy U

shy X+ V

shy U

shy Y= 2

shy P

shy X+

Pr

Ra

rshy 2U

shy X 2+

shy 2U

shy Y 2

sup3 acute (2)

shy V

shy t+ U

shy V

shy X+ V

shy V

shy Y= 2

shy P

shy Y+ H +

Pr

Ra

rshy 2V

shy X 2+

shy 2V

shy Y 2

sup3 acute (3)

shy H

shy t+ U

shy H

shy X+ V

shy H

shy Y=

1PrRa

p shy 2H

shy X 2+

shy 2H

shy Y 2

sup3 acute(4)

Equations (1)-(4) are put into their dimensionless forms by scaling differentlengths with average inter wall spacing between the top and the bottom walls(d) velocity components with reference velocity (V0) which is equal to(gbDTd)12 pressure with rV 2

0 time with dV0 The dimensionless temperaturecan be deregned as H = (T 2 T yen ) DT where DT is equal to (TH2 TC)

Boundary and initial conditionsFigure 1 shows the computational grid structure and the geometry underconsideration in the present investigation along with the coordinate frameworkand different boundary conditions The surface shape of the wavy wallsfollows the proregle given in equations (5) and (6) The hot and cold walltemperatures are TH and TC respectively Initial macruid temperature inside thecavity is taken as T yen The initial temperature of the adiabatic walls is set equalto the initial macruid temperature (T yen ) inside the cavity The gravity accelerationg is acted downwards No slip boundary condition is applied for velocitycomponents at each wall For the present investigation boundary conditionsare summarized by the following equations

Figure 1Schematic representation

of the computationalgeometry under

consideration withboundary conditions

Effect of surfacewaviness and

aspect ratio

1101

0 X 1 and

0 Y l[1 2 cos (2pX ) ] U = V = 0 H = 1(5)

0 X 1 and

A Y A + l[1 2 cos (2pX )] U = V = 0 H = 0(6)

X = 0 and 0 Y A U = V = 0shy H

shy n= 0 (7)

X = 1 and 0 Y A U = V = 0shy H

shy n= 0 (8)

where X (= XA) and Y (= YA) are modireged dimensionless horizontal andvertical lengths

Numerical methodologyTo conduct a numerical simulation for the thermomacruid dynamics regelds weused the technique similar to that Hortmann et al (1990) based on theregnite-volume method as described in Ferziger and Peric (1996) Theregnite-volume method starts from the conservation equation in integral form asZ

S

rf v Ccedil n dS =

Z

S

Ggradf Ccedil n dS +

Z

V

qf dV (9)

where f is any variable G is the diffusivity for the quantity f and qf is thesource or sink of f

The solution domain is subdivided into a regnite number of contiguousquadrilateral control volumes (CV) The CVs are deregned by coordinates of theirvertices which are assumed to be connected by straight lines This simple formis possible due to the fact that the equations contain no curvature terms andonly the projections of the CV faces onto the Cartesian coordinates surfaces arerequired in the course of discretization as demonstrated below All thedependent variables solved for and all macruid properties are stored in the CVcenter (collocated arrangement) A suitable spatial distribution of dependentvariables is assumed and the conservation equations (1)-(4) are applied to eachCV leading to a system of non-linear algebraic equations The main steps ofdiscretization procedure to calculate convection and diffusion macruxes and sourceterms are outlined below

The mass macrux through the cell face ordfeordm (Figure 2) is evaluated as

Ccedilme =

Z

Se

rV dS lt (rV)eS1e = re USx1 + VSy

1

iexcl cent (10)

HFF138

1102

where S1e is the surface vector representing the area of the cell face (x = const)and Sx

1e and Sy1e denotes its Cartesian components These are given in terms of

the CV vertex coordinates as follows

Sx1e = (yn 2 ys) and S y

1e = 2 (xn 2 xs) (11)

The mean cell face velocity components Ue and Ve are obtained byinterpolating neighborsrsquo nodal values in a way which ensures the stability ofgrid scheme The convection macrux of any variable f can be expressed as

FCe =

Z

Se

rfV dS lt (rfV )eS1e = Ccedilmefe (12)

The diffusion macrux of f is calculated as

FDe = 2

Z

Se

Gf grad f dS lt 2 (Gf grad f)eS1e (13)

By expressing the gradient of f at the cell face center ordfeordm which is taken here torepresent the mean value over the whole cell face through the derivatives inx and Z directions (Figure 2) and by discretizing these derivatives with CDSthe following expression results

FDe lt Gje

S1e Ccedil PE[(fE 2 fP )(S1e Ccedil S1e) + (fn 2 fs)e(S1e Ccedil S2e)] (14)

where PE is the vector representing the distance from P to E directed towardsE S2e is the surface vector orthogonal to PE and directed outwards positive Z

Figure 2A typical 2D-control

volume and the notationfor a Cartesian grid

Effect of surfacewaviness and

aspect ratio

1103

coordinate (Figure 2) representing the area in the surface Z = 0 bounded byP and E Its x and y components are

Sx2e = 2 ( yE 2 yP ) and Sy

2e = (xE 2 xP ) (15)

The volumetric source term is integrated by simply multiplying the speciregcsource at the control volume center P (which is assumed to represent the meanvalue over the whole cell) with the cell volume ie

Qqf =

Z

V

qf dV lt (qf)PDV (16)

The pressure term in the momentum equations are treated as body force andmay be regarded as pressure sources for the Cartesian velocity componentsThey are evaluated as

Q pui

= 2Z

S

pi i dS = 2Z

V

(grad pi i) dV

lt 2 [( pe 2 pw)S1P + ( pn 2 ps)S2p] Ccedil ii (17)

where the surface vectors S1p and S2p represent the area of the CV crosssection at x = 0 and Z = 0 respectively Since CVrsquos are bounded by straightlines these two vectors can be expressed through the CV surface vectors egS1P = (S1e + S1w) 2 Terms in the momentum equations not featuring inequation (9) are discretized using the same approach and added to the sourceterm After summing up all cell face macruxes and sources the discretizedtransport equation reduces to the following algebraic equation

APfP +nb

XAnbfnb = Qf (18)

where the coefregcients Anb contains the convective and diffusive macruxcontributions and Qf represents the source term The system of equations issolved by using Stonersquos SIP solver

Discretized momentum equations lead to an algebraic equation system forvelocity components U and V where pressure temperature and macruid propertiesare taken from the previous iteration except the regrst iteration where initialconditions are applied These linear equation systems are solved iteratively(inner iteration) to obtain an improved estimate of velocity The improvedvelocity regeld is then used to estimate new mass macruxes which satisfy thecontinuity equation Pressure-correction equation is then solved using the samelinear equation solver and to the same tolerance Energy equation is then

HFF138

1104

solved in the same manner to obtain better estimate of new solution Then theabove procedure is repeated for a new time step For time marching weselected Three Time Level Method which is a fully implicit scheme of secondorder accurate Iteration is continued until the difference between the twoconsecutive regeld values of variables is less than or equal to 10 2 5

AccuracyIn the present investigation regve combinations (20 pound10 40 pound20 80 pound40160 pound80 and 320 pound160) of control volumes are used to test the effect of gridsize on the accuracy of the predicted results Figure 3 shows the distribution ofaverage Nusselt numbers of the hot wall as a function of grid sizes for fourdifferent Rayleigh numbers It is clear from the reggure that at lower Rayleighnumber average Nusselt number is almost independent of grid sizes At higherRayleigh numbers the two higher grid sizes (160 pound 80 and 320 pound160) givealmost the same result It is well known that the high mesh reregnement alwaysprovides better result in the regnite-volume method The main disadvantage intaking higher mesh number is the increase in calculation time which can bereduced by using a higher speed of Pentium processor Our goal was to getbetter results Thus throughout this study the results are presented for 320 pound160 CVsrsquo for better accuracy Predicted results of average Nusselt numbers fora square cavity (A = 1 0 l = 0) with the same boundary conditions arecompared with the results given by Bejan (1984) and Ozisik (1985)Comparisons are shown in Figure 4 where average Nusselt number isplotted as a function of Rayleigh number Predicted results show a very goodagreement with the result of Ozisic at lower Rayleigh number but differ fromBejanrsquos prediction At higher Rayleigh number present prediction shows betteragreement with Bejan compared to Ozisicrsquos results

Figure 3Variation of averageNusselt number as a

function of number of CVfor A = 0 25 and

l = 0 05

Effect of surfacewaviness and

aspect ratio

1105

Results and discussionsFlow and thermal regeldThe macrow pattern inside the wavy enclosure and the temperature proregle arepresented in terms of streamlines and isothermal lines in Figure 5(a)-(d) forA = 0 25 l = 0 10 at four selected Rayleigh numbers At low Rayleighnumber when convection current inside the cavity is comparatively weak macruidstream near the hot wall tends to move towards the centerline or crest of thecavity were two streams from the opposite direction mix and rise upwardsTwo counter-rotating vortices with small cores are observed on the macrow regeldwere bi-cellular macrow pattern divides the cavity into two symmetric parts withrespect to the centerline (X = 0 5) of the cavity Due to the uniformtemperature gradient at low Rayleigh number (= 103) circulation inside thecavity is very weak Convection is less prominent at this Rayleigh number andheat transfer is mainly dominated by conduction The basic difference betweenthe isotherms in a wavy cavity with other geometry is their shape Isothermallines are nearly parallel to each other and follow the geometry of the wavysurfaces A further increase of Rayleigh numbers increases the circulationstrength inside the cavity Ra = 104 is also characterized by the bi-cellular macrowpattern but thermal regeld is completely different compared to Ra = 103 Hereuniform temperature proregle is changed and three high gradient spot isobserved due to rapid circulation of macruid inside the cavity Isotherms turn up(convective distortion) towards the cold wall due to the strong inmacruence of theconvection current At Ra = 105 multi-cellular macrow appears with four vorticesDue to comparatively high buoyant effect another vertical stream of the macruidappears near the adiabatic walls A periodic swirling of the isothermal linesappear at Ra = 105 due to the appearance of multi-cellular macrow Two highgradient spots appear near the bottom wall where isothermal lines areconcentrated Heat transfer rate is higher in magnitude at that spots

Figure 4Validation of averageNusselt number forA = 0 25 and l = 0 0with other referencework

HFF138

1106

Further increase of Rayleigh number increases the strength of the vorticeswhere four-cell multi-cellular macrow pattern turns into two cell bi-cellular patternThis bi-cellular macrow pattern is qualitatively different from the macrow pattern atlow Rayleigh number Periodic swirling nature of isothermal lines disappearsdue to the occurrence of reverse transition in macrow regeld Hot spots almostoccupy the bottom wall of the cavity

Transition of cell modeThe problem of the onset of hydrodynamic and thermal instabilities in thehorizontal layers of macruid heated from bellow is well suited to illustrate themany facets mathematical and physical of the general theory ofhydrodynamic stability Specially many aspects of the stability (orinstability) of the macruid-layer trapped between the two rigid walls (bottom hotand top cold) have been treated by many researchers over the last 100 years (forexample see Chandrasekhar 1961) The transition of cell mode of the macruidinside two rigid walls (in Rayleigh-Benard convection) is also a well establishedproblem However a mathematical analysis to predict the transition parameter

Figure 5Streamlines (top)

and isothermal lines(bottom) for A = 0 25

and l = 0 10

Effect of surfacewaviness and

aspect ratio

1107

(for example the critical Rayleigh number) and the physical mechanism is anextremely challenging task in the case of wavy walled enclosure under theRayleigh-Benard convection like situation due to the presence of two additionalparameters (A and l) Instead of performing a linear stability analysis wemodireged our numerical algorithm for post processing (a programme written inFORTRAN77 and connected to the software Tecplot 8) to predict the transitionbetween unicellular macrow to bicellular or multicellular macrow Initially forconstant l and A the macrow solver is executed for a range of Rayleigh numberskeeping a considerably large gap between the two consecutive Rayleighnumbers (50-100 depending on the value l) Once we get a rough picture of thetransition of cell mode we re-execute the macrow solver for a small range ofRayleigh numbers keeping a comparatively small gap between the twoconsecutive Rayleigh numbers (1 5) For example when l = 0 01 and A =0 25 the magnitude of the starting Rayleigh number for the macrow solver is1700 Note that the regrst critical Rayleigh number for the Rayleigh-Benardconvection with rigid walls is 1708 (Chandrasekhar 1961) Figure 6(a)-(h)shows the cell pattern for l = 0 01 and A = 0 25 for eight selected values ofRayleigh number A symmetric bicellular macrow pattern characterizes the macrowreglled inside the cavity until the value of Rayleigh number reaches 1905 Twoadditional cells appear near the adiabatic walls (Figure 6(d)) at which the

Figure 6Variation of cell patterninside a cavity withl = 0 01 and A = 0 25for different Rayleighnumber

HFF138

1108

Rayleigh number changes the bicellular macrow into a four-cell multicellular macrowThese cells grow in size with the increasing Rayleigh number but growth rateof the cells becomes slower when Ra approaches 104 A second transition ofthe cell mode is observed near Ra = 106 The overall cell modes and theirtransitions are shown in Figure 7 Note that a non-uniform scale is selected forRayleigh numbers in the abscissa of the Figure 7 for convenience Howeverthis scenario is completely different for l = 0 125 as shown in Figures 8 and 9The symmetric bicellular macrow changes into an asymmetric tricellular macrowaround Ra = 13 995 This tricellular macrow pattern remains unchanged for asmall interval of Rayleigh number (Ra = 13 995 plusmn 14 877) Then a secondtransition makes it to a four-cell multicellular macrow pattern A reverse transitionoccurs around Ra = 106 The overall cell modes and their transitions areshown in Figure 9 for this case

Heat transferHeat transfer is calculated in terms of local (NuL) and average (Nuav) Nusseltnumbers using the following equations

NuL =dk

hL =dk

k

DT

dT

dn

sup3 acute

w

raquo frac14=

dH

dn

sup3 acute

w

(19)

Nuav =1

S

Z S

0

NuL ds (20a)

S =

Z 1

0

1 +dY

dX

sup3 acute2s

dX =EllipticE 2p

2 l2

pplusmn sup2

p (20b)

Figure 7Transition of the cell

mode as a functionof Rayleigh number forl = 0 01 and A = 0 25

Effect of surfacewaviness and

aspect ratio

1109

where the special function ordfEllipticEordm is the elliptic integral of second kind(Abramowitz and Stegun 1965) Local Nusselt number distribution ispresented in Figures 10 and 11 Average Nusselt number distribution ispresented in Figures 12-14 Detail discussions on heat transfer is presented inthe following two paragraphs

Local Nusselt number distribution along the hot wavy wall is shown inFigure 10 for a constant Rayleigh number and four selected surface waviness

Figure 8Variation of cell patterninside a cavity withl = 0 125 and A = 0 25for different Rayleighnumber

HFF138

1110

(l = 0 0 005 0075 01) for an aspect ratio A = 0 25 At Ra = 104 heattransfer rate shows true periodic nature with respect to the axial distance alongthe length of the bottom wall for l = 0 0 where three peaks represent not onlythe three spots of highest temperature gradient but also characterize threepairs of cell generated due to the macruid motion inside the cavity However forl = 0 05 0075 01 local Nusselt number distribution changed signiregcantlydue to low temperature gradient at the midpoint of the cavity In that case heat

Figure 9Transition of the cell

mode as a functionof Rayleigh number for

l = 0 125 and A = 0 25

Figure 10Change of the local

Nusselt number alongthe length of the bottom

wave

Effect of surfacewaviness and

aspect ratio

1111

transfer rate increases up to the X = 0 15 and falls along the wavy wallgradually up to the midpoint (X = 0 5) of the cavity This nature is similarfor other surface waviness with a slight variation in magnitudes Completelydifferent picture is observed at Ra = 105 For l = 0 0 three peaks againrepresent the three spots of highest temperature gradient with a highertemperature gradient at the mid point of the wave

Figure 12Nusselt number as afunction of Rayleighnumber for differentsurface waviness fortwo regxed aspect ratio(a) A = 0 25 and(b) A = 0 50

Figure 11Change of the localNusselt number alongthe length of the bottomwave

HFF138

1112

Figure 11 depicts the extent to which the local Nusselt number distribution isaffected by changes in the aspect ratio We observed that at Ra = 104 twopeaks move towards the center of the cavity with the increase of aspect ratioSimilar pattern was observed at Ra = 106 but the magnitude of the two peak ishigher It is also noticed that heat transfer rate is faster near the straight walland decrease rapidly with the increase of X Whatever be the values of aspectratio heat transfer rate is always minimum at the mid point of the bottom wall

Local Nusselt numbers are integrated to calculate the average value of theNusselt number according to equation (20) Average Nusselt number is plottedas a function of Rayleigh number in Figure 12 for two regxed aspect ratiosA = 0 25 and 050 for different surface waviness At lower Rayleigh number

Figure 13Nusselt number as afunction of Rayleighnumber for different

aspect ratio for two regxedsurface waviness(a) l = 0 05 and

(b) l = 0 10

Figure 14Nusselt number as a

function of surfacewaviness for regxedRayleigh number

(Ra = 5 pound104)Two lines correspond

to two different aspectratio of the enclosure asindicated in the legend

Effect of surfacewaviness and

aspect ratio

1113

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

HFF138

1114

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

transfer and macruid macrow inside an annular space using a non-orthogonalcoordinate transformation In their study it was reported that both thelocal and average heat transfer increase with the increase of protuberanceamplitude and Rayleigh number and decreasing Prandtl number Yao(1983) presented the near wall characteristics of macrow and thermal regeld of avertical wavy wall Saidi et al (1987) presented numerical and experimentalresults of macrow over and heat transfer from a sinusoidal cavity Theyreported that the total heat exchange between the wavy wall of the cavityand the macrowing macruid was reduced by the presence of vortex The vortexplays the role of a thermal screen which creates a large region of uniformtemperature in the bottom of the cavity Asako and Faghri (1987) andMahmud et al (2001) gave a regnite-volume prediction of heat transfer andmacruid macrow characteristics inside a wavy walled duct and tube respectivelyLage and Bejan (1990) documented heat transfer results near a periodically(timewise and spatial) stretching wall Hossain and Rees (1999) Moulic andYao (1989) and Rees and Pop (1995) presented similar solutions for naturalconvection macrow near wavy surface at different boundary conditions Aydinet al (1999) Elsherbiny (1996) Hamady et al (1989) Ozoe et al (1975) andSundstrom and Kimura (1996) presented results of heat transfercharacteristics inside the rectangular enclosures at different aspect ratiosand orientations without surface waviness

In this study we have presented numerical results based on theregnite-volume analysis for laminar natural convection inside a cavity whichconsists of two wavy surfaces with different amplitude-wavelength ratiosand aspect ratios Rate of heat transfer in terms of local and global Nusseltnumbers are calculated for different Rayleigh numbers Flow and thermalregelds are analyzed by parametric presentation of streamlines and isothermallines

Mathematical modelingIn this section we present the mathematical formulation of the problemconsidered in terms of the continuity momentum and energy equations andthe pertinent boundary conditions for a wavy domain containing two wavyand two macrat surfaces We begin by considering two-dimensional laminarnatural convection of an incompressible and Newtonian macruid in a cavityas shown in Figure 1 Distance between the two straight walls is L twowavy walls is d and a is the amplitude of the wavy walls Modeling themacrow as ordfBoussinesq-incompressibleordm to take into account the couplingbetween the energy and momentum equations we regard the density r asconstant everywhere except in the buoyancy term of momentum equations(equation (3)) Correspondingly the non-dimensional equations governingthe conservation of mass momentum and energy in the cavity of Figure 1 areas follows

