international journal of heat and mass...

International Journal of Heat and Mass Transfer 126 (2018) 95–108

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Evolution to chaotic natural convection in a horizontal annulus with aninternally slotted circle

https://doi.org/10.1016/j.ijheatmasstransfer.2018.06.0070017-9310/� 2018 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (Y. Zhang).

M. Zhao a, D.M. Yu a, Yuwen Zhang b,⇑aCollege of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, ChinabDepartment of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 8 December 2017Received in revised form 8 April 2018Accepted 2 June 2018

Keywords:Numerical simulationTurbulent transportAnnulusChaosLBM

The characteristics of transition from laminar to chaotic natural convection in a two-dimensional hori-zontal annulus with an internally slotted circle is analyzed using Lattice Boltzmann method (LBM).The aim of this paper is to identify the route(s) to chaos, and to illustrate the dynamical response ofthe flow with the change of the control parameter (Ra). The results obtained for a range of theRayleigh number, Ra, from 5 � 103 to 2 � 106 at Pr = 0.71, and the slot degree, Sf, from 0.1 to 0.4. Thenumerical results show that slot ratio, slot configuration, and Rayleigh number are influential to oscilla-tion phenomenon in this model; the flow inside the annulus may be: (1) a stable base-two-cells regime,(2) a multi-cellular flow with four-stable-symmetrical-secondary cells regime, (3) a multi-cellular flowwith four-oscillatory-secondary cells regime, and (4) an asymmetrical oscillation regime. The results alsoshow that the oscillatory flow undergoes several bifurcations and ultimately evolves to a chaotic flowafter the first bifurcation. In addition, certain features of nonlinear dynamical systems like bifurcation,self-sustained oscillations are also observed. The simulation results also show that slot degree Sf is rele-vant to oscillations. Furthermore, with the larger slotted ratio, the flow is more unstable, and the config-uration with top and bottom slot seems to be the most unstable among the given four models.

� 2018 Elsevier Ltd. All rights reserved.

1. Introduction

In the recent years, there has been a growth of interest in thebehavior of chaos dynamic systems. Chaos theory is defined asthe qualitative study of unstable aperiodic behavior in determinis-tic nonlinear systems. The particular interest is in how a determin-istic system behaves in a complicated way (what we couldconventionally call ‘‘random” or ‘‘stochastic” behavior, now wecould call ‘‘chaotic” behavior) owing to some element of non-linearity [1]. In addition, an important route to chaos is by meansof a cascade of period-doubling bifurcations. This route has univer-sal properties, and is observed in real dynamic systems, mostimportantly (from our point of view) in certain types of thermallygenerated turbulence (i. e. natural convection instability). Now theidea that the ‘‘soft” or ‘‘weak” turbulence can be connected tochaos theories based on a small number of degrees of freedom isbeing accepted by the researchers.

Transitions to oscillatory or chaotic convections are very inter-esting phenomena, and researches to clarify the route(s) to turbu-lent convections are in progress. The instability of natural

convections has attracted wide attentions in the past decades,due to a desire to improve the phenomenological understandingof natural convection, and the pressing need for numerical modelscapable of predicting the corresponding flow structures and relatedheat transfer processes. A great deal of literature relevant to natu-ral convection has concentrated on the transition process to unsta-ble periodic flow and route to chaos. Benouaguef et al. [2] studiedthe unstable natural convection in an air-filled square enclosure.The temporal evolution of the hot global Nusselt number and theattractors in a space trajectory were plotted, and the effects ofthe Rayleigh number on the route to chaos were discussed. Pao-lucci and Chenoweth [3] numerically studied the transition fromlaminar to chaotic flow in a differentially heated vertical cavity.They obtained the critical Rayleigh number as a function of aspectratio and developed expressions