HFF138

1100

shy U

shy X+

shy V

shy Y= 0 (1)

shy U

shy t+ U

shy U

shy X+ V

shy U

shy Y= 2

shy P

shy X+

Pr

Ra

rshy 2U

shy X 2+

shy 2U

shy Y 2

sup3 acute (2)

shy V

shy t+ U

shy V

shy X+ V

shy V

shy Y= 2

shy P

shy Y+ H +

Pr

Ra

rshy 2V

shy X 2+

shy 2V

shy Y 2

sup3 acute (3)

shy H

shy t+ U

shy H

shy X+ V

shy H

shy Y=

1PrRa

p shy 2H

shy X 2+

shy 2H

shy Y 2

sup3 acute(4)

Equations (1)-(4) are put into their dimensionless forms by scaling differentlengths with average inter wall spacing between the top and the bottom walls(d) velocity components with reference velocity (V0) which is equal to(gbDTd)12 pressure with rV 2

0 time with dV0 The dimensionless temperaturecan be deregned as H = (T 2 T yen ) DT where DT is equal to (TH2 TC)

Boundary and initial conditionsFigure 1 shows the computational grid structure and the geometry underconsideration in the present investigation along with the coordinate frameworkand different boundary conditions The surface shape of the wavy wallsfollows the proregle given in equations (5) and (6) The hot and cold walltemperatures are TH and TC respectively Initial macruid temperature inside thecavity is taken as T yen The initial temperature of the adiabatic walls is set equalto the initial macruid temperature (T yen ) inside the cavity The gravity accelerationg is acted downwards No slip boundary condition is applied for velocitycomponents at each wall For the present investigation boundary conditionsare summarized by the following equations

Figure 1Schematic representation

of the computationalgeometry under

consideration withboundary conditions

Effect of surfacewaviness and

aspect ratio

1101

0 X 1 and

0 Y l[1 2 cos (2pX ) ] U = V = 0 H = 1(5)

0 X 1 and

A Y A + l[1 2 cos (2pX )] U = V = 0 H = 0(6)

X = 0 and 0 Y A U = V = 0shy H

shy n= 0 (7)

X = 1 and 0 Y A U = V = 0shy H

shy n= 0 (8)

where X (= XA) and Y (= YA) are modireged dimensionless horizontal andvertical lengths

Numerical methodologyTo conduct a numerical simulation for the thermomacruid dynamics regelds weused the technique similar to that Hortmann et al (1990) based on theregnite-volume method as described in Ferziger and Peric (1996) Theregnite-volume method starts from the conservation equation in integral form asZ

S

rf v Ccedil n dS =

Z

S

Ggradf Ccedil n dS +

Z

V

qf dV (9)

where f is any variable G is the diffusivity for the quantity f and qf is thesource or sink of f

The solution domain is subdivided into a regnite number of contiguousquadrilateral control volumes (CV) The CVs are deregned by coordinates of theirvertices which are assumed to be connected by straight lines This simple formis possible due to the fact that the equations contain no curvature terms andonly the projections of the CV faces onto the Cartesian coordinates surfaces arerequired in the course of discretization as demonstrated below All thedependent variables solved for and all macruid properties are stored in the CVcenter (collocated arrangement) A suitable spatial distribution of dependentvariables is assumed and the conservation equations (1)-(4) are applied to eachCV leading to a system of non-linear algebraic equations The main steps ofdiscretization procedure to calculate convection and diffusion macruxes and sourceterms are outlined below

The mass macrux through the cell face ordfeordm (Figure 2) is evaluated as

Ccedilme =

Z

Se

rV dS lt (rV)eS1e = re USx1 + VSy

1

iexcl cent (10)

HFF138

1102

where S1e is the surface vector representing the area of the cell face (x = const)and Sx

1e and Sy1e denotes its Cartesian components These are given in terms of

the CV vertex coordinates as follows

Sx1e = (yn 2 ys) and S y

1e = 2 (xn 2 xs) (11)

The mean cell face velocity components Ue and Ve are obtained byinterpolating neighborsrsquo nodal values in a way which ensures the stability ofgrid scheme The convection macrux of any variable f can be expressed as

FCe =

Z

Se

rfV dS lt (rfV )eS1e = Ccedilmefe (12)

The diffusion macrux of f is calculated as

FDe = 2

Z

Se

Gf grad f dS lt 2 (Gf grad f)eS1e (13)

By expressing the gradient of f at the cell face center ordfeordm which is taken here torepresent the mean value over the whole cell face through the derivatives inx and Z directions (Figure 2) and by discretizing these derivatives with CDSthe following expression results

FDe lt Gje

S1e Ccedil PE[(fE 2 fP )(S1e Ccedil S1e) + (fn 2 fs)e(S1e Ccedil S2e)] (14)

where PE is the vector representing the distance from P to E directed towardsE S2e is the surface vector orthogonal to PE and directed outwards positive Z

Figure 2A typical 2D-control

volume and the notationfor a Cartesian grid

Effect of surfacewaviness and

aspect ratio

1103

coordinate (Figure 2) representing the area in the surface Z = 0 bounded byP and E Its x and y components are

Sx2e = 2 ( yE 2 yP ) and Sy

2e = (xE 2 xP ) (15)

The volumetric source term is integrated by simply multiplying the speciregcsource at the control volume center P (which is assumed to represent the meanvalue over the whole cell) with the cell volume ie

Qqf =

Z

V

qf dV lt (qf)PDV (16)

The pressure term in the momentum equations are treated as body force andmay be regarded as pressure sources for the Cartesian velocity componentsThey are evaluated as

Q pui

= 2Z

S

pi i dS = 2Z

V

(grad pi i) dV

lt 2 [( pe 2 pw)S1P + ( pn 2 ps)S2p] Ccedil ii (17)

where the surface vectors S1p and S2p represent the area of the CV crosssection at x = 0 and Z = 0 respectively Since CVrsquos are bounded by straightlines these two vectors can be expressed through the CV surface vectors egS1P = (S1e + S1w) 2 Terms in the momentum equations not featuring inequation (9) are discretized using the same approach and added to the sourceterm After summing up all cell face macruxes and sources the discretizedtransport equation reduces to the following algebraic equation

APfP +nb

XAnbfnb = Qf (18)

where the coefregcients Anb contains the convective and diffusive macruxcontributions and Qf represents the source term The system of equations issolved by using Stonersquos SIP solver

Discretized momentum equations lead to an algebraic equation system forvelocity components U and V where pressure temperature and macruid propertiesare taken from the previous iteration except the regrst iteration where initialconditions are applied These linear equation systems are solved iteratively(inner iteration) to obtain an improved estimate of velocity The improvedvelocity regeld is then used to estimate new mass macruxes which satisfy thecontinuity equation Pressure-correction equation is then solved using the samelinear equation solver and to the same tolerance Energy equation is then

HFF138

1104

solved in the same manner to obtain better estimate of new solution Then theabove procedure is repeated for a new time step For time marching weselected Three Time Level Method which is a fully implicit scheme of secondorder accurate Iteration is continued until the difference between the twoconsecutive regeld values of variables is less than or equal to 10 2 5

AccuracyIn the present investigation regve combinations (20 pound10 40 pound20 80 pound40160 pound80 and 320 pound160) of control volumes are used to test the effect of gridsize on the accuracy of the predicted results Figure 3 shows the distribution ofaverage Nusselt numbers of the hot wall as a function of grid sizes for fourdifferent Rayleigh numbers It is clear from the reggure that at lower Rayleighnumber average Nusselt number is almost independent of grid sizes At higherRayleigh numbers the two higher grid sizes (160 pound 80 and 320 pound160) givealmost the same result It is well known that the high mesh reregnement alwaysprovides better result in the regnite-volume method The main disadvantage intaking higher mesh number is the increase in calculation time which can bereduced by using a higher speed of Pentium processor Our goal was to getbetter results Thus throughout this study the results are presented for 320 pound160 CVsrsquo for better accuracy Predicted results of average Nusselt numbers fora square cavity (A = 1 0 l = 0) with the same boundary conditions arecompared with the results given by Bejan (1984) and Ozisik (1985)Comparisons are shown in Figure 4 where average Nusselt number isplotted as a function of Rayleigh number Predicted results show a very goodagreement with the result of Ozisic at lower Rayleigh number but differ fromBejanrsquos prediction At higher Rayleigh number present prediction shows betteragreement with Bejan compared to Ozisicrsquos results

Figure 3Variation of averageNusselt number as a

function of number of CVfor A = 0 25 and

l = 0 05

Effect of surfacewaviness and

aspect ratio

1105

Results and discussionsFlow and thermal regeldThe macrow pattern inside the wavy enclosure and the temperature proregle arepresented in terms of streamlines and isothermal lines in Figure 5(a)-(d) forA = 0 25 l = 0 10 at four selected Rayleigh numbers At low Rayleighnumber when convection current inside the cavity is comparatively weak macruidstream near the hot wall tends to move towards the centerline or crest of thecavity were two streams from the opposite direction mix and rise upwardsTwo counter-rotating vortices with small cores are observed on the macrow regeldwere bi-cellular macrow pattern divides the cavity into two symmetric parts withrespect to the centerline (X = 0 5) of the cavity Due to the uniformtemperature gradient at low Rayleigh number (= 103) circulation inside thecavity is very weak Convection is less prominent at this Rayleigh number andheat transfer is mainly dominated by conduction The basic difference betweenthe isotherms in a wavy cavity with other geometry is their shape Isothermallines are nearly parallel to each other and follow the geometry of the wavysurfaces A further increase of Rayleigh numbers increases the circulationstrength inside the cavity Ra = 104 is also characterized by the bi-cellular macrowpattern but thermal regeld is completely different compared to Ra = 103 Hereuniform temperature proregle is changed and three high gradient spot isobserved due to rapid circulation of macruid inside the cavity Isotherms turn up(convective distortion) towards the cold wall due to the strong inmacruence of theconvection current At Ra = 105 multi-cellular macrow appears with four vorticesDue to comparatively high buoyant effect another vertical stream of the macruidappears near the adiabatic walls A periodic swirling of the isothermal linesappear at Ra = 105 due to the appearance of multi-cellular macrow Two highgradient spots appear near the bottom wall where isothermal lines areconcentrated Heat transfer rate is higher in magnitude at that spots

Figure 4Validation of averageNusselt number forA = 0 25 and l = 0 0with other referencework

HFF138

1106

Further increase of Rayleigh number increases the strength of the vorticeswhere four-cell multi-cellular macrow pattern turns into two cell bi-cellular patternThis bi-cellular macrow pattern is qualitatively different from the macrow pattern atlow Rayleigh number Periodic swirling nature of isothermal lines disappearsdue to the occurrence of reverse transition in macrow regeld Hot spots almostoccupy the bottom wall of the cavity

Transition of cell modeThe problem of the onset of hydrodynamic and thermal instabilities in thehorizontal layers of macruid heated from bellow is well suited to illustrate themany facets mathematical and physical of the general theory ofhydrodynamic stability Specially many aspects of the stability (orinstability) of the macruid-layer trapped between the two rigid walls (bottom hotand top cold) have been treated by many researchers over the last 100 years (forexample see Chandrasekhar 1961) The transition of cell mode of the macruidinside two rigid walls (in Rayleigh-Benard convection) is also a well establishedproblem However a mathematical analysis to predict the transition parameter

Figure 5Streamlines (top)

and isothermal lines(bottom) for A = 0 25

and l = 0 10

Effect of surfacewaviness and

aspect ratio

1107

(for example the critical Rayleigh number) and the physical mechanism is anextremely challenging task in the case of wavy walled enclosure under theRayleigh-Benard convection like situation due to the presence of two additionalparameters (A and l) Instead of performing a linear stability analysis wemodireged our numerical algorithm for post processing (a programme written inFORTRAN77 and connected to the software Tecplot 8) to predict the transitionbetween unicellular macrow to bicellular or multicellular macrow Initially forconstant l and A the macrow solver is executed for a range of Rayleigh numberskeeping a considerably large gap between the two consecutive Rayleighnumbers (50-100 depending on the value l) Once we get a rough picture of thetransition of cell mode we re-execute the macrow solver for a small range ofRayleigh numbers keeping a comparatively small gap between the twoconsecutive Rayleigh numbers (1 5) For example when l = 0 01 and A =0 25 the magnitude of the starting Rayleigh number for the macrow solver is1700 Note that the regrst critical Rayleigh number for the Rayleigh-Benardconvection with rigid walls is 1708 (Chandrasekhar 1961) Figure 6(a)-(h)shows the cell pattern for l = 0 01 and A = 0 25 for eight selected values ofRayleigh number A symmetric bicellular macrow pattern characterizes the macrowreglled inside the cavity until the value of Rayleigh number reaches 1905 Twoadditional cells appear near the adiabatic walls (Figure 6(d)) at which the

Figure 6Variation of cell patterninside a cavity withl = 0 01 and A = 0 25for different Rayleighnumber

HFF138

1108

Rayleigh number changes the bicellular macrow into a four-cell multicellular macrowThese cells grow in size with the increasing Rayleigh number but growth rateof the cells becomes slower when Ra approaches 104 A second transition ofthe cell mode is observed near Ra = 106 The overall cell modes and theirtransitions are shown in Figure 7 Note that a non-uniform scale is selected forRayleigh numbers in the abscissa of the Figure 7 for convenience Howeverthis scenario is completely different for l = 0 125 as shown in Figures 8 and 9The symmetric bicellular macrow changes into an asymmetric tricellular macrowaround Ra = 13 995 This tricellular macrow pattern remains unchanged for asmall interval of Rayleigh number (Ra = 13 995 plusmn 14 877) Then a secondtransition makes it to a four-cell multicellular macrow pattern A reverse transitionoccurs around Ra = 106 The overall cell modes and their transitions areshown in Figure 9 for this case

Heat transferHeat transfer is calculated in terms of local (NuL) and average (Nuav) Nusseltnumbers using the following equations

NuL =dk

hL =dk

k

DT

dT

dn

sup3 acute

w

raquo frac14=

dH

dn

sup3 acute

w

(19)

Nuav =1

S

Z S

0

NuL ds (20a)

S =

Z 1

0

1 +dY

dX

sup3 acute2s

dX =EllipticE 2p

2 l2

pplusmn sup2

p (20b)

Figure 7Transition of the cell

mode as a functionof Rayleigh number forl = 0 01 and A = 0 25

Effect of surfacewaviness and

aspect ratio

1109

where the special function ordfEllipticEordm is the elliptic integral of second kind(Abramowitz and Stegun 1965) Local Nusselt number distribution ispresented in Figures 10 and 11 Average Nusselt number distribution ispresented in Figures 12-14 Detail discussions on heat transfer is presented inthe following two paragraphs

Local Nusselt number distribution along the hot wavy wall is shown inFigure 10 for a constant Rayleigh number and four selected surface waviness

Figure 8Variation of cell patterninside a cavity withl = 0 125 and A = 0 25for different Rayleighnumber

HFF138

1110

(l = 0 0 005 0075 01) for an aspect ratio A = 0 25 At Ra = 104 heattransfer rate shows true periodic nature with respect to the axial distance alongthe length of the bottom wall for l = 0 0 where three peaks represent not onlythe three spots of highest temperature gradient but also characterize threepairs of cell generated due to the macruid motion inside the cavity However forl = 0 05 0075 01 local Nusselt number distribution changed signiregcantlydue to low temperature gradient at the midpoint of the cavity In that case heat

Figure 9Transition of the cell

mode as a functionof Rayleigh number for

l = 0 125 and A = 0 25

Figure 10Change of the local

Nusselt number alongthe length of the bottom

wave

Effect of surfacewaviness and

aspect ratio

1111

transfer rate increases up to the X = 0 15 and falls along the wavy wallgradually up to the midpoint (X = 0 5) of the cavity This nature is similarfor other surface waviness with a slight variation in magnitudes Completelydifferent picture is observed at Ra = 105 For l = 0 0 three peaks againrepresent the three spots of highest temperature gradient with a highertemperature gradient at the mid point of the wave

Figure 12Nusselt number as afunction of Rayleighnumber for differentsurface waviness fortwo regxed aspect ratio(a) A = 0 25 and(b) A = 0 50

Figure 11Change of the localNusselt number alongthe length of the bottomwave

HFF138

1112

Figure 11 depicts the extent to which the local Nusselt number distribution isaffected by changes in the aspect ratio We observed that at Ra = 104 twopeaks move towards the center of the cavity with the increase of aspect ratioSimilar pattern was observed at Ra = 106 but the magnitude of the two peak ishigher It is also noticed that heat transfer rate is faster near the straight walland decrease rapidly with the increase of X Whatever be the values of aspectratio heat transfer rate is always minimum at the mid point of the bottom wall

Local Nusselt numbers are integrated to calculate the average value of theNusselt number according to equation (20) Average Nusselt number is plottedas a function of Rayleigh number in Figure 12 for two regxed aspect ratiosA = 0 25 and 050 for different surface waviness At lower Rayleigh number

Figure 13Nusselt number as afunction of Rayleighnumber for different

aspect ratio for two regxedsurface waviness(a) l = 0 05 and

(b) l = 0 10

Figure 14Nusselt number as a

function of surfacewaviness for regxedRayleigh number

(Ra = 5 pound104)Two lines correspond

to two different aspectratio of the enclosure asindicated in the legend

Effect of surfacewaviness and

aspect ratio

1113

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

HFF138

1114

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

shy U

shy X+

shy V

shy Y= 0 (1)

shy U

shy t+ U

shy U

shy X+ V

shy U

shy Y= 2

shy P

shy X+

Pr

Ra

rshy 2U

shy X 2+

shy 2U

shy Y 2

sup3 acute (2)

shy V

shy t+ U

shy V

shy X+ V

shy V

shy Y= 2

shy P

shy Y+ H +

Pr

Ra

rshy 2V

shy X 2+

shy 2V

shy Y 2

sup3 acute (3)

shy H

shy t+ U

shy H

shy X+ V

shy H

shy Y=

1PrRa

p shy 2H

shy X 2+

shy 2H

shy Y 2

sup3 acute(4)

Equations (1)-(4) are put into their dimensionless forms by scaling differentlengths with average inter wall spacing between the top and the bottom walls(d) velocity components with reference velocity (V0) which is equal to(gbDTd)12 pressure with rV 2

0 time with dV0 The dimensionless temperaturecan be deregned as H = (T 2 T yen ) DT where DT is equal to (TH2 TC)

Boundary and initial conditionsFigure 1 shows the computational grid structure and the geometry underconsideration in the present investigation along with the coordinate frameworkand different boundary conditions The surface shape of the wavy wallsfollows the proregle given in equations (5) and (6) The hot and cold walltemperatures are TH and TC respectively Initial macruid temperature inside thecavity is taken as T yen The initial temperature of the adiabatic walls is set equalto the initial macruid temperature (T yen ) inside the cavity The gravity accelerationg is acted downwards No slip boundary condition is applied for velocitycomponents at each wall For the present investigation boundary conditionsare summarized by the following equations

Figure 1Schematic representation

of the computationalgeometry under

consideration withboundary conditions

Effect of surfacewaviness and

aspect ratio

1101

0 X 1 and

0 Y l[1 2 cos (2pX ) ] U = V = 0 H = 1(5)

0 X 1 and

A Y A + l[1 2 cos (2pX )] U = V = 0 H = 0(6)

X = 0 and 0 Y A U = V = 0shy H

shy n= 0 (7)

X = 1 and 0 Y A U = V = 0shy H

shy n= 0 (8)

where X (= XA) and Y (= YA) are modireged dimensionless horizontal andvertical lengths

Numerical methodologyTo conduct a numerical simulation for the thermomacruid dynamics regelds weused the technique similar to that Hortmann et al (1990) based on theregnite-volume method as described in Ferziger and Peric (1996) Theregnite-volume method starts from the conservation equation in integral form asZ

S

rf v Ccedil n dS =

Z

S

Ggradf Ccedil n dS +

Z

V

qf dV (9)

where f is any variable G is the diffusivity for the quantity f and qf is thesource or sink of f

The solution domain is subdivided into a regnite number of contiguousquadrilateral control volumes (CV) The CVs are deregned by coordinates of theirvertices which are assumed to be connected by straight lines This simple formis possible due to the fact that the equations contain no curvature terms andonly the projections of the CV faces onto the Cartesian coordinates surfaces arerequired in the course of discretization as demonstrated below All thedependent variables solved for and all macruid properties are stored in the CVcenter (collocated arrangement) A suitable spatial distribution of dependentvariables is assumed and the conservation equations (1)-(4) are applied to eachCV leading to a system of non-linear algebraic equations The main steps ofdiscretization procedure to calculate convection and diffusion macruxes and sourceterms are outlined below

The mass macrux through the cell face ordfeordm (Figure 2) is evaluated as

Ccedilme =

Z

Se

rV dS lt (rV)eS1e = re USx1 + VSy

1

iexcl cent (10)

HFF138

1102

where S1e is the surface vector representing the area of the cell face (x = const)and Sx

1e and Sy1e denotes its Cartesian components These are given in terms of

the CV vertex coordinates as follows

Sx1e = (yn 2 ys) and S y

1e = 2 (xn 2 xs) (11)

The mean cell face velocity components Ue and Ve are obtained byinterpolating neighborsrsquo nodal values in a way which ensures the stability ofgrid scheme The convection macrux of any variable f can be expressed as

FCe =

Z

Se

rfV dS lt (rfV )eS1e = Ccedilmefe (12)

The diffusion macrux of f is calculated as

FDe = 2

Z

Se

Gf grad f dS lt 2 (Gf grad f)eS1e (13)

By expressing the gradient of f at the cell face center ordfeordm which is taken here torepresent the mean value over the whole cell face through the derivatives inx and Z directions (Figure 2) and by discretizing these derivatives with CDSthe following expression results

FDe lt Gje

S1e Ccedil PE[(fE 2 fP )(S1e Ccedil S1e) + (fn 2 fs)e(S1e Ccedil S2e)] (14)

where PE is the vector representing the distance from P to E directed towardsE S2e is the surface vector orthogonal to PE and directed outwards positive Z

Figure 2A typical 2D-control

volume and the notationfor a Cartesian grid

Effect of surfacewaviness and

aspect ratio

1103

coordinate (Figure 2) representing the area in the surface Z = 0 bounded byP and E Its x and y components are

Sx2e = 2 ( yE 2 yP ) and Sy

2e = (xE 2 xP ) (15)

The volumetric source term is integrated by simply multiplying the speciregcsource at the control volume center P (which is assumed to represent the meanvalue over the whole cell) with the cell volume ie

Qqf =

Z

V

qf dV lt (qf)PDV (16)

The pressure term in the momentum equations are treated as body force andmay be regarded as pressure sources for the Cartesian velocity componentsThey are evaluated as

Q pui

= 2Z

S

pi i dS = 2Z

V

(grad pi i) dV

lt 2 [( pe 2 pw)S1P + ( pn 2 ps)S2p] Ccedil ii (17)

where the surface vectors S1p and S2p represent the area of the CV crosssection at x = 0 and Z = 0 respectively Since CVrsquos are bounded by straightlines these two vectors can be expressed through the CV surface vectors egS1P = (S1e + S1w) 2 Terms in the momentum equations not featuring inequation (9) are discretized using the same approach and added to the sourceterm After summing up all cell face macruxes and sources the discretizedtransport equation reduces to the following algebraic equation

APfP +nb

XAnbfnb = Qf (18)

where the coefregcients Anb contains the convective and diffusive macruxcontributions and Qf represents the source term The system of equations issolved by using Stonersquos SIP solver

Discretized momentum equations lead to an algebraic equation system forvelocity components U and V where pressure temperature and macruid propertiesare taken from the previous iteration except the regrst iteration where initialconditions are applied These linear equation systems are solved iteratively(inner iteration) to obtain an improved estimate of velocity The improvedvelocity regeld is then used to estimate new mass macruxes which satisfy thecontinuity equation Pressure-correction equation is then solved using the samelinear equation solver and to the same tolerance Energy equation is then

HFF138

1104

solved in the same manner to obtain better estimate of new solution Then theabove procedure is repeated for a new time step For time marching weselected Three Time Level Method which is a fully implicit scheme of secondorder accurate Iteration is continued until the difference between the twoconsecutive regeld values of variables is less than or equal to 10 2 5

AccuracyIn the present investigation regve combinations (20 pound10 40 pound20 80 pound40160 pound80 and 320 pound160) of control volumes are used to test the effect of gridsize on the accuracy of the predicted results Figure 3 shows the distribution ofaverage Nusselt numbers of the hot wall as a function of grid sizes for fourdifferent Rayleigh numbers It is clear from the reggure that at lower Rayleighnumber average Nusselt number is almost independent of grid sizes At higherRayleigh numbers the two higher grid sizes (160 pound 80 and 320 pound160) givealmost the same result It is well known that the high mesh reregnement alwaysprovides better result in the regnite-volume method The main disadvantage intaking higher mesh number is the increase in calculation time which can bereduced by using a higher speed of Pentium processor Our goal was to getbetter results Thus throughout this study the results are presented for 320 pound160 CVsrsquo for better accuracy Predicted results of average Nusselt numbers fora square cavity (A = 1 0 l = 0) with the same boundary conditions arecompared with the results given by Bejan (1984) and Ozisik (1985)Comparisons are shown in Figure 4 where average Nusselt number isplotted as a function of Rayleigh number Predicted results show a very goodagreement with the result of Ozisic at lower Rayleigh number but differ fromBejanrsquos prediction At higher Rayleigh number present prediction shows betteragreement with Bejan compared to Ozisicrsquos results

Figure 3Variation of averageNusselt number as a

function of number of CVfor A = 0 25 and

l = 0 05

Effect of surfacewaviness and

aspect ratio

1105

Results and discussionsFlow and thermal regeldThe macrow pattern inside the wavy enclosure and the temperature proregle arepresented in terms of streamlines and isothermal lines in Figure 5(a)-(d) forA = 0 25 l = 0 10 at four selected Rayleigh numbers At low Rayleighnumber when convection current inside the cavity is comparatively weak macruidstream near the hot wall tends to move towards the centerline or crest of thecavity were two streams from the opposite direction mix and rise upwardsTwo counter-rotating vortices with small cores are observed on the macrow regeldwere bi-cellular macrow pattern divides the cavity into two symmetric parts withrespect to the centerline (X = 0 5) of the cavity Due to the uniformtemperature gradient at low Rayleigh number (= 103) circulation inside thecavity is very weak Convection is less prominent at this Rayleigh number andheat transfer is mainly dominated by conduction The basic difference betweenthe isotherms in a wavy cavity with other geometry is their shape Isothermallines are nearly parallel to each other and follow the geometry of the wavysurfaces A further increase of Rayleigh numbers increases the circulationstrength inside the cavity Ra = 104 is also characterized by the bi-cellular macrowpattern but thermal regeld is completely different compared to Ra = 103 Hereuniform temperature proregle is changed and three high gradient spot isobserved due to rapid circulation of macruid inside the cavity Isotherms turn up(convective distortion) towards the cold wall due to the strong inmacruence of theconvection current At Ra = 105 multi-cellular macrow appears with four vorticesDue to comparatively high buoyant effect another vertical stream of the macruidappears near the adiabatic walls A periodic swirling of the isothermal linesappear at Ra = 105 due to the appearance of multi-cellular macrow Two highgradient spots appear near the bottom wall where isothermal lines areconcentrated Heat transfer rate is higher in magnitude at that spots

Figure 4Validation of averageNusselt number forA = 0 25 and l = 0 0with other referencework

HFF138

1106

Further increase of Rayleigh number increases the strength of the vorticeswhere four-cell multi-cellular macrow pattern turns into two cell bi-cellular patternThis bi-cellular macrow pattern is qualitatively different from the macrow pattern atlow Rayleigh number Periodic swirling nature of isothermal lines disappearsdue to the occurrence of reverse transition in macrow regeld Hot spots almostoccupy the bottom wall of the cavity

Transition of cell modeThe problem of the onset of hydrodynamic and thermal instabilities in thehorizontal layers of macruid heated from bellow is well suited to illustrate themany facets mathematical and physical of the general theory ofhydrodynamic stability Specially many aspects of the stability (orinstability) of the macruid-layer trapped between the two rigid walls (bottom hotand top cold) have been treated by many researchers over the last 100 years (forexample see Chandrasekhar 1961) The transition of cell mode of the macruidinside two rigid walls (in Rayleigh-Benard convection) is also a well establishedproblem However a mathematical analysis to predict the transition parameter

Figure 5Streamlines (top)

and isothermal lines(bottom) for A = 0 25

and l = 0 10

Effect of surfacewaviness and

aspect ratio

1107

(for example the critical Rayleigh number) and the physical mechanism is anextremely challenging task in the case of wavy walled enclosure under theRayleigh-Benard convection like situation due to the presence of two additionalparameters (A and l) Instead of performing a linear stability analysis wemodireged our numerical algorithm for post processing (a programme written inFORTRAN77 and connected to the software Tecplot 8) to predict the transitionbetween unicellular macrow to bicellular or multicellular macrow Initially forconstant l and A the macrow solver is executed for a range of Rayleigh numberskeeping a considerably large gap between the two consecutive Rayleighnumbers (50-100 depending on the value l) Once we get a rough picture of thetransition of cell mode we re-execute the macrow solver for a small range ofRayleigh numbers keeping a comparatively small gap between the twoconsecutive Rayleigh numbers (1 5) For example when l = 0 01 and A =0 25 the magnitude of the starting Rayleigh number for the macrow solver is1700 Note that the regrst critical Rayleigh number for the Rayleigh-Benardconvection with rigid walls is 1708 (Chandrasekhar 1961) Figure 6(a)-(h)shows the cell pattern for l = 0 01 and A = 0 25 for eight selected values ofRayleigh number A symmetric bicellular macrow pattern characterizes the macrowreglled inside the cavity until the value of Rayleigh number reaches 1905 Twoadditional cells appear near the adiabatic walls (Figure 6(d)) at which the

Figure 6Variation of cell patterninside a cavity withl = 0 01 and A = 0 25for different Rayleighnumber

HFF138

1108

Rayleigh number changes the bicellular macrow into a four-cell multicellular macrowThese cells grow in size with the increasing Rayleigh number but growth rateof the cells becomes slower when Ra approaches 104 A second transition ofthe cell mode is observed near Ra = 106 The overall cell modes and theirtransitions are shown in Figure 7 Note that a non-uniform scale is selected forRayleigh numbers in the abscissa of the Figure 7 for convenience Howeverthis scenario is completely different for l = 0 125 as shown in Figures 8 and 9The symmetric bicellular macrow changes into an asymmetric tricellular macrowaround Ra = 13 995 This tricellular macrow pattern remains unchanged for asmall interval of Rayleigh number (Ra = 13 995 plusmn 14 877) Then a secondtransition makes it to a four-cell multicellular macrow pattern A reverse transitionoccurs around Ra = 106 The overall cell modes and their transitions areshown in Figure 9 for this case

Heat transferHeat transfer is calculated in terms of local (NuL) and average (Nuav) Nusseltnumbers using the following equations

NuL =dk

hL =dk

k

DT

dT

dn

sup3 acute

w

raquo frac14=

dH

dn

sup3 acute

w

(19)

Nuav =1

S

Z S

0

NuL ds (20a)

S =

Z 1

0

1 +dY

dX

sup3 acute2s

dX =EllipticE 2p

2 l2

pplusmn sup2

p (20b)

Figure 7Transition of the cell

mode as a functionof Rayleigh number forl = 0 01 and A = 0 25

Effect of surfacewaviness and

aspect ratio

1109

where the special function ordfEllipticEordm is the elliptic integral of second kind(Abramowitz and Stegun 1965) Local Nusselt number distribution ispresented in Figures 10 and 11 Average Nusselt number distribution ispresented in Figures 12-14 Detail discussions on heat transfer is presented inthe following two paragraphs

Local Nusselt number distribution along the hot wavy wall is shown inFigure 10 for a constant Rayleigh number and four selected surface waviness

Figure 8Variation of cell patterninside a cavity withl = 0 125 and A = 0 25for different Rayleighnumber

HFF138

1110

(l = 0 0 005 0075 01) for an aspect ratio A = 0 25 At Ra = 104 heattransfer rate shows true periodic nature with respect to the axial distance alongthe length of the bottom wall for l = 0 0 where three peaks represent not onlythe three spots of highest temperature gradient but also characterize threepairs of cell generated due to the macruid motion inside the cavity However forl = 0 05 0075 01 local Nusselt number distribution changed signiregcantlydue to low temperature gradient at the midpoint of the cavity In that case heat

Figure 9Transition of the cell

mode as a functionof Rayleigh number for

l = 0 125 and A = 0 25

Figure 10Change of the local

Nusselt number alongthe length of the bottom

wave

Effect of surfacewaviness and

aspect ratio

1111

transfer rate increases up to the X = 0 15 and falls along the wavy wallgradually up to the midpoint (X = 0 5) of the cavity This nature is similarfor other surface waviness with a slight variation in magnitudes Completelydifferent picture is observed at Ra = 105 For l = 0 0 three peaks againrepresent the three spots of highest temperature gradient with a highertemperature gradient at the mid point of the wave

Figure 12Nusselt number as afunction of Rayleighnumber for differentsurface waviness fortwo regxed aspect ratio(a) A = 0 25 and(b) A = 0 50

Figure 11Change of the localNusselt number alongthe length of the bottomwave

HFF138

1112

Figure 11 depicts the extent to which the local Nusselt number distribution isaffected by changes in the aspect ratio We observed that at Ra = 104 twopeaks move towards the center of the cavity with the increase of aspect ratioSimilar pattern was observed at Ra = 106 but the magnitude of the two peak ishigher It is also noticed that heat transfer rate is faster near the straight walland decrease rapidly with the increase of X Whatever be the values of aspectratio heat transfer rate is always minimum at the mid point of the bottom wall

Local Nusselt numbers are integrated to calculate the average value of theNusselt number according to equation (20) Average Nusselt number is plottedas a function of Rayleigh number in Figure 12 for two regxed aspect ratiosA = 0 25 and 050 for different surface waviness At lower Rayleigh number

Figure 13Nusselt number as afunction of Rayleighnumber for different

aspect ratio for two regxedsurface waviness(a) l = 0 05 and

(b) l = 0 10

Figure 14Nusselt number as a

function of surfacewaviness for regxedRayleigh number

(Ra = 5 pound104)Two lines correspond

to two different aspectratio of the enclosure asindicated in the legend

Effect of surfacewaviness and

aspect ratio

1113

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

HFF138

1114

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

0 X 1 and

0 Y l[1 2 cos (2pX ) ] U = V = 0 H = 1(5)

0 X 1 and

A Y A + l[1 2 cos (2pX )] U = V = 0 H = 0(6)

X = 0 and 0 Y A U = V = 0shy H

shy n= 0 (7)

X = 1 and 0 Y A U = V = 0shy H

shy n= 0 (8)

where X (= XA) and Y (= YA) are modireged dimensionless horizontal andvertical lengths

Numerical methodologyTo conduct a numerical simulation for the thermomacruid dynamics regelds weused the technique similar to that Hortmann et al (1990) based on theregnite-volume method as described in Ferziger and Peric (1996) Theregnite-volume method starts from the conservation equation in integral form asZ

S

rf v Ccedil n dS =

Z

S

Ggradf Ccedil n dS +

Z

V

qf dV (9)

where f is any variable G is the diffusivity for the quantity f and qf is thesource or sink of f

The solution domain is subdivided into a regnite number of contiguousquadrilateral control volumes (CV) The CVs are deregned by coordinates of theirvertices which are assumed to be connected by straight lines This simple formis possible due to the fact that the equations contain no curvature terms andonly the projections of the CV faces onto the Cartesian coordinates surfaces arerequired in the course of discretization as demonstrated below All thedependent variables solved for and all macruid properties are stored in the CVcenter (collocated arrangement) A suitable spatial distribution of dependentvariables is assumed and the conservation equations (1)-(4) are applied to eachCV leading to a system of non-linear algebraic equations The main steps ofdiscretization procedure to calculate convection and diffusion macruxes and sourceterms are outlined below

The mass macrux through the cell face ordfeordm (Figure 2) is evaluated as

Ccedilme =

Z

Se

rV dS lt (rV)eS1e = re USx1 + VSy

1

iexcl cent (10)

HFF138

1102

where S1e is the surface vector representing the area of the cell face (x = const)and Sx

1e and Sy1e denotes its Cartesian components These are given in terms of

the CV vertex coordinates as follows

Sx1e = (yn 2 ys) and S y

1e = 2 (xn 2 xs) (11)

The mean cell face velocity components Ue and Ve are obtained byinterpolating neighborsrsquo nodal values in a way which ensures the stability ofgrid scheme The convection macrux of any variable f can be expressed as

FCe =

Z

Se

rfV dS lt (rfV )eS1e = Ccedilmefe (12)

The diffusion macrux of f is calculated as

FDe = 2

Z

Se

Gf grad f dS lt 2 (Gf grad f)eS1e (13)

By expressing the gradient of f at the cell face center ordfeordm which is taken here torepresent the mean value over the whole cell face through the derivatives inx and Z directions (Figure 2) and by discretizing these derivatives with CDSthe following expression results

FDe lt Gje

S1e Ccedil PE[(fE 2 fP )(S1e Ccedil S1e) + (fn 2 fs)e(S1e Ccedil S2e)] (14)

where PE is the vector representing the distance from P to E directed towardsE S2e is the surface vector orthogonal to PE and directed outwards positive Z

Figure 2A typical 2D-control

volume and the notationfor a Cartesian grid

Effect of surfacewaviness and

aspect ratio

1103

coordinate (Figure 2) representing the area in the surface Z = 0 bounded byP and E Its x and y components are

Sx2e = 2 ( yE 2 yP ) and Sy

2e = (xE 2 xP ) (15)

The volumetric source term is integrated by simply multiplying the speciregcsource at the control volume center P (which is assumed to represent the meanvalue over the whole cell) with the cell volume ie

Qqf =

Z

V

qf dV lt (qf)PDV (16)

The pressure term in the momentum equations are treated as body force andmay be regarded as pressure sources for the Cartesian velocity componentsThey are evaluated as

Q pui

= 2Z

S

pi i dS = 2Z

V

(grad pi i) dV

lt 2 [( pe 2 pw)S1P + ( pn 2 ps)S2p] Ccedil ii (17)

where the surface vectors S1p and S2p represent the area of the CV crosssection at x = 0 and Z = 0 respectively Since CVrsquos are bounded by straightlines these two vectors can be expressed through the CV surface vectors egS1P = (S1e + S1w) 2 Terms in the momentum equations not featuring inequation (9) are discretized using the same approach and added to the sourceterm After summing up all cell face macruxes and sources the discretizedtransport equation reduces to the following algebraic equation

APfP +nb

XAnbfnb = Qf (18)

where the coefregcients Anb contains the convective and diffusive macruxcontributions and Qf represents the source term The system of equations issolved by using Stonersquos SIP solver

Discretized momentum equations lead to an algebraic equation system forvelocity components U and V where pressure temperature and macruid propertiesare taken from the previous iteration except the regrst iteration where initialconditions are applied These linear equation systems are solved iteratively(inner iteration) to obtain an improved estimate of velocity The improvedvelocity regeld is then used to estimate new mass macruxes which satisfy thecontinuity equation Pressure-correction equation is then solved using the samelinear equation solver and to the same tolerance Energy equation is then

HFF138

1104

solved in the same manner to obtain better estimate of new solution Then theabove procedure is repeated for a new time step For time marching weselected Three Time Level Method which is a fully implicit scheme of secondorder accurate Iteration is continued until the difference between the twoconsecutive regeld values of variables is less than or equal to 10 2 5

AccuracyIn the present investigation regve combinations (20 pound10 40 pound20 80 pound40160 pound80 and 320 pound160) of control volumes are used to test the effect of gridsize on the accuracy of the predicted results Figure 3 shows the distribution ofaverage Nusselt numbers of the hot wall as a function of grid sizes for fourdifferent Rayleigh numbers It is clear from the reggure that at lower Rayleighnumber average Nusselt number is almost independent of grid sizes At higherRayleigh numbers the two higher grid sizes (160 pound 80 and 320 pound160) givealmost the same result It is well known that the high mesh reregnement alwaysprovides better result in the regnite-volume method The main disadvantage intaking higher mesh number is the increase in calculation time which can bereduced by using a higher speed of Pentium processor Our goal was to getbetter results Thus throughout this study the results are presented for 320 pound160 CVsrsquo for better accuracy Predicted results of average Nusselt numbers fora square cavity (A = 1 0 l = 0) with the same boundary conditions arecompared with the results given by Bejan (1984) and Ozisik (1985)Comparisons are shown in Figure 4 where average Nusselt number isplotted as a function of Rayleigh number Predicted results show a very goodagreement with the result of Ozisic at lower Rayleigh number but differ fromBejanrsquos prediction At higher Rayleigh number present prediction shows betteragreement with Bejan compared to Ozisicrsquos results

Figure 3Variation of averageNusselt number as a

function of number of CVfor A = 0 25 and

l = 0 05

Effect of surfacewaviness and

aspect ratio

1105

Results and discussionsFlow and thermal regeldThe macrow pattern inside the wavy enclosure and the temperature proregle arepresented in terms of streamlines and isothermal lines in Figure 5(a)-(d) forA = 0 25 l = 0 10 at four selected Rayleigh numbers At low Rayleighnumber when convection current inside the cavity is comparatively weak macruidstream near the hot wall tends to move towards the centerline or crest of thecavity were two streams from the opposite direction mix and rise upwardsTwo counter-rotating vortices with small cores are observed on the macrow regeldwere bi-cellular macrow pattern divides the cavity into two symmetric parts withrespect to the centerline (X = 0 5) of the cavity Due to the uniformtemperature gradient at low Rayleigh number (= 103) circulation inside thecavity is very weak Convection is less prominent at this Rayleigh number andheat transfer is mainly dominated by conduction The basic difference betweenthe isotherms in a wavy cavity with other geometry is their shape Isothermallines are nearly parallel to each other and follow the geometry of the wavysurfaces A further increase of Rayleigh numbers increases the circulationstrength inside the cavity Ra = 104 is also characterized by the bi-cellular macrowpattern but thermal regeld is completely different compared to Ra = 103 Hereuniform temperature proregle is changed and three high gradient spot isobserved due to rapid circulation of macruid inside the cavity Isotherms turn up(convective distortion) towards the cold wall due to the strong inmacruence of theconvection current At Ra = 105 multi-cellular macrow appears with four vorticesDue to comparatively high buoyant effect another vertical stream of the macruidappears near the adiabatic walls A periodic swirling of the isothermal linesappear at Ra = 105 due to the appearance of multi-cellular macrow Two highgradient spots appear near the bottom wall where isothermal lines areconcentrated Heat transfer rate is higher in magnitude at that spots

Figure 4Validation of averageNusselt number forA = 0 25 and l = 0 0with other referencework

HFF138

1106

Further increase of Rayleigh number increases the strength of the vorticeswhere four-cell multi-cellular macrow pattern turns into two cell bi-cellular patternThis bi-cellular macrow pattern is qualitatively different from the macrow pattern atlow Rayleigh number Periodic swirling nature of isothermal lines disappearsdue to the occurrence of reverse transition in macrow regeld Hot spots almostoccupy the bottom wall of the cavity

Transition of cell modeThe problem of the onset of hydrodynamic and thermal instabilities in thehorizontal layers of macruid heated from bellow is well suited to illustrate themany facets mathematical and physical of the general theory ofhydrodynamic stability Specially many aspects of the stability (orinstability) of the macruid-layer trapped between the two rigid walls (bottom hotand top cold) have been treated by many researchers over the last 100 years (forexample see Chandrasekhar 1961) The transition of cell mode of the macruidinside two rigid walls (in Rayleigh-Benard convection) is also a well establishedproblem However a mathematical analysis to predict the transition parameter

Figure 5Streamlines (top)

and isothermal lines(bottom) for A = 0 25

and l = 0 10

Effect of surfacewaviness and

aspect ratio

1107

(for example the critical Rayleigh number) and the physical mechanism is anextremely challenging task in the case of wavy walled enclosure under theRayleigh-Benard convection like situation due to the presence of two additionalparameters (A and l) Instead of performing a linear stability analysis wemodireged our numerical algorithm for post processing (a programme written inFORTRAN77 and connected to the software Tecplot 8) to predict the transitionbetween unicellular macrow to bicellular or multicellular macrow Initially forconstant l and A the macrow solver is executed for a range of Rayleigh numberskeeping a considerably large gap between the two consecutive Rayleighnumbers (50-100 depending on the value l) Once we get a rough picture of thetransition of cell mode we re-execute the macrow solver for a small range ofRayleigh numbers keeping a comparatively small gap between the twoconsecutive Rayleigh numbers (1 5) For example when l = 0 01 and A =0 25 the magnitude of the starting Rayleigh number for the macrow solver is1700 Note that the regrst critical Rayleigh number for the Rayleigh-Benardconvection with rigid walls is 1708 (Chandrasekhar 1961) Figure 6(a)-(h)shows the cell pattern for l = 0 01 and A = 0 25 for eight selected values ofRayleigh number A symmetric bicellular macrow pattern characterizes the macrowreglled inside the cavity until the value of Rayleigh number reaches 1905 Twoadditional cells appear near the adiabatic walls (Figure 6(d)) at which the

Figure 6Variation of cell patterninside a cavity withl = 0 01 and A = 0 25for different Rayleighnumber

HFF138

1108

Rayleigh number changes the bicellular macrow into a four-cell multicellular macrowThese cells grow in size with the increasing Rayleigh number but growth rateof the cells becomes slower when Ra approaches 104 A second transition ofthe cell mode is observed near Ra = 106 The overall cell modes and theirtransitions are shown in Figure 7 Note that a non-uniform scale is selected forRayleigh numbers in the abscissa of the Figure 7 for convenience Howeverthis scenario is completely different for l = 0 125 as shown in Figures 8 and 9The symmetric bicellular macrow changes into an asymmetric tricellular macrowaround Ra = 13 995 This tricellular macrow pattern remains unchanged for asmall interval of Rayleigh number (Ra = 13 995 plusmn 14 877) Then a secondtransition makes it to a four-cell multicellular macrow pattern A reverse transitionoccurs around Ra = 106 The overall cell modes and their transitions areshown in Figure 9 for this case

Heat transferHeat transfer is calculated in terms of local (NuL) and average (Nuav) Nusseltnumbers using the following equations

NuL =dk

hL =dk

k

DT

dT

dn

sup3 acute

w

raquo frac14=

dH

dn

sup3 acute

w

(19)

Nuav =1

S

Z S

0

NuL ds (20a)

S =

Z 1

0

1 +dY

dX

sup3 acute2s

dX =EllipticE 2p

2 l2

pplusmn sup2

p (20b)

Figure 7Transition of the cell

mode as a functionof Rayleigh number forl = 0 01 and A = 0 25

Effect of surfacewaviness and

aspect ratio

1109

where the special function ordfEllipticEordm is the elliptic integral of second kind(Abramowitz and Stegun 1965) Local Nusselt number distribution ispresented in Figures 10 and 11 Average Nusselt number distribution ispresented in Figures 12-14 Detail discussions on heat transfer is presented inthe following two paragraphs

Local Nusselt number distribution along the hot wavy wall is shown inFigure 10 for a constant Rayleigh number and four selected surface waviness

Figure 8Variation of cell patterninside a cavity withl = 0 125 and A = 0 25for different Rayleighnumber

HFF138

1110

(l = 0 0 005 0075 01) for an aspect ratio A = 0 25 At Ra = 104 heattransfer rate shows true periodic nature with respect to the axial distance alongthe length of the bottom wall for l = 0 0 where three peaks represent not onlythe three spots of highest temperature gradient but also characterize threepairs of cell generated due to the macruid motion inside the cavity However forl = 0 05 0075 01 local Nusselt number distribution changed signiregcantlydue to low temperature gradient at the midpoint of the cavity In that case heat

Figure 9Transition of the cell

mode as a functionof Rayleigh number for

l = 0 125 and A = 0 25

Figure 10Change of the local

Nusselt number alongthe length of the bottom

wave

Effect of surfacewaviness and

aspect ratio

1111

transfer rate increases up to the X = 0 15 and falls along the wavy wallgradually up to the midpoint (X = 0 5) of the cavity This nature is similarfor other surface waviness with a slight variation in magnitudes Completelydifferent picture is observed at Ra = 105 For l = 0 0 three peaks againrepresent the three spots of highest temperature gradient with a highertemperature gradient at the mid point of the wave

Figure 12Nusselt number as afunction of Rayleighnumber for differentsurface waviness fortwo regxed aspect ratio(a) A = 0 25 and(b) A = 0 50

Figure 11Change of the localNusselt number alongthe length of the bottomwave

HFF138

1112

Figure 11 depicts the extent to which the local Nusselt number distribution isaffected by changes in the aspect ratio We observed that at Ra = 104 twopeaks move towards the center of the cavity with the increase of aspect ratioSimilar pattern was observed at Ra = 106 but the magnitude of the two peak ishigher It is also noticed that heat transfer rate is faster near the straight walland decrease rapidly with the increase of X Whatever be the values of aspectratio heat transfer rate is always minimum at the mid point of the bottom wall

Local Nusselt numbers are integrated to calculate the average value of theNusselt number according to equation (20) Average Nusselt number is plottedas a function of Rayleigh number in Figure 12 for two regxed aspect ratiosA = 0 25 and 050 for different surface waviness At lower Rayleigh number

Figure 13Nusselt number as afunction of Rayleighnumber for different

aspect ratio for two regxedsurface waviness(a) l = 0 05 and

(b) l = 0 10

Figure 14Nusselt number as a

function of surfacewaviness for regxedRayleigh number

(Ra = 5 pound104)Two lines correspond

to two different aspectratio of the enclosure asindicated in the legend

Effect of surfacewaviness and

aspect ratio

1113

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

HFF138

1114

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

where S1e is the surface vector representing the area of the cell face (x = const)and Sx

1e and Sy1e denotes its Cartesian components These are given in terms of

the CV vertex coordinates as follows

Sx1e = (yn 2 ys) and S y

1e = 2 (xn 2 xs) (11)

The mean cell face velocity components Ue and Ve are obtained byinterpolating neighborsrsquo nodal values in a way which ensures the stability ofgrid scheme The convection macrux of any variable f can be expressed as

FCe =

Z

Se

rfV dS lt (rfV )eS1e = Ccedilmefe (12)

The diffusion macrux of f is calculated as

FDe = 2

Z

Se

Gf grad f dS lt 2 (Gf grad f)eS1e (13)

By expressing the gradient of f at the cell face center ordfeordm which is taken here torepresent the mean value over the whole cell face through the derivatives inx and Z directions (Figure 2) and by discretizing these derivatives with CDSthe following expression results

FDe lt Gje

S1e Ccedil PE[(fE 2 fP )(S1e Ccedil S1e) + (fn 2 fs)e(S1e Ccedil S2e)] (14)

where PE is the vector representing the distance from P to E directed towardsE S2e is the surface vector orthogonal to PE and directed outwards positive Z

Figure 2A typical 2D-control

volume and the notationfor a Cartesian grid

Effect of surfacewaviness and

aspect ratio

1103

coordinate (Figure 2) representing the area in the surface Z = 0 bounded byP and E Its x and y components are

Sx2e = 2 ( yE 2 yP ) and Sy

2e = (xE 2 xP ) (15)

The volumetric source term is integrated by simply multiplying the speciregcsource at the control volume center P (which is assumed to represent the meanvalue over the whole cell) with the cell volume ie

Qqf =

Z

V

qf dV lt (qf)PDV (16)

The pressure term in the momentum equations are treated as body force andmay be regarded as pressure sources for the Cartesian velocity componentsThey are evaluated as

Q pui

= 2Z

S

pi i dS = 2Z

V

(grad pi i) dV

lt 2 [( pe 2 pw)S1P + ( pn 2 ps)S2p] Ccedil ii (17)

where the surface vectors S1p and S2p represent the area of the CV crosssection at x = 0 and Z = 0 respectively Since CVrsquos are bounded by straightlines these two vectors can be expressed through the CV surface vectors egS1P = (S1e + S1w) 2 Terms in the momentum equations not featuring inequation (9) are discretized using the same approach and added to the sourceterm After summing up all cell face macruxes and sources the discretizedtransport equation reduces to the following algebraic equation

APfP +nb

XAnbfnb = Qf (18)

where the coefregcients Anb contains the convective and diffusive macruxcontributions and Qf represents the source term The system of equations issolved by using Stonersquos SIP solver

Discretized momentum equations lead to an algebraic equation system forvelocity components U and V where pressure temperature and macruid propertiesare taken from the previous iteration except the regrst iteration where initialconditions are applied These linear equation systems are solved iteratively(inner iteration) to obtain an improved estimate of velocity The improvedvelocity regeld is then used to estimate new mass macruxes which satisfy thecontinuity equation Pressure-correction equation is then solved using the samelinear equation solver and to the same tolerance Energy equation is then

HFF138

1104

solved in the same manner to obtain better estimate of new solution Then theabove procedure is repeated for a new time step For time marching weselected Three Time Level Method which is a fully implicit scheme of secondorder accurate Iteration is continued until the difference between the twoconsecutive regeld values of variables is less than or equal to 10 2 5

AccuracyIn the present investigation regve combinations (20 pound10 40 pound20 80 pound40160 pound80 and 320 pound160) of control volumes are used to test the effect of gridsize on the accuracy of the predicted results Figure 3 shows the distribution ofaverage Nusselt numbers of the hot wall as a function of grid sizes for fourdifferent Rayleigh numbers It is clear from the reggure that at lower Rayleighnumber average Nusselt number is almost independent of grid sizes At higherRayleigh numbers the two higher grid sizes (160 pound 80 and 320 pound160) givealmost the same result It is well known that the high mesh reregnement alwaysprovides better result in the regnite-volume method The main disadvantage intaking higher mesh number is the increase in calculation time which can bereduced by using a higher speed of Pentium processor Our goal was to getbetter results Thus throughout this study the results are presented for 320 pound160 CVsrsquo for better accuracy Predicted results of average Nusselt numbers fora square cavity (A = 1 0 l = 0) with the same boundary conditions arecompared with the results given by Bejan (1984) and Ozisik (1985)Comparisons are shown in Figure 4 where average Nusselt number isplotted as a function of Rayleigh number Predicted results show a very goodagreement with the result of Ozisic at lower Rayleigh number but differ fromBejanrsquos prediction At higher Rayleigh number present prediction shows betteragreement with Bejan compared to Ozisicrsquos results

Figure 3Variation of averageNusselt number as a

function of number of CVfor A = 0 25 and

l = 0 05

Effect of surfacewaviness and

aspect ratio

1105

Results and discussionsFlow and thermal regeldThe macrow pattern inside the wavy enclosure and the temperature proregle arepresented in terms of streamlines and isothermal lines in Figure 5(a)-(d) forA = 0 25 l = 0 10 at four selected Rayleigh numbers At low Rayleighnumber when convection current inside the cavity is comparatively weak macruidstream near the hot wall tends to move towards the centerline or crest of thecavity were two streams from the opposite direction mix and rise upwardsTwo counter-rotating vortices with small cores are observed on the macrow regeldwere bi-cellular macrow pattern divides the cavity into two symmetric parts withrespect to the centerline (X = 0 5) of the cavity Due to the uniformtemperature gradient at low Rayleigh number (= 103) circulation inside thecavity is very weak Convection is less prominent at this Rayleigh number andheat transfer is mainly dominated by conduction The basic difference betweenthe isotherms in a wavy cavity with other geometry is their shape Isothermallines are nearly parallel to each other and follow the geometry of the wavysurfaces A further increase of Rayleigh numbers increases the circulationstrength inside the cavity Ra = 104 is also characterized by the bi-cellular macrowpattern but thermal regeld is completely different compared to Ra = 103 Hereuniform temperature proregle is changed and three high gradient spot isobserved due to rapid circulation of macruid inside the cavity Isotherms turn up(convective distortion) towards the cold wall due to the strong inmacruence of theconvection current At Ra = 105 multi-cellular macrow appears with four vorticesDue to comparatively high buoyant effect another vertical stream of the macruidappears near the adiabatic walls A periodic swirling of the isothermal linesappear at Ra = 105 due to the appearance of multi-cellular macrow Two highgradient spots appear near the bottom wall where isothermal lines areconcentrated Heat transfer rate is higher in magnitude at that spots

Figure 4Validation of averageNusselt number forA = 0 25 and l = 0 0with other referencework

HFF138

1106

Further increase of Rayleigh number increases the strength of the vorticeswhere four-cell multi-cellular macrow pattern turns into two cell bi-cellular patternThis bi-cellular macrow pattern is qualitatively different from the macrow pattern atlow Rayleigh number Periodic swirling nature of isothermal lines disappearsdue to the occurrence of reverse transition in macrow regeld Hot spots almostoccupy the bottom wall of the cavity

Transition of cell modeThe problem of the onset of hydrodynamic and thermal instabilities in thehorizontal layers of macruid heated from bellow is well suited to illustrate themany facets mathematical and physical of the general theory ofhydrodynamic stability Specially many aspects of the stability (orinstability) of the macruid-layer trapped between the two rigid walls (bottom hotand top cold) have been treated by many researchers over the last 100 years (forexample see Chandrasekhar 1961) The transition of cell mode of the macruidinside two rigid walls (in Rayleigh-Benard convection) is also a well establishedproblem However a mathematical analysis to predict the transition parameter

Figure 5Streamlines (top)

and isothermal lines(bottom) for A = 0 25

and l = 0 10

Effect of surfacewaviness and

aspect ratio

1107

(for example the critical Rayleigh number) and the physical mechanism is anextremely challenging task in the case of wavy walled enclosure under theRayleigh-Benard convection like situation due to the presence of two additionalparameters (A and l) Instead of performing a linear stability analysis wemodireged our numerical algorithm for post processing (a programme written inFORTRAN77 and connected to the software Tecplot 8) to predict the transitionbetween unicellular macrow to bicellular or multicellular macrow Initially forconstant l and A the macrow solver is executed for a range of Rayleigh numberskeeping a considerably large gap between the two consecutive Rayleighnumbers (50-100 depending on the value l) Once we get a rough picture of thetransition of cell mode we re-execute the macrow solver for a small range ofRayleigh numbers keeping a comparatively small gap between the twoconsecutive Rayleigh numbers (1 5) For example when l = 0 01 and A =0 25 the magnitude of the starting Rayleigh number for the macrow solver is1700 Note that the regrst critical Rayleigh number for the Rayleigh-Benardconvection with rigid walls is 1708 (Chandrasekhar 1961) Figure 6(a)-(h)shows the cell pattern for l = 0 01 and A = 0 25 for eight selected values ofRayleigh number A symmetric bicellular macrow pattern characterizes the macrowreglled inside the cavity until the value of Rayleigh number reaches 1905 Twoadditional cells appear near the adiabatic walls (Figure 6(d)) at which the

Figure 6Variation of cell patterninside a cavity withl = 0 01 and A = 0 25for different Rayleighnumber

HFF138

1108

Rayleigh number changes the bicellular macrow into a four-cell multicellular macrowThese cells grow in size with the increasing Rayleigh number but growth rateof the cells becomes slower when Ra approaches 104 A second transition ofthe cell mode is observed near Ra = 106 The overall cell modes and theirtransitions are shown in Figure 7 Note that a non-uniform scale is selected forRayleigh numbers in the abscissa of the Figure 7 for convenience Howeverthis scenario is completely different for l = 0 125 as shown in Figures 8 and 9The symmetric bicellular macrow changes into an asymmetric tricellular macrowaround Ra = 13 995 This tricellular macrow pattern remains unchanged for asmall interval of Rayleigh number (Ra = 13 995 plusmn 14 877) Then a secondtransition makes it to a four-cell multicellular macrow pattern A reverse transitionoccurs around Ra = 106 The overall cell modes and their transitions areshown in Figure 9 for this case

Heat transferHeat transfer is calculated in terms of local (NuL) and average (Nuav) Nusseltnumbers using the following equations

NuL =dk

hL =dk

k

DT

dT

dn

sup3 acute

w

raquo frac14=

dH

dn

sup3 acute

w

(19)

Nuav =1

S

Z S

0

NuL ds (20a)

S =

Z 1

0

1 +dY

dX

sup3 acute2s

dX =EllipticE 2p

2 l2

pplusmn sup2

p (20b)

Figure 7Transition of the cell

mode as a functionof Rayleigh number forl = 0 01 and A = 0 25

Effect of surfacewaviness and

aspect ratio

1109

where the special function ordfEllipticEordm is the elliptic integral of second kind(Abramowitz and Stegun 1965) Local Nusselt number distribution ispresented in Figures 10 and 11 Average Nusselt number distribution ispresented in Figures 12-14 Detail discussions on heat transfer is presented inthe following two paragraphs

Local Nusselt number distribution along the hot wavy wall is shown inFigure 10 for a constant Rayleigh number and four selected surface waviness

Figure 8Variation of cell patterninside a cavity withl = 0 125 and A = 0 25for different Rayleighnumber

HFF138

1110

(l = 0 0 005 0075 01) for an aspect ratio A = 0 25 At Ra = 104 heattransfer rate shows true periodic nature with respect to the axial distance alongthe length of the bottom wall for l = 0 0 where three peaks represent not onlythe three spots of highest temperature gradient but also characterize threepairs of cell generated due to the macruid motion inside the cavity However forl = 0 05 0075 01 local Nusselt number distribution changed signiregcantlydue to low temperature gradient at the midpoint of the cavity In that case heat

Figure 9Transition of the cell

mode as a functionof Rayleigh number for

l = 0 125 and A = 0 25

Figure 10Change of the local

Nusselt number alongthe length of the bottom

wave

Effect of surfacewaviness and

aspect ratio

1111

transfer rate increases up to the X = 0 15 and falls along the wavy wallgradually up to the midpoint (X = 0 5) of the cavity This nature is similarfor other surface waviness with a slight variation in magnitudes Completelydifferent picture is observed at Ra = 105 For l = 0 0 three peaks againrepresent the three spots of highest temperature gradient with a highertemperature gradient at the mid point of the wave

Figure 12Nusselt number as afunction of Rayleighnumber for differentsurface waviness fortwo regxed aspect ratio(a) A = 0 25 and(b) A = 0 50

Figure 11Change of the localNusselt number alongthe length of the bottomwave

HFF138

1112

Figure 11 depicts the extent to which the local Nusselt number distribution isaffected by changes in the aspect ratio We observed that at Ra = 104 twopeaks move towards the center of the cavity with the increase of aspect ratioSimilar pattern was observed at Ra = 106 but the magnitude of the two peak ishigher It is also noticed that heat transfer rate is faster near the straight walland decrease rapidly with the increase of X Whatever be the values of aspectratio heat transfer rate is always minimum at the mid point of the bottom wall

Local Nusselt numbers are integrated to calculate the average value of theNusselt number according to equation (20) Average Nusselt number is plottedas a function of Rayleigh number in Figure 12 for two regxed aspect ratiosA = 0 25 and 050 for different surface waviness At lower Rayleigh number

Figure 13Nusselt number as afunction of Rayleighnumber for different

aspect ratio for two regxedsurface waviness(a) l = 0 05 and

(b) l = 0 10

Figure 14Nusselt number as a

function of surfacewaviness for regxedRayleigh number

(Ra = 5 pound104)Two lines correspond

to two different aspectratio of the enclosure asindicated in the legend

Effect of surfacewaviness and

aspect ratio

1113

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

HFF138

1114

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

coordinate (Figure 2) representing the area in the surface Z = 0 bounded byP and E Its x and y components are

Sx2e = 2 ( yE 2 yP ) and Sy

2e = (xE 2 xP ) (15)

The volumetric source term is integrated by simply multiplying the speciregcsource at the control volume center P (which is assumed to represent the meanvalue over the whole cell) with the cell volume ie

Qqf =

Z

V

qf dV lt (qf)PDV (16)

The pressure term in the momentum equations are treated as body force andmay be regarded as pressure sources for the Cartesian velocity componentsThey are evaluated as

Q pui

= 2Z

S

pi i dS = 2Z

V

(grad pi i) dV

lt 2 [( pe 2 pw)S1P + ( pn 2 ps)S2p] Ccedil ii (17)

where the surface vectors S1p and S2p represent the area of the CV crosssection at x = 0 and Z = 0 respectively Since CVrsquos are bounded by straightlines these two vectors can be expressed through the CV surface vectors egS1P = (S1e + S1w) 2 Terms in the momentum equations not featuring inequation (9) are discretized using the same approach and added to the sourceterm After summing up all cell face macruxes and sources the discretizedtransport equation reduces to the following algebraic equation

APfP +nb

XAnbfnb = Qf (18)

where the coefregcients Anb contains the convective and diffusive macruxcontributions and Qf represents the source term The system of equations issolved by using Stonersquos SIP solver

Discretized momentum equations lead to an algebraic equation system forvelocity components U and V where pressure temperature and macruid propertiesare taken from the previous iteration except the regrst iteration where initialconditions are applied These linear equation systems are solved iteratively(inner iteration) to obtain an improved estimate of velocity The improvedvelocity regeld is then used to estimate new mass macruxes which satisfy thecontinuity equation Pressure-correction equation is then solved using the samelinear equation solver and to the same tolerance Energy equation is then

HFF138

1104

solved in the same manner to obtain better estimate of new solution Then theabove procedure is repeated for a new time step For time marching weselected Three Time Level Method which is a fully implicit scheme of secondorder accurate Iteration is continued until the difference between the twoconsecutive regeld values of variables is less than or equal to 10 2 5

AccuracyIn the present investigation regve combinations (20 pound10 40 pound20 80 pound40160 pound80 and 320 pound160) of control volumes are used to test the effect of gridsize on the accuracy of the predicted results Figure 3 shows the distribution ofaverage Nusselt numbers of the hot wall as a function of grid sizes for fourdifferent Rayleigh numbers It is clear from the reggure that at lower Rayleighnumber average Nusselt number is almost independent of grid sizes At higherRayleigh numbers the two higher grid sizes (160 pound 80 and 320 pound160) givealmost the same result It is well known that the high mesh reregnement alwaysprovides better result in the regnite-volume method The main disadvantage intaking higher mesh number is the increase in calculation time which can bereduced by using a higher speed of Pentium processor Our goal was to getbetter results Thus throughout this study the results are presented for 320 pound160 CVsrsquo for better accuracy Predicted results of average Nusselt numbers fora square cavity (A = 1 0 l = 0) with the same boundary conditions arecompared with the results given by Bejan (1984) and Ozisik (1985)Comparisons are shown in Figure 4 where average Nusselt number isplotted as a function of Rayleigh number Predicted results show a very goodagreement with the result of Ozisic at lower Rayleigh number but differ fromBejanrsquos prediction At higher Rayleigh number present prediction shows betteragreement with Bejan compared to Ozisicrsquos results

Figure 3Variation of averageNusselt number as a

function of number of CVfor A = 0 25 and

l = 0 05

Effect of surfacewaviness and

aspect ratio

1105

Results and discussionsFlow and thermal regeldThe macrow pattern inside the wavy enclosure and the temperature proregle arepresented in terms of streamlines and isothermal lines in Figure 5(a)-(d) forA = 0 25 l = 0 10 at four selected Rayleigh numbers At low Rayleighnumber when convection current inside the cavity is comparatively weak macruidstream near the hot wall tends to move towards the centerline or crest of thecavity were two streams from the opposite direction mix and rise upwardsTwo counter-rotating vortices with small cores are observed on the macrow regeldwere bi-cellular macrow pattern divides the cavity into two symmetric parts withrespect to the centerline (X = 0 5) of the cavity Due to the uniformtemperature gradient at low Rayleigh number (= 103) circulation inside thecavity is very weak Convection is less prominent at this Rayleigh number andheat transfer is mainly dominated by conduction The basic difference betweenthe isotherms in a wavy cavity with other geometry is their shape Isothermallines are nearly parallel to each other and follow the geometry of the wavysurfaces A further increase of Rayleigh numbers increases the circulationstrength inside the cavity Ra = 104 is also characterized by the bi-cellular macrowpattern but thermal regeld is completely different compared to Ra = 103 Hereuniform temperature proregle is changed and three high gradient spot isobserved due to rapid circulation of macruid inside the cavity Isotherms turn up(convective distortion) towards the cold wall due to the strong inmacruence of theconvection current At Ra = 105 multi-cellular macrow appears with four vorticesDue to comparatively high buoyant effect another vertical stream of the macruidappears near the adiabatic walls A periodic swirling of the isothermal linesappear at Ra = 105 due to the appearance of multi-cellular macrow Two highgradient spots appear near the bottom wall where isothermal lines areconcentrated Heat transfer rate is higher in magnitude at that spots

Figure 4Validation of averageNusselt number forA = 0 25 and l = 0 0with other referencework

HFF138

1106

Further increase of Rayleigh number increases the strength of the vorticeswhere four-cell multi-cellular macrow pattern turns into two cell bi-cellular patternThis bi-cellular macrow pattern is qualitatively different from the macrow pattern atlow Rayleigh number Periodic swirling nature of isothermal lines disappearsdue to the occurrence of reverse transition in macrow regeld Hot spots almostoccupy the bottom wall of the cavity

Transition of cell modeThe problem of the onset of hydrodynamic and thermal instabilities in thehorizontal layers of macruid heated from bellow is well suited to illustrate themany facets mathematical and physical of the general theory ofhydrodynamic stability Specially many aspects of the stability (orinstability) of the macruid-layer trapped between the two rigid walls (bottom hotand top cold) have been treated by many researchers over the last 100 years (forexample see Chandrasekhar 1961) The transition of cell mode of the macruidinside two rigid walls (in Rayleigh-Benard convection) is also a well establishedproblem However a mathematical analysis to predict the transition parameter

Figure 5Streamlines (top)

and isothermal lines(bottom) for A = 0 25

and l = 0 10

Effect of surfacewaviness and

aspect ratio

1107

(for example the critical Rayleigh number) and the physical mechanism is anextremely challenging task in the case of wavy walled enclosure under theRayleigh-Benard convection like situation due to the presence of two additionalparameters (A and l) Instead of performing a linear stability analysis wemodireged our numerical algorithm for post processing (a programme written inFORTRAN77 and connected to the software Tecplot 8) to predict the transitionbetween unicellular macrow to bicellular or multicellular macrow Initially forconstant l and A the macrow solver is executed for a range of Rayleigh numberskeeping a considerably large gap between the two consecutive Rayleighnumbers (50-100 depending on the value l) Once we get a rough picture of thetransition of cell mode we re-execute the macrow solver for a small range ofRayleigh numbers keeping a comparatively small gap between the twoconsecutive Rayleigh numbers (1 5) For example when l = 0 01 and A =0 25 the magnitude of the starting Rayleigh number for the macrow solver is1700 Note that the regrst critical Rayleigh number for the Rayleigh-Benardconvection with rigid walls is 1708 (Chandrasekhar 1961) Figure 6(a)-(h)shows the cell pattern for l = 0 01 and A = 0 25 for eight selected values ofRayleigh number A symmetric bicellular macrow pattern characterizes the macrowreglled inside the cavity until the value of Rayleigh number reaches 1905 Twoadditional cells appear near the adiabatic walls (Figure 6(d)) at which the

Figure 6Variation of cell patterninside a cavity withl = 0 01 and A = 0 25for different Rayleighnumber

HFF138

1108

Rayleigh number changes the bicellular macrow into a four-cell multicellular macrowThese cells grow in size with the increasing Rayleigh number but growth rateof the cells becomes slower when Ra approaches 104 A second transition ofthe cell mode is observed near Ra = 106 The overall cell modes and theirtransitions are shown in Figure 7 Note that a non-uniform scale is selected forRayleigh numbers in the abscissa of the Figure 7 for convenience Howeverthis scenario is completely different for l = 0 125 as shown in Figures 8 and 9The symmetric bicellular macrow changes into an asymmetric tricellular macrowaround Ra = 13 995 This tricellular macrow pattern remains unchanged for asmall interval of Rayleigh number (Ra = 13 995 plusmn 14 877) Then a secondtransition makes it to a four-cell multicellular macrow pattern A reverse transitionoccurs around Ra = 106 The overall cell modes and their transitions areshown in Figure 9 for this case

Heat transferHeat transfer is calculated in terms of local (NuL) and average (Nuav) Nusseltnumbers using the following equations

NuL =dk

hL =dk

k

DT

dT

dn

sup3 acute

w

raquo frac14=

dH

dn

sup3 acute

w

(19)

Nuav =1

S

Z S

0

NuL ds (20a)

S =

Z 1

0

1 +dY

dX

sup3 acute2s

dX =EllipticE 2p

2 l2

pplusmn sup2

p (20b)

Figure 7Transition of the cell

mode as a functionof Rayleigh number forl = 0 01 and A = 0 25

Effect of surfacewaviness and

aspect ratio

1109

where the special function ordfEllipticEordm is the elliptic integral of second kind(Abramowitz and Stegun 1965) Local Nusselt number distribution ispresented in Figures 10 and 11 Average Nusselt number distribution ispresented in Figures 12-14 Detail discussions on heat transfer is presented inthe following two paragraphs

Local Nusselt number distribution along the hot wavy wall is shown inFigure 10 for a constant Rayleigh number and four selected surface waviness

Figure 8Variation of cell patterninside a cavity withl = 0 125 and A = 0 25for different Rayleighnumber

HFF138

1110

(l = 0 0 005 0075 01) for an aspect ratio A = 0 25 At Ra = 104 heattransfer rate shows true periodic nature with respect to the axial distance alongthe length of the bottom wall for l = 0 0 where three peaks represent not onlythe three spots of highest temperature gradient but also characterize threepairs of cell generated due to the macruid motion inside the cavity However forl = 0 05 0075 01 local Nusselt number distribution changed signiregcantlydue to low temperature gradient at the midpoint of the cavity In that case heat

Figure 9Transition of the cell

mode as a functionof Rayleigh number for

l = 0 125 and A = 0 25

Figure 10Change of the local

Nusselt number alongthe length of the bottom

wave

Effect of surfacewaviness and

aspect ratio

1111

transfer rate increases up to the X = 0 15 and falls along the wavy wallgradually up to the midpoint (X = 0 5) of the cavity This nature is similarfor other surface waviness with a slight variation in magnitudes Completelydifferent picture is observed at Ra = 105 For l = 0 0 three peaks againrepresent the three spots of highest temperature gradient with a highertemperature gradient at the mid point of the wave

Figure 12Nusselt number as afunction of Rayleighnumber for differentsurface waviness fortwo regxed aspect ratio(a) A = 0 25 and(b) A = 0 50

Figure 11Change of the localNusselt number alongthe length of the bottomwave

HFF138

1112

Figure 11 depicts the extent to which the local Nusselt number distribution isaffected by changes in the aspect ratio We observed that at Ra = 104 twopeaks move towards the center of the cavity with the increase of aspect ratioSimilar pattern was observed at Ra = 106 but the magnitude of the two peak ishigher It is also noticed that heat transfer rate is faster near the straight walland decrease rapidly with the increase of X Whatever be the values of aspectratio heat transfer rate is always minimum at the mid point of the bottom wall

Local Nusselt numbers are integrated to calculate the average value of theNusselt number according to equation (20) Average Nusselt number is plottedas a function of Rayleigh number in Figure 12 for two regxed aspect ratiosA = 0 25 and 050 for different surface waviness At lower Rayleigh number

Figure 13Nusselt number as afunction of Rayleighnumber for different

aspect ratio for two regxedsurface waviness(a) l = 0 05 and

(b) l = 0 10

Figure 14Nusselt number as a

function of surfacewaviness for regxedRayleigh number

(Ra = 5 pound104)Two lines correspond

to two different aspectratio of the enclosure asindicated in the legend

Effect of surfacewaviness and

aspect ratio

1113

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

HFF138

1114

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

solved in the same manner to obtain better estimate of new solution Then theabove procedure is repeated for a new time step For time marching weselected Three Time Level Method which is a fully implicit scheme of secondorder accurate Iteration is continued until the difference between the twoconsecutive regeld values of variables is less than or equal to 10 2 5

AccuracyIn the present investigation regve combinations (20 pound10 40 pound20 80 pound40160 pound80 and 320 pound160) of control volumes are used to test the effect of gridsize on the accuracy of the predicted results Figure 3 shows the distribution ofaverage Nusselt numbers of the hot wall as a function of grid sizes for fourdifferent Rayleigh numbers It is clear from the reggure that at lower Rayleighnumber average Nusselt number is almost independent of grid sizes At higherRayleigh numbers the two higher grid sizes (160 pound 80 and 320 pound160) givealmost the same result It is well known that the high mesh reregnement alwaysprovides better result in the regnite-volume method The main disadvantage intaking higher mesh number is the increase in calculation time which can bereduced by using a higher speed of Pentium processor Our goal was to getbetter results Thus throughout this study the results are presented for 320 pound160 CVsrsquo for better accuracy Predicted results of average Nusselt numbers fora square cavity (A = 1 0 l = 0) with the same boundary conditions arecompared with the results given by Bejan (1984) and Ozisik (1985)Comparisons are shown in Figure 4 where average Nusselt number isplotted as a function of Rayleigh number Predicted results show a very goodagreement with the result of Ozisic at lower Rayleigh number but differ fromBejanrsquos prediction At higher Rayleigh number present prediction shows betteragreement with Bejan compared to Ozisicrsquos results

Figure 3Variation of averageNusselt number as a

function of number of CVfor A = 0 25 and

l = 0 05

Effect of surfacewaviness and

aspect ratio

1105

Results and discussionsFlow and thermal regeldThe macrow pattern inside the wavy enclosure and the temperature proregle arepresented in terms of streamlines and isothermal lines in Figure 5(a)-(d) forA = 0 25 l = 0 10 at four selected Rayleigh numbers At low Rayleighnumber when convection current inside the cavity is comparatively weak macruidstream near the hot wall tends to move towards the centerline or crest of thecavity were two streams from the opposite direction mix and rise upwardsTwo counter-rotating vortices with small cores are observed on the macrow regeldwere bi-cellular macrow pattern divides the cavity into two symmetric parts withrespect to the centerline (X = 0 5) of the cavity Due to the uniformtemperature gradient at low Rayleigh number (= 103) circulation inside thecavity is very weak Convection is less prominent at this Rayleigh number andheat transfer is mainly dominated by conduction The basic difference betweenthe isotherms in a wavy cavity with other geometry is their shape Isothermallines are nearly parallel to each other and follow the geometry of the wavysurfaces A further increase of Rayleigh numbers increases the circulationstrength inside the cavity Ra = 104 is also characterized by the bi-cellular macrowpattern but thermal regeld is completely different compared to Ra = 103 Hereuniform temperature proregle is changed and three high gradient spot isobserved due to rapid circulation of macruid inside the cavity Isotherms turn up(convective distortion) towards the cold wall due to the strong inmacruence of theconvection current At Ra = 105 multi-cellular macrow appears with four vorticesDue to comparatively high buoyant effect another vertical stream of the macruidappears near the adiabatic walls A periodic swirling of the isothermal linesappear at Ra = 105 due to the appearance of multi-cellular macrow Two highgradient spots appear near the bottom wall where isothermal lines areconcentrated Heat transfer rate is higher in magnitude at that spots

Figure 4Validation of averageNusselt number forA = 0 25 and l = 0 0with other referencework

HFF138

1106

Further increase of Rayleigh number increases the strength of the vorticeswhere four-cell multi-cellular macrow pattern turns into two cell bi-cellular patternThis bi-cellular macrow pattern is qualitatively different from the macrow pattern atlow Rayleigh number Periodic swirling nature of isothermal lines disappearsdue to the occurrence of reverse transition in macrow regeld Hot spots almostoccupy the bottom wall of the cavity

Transition of cell modeThe problem of the onset of hydrodynamic and thermal instabilities in thehorizontal layers of macruid heated from bellow is well suited to illustrate themany facets mathematical and physical of the general theory ofhydrodynamic stability Specially many aspects of the stability (orinstability) of the macruid-layer trapped between the two rigid walls (bottom hotand top cold) have been treated by many researchers over the last 100 years (forexample see Chandrasekhar 1961) The transition of cell mode of the macruidinside two rigid walls (in Rayleigh-Benard convection) is also a well establishedproblem However a mathematical analysis to predict the transition parameter

Figure 5Streamlines (top)

and isothermal lines(bottom) for A = 0 25

and l = 0 10

Effect of surfacewaviness and

aspect ratio

1107

(for example the critical Rayleigh number) and the physical mechanism is anextremely challenging task in the case of wavy walled enclosure under theRayleigh-Benard convection like situation due to the presence of two additionalparameters (A and l) Instead of performing a linear stability analysis wemodireged our numerical algorithm for post processing (a programme written inFORTRAN77 and connected to the software Tecplot 8) to predict the transitionbetween unicellular macrow to bicellular or multicellular macrow Initially forconstant l and A the macrow solver is executed for a range of Rayleigh numberskeeping a considerably large gap between the two consecutive Rayleighnumbers (50-100 depending on the value l) Once we get a rough picture of thetransition of cell mode we re-execute the macrow solver for a small range ofRayleigh numbers keeping a comparatively small gap between the twoconsecutive Rayleigh numbers (1 5) For example when l = 0 01 and A =0 25 the magnitude of the starting Rayleigh number for the macrow solver is1700 Note that the regrst critical Rayleigh number for the Rayleigh-Benardconvection with rigid walls is 1708 (Chandrasekhar 1961) Figure 6(a)-(h)shows the cell pattern for l = 0 01 and A = 0 25 for eight selected values ofRayleigh number A symmetric bicellular macrow pattern characterizes the macrowreglled inside the cavity until the value of Rayleigh number reaches 1905 Twoadditional cells appear near the adiabatic walls (Figure 6(d)) at which the

Figure 6Variation of cell patterninside a cavity withl = 0 01 and A = 0 25for different Rayleighnumber

HFF138

1108

Rayleigh number changes the bicellular macrow into a four-cell multicellular macrowThese cells grow in size with the increasing Rayleigh number but growth rateof the cells becomes slower when Ra approaches 104 A second transition ofthe cell mode is observed near Ra = 106 The overall cell modes and theirtransitions are shown in Figure 7 Note that a non-uniform scale is selected forRayleigh numbers in the abscissa of the Figure 7 for convenience Howeverthis scenario is completely different for l = 0 125 as shown in Figures 8 and 9The symmetric bicellular macrow changes into an asymmetric tricellular macrowaround Ra = 13 995 This tricellular macrow pattern remains unchanged for asmall interval of Rayleigh number (Ra = 13 995 plusmn 14 877) Then a secondtransition makes it to a four-cell multicellular macrow pattern A reverse transitionoccurs around Ra = 106 The overall cell modes and their transitions areshown in Figure 9 for this case

Heat transferHeat transfer is calculated in terms of local (NuL) and average (Nuav) Nusseltnumbers using the following equations

NuL =dk

hL =dk

k

DT

dT

dn

sup3 acute

w

raquo frac14=

dH

dn

sup3 acute

w

(19)

Nuav =1

S

Z S

0

NuL ds (20a)

S =

Z 1

0

1 +dY

dX

sup3 acute2s

dX =EllipticE 2p

2 l2

pplusmn sup2

p (20b)

Figure 7Transition of the cell

mode as a functionof Rayleigh number forl = 0 01 and A = 0 25

Effect of surfacewaviness and

aspect ratio

1109

where the special function ordfEllipticEordm is the elliptic integral of second kind(Abramowitz and Stegun 1965) Local Nusselt number distribution ispresented in Figures 10 and 11 Average Nusselt number distribution ispresented in Figures 12-14 Detail discussions on heat transfer is presented inthe following two paragraphs

Local Nusselt number distribution along the hot wavy wall is shown inFigure 10 for a constant Rayleigh number and four selected surface waviness

Figure 8Variation of cell patterninside a cavity withl = 0 125 and A = 0 25for different Rayleighnumber

HFF138

1110

(l = 0 0 005 0075 01) for an aspect ratio A = 0 25 At Ra = 104 heattransfer rate shows true periodic nature with respect to the axial distance alongthe length of the bottom wall for l = 0 0 where three peaks represent not onlythe three spots of highest temperature gradient but also characterize threepairs of cell generated due to the macruid motion inside the cavity However forl = 0 05 0075 01 local Nusselt number distribution changed signiregcantlydue to low temperature gradient at the midpoint of the cavity In that case heat

Figure 9Transition of the cell

mode as a functionof Rayleigh number for

l = 0 125 and A = 0 25

Figure 10Change of the local

Nusselt number alongthe length of the bottom

wave

Effect of surfacewaviness and

aspect ratio

1111

transfer rate increases up to the X = 0 15 and falls along the wavy wallgradually up to the midpoint (X = 0 5) of the cavity This nature is similarfor other surface waviness with a slight variation in magnitudes Completelydifferent picture is observed at Ra = 105 For l = 0 0 three peaks againrepresent the three spots of highest temperature gradient with a highertemperature gradient at the mid point of the wave

Figure 12Nusselt number as afunction of Rayleighnumber for differentsurface waviness fortwo regxed aspect ratio(a) A = 0 25 and(b) A = 0 50

Figure 11Change of the localNusselt number alongthe length of the bottomwave

HFF138

1112

Figure 11 depicts the extent to which the local Nusselt number distribution isaffected by changes in the aspect ratio We observed that at Ra = 104 twopeaks move towards the center of the cavity with the increase of aspect ratioSimilar pattern was observed at Ra = 106 but the magnitude of the two peak ishigher It is also noticed that heat transfer rate is faster near the straight walland decrease rapidly with the increase of X Whatever be the values of aspectratio heat transfer rate is always minimum at the mid point of the bottom wall

Local Nusselt numbers are integrated to calculate the average value of theNusselt number according to equation (20) Average Nusselt number is plottedas a function of Rayleigh number in Figure 12 for two regxed aspect ratiosA = 0 25 and 050 for different surface waviness At lower Rayleigh number

Figure 13Nusselt number as afunction of Rayleighnumber for different

aspect ratio for two regxedsurface waviness(a) l = 0 05 and

(b) l = 0 10

Figure 14Nusselt number as a

function of surfacewaviness for regxedRayleigh number

(Ra = 5 pound104)Two lines correspond

to two different aspectratio of the enclosure asindicated in the legend

Effect of surfacewaviness and

aspect ratio

1113

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

HFF138

1114

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

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macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

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Results and discussionsFlow and thermal regeldThe macrow pattern inside the wavy enclosure and the temperature proregle arepresented in terms of streamlines and isothermal lines in Figure 5(a)-(d) forA = 0 25 l = 0 10 at four selected Rayleigh numbers At low Rayleighnumber when convection current inside the cavity is comparatively weak macruidstream near the hot wall tends to move towards the centerline or crest of thecavity were two streams from the opposite direction mix and rise upwardsTwo counter-rotating vortices with small cores are observed on the macrow regeldwere bi-cellular macrow pattern divides the cavity into two symmetric parts withrespect to the centerline (X = 0 5) of the cavity Due to the uniformtemperature gradient at low Rayleigh number (= 103) circulation inside thecavity is very weak Convection is less prominent at this Rayleigh number andheat transfer is mainly dominated by conduction The basic difference betweenthe isotherms in a wavy cavity with other geometry is their shape Isothermallines are nearly parallel to each other and follow the geometry of the wavysurfaces A further increase of Rayleigh numbers increases the circulationstrength inside the cavity Ra = 104 is also characterized by the bi-cellular macrowpattern but thermal regeld is completely different compared to Ra = 103 Hereuniform temperature proregle is changed and three high gradient spot isobserved due to rapid circulation of macruid inside the cavity Isotherms turn up(convective distortion) towards the cold wall due to the strong inmacruence of theconvection current At Ra = 105 multi-cellular macrow appears with four vorticesDue to comparatively high buoyant effect another vertical stream of the macruidappears near the adiabatic walls A periodic swirling of the isothermal linesappear at Ra = 105 due to the appearance of multi-cellular macrow Two highgradient spots appear near the bottom wall where isothermal lines areconcentrated Heat transfer rate is higher in magnitude at that spots

Figure 4Validation of averageNusselt number forA = 0 25 and l = 0 0with other referencework

HFF138

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Further increase of Rayleigh number increases the strength of the vorticeswhere four-cell multi-cellular macrow pattern turns into two cell bi-cellular patternThis bi-cellular macrow pattern is qualitatively different from the macrow pattern atlow Rayleigh number Periodic swirling nature of isothermal lines disappearsdue to the occurrence of reverse transition in macrow regeld Hot spots almostoccupy the bottom wall of the cavity

Transition of cell modeThe problem of the onset of hydrodynamic and thermal instabilities in thehorizontal layers of macruid heated from bellow is well suited to illustrate themany facets mathematical and physical of the general theory ofhydrodynamic stability Specially many aspects of the stability (orinstability) of the macruid-layer trapped between the two rigid walls (bottom hotand top cold) have been treated by many researchers over the last 100 years (forexample see Chandrasekhar 1961) The transition of cell mode of the macruidinside two rigid walls (in Rayleigh-Benard convection) is also a well establishedproblem However a mathematical analysis to predict the transition parameter

Figure 5Streamlines (top)

and isothermal lines(bottom) for A = 0 25

and l = 0 10

Effect of surfacewaviness and

aspect ratio

1107

(for example the critical Rayleigh number) and the physical mechanism is anextremely challenging task in the case of wavy walled enclosure under theRayleigh-Benard convection like situation due to the presence of two additionalparameters (A and l) Instead of performing a linear stability analysis wemodireged our numerical algorithm for post processing (a programme written inFORTRAN77 and connected to the software Tecplot 8) to predict the transitionbetween unicellular macrow to bicellular or multicellular macrow Initially forconstant l and A the macrow solver is executed for a range of Rayleigh numberskeeping a considerably large gap between the two consecutive Rayleighnumbers (50-100 depending on the value l) Once we get a rough picture of thetransition of cell mode we re-execute the macrow solver for a small range ofRayleigh numbers keeping a comparatively small gap between the twoconsecutive Rayleigh numbers (1 5) For example when l = 0 01 and A =0 25 the magnitude of the starting Rayleigh number for the macrow solver is1700 Note that the regrst critical Rayleigh number for the Rayleigh-Benardconvection with rigid walls is 1708 (Chandrasekhar 1961) Figure 6(a)-(h)shows the cell pattern for l = 0 01 and A = 0 25 for eight selected values ofRayleigh number A symmetric bicellular macrow pattern characterizes the macrowreglled inside the cavity until the value of Rayleigh number reaches 1905 Twoadditional cells appear near the adiabatic walls (Figure 6(d)) at which the

Figure 6Variation of cell patterninside a cavity withl = 0 01 and A = 0 25for different Rayleighnumber

HFF138

1108

Rayleigh number changes the bicellular macrow into a four-cell multicellular macrowThese cells grow in size with the increasing Rayleigh number but growth rateof the cells becomes slower when Ra approaches 104 A second transition ofthe cell mode is observed near Ra = 106 The overall cell modes and theirtransitions are shown in Figure 7 Note that a non-uniform scale is selected forRayleigh numbers in the abscissa of the Figure 7 for convenience Howeverthis scenario is completely different for l = 0 125 as shown in Figures 8 and 9The symmetric bicellular macrow changes into an asymmetric tricellular macrowaround Ra = 13 995 This tricellular macrow pattern remains unchanged for asmall interval of Rayleigh number (Ra = 13 995 plusmn 14 877) Then a secondtransition makes it to a four-cell multicellular macrow pattern A reverse transitionoccurs around Ra = 106 The overall cell modes and their transitions areshown in Figure 9 for this case

Heat transferHeat transfer is calculated in terms of local (NuL) and average (Nuav) Nusseltnumbers using the following equations

NuL =dk

hL =dk

k

DT

dT

dn

sup3 acute

w

raquo frac14=

dH

dn

sup3 acute

w

(19)

Nuav =1

S

Z S

0

NuL ds (20a)

S =

Z 1

0

1 +dY

dX

sup3 acute2s

dX =EllipticE 2p

2 l2

pplusmn sup2

p (20b)

Figure 7Transition of the cell

mode as a functionof Rayleigh number forl = 0 01 and A = 0 25

Effect of surfacewaviness and

aspect ratio

1109

where the special function ordfEllipticEordm is the elliptic integral of second kind(Abramowitz and Stegun 1965) Local Nusselt number distribution ispresented in Figures 10 and 11 Average Nusselt number distribution ispresented in Figures 12-14 Detail discussions on heat transfer is presented inthe following two paragraphs

Local Nusselt number distribution along the hot wavy wall is shown inFigure 10 for a constant Rayleigh number and four selected surface waviness

Figure 8Variation of cell patterninside a cavity withl = 0 125 and A = 0 25for different Rayleighnumber

HFF138

1110

(l = 0 0 005 0075 01) for an aspect ratio A = 0 25 At Ra = 104 heattransfer rate shows true periodic nature with respect to the axial distance alongthe length of the bottom wall for l = 0 0 where three peaks represent not onlythe three spots of highest temperature gradient but also characterize threepairs of cell generated due to the macruid motion inside the cavity However forl = 0 05 0075 01 local Nusselt number distribution changed signiregcantlydue to low temperature gradient at the midpoint of the cavity In that case heat

Figure 9Transition of the cell

mode as a functionof Rayleigh number for

l = 0 125 and A = 0 25

Figure 10Change of the local

Nusselt number alongthe length of the bottom

wave

Effect of surfacewaviness and

aspect ratio

1111

transfer rate increases up to the X = 0 15 and falls along the wavy wallgradually up to the midpoint (X = 0 5) of the cavity This nature is similarfor other surface waviness with a slight variation in magnitudes Completelydifferent picture is observed at Ra = 105 For l = 0 0 three peaks againrepresent the three spots of highest temperature gradient with a highertemperature gradient at the mid point of the wave

Figure 12Nusselt number as afunction of Rayleighnumber for differentsurface waviness fortwo regxed aspect ratio(a) A = 0 25 and(b) A = 0 50

Figure 11Change of the localNusselt number alongthe length of the bottomwave

HFF138

1112

Figure 11 depicts the extent to which the local Nusselt number distribution isaffected by changes in the aspect ratio We observed that at Ra = 104 twopeaks move towards the center of the cavity with the increase of aspect ratioSimilar pattern was observed at Ra = 106 but the magnitude of the two peak ishigher It is also noticed that heat transfer rate is faster near the straight walland decrease rapidly with the increase of X Whatever be the values of aspectratio heat transfer rate is always minimum at the mid point of the bottom wall

Local Nusselt numbers are integrated to calculate the average value of theNusselt number according to equation (20) Average Nusselt number is plottedas a function of Rayleigh number in Figure 12 for two regxed aspect ratiosA = 0 25 and 050 for different surface waviness At lower Rayleigh number

Figure 13Nusselt number as afunction of Rayleighnumber for different

aspect ratio for two regxedsurface waviness(a) l = 0 05 and

(b) l = 0 10

Figure 14Nusselt number as a

function of surfacewaviness for regxedRayleigh number

(Ra = 5 pound104)Two lines correspond

to two different aspectratio of the enclosure asindicated in the legend

Effect of surfacewaviness and

aspect ratio

1113

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

HFF138

1114

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

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macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

Further increase of Rayleigh number increases the strength of the vorticeswhere four-cell multi-cellular macrow pattern turns into two cell bi-cellular patternThis bi-cellular macrow pattern is qualitatively different from the macrow pattern atlow Rayleigh number Periodic swirling nature of isothermal lines disappearsdue to the occurrence of reverse transition in macrow regeld Hot spots almostoccupy the bottom wall of the cavity

Transition of cell modeThe problem of the onset of hydrodynamic and thermal instabilities in thehorizontal layers of macruid heated from bellow is well suited to illustrate themany facets mathematical and physical of the general theory ofhydrodynamic stability Specially many aspects of the stability (orinstability) of the macruid-layer trapped between the two rigid walls (bottom hotand top cold) have been treated by many researchers over the last 100 years (forexample see Chandrasekhar 1961) The transition of cell mode of the macruidinside two rigid walls (in Rayleigh-Benard convection) is also a well establishedproblem However a mathematical analysis to predict the transition parameter

Figure 5Streamlines (top)

and isothermal lines(bottom) for A = 0 25

and l = 0 10

Effect of surfacewaviness and

aspect ratio

1107

(for example the critical Rayleigh number) and the physical mechanism is anextremely challenging task in the case of wavy walled enclosure under theRayleigh-Benard convection like situation due to the presence of two additionalparameters (A and l) Instead of performing a linear stability analysis wemodireged our numerical algorithm for post processing (a programme written inFORTRAN77 and connected to the software Tecplot 8) to predict the transitionbetween unicellular macrow to bicellular or multicellular macrow Initially forconstant l and A the macrow solver is executed for a range of Rayleigh numberskeeping a considerably large gap between the two consecutive Rayleighnumbers (50-100 depending on the value l) Once we get a rough picture of thetransition of cell mode we re-execute the macrow solver for a small range ofRayleigh numbers keeping a comparatively small gap between the twoconsecutive Rayleigh numbers (1 5) For example when l = 0 01 and A =0 25 the magnitude of the starting Rayleigh number for the macrow solver is1700 Note that the regrst critical Rayleigh number for the Rayleigh-Benardconvection with rigid walls is 1708 (Chandrasekhar 1961) Figure 6(a)-(h)shows the cell pattern for l = 0 01 and A = 0 25 for eight selected values ofRayleigh number A symmetric bicellular macrow pattern characterizes the macrowreglled inside the cavity until the value of Rayleigh number reaches 1905 Twoadditional cells appear near the adiabatic walls (Figure 6(d)) at which the

Figure 6Variation of cell patterninside a cavity withl = 0 01 and A = 0 25for different Rayleighnumber

HFF138

1108

Rayleigh number changes the bicellular macrow into a four-cell multicellular macrowThese cells grow in size with the increasing Rayleigh number but growth rateof the cells becomes slower when Ra approaches 104 A second transition ofthe cell mode is observed near Ra = 106 The overall cell modes and theirtransitions are shown in Figure 7 Note that a non-uniform scale is selected forRayleigh numbers in the abscissa of the Figure 7 for convenience Howeverthis scenario is completely different for l = 0 125 as shown in Figures 8 and 9The symmetric bicellular macrow changes into an asymmetric tricellular macrowaround Ra = 13 995 This tricellular macrow pattern remains unchanged for asmall interval of Rayleigh number (Ra = 13 995 plusmn 14 877) Then a secondtransition makes it to a four-cell multicellular macrow pattern A reverse transitionoccurs around Ra = 106 The overall cell modes and their transitions areshown in Figure 9 for this case

Heat transferHeat transfer is calculated in terms of local (NuL) and average (Nuav) Nusseltnumbers using the following equations

NuL =dk

hL =dk

k

DT

dT

dn

sup3 acute

w

raquo frac14=

dH

dn

sup3 acute

w

(19)

Nuav =1

S

Z S

0

NuL ds (20a)

S =

Z 1

0

1 +dY

dX

sup3 acute2s

dX =EllipticE 2p

2 l2

pplusmn sup2

p (20b)

Figure 7Transition of the cell

mode as a functionof Rayleigh number forl = 0 01 and A = 0 25

Effect of surfacewaviness and

aspect ratio

1109

where the special function ordfEllipticEordm is the elliptic integral of second kind(Abramowitz and Stegun 1965) Local Nusselt number distribution ispresented in Figures 10 and 11 Average Nusselt number distribution ispresented in Figures 12-14 Detail discussions on heat transfer is presented inthe following two paragraphs

Local Nusselt number distribution along the hot wavy wall is shown inFigure 10 for a constant Rayleigh number and four selected surface waviness

Figure 8Variation of cell patterninside a cavity withl = 0 125 and A = 0 25for different Rayleighnumber

HFF138

1110

(l = 0 0 005 0075 01) for an aspect ratio A = 0 25 At Ra = 104 heattransfer rate shows true periodic nature with respect to the axial distance alongthe length of the bottom wall for l = 0 0 where three peaks represent not onlythe three spots of highest temperature gradient but also characterize threepairs of cell generated due to the macruid motion inside the cavity However forl = 0 05 0075 01 local Nusselt number distribution changed signiregcantlydue to low temperature gradient at the midpoint of the cavity In that case heat

Figure 9Transition of the cell

mode as a functionof Rayleigh number for

l = 0 125 and A = 0 25

Figure 10Change of the local

Nusselt number alongthe length of the bottom

wave

Effect of surfacewaviness and

aspect ratio

1111

transfer rate increases up to the X = 0 15 and falls along the wavy wallgradually up to the midpoint (X = 0 5) of the cavity This nature is similarfor other surface waviness with a slight variation in magnitudes Completelydifferent picture is observed at Ra = 105 For l = 0 0 three peaks againrepresent the three spots of highest temperature gradient with a highertemperature gradient at the mid point of the wave

Figure 12Nusselt number as afunction of Rayleighnumber for differentsurface waviness fortwo regxed aspect ratio(a) A = 0 25 and(b) A = 0 50

Figure 11Change of the localNusselt number alongthe length of the bottomwave

HFF138

1112

Figure 11 depicts the extent to which the local Nusselt number distribution isaffected by changes in the aspect ratio We observed that at Ra = 104 twopeaks move towards the center of the cavity with the increase of aspect ratioSimilar pattern was observed at Ra = 106 but the magnitude of the two peak ishigher It is also noticed that heat transfer rate is faster near the straight walland decrease rapidly with the increase of X Whatever be the values of aspectratio heat transfer rate is always minimum at the mid point of the bottom wall

Local Nusselt numbers are integrated to calculate the average value of theNusselt number according to equation (20) Average Nusselt number is plottedas a function of Rayleigh number in Figure 12 for two regxed aspect ratiosA = 0 25 and 050 for different surface waviness At lower Rayleigh number

Figure 13Nusselt number as afunction of Rayleighnumber for different

aspect ratio for two regxedsurface waviness(a) l = 0 05 and

(b) l = 0 10

Figure 14Nusselt number as a

function of surfacewaviness for regxedRayleigh number

(Ra = 5 pound104)Two lines correspond

to two different aspectratio of the enclosure asindicated in the legend

Effect of surfacewaviness and

aspect ratio

1113

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

HFF138

1114

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

(for example the critical Rayleigh number) and the physical mechanism is anextremely challenging task in the case of wavy walled enclosure under theRayleigh-Benard convection like situation due to the presence of two additionalparameters (A and l) Instead of performing a linear stability analysis wemodireged our numerical algorithm for post processing (a programme written inFORTRAN77 and connected to the software Tecplot 8) to predict the transitionbetween unicellular macrow to bicellular or multicellular macrow Initially forconstant l and A the macrow solver is executed for a range of Rayleigh numberskeeping a considerably large gap between the two consecutive Rayleighnumbers (50-100 depending on the value l) Once we get a rough picture of thetransition of cell mode we re-execute the macrow solver for a small range ofRayleigh numbers keeping a comparatively small gap between the twoconsecutive Rayleigh numbers (1 5) For example when l = 0 01 and A =0 25 the magnitude of the starting Rayleigh number for the macrow solver is1700 Note that the regrst critical Rayleigh number for the Rayleigh-Benardconvection with rigid walls is 1708 (Chandrasekhar 1961) Figure 6(a)-(h)shows the cell pattern for l = 0 01 and A = 0 25 for eight selected values ofRayleigh number A symmetric bicellular macrow pattern characterizes the macrowreglled inside the cavity until the value of Rayleigh number reaches 1905 Twoadditional cells appear near the adiabatic walls (Figure 6(d)) at which the

Figure 6Variation of cell patterninside a cavity withl = 0 01 and A = 0 25for different Rayleighnumber

HFF138

1108

Rayleigh number changes the bicellular macrow into a four-cell multicellular macrowThese cells grow in size with the increasing Rayleigh number but growth rateof the cells becomes slower when Ra approaches 104 A second transition ofthe cell mode is observed near Ra = 106 The overall cell modes and theirtransitions are shown in Figure 7 Note that a non-uniform scale is selected forRayleigh numbers in the abscissa of the Figure 7 for convenience Howeverthis scenario is completely different for l = 0 125 as shown in Figures 8 and 9The symmetric bicellular macrow changes into an asymmetric tricellular macrowaround Ra = 13 995 This tricellular macrow pattern remains unchanged for asmall interval of Rayleigh number (Ra = 13 995 plusmn 14 877) Then a secondtransition makes it to a four-cell multicellular macrow pattern A reverse transitionoccurs around Ra = 106 The overall cell modes and their transitions areshown in Figure 9 for this case

Heat transferHeat transfer is calculated in terms of local (NuL) and average (Nuav) Nusseltnumbers using the following equations

NuL =dk

hL =dk

k

DT

dT

dn

sup3 acute

w

raquo frac14=

dH

dn

sup3 acute

w

(19)

Nuav =1

S

Z S

0

NuL ds (20a)

S =

Z 1

0

1 +dY

dX

sup3 acute2s

dX =EllipticE 2p

2 l2

pplusmn sup2

p (20b)

Figure 7Transition of the cell

mode as a functionof Rayleigh number forl = 0 01 and A = 0 25

Effect of surfacewaviness and

aspect ratio

1109

where the special function ordfEllipticEordm is the elliptic integral of second kind(Abramowitz and Stegun 1965) Local Nusselt number distribution ispresented in Figures 10 and 11 Average Nusselt number distribution ispresented in Figures 12-14 Detail discussions on heat transfer is presented inthe following two paragraphs

Local Nusselt number distribution along the hot wavy wall is shown inFigure 10 for a constant Rayleigh number and four selected surface waviness

Figure 8Variation of cell patterninside a cavity withl = 0 125 and A = 0 25for different Rayleighnumber

HFF138

1110

(l = 0 0 005 0075 01) for an aspect ratio A = 0 25 At Ra = 104 heattransfer rate shows true periodic nature with respect to the axial distance alongthe length of the bottom wall for l = 0 0 where three peaks represent not onlythe three spots of highest temperature gradient but also characterize threepairs of cell generated due to the macruid motion inside the cavity However forl = 0 05 0075 01 local Nusselt number distribution changed signiregcantlydue to low temperature gradient at the midpoint of the cavity In that case heat

Figure 9Transition of the cell

mode as a functionof Rayleigh number for

l = 0 125 and A = 0 25

Figure 10Change of the local

Nusselt number alongthe length of the bottom

wave

Effect of surfacewaviness and

aspect ratio

1111

transfer rate increases up to the X = 0 15 and falls along the wavy wallgradually up to the midpoint (X = 0 5) of the cavity This nature is similarfor other surface waviness with a slight variation in magnitudes Completelydifferent picture is observed at Ra = 105 For l = 0 0 three peaks againrepresent the three spots of highest temperature gradient with a highertemperature gradient at the mid point of the wave

Figure 12Nusselt number as afunction of Rayleighnumber for differentsurface waviness fortwo regxed aspect ratio(a) A = 0 25 and(b) A = 0 50

Figure 11Change of the localNusselt number alongthe length of the bottomwave

HFF138

1112

Figure 11 depicts the extent to which the local Nusselt number distribution isaffected by changes in the aspect ratio We observed that at Ra = 104 twopeaks move towards the center of the cavity with the increase of aspect ratioSimilar pattern was observed at Ra = 106 but the magnitude of the two peak ishigher It is also noticed that heat transfer rate is faster near the straight walland decrease rapidly with the increase of X Whatever be the values of aspectratio heat transfer rate is always minimum at the mid point of the bottom wall

Local Nusselt numbers are integrated to calculate the average value of theNusselt number according to equation (20) Average Nusselt number is plottedas a function of Rayleigh number in Figure 12 for two regxed aspect ratiosA = 0 25 and 050 for different surface waviness At lower Rayleigh number

Figure 13Nusselt number as afunction of Rayleighnumber for different

aspect ratio for two regxedsurface waviness(a) l = 0 05 and

(b) l = 0 10

Figure 14Nusselt number as a

function of surfacewaviness for regxedRayleigh number

(Ra = 5 pound104)Two lines correspond

to two different aspectratio of the enclosure asindicated in the legend

Effect of surfacewaviness and

aspect ratio

1113

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

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cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

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generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

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macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

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Rayleigh number changes the bicellular macrow into a four-cell multicellular macrowThese cells grow in size with the increasing Rayleigh number but growth rateof the cells becomes slower when Ra approaches 104 A second transition ofthe cell mode is observed near Ra = 106 The overall cell modes and theirtransitions are shown in Figure 7 Note that a non-uniform scale is selected forRayleigh numbers in the abscissa of the Figure 7 for convenience Howeverthis scenario is completely different for l = 0 125 as shown in Figures 8 and 9The symmetric bicellular macrow changes into an asymmetric tricellular macrowaround Ra = 13 995 This tricellular macrow pattern remains unchanged for asmall interval of Rayleigh number (Ra = 13 995 plusmn 14 877) Then a secondtransition makes it to a four-cell multicellular macrow pattern A reverse transitionoccurs around Ra = 106 The overall cell modes and their transitions areshown in Figure 9 for this case

Heat transferHeat transfer is calculated in terms of local (NuL) and average (Nuav) Nusseltnumbers using the following equations

NuL =dk

hL =dk

k

DT

dT

dn

sup3 acute

w

raquo frac14=

dH

dn

sup3 acute

w

(19)

Nuav =1

S

Z S

0

NuL ds (20a)

S =

Z 1

0

1 +dY

dX

sup3 acute2s

dX =EllipticE 2p

2 l2

pplusmn sup2

p (20b)

Figure 7Transition of the cell

mode as a functionof Rayleigh number forl = 0 01 and A = 0 25

Effect of surfacewaviness and

aspect ratio

1109

where the special function ordfEllipticEordm is the elliptic integral of second kind(Abramowitz and Stegun 1965) Local Nusselt number distribution ispresented in Figures 10 and 11 Average Nusselt number distribution ispresented in Figures 12-14 Detail discussions on heat transfer is presented inthe following two paragraphs

Local Nusselt number distribution along the hot wavy wall is shown inFigure 10 for a constant Rayleigh number and four selected surface waviness

Figure 8Variation of cell patterninside a cavity withl = 0 125 and A = 0 25for different Rayleighnumber

HFF138

1110

(l = 0 0 005 0075 01) for an aspect ratio A = 0 25 At Ra = 104 heattransfer rate shows true periodic nature with respect to the axial distance alongthe length of the bottom wall for l = 0 0 where three peaks represent not onlythe three spots of highest temperature gradient but also characterize threepairs of cell generated due to the macruid motion inside the cavity However forl = 0 05 0075 01 local Nusselt number distribution changed signiregcantlydue to low temperature gradient at the midpoint of the cavity In that case heat

Figure 9Transition of the cell

mode as a functionof Rayleigh number for

l = 0 125 and A = 0 25

Figure 10Change of the local

Nusselt number alongthe length of the bottom

wave

Effect of surfacewaviness and

aspect ratio

1111

transfer rate increases up to the X = 0 15 and falls along the wavy wallgradually up to the midpoint (X = 0 5) of the cavity This nature is similarfor other surface waviness with a slight variation in magnitudes Completelydifferent picture is observed at Ra = 105 For l = 0 0 three peaks againrepresent the three spots of highest temperature gradient with a highertemperature gradient at the mid point of the wave

Figure 12Nusselt number as afunction of Rayleighnumber for differentsurface waviness fortwo regxed aspect ratio(a) A = 0 25 and(b) A = 0 50

Figure 11Change of the localNusselt number alongthe length of the bottomwave

HFF138

1112

Figure 11 depicts the extent to which the local Nusselt number distribution isaffected by changes in the aspect ratio We observed that at Ra = 104 twopeaks move towards the center of the cavity with the increase of aspect ratioSimilar pattern was observed at Ra = 106 but the magnitude of the two peak ishigher It is also noticed that heat transfer rate is faster near the straight walland decrease rapidly with the increase of X Whatever be the values of aspectratio heat transfer rate is always minimum at the mid point of the bottom wall

Local Nusselt numbers are integrated to calculate the average value of theNusselt number according to equation (20) Average Nusselt number is plottedas a function of Rayleigh number in Figure 12 for two regxed aspect ratiosA = 0 25 and 050 for different surface waviness At lower Rayleigh number

Figure 13Nusselt number as afunction of Rayleighnumber for different

aspect ratio for two regxedsurface waviness(a) l = 0 05 and

(b) l = 0 10

Figure 14Nusselt number as a

function of surfacewaviness for regxedRayleigh number

(Ra = 5 pound104)Two lines correspond

to two different aspectratio of the enclosure asindicated in the legend

Effect of surfacewaviness and

aspect ratio

1113

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

HFF138

1114

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

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where the special function ordfEllipticEordm is the elliptic integral of second kind(Abramowitz and Stegun 1965) Local Nusselt number distribution ispresented in Figures 10 and 11 Average Nusselt number distribution ispresented in Figures 12-14 Detail discussions on heat transfer is presented inthe following two paragraphs

Local Nusselt number distribution along the hot wavy wall is shown inFigure 10 for a constant Rayleigh number and four selected surface waviness

Figure 8Variation of cell patterninside a cavity withl = 0 125 and A = 0 25for different Rayleighnumber

HFF138

1110

(l = 0 0 005 0075 01) for an aspect ratio A = 0 25 At Ra = 104 heattransfer rate shows true periodic nature with respect to the axial distance alongthe length of the bottom wall for l = 0 0 where three peaks represent not onlythe three spots of highest temperature gradient but also characterize threepairs of cell generated due to the macruid motion inside the cavity However forl = 0 05 0075 01 local Nusselt number distribution changed signiregcantlydue to low temperature gradient at the midpoint of the cavity In that case heat

Figure 9Transition of the cell

mode as a functionof Rayleigh number for

l = 0 125 and A = 0 25

Figure 10Change of the local

Nusselt number alongthe length of the bottom

wave

Effect of surfacewaviness and

aspect ratio

1111

transfer rate increases up to the X = 0 15 and falls along the wavy wallgradually up to the midpoint (X = 0 5) of the cavity This nature is similarfor other surface waviness with a slight variation in magnitudes Completelydifferent picture is observed at Ra = 105 For l = 0 0 three peaks againrepresent the three spots of highest temperature gradient with a highertemperature gradient at the mid point of the wave

Figure 12Nusselt number as afunction of Rayleighnumber for differentsurface waviness fortwo regxed aspect ratio(a) A = 0 25 and(b) A = 0 50

Figure 11Change of the localNusselt number alongthe length of the bottomwave

HFF138

1112

Figure 11 depicts the extent to which the local Nusselt number distribution isaffected by changes in the aspect ratio We observed that at Ra = 104 twopeaks move towards the center of the cavity with the increase of aspect ratioSimilar pattern was observed at Ra = 106 but the magnitude of the two peak ishigher It is also noticed that heat transfer rate is faster near the straight walland decrease rapidly with the increase of X Whatever be the values of aspectratio heat transfer rate is always minimum at the mid point of the bottom wall

Local Nusselt numbers are integrated to calculate the average value of theNusselt number according to equation (20) Average Nusselt number is plottedas a function of Rayleigh number in Figure 12 for two regxed aspect ratiosA = 0 25 and 050 for different surface waviness At lower Rayleigh number

Figure 13Nusselt number as afunction of Rayleighnumber for different

aspect ratio for two regxedsurface waviness(a) l = 0 05 and

(b) l = 0 10

Figure 14Nusselt number as a

function of surfacewaviness for regxedRayleigh number

(Ra = 5 pound104)Two lines correspond

to two different aspectratio of the enclosure asindicated in the legend

Effect of surfacewaviness and

aspect ratio

1113

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

HFF138

1114

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

(l = 0 0 005 0075 01) for an aspect ratio A = 0 25 At Ra = 104 heattransfer rate shows true periodic nature with respect to the axial distance alongthe length of the bottom wall for l = 0 0 where three peaks represent not onlythe three spots of highest temperature gradient but also characterize threepairs of cell generated due to the macruid motion inside the cavity However forl = 0 05 0075 01 local Nusselt number distribution changed signiregcantlydue to low temperature gradient at the midpoint of the cavity In that case heat

Figure 9Transition of the cell

mode as a functionof Rayleigh number for

l = 0 125 and A = 0 25

Figure 10Change of the local

Nusselt number alongthe length of the bottom

wave

Effect of surfacewaviness and

aspect ratio

1111

transfer rate increases up to the X = 0 15 and falls along the wavy wallgradually up to the midpoint (X = 0 5) of the cavity This nature is similarfor other surface waviness with a slight variation in magnitudes Completelydifferent picture is observed at Ra = 105 For l = 0 0 three peaks againrepresent the three spots of highest temperature gradient with a highertemperature gradient at the mid point of the wave

Figure 12Nusselt number as afunction of Rayleighnumber for differentsurface waviness fortwo regxed aspect ratio(a) A = 0 25 and(b) A = 0 50

Figure 11Change of the localNusselt number alongthe length of the bottomwave

HFF138

1112

Figure 11 depicts the extent to which the local Nusselt number distribution isaffected by changes in the aspect ratio We observed that at Ra = 104 twopeaks move towards the center of the cavity with the increase of aspect ratioSimilar pattern was observed at Ra = 106 but the magnitude of the two peak ishigher It is also noticed that heat transfer rate is faster near the straight walland decrease rapidly with the increase of X Whatever be the values of aspectratio heat transfer rate is always minimum at the mid point of the bottom wall

Local Nusselt numbers are integrated to calculate the average value of theNusselt number according to equation (20) Average Nusselt number is plottedas a function of Rayleigh number in Figure 12 for two regxed aspect ratiosA = 0 25 and 050 for different surface waviness At lower Rayleigh number

Figure 13Nusselt number as afunction of Rayleighnumber for different

aspect ratio for two regxedsurface waviness(a) l = 0 05 and

(b) l = 0 10

Figure 14Nusselt number as a

function of surfacewaviness for regxedRayleigh number

(Ra = 5 pound104)Two lines correspond

to two different aspectratio of the enclosure asindicated in the legend

Effect of surfacewaviness and

aspect ratio

1113

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

HFF138

1114

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

transfer rate increases up to the X = 0 15 and falls along the wavy wallgradually up to the midpoint (X = 0 5) of the cavity This nature is similarfor other surface waviness with a slight variation in magnitudes Completelydifferent picture is observed at Ra = 105 For l = 0 0 three peaks againrepresent the three spots of highest temperature gradient with a highertemperature gradient at the mid point of the wave

Figure 12Nusselt number as afunction of Rayleighnumber for differentsurface waviness fortwo regxed aspect ratio(a) A = 0 25 and(b) A = 0 50

Figure 11Change of the localNusselt number alongthe length of the bottomwave

HFF138

1112

Figure 11 depicts the extent to which the local Nusselt number distribution isaffected by changes in the aspect ratio We observed that at Ra = 104 twopeaks move towards the center of the cavity with the increase of aspect ratioSimilar pattern was observed at Ra = 106 but the magnitude of the two peak ishigher It is also noticed that heat transfer rate is faster near the straight walland decrease rapidly with the increase of X Whatever be the values of aspectratio heat transfer rate is always minimum at the mid point of the bottom wall

Local Nusselt numbers are integrated to calculate the average value of theNusselt number according to equation (20) Average Nusselt number is plottedas a function of Rayleigh number in Figure 12 for two regxed aspect ratiosA = 0 25 and 050 for different surface waviness At lower Rayleigh number

Figure 13Nusselt number as afunction of Rayleighnumber for different

aspect ratio for two regxedsurface waviness(a) l = 0 05 and

(b) l = 0 10

Figure 14Nusselt number as a

function of surfacewaviness for regxedRayleigh number

(Ra = 5 pound104)Two lines correspond

to two different aspectratio of the enclosure asindicated in the legend

Effect of surfacewaviness and

aspect ratio

1113

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

HFF138

1114

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

Figure 11 depicts the extent to which the local Nusselt number distribution isaffected by changes in the aspect ratio We observed that at Ra = 104 twopeaks move towards the center of the cavity with the increase of aspect ratioSimilar pattern was observed at Ra = 106 but the magnitude of the two peak ishigher It is also noticed that heat transfer rate is faster near the straight walland decrease rapidly with the increase of X Whatever be the values of aspectratio heat transfer rate is always minimum at the mid point of the bottom wall

Local Nusselt numbers are integrated to calculate the average value of theNusselt number according to equation (20) Average Nusselt number is plottedas a function of Rayleigh number in Figure 12 for two regxed aspect ratiosA = 0 25 and 050 for different surface waviness At lower Rayleigh number

Figure 13Nusselt number as afunction of Rayleighnumber for different

aspect ratio for two regxedsurface waviness(a) l = 0 05 and

(b) l = 0 10

Figure 14Nusselt number as a

function of surfacewaviness for regxedRayleigh number

(Ra = 5 pound104)Two lines correspond

to two different aspectratio of the enclosure asindicated in the legend

Effect of surfacewaviness and

aspect ratio

1113

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

HFF138

1114

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

the change of average heat transfer is almost negligible However at higherRayleigh number this effect is quite signiregcant At higher surface wavinessratio heat transfer rate is higher at higher Rayleigh number when surfacewaviness ratio increase from zero to other values and then further increases ofsurface waviness ratio which shows a negligible effect on average heat transferrate But for A = 0 5 different scenario is observed at higher Rayleigh numberthough the similar trend is observed at lower higher Rayleigh number Hereheat transfer rate increases with the increase of surface waviness ratio whichincrease from zero to other values

Effect of aspect ratioFigure 13 depicts the variation of average Nusselt number as a function of theRayleigh number for three different aspect ratios A = 0 25 0375 and 05 anda constant surface waviness Two sets of results are shown thosecorresponding to l = 0 05 in Figure 13(a) and l = 0 10 in Figure 13(b)From Figure 13(a) it is clear that the change of heat transfer rate is negligiblewith changes of aspect ratio of a wavy enclosure A tiny increase of heattransfer rate is obtained at higher aspect ratio For l = 0 10 similar trend isobserved with slightly better variation of heat transfer with aspect ratioMagnitudes of average Nusselt number are slightly higher for higher aspectratio at higher Rayleigh number

Effect of surface wavinessFigure 14 shows the variation of average Nusselt number for a hot wall as afunction of surface waviness (l) for Ra = 5 pound104 Four lines correspond to fourdifferent aspect ratios of the enclosure as indicated in the legend wheresymbols show the data points and lines are the best regtted curve We observecompletely a different picture for different aspect ratios at lower surfacewaviness ratio At A = 0 25 heat transfer rate decrease rapidly with theincrease of surface waviness but after a certain waviness (l $ 0 2) heattransfer rate increases At l = 0 2 we observed the minimum heat transfercompared to other values of surface waviness Completely different nature isobserved on the heat transfer rate for A = 0 5 When surface wavinessincreases from zero to another value heat transfer also increases followingwhich the negligible variation is obtained for 0 05 l 0 1 Further increaseof surface waviness causes the decrease of heat transfer rate up to l lt 0 2after which the heat transfer rate again goes up

Axial velocity proreglesFor three selected Rayleigh numbers (= 104 105 and 106) axial velocity proreglesare shown in Figure 15(a)-(c) at six locations along the axial direction (X) andfor a constant surface waviness (l = 0 1) Velocity proregles are symmetric inthe vertical centerline (X = 0 5) of the cavity for all the values of the Rayleighnumber We only concentrate about our discussions on the left portion of the

HFF138

1114

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

cavity about the vertical centerline Vertical line (dash-dot-dot) with eachproregles indicate a reference line where axial velocity is zero At Ra = 104

(Figure 15(a)) an S-shaped velocity proregle resembles to the sinusoidaldistribution of the velocity along the vertical direction which is a characteristicdistribution of the velocity for shallow cavities Velocity proregles support thebi-cellular macrow pattern (see streamlines in Figure 5(b)) inside the cavity at thisRayleigh number Magnitude of the velocity decreases towards the adiabaticwalls and the vertical centerline of the cavity Appearance of multi-cellular macrow(see streamlines in Figure 5(c)) dramatically changes the pattern of axial

Figure 15Axial velocity proregles atX = 0 125 025 0375

050 0625 075 and 0875for Ra = 104 105 and

106 and l = 0 10

Effect of surfacewaviness and

aspect ratio

1115

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

velocity proregle as shown in Figure 15(b) Considering the left half of the cavitythe vortex near the adiabatic wall is rotating clockwise and near the verticalsymmetry line it is in the counterclockwise direction Locations X = 0 125and 0375 coincide with the clockwise and counterclockwise vorticesrespectively This is the main reason for the opposite nature of proregles atthese two locations However most of the upper parts of the location X = 0 25(see also Figure 5(c)) are occupied by the counterclockwise vortex Only a smallpart near the bottom wall is shared by the clockwise vortex This commonsharing by two vortices occurs due to the presence of the wavy nature of the topand the bottom walls which is absent (no sharing) in the case of the cavity withmacrat walls Axial velocities at the upper part of the location X = 0 25 form theusual S-shaped proregle A small portion near the bottom wall shows negativevelocities due to the interference of the clockwise vortex Flow at Ra = 106 isbi-cellular (see Figure 5(d)) and this is qualitatively different from the macrowfeature at Ra = 104 (which is also bi-cellular in nature) This is the boundarylayer regime of the macrow and velocity proregles which are characterized by highnear-wall-gradient along with almost zero velocity at the center of the cavity

Heat transfer irreversibilityAny heat transfer problem is always accompanied by entropy generationhence by the one-way destruction of available work (Bejan 1996) Entropygeneration is related to the thermodynamic imperfection of a real device and inthe long run is a real cause for the inherent irreversibility of such devicesTherefore it makes good engineering sense to focus on the irreversibility ofheat transfer processes and try to understand the function of the entropygeneration mechanism The starting point of the irreversibility analysis is aperfect description of the local entropy generation rate and according to Bejan(1996) the local rate of entropy generation can be written as

CcedilS gen =

k

T20

shy T

shy x

sup3 acute2

+shy T

shy y

sup3 acute2

+mT0

2shy u

shy x

sup3 acute2

+shy v

shy y

sup3 acute2( )

+shy u

shy y+

shy v

shy x

sup3 acute2

(21)

where T0 is the reference temperature Using the same characteristicparameters already used for the dimensionless purpose the non-dimensionalform of equation (21) is

N S =shy H

shy X

sup3 acute2

+shy H

shy Y

sup3 acute2

+Br

O2

shy U

shy X

sup3 acute2

+shy V

shy Y

sup3 acute2( )

+shy U

shy Y+

shy V

shy Y

sup3 acute2

(22)

where NS Br and O represent entropy generation number the Brinkmannumber and dimensionless temperature difference respectively Entropy

HFF138

1116

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

generation number (NS) is a ratio between the local entropy generation rate( CcedilS

gen) and a characteristic entropy transfer rate ( CcedilS 0 ) which is shown in

the following equation

NS =CcedilS

gen

CcedilS 0

where CcedilS 0 =

k(DT)2

d2T20

(23)

The dimensionless form of the entropy generation rate (NS) given in equation(22) consists of two parts The regrst part (regrst square bracketed term at theright-hand side of equation (22)) is the irreversibility due to regnite temperaturegradient and generally termed as heat transfer irreversibility (HTI) The secondpart is the contribution of macruid friction irreversibility (FFI) to entropygeneration which can be calculated from the second square bracketed termThe overall entropy generation for a particular problem is an internalcompetition between HTI and FFI Usually free convection problems at lowand moderate Rayleigh numbers are dominated by the HTI (discussed later indetail) Entropy generation number (NS) is good for generating entropygeneration proregles or maps but fails to give any idea whether macruid friction orheat transfer dominates Bejan (1979) proposed irreversibility distribution ratio(F ) which is the ratio between FFI and HTI As an alternative irreversibilitydistribution parameter Paoletti et al (1989) deregned Bejan number (Be) whichis the ratio of HTI to the total entropy generation Mathematically Bejannumber becomes

Be =HTI

HTI + FFI=

1

1 + F(24)

Bejan number ranges from 0 to 1 Accordingly Be = 1 is the limit at which theHTI dominates Be = 0 is the opposite limit at which the irreversibility isdominated by macruid friction effects and Be = 1 2 is the case in which the heattransfer and macruid friction entropy generation rates are equal

For a regxed aspect ratio (A = 0 25) contours of Bejan numbers are presentedin Figure 16(a)-(f) for Ra = 104 and 105 and l = 0 0 005 and 01 Whatever thevalues of Ra l and A be regions near the top and the bottom walls always actas strong concentrators of HTI The maximum value of Bejan contour (Bemax)appears in the immediate neighborhood of the top and the bottom walls AtRa = 104 and l = 0 0 (Figure 16(a)) HTI is characterized by three spots ofordfHighly-Concentrated-Bejan-Contours (HCBC)ordm near the top wall and two spotsnear the bottom wall To understand HTI as shown by Bejan contour inFigure 16(a) in detail regrst focus on Figure 17 (we will go throughFigure 16(a)-(f) next) To ease our understanding about Figure 16(a) weintroduce regeld of streamlines and isothermal lines along with the constantBejan number contours in Figure 17 We will regrst focus our attention on thethree spots marked by ellipse (E1 E2 and E3) This is a case of multicellular

Effect of surfacewaviness and

aspect ratio

1117

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

HFF138

1120

Figure 16Contours of Bejannumber for A = 0 25

Figure 17A combined plot ofstreamlines isothermallines and constantBejan number contourfor Ra = 104 l = 0 0and A = 0 25

HFF138

1118

macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

Effect of surfacewaviness and

aspect ratio

1119

a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

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macrow with four vortices Dashed lines indicate the clockwise rotation and solidlines indicate the counterclockwise rotation Fluid region marked by E1 isforced upward direction At this Rayleigh number (= 104) convection current iswell set and strong enough to cause the swirl (convective distortion) inisothermal lines Isothermal lines are heavily concentrated inside the regionspotted by E2 This is the region with high temperature gradient which causeshigh HTI (E3) Next focus on the spots marked by circle (C1 C2 and C3)Obviously two downward streams of macruid (C1) cause very low temperaturegradient near the top wall (C2) and which leaves a region of low HTI (C3)A spatially periodic region extended along the horizontal centerline of thecavity shows very low irreversibility (HTI) due the lower temperaturegradients Such region is deregned as an idle region for irreversibility (Mahmudand Fraser 2002a b) A thin extension of the idle region for irreversibilityexists between the two consecutive spots of HCBC at the top and the bottomwalls

At Ra = 105 (Figure 16(b)) spots of HCBC elongates along the horizontaldirection and at the same time concentrates more towards the walls Now threespots of HCBC exist near the top wall and two near the bottom wall Idle regionfor irreversibility occupies more area inside the enclosure An introduction ofthe surface undulant dramatically changes the characteristic features of theHTI inside the enclosure Three spots of HCBC at the top wavy wall and two atthe bottom wavy wall are observed at Ra = 104 for l = 0 05 (Figure 16(c))Focusing regrst on the top wavy wall the extension of HTI is higher near themiddle part of the wall than the part near the adiabatic walls However at thebottom wavy wall zones of high HTI are symmetric (and also equal in size)with respect to the vertical centerline of the cavity For l = 0 1 (Figure 16(e))spots of HCBC occupy slightly more region near the top and bottom wavy wallswhen compared with the case l = 0 05 For Ra = 105 (Figure 16(d) and (f))HTI scenarios remain the same as in the case of Ra = 104 but now spots ofHCBC concentrate more towards the wavy walls

ConclusionLaminar steady natural convection heat transfer and macruid macrow inside a wavyenclosure with two wavy walls and two straight walls are investigatednumerically The effect of aspect ratio and surface waviness on heat transferand HTI is tested at different Rayleigh numbers Aspect ratio does not play anyimportant role on the heat transfer inside a wavy enclosure of regxed surfacewaviness but when surface waviness changes from zero to a certain valueaspect ratio dominates the heat transfer characteristics The lower the surfacewaviness the higher is the heat transfer for lower aspect ratio but thisstatement macripped for higher aspect ratio At higher aspect ratio heat transferincreases with the increase of surface waviness For a constant Rayleighnumber heat transfer falls gradually with an increasing surface waviness up to

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a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

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a certain value of surface waviness above which heat transfer increases againfor low aspect ratio Whereas for higher aspect ratio heat transfer graduallyincreases with an increase of surface waviness up to a certain value of surfacewaviness and subsequently heat transfer falls gradually upto certain wavinessabove which heat transfer increases again like the other case

References

Abramowitz M and Stegun I (1965) ordfElliptic Integralsordm Handbook of Mathematical FunctionsChapter 17 Dover Publications New York

Adjlout L Imine O Azzi A and Belkadi M (2002) ordfLaminar natural convection in an inclinedcavity with a wavy wallordm Int J Heat Mass Transfer Vol 45 pp 2141-52

Asako Y and Faghri M (1987) ordfFinite volume solution for laminar macrow and heat transfer in acorrugated ductordm J Heat Transfer Vol 109 pp 627-34

Aydin O Unal A and Ayhan T (1999) ordfA numerical study on buoyancy-driven macrow in aninclined square enclosure heated and cooled on adjacent wallsordm Numerical Heat Transfer(Part A) Vol 36 pp 585-899

Bejan A (1979) ordfA study of entropy generation in fundamental convective heat transferordm J HeatTransfer Vol 101 pp 718-25

Bejan A (1984) Convective Heat Transfer Wiley New York

Bejan A (1996) Entropy Generation Minimization CRC Press New York

Chandrasekhar S (1961) Hydrodynamic and Hydromagnetic Stability Oxford University PressLondon

Chen CL and Cheng CH (2002) ordfBuoyancy-induced macrow and convective heat transfer in aninclined arc-shape enclosureordm Int J Heat Fluid Flow Vol 23 pp 823-30

Das PK and Mahmud S (2003) ordfNumerical investigation of natural convection inside a wavyenclosureordm Int J Thermal Sciences Vol 42 pp 397-406

Elsherbiny SM (1996) ordfFree convection in inclined air layers heated from aboveordm Int J HeatMass Transfer Vol 39 pp 3925-30

Ferziger J and Peric M (1996) Computational Methods for Fluid Dynamics Springer VerlagBerlin

Hadjadj A and Kyal ME (1999) ordfEffect of two sinusoidal protuberances on natural convectionin a vertical annulusordm Numerical Heat Transfer (Part A) Vol 36 pp 273-89

Hamady FJ Lloyd JR Yang HQ and Yang KT (1989) ordfStudy of local naturalconvection heat transfer in an inclined enclosureordm Int J Heat Mass Transfer Vol 32pp 1697-708

Hortmann M Peric M and Scheuerer G (1990) ordfFinite volume multigrid prediction of laminarnatural convection Benchmark solutionsordm Int J Numer Methods Fluids Vol 11pp 189-207

Hossain MA and Rees DAS (1999) ordfCombined heat and mass transfer in natural convectionmacrow from a vertical wavy surfaceordm Acta Mechanca Vol 136 pp 133-41

Kumar BVR (2000) ordfA study of free convection induced by a vertical wavy surface with heatmacrux in a porous enclosureordm Numerical Heat Transfer (Part A) Vol 37 pp 493-510

Lage JL and Bejan A (1990) ordfConvection from a periodically stretching plane wallordm J HeatTransfer Vol 112 pp 92-9

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Mahmud S Islam AKMS and Mamun MAH (2002b) ordfSeparation characteristics of macruidmacrow in a pipe with wavy surfaceordm Int J Appl Mechanics and Engineering Vol 7pp 1255-70

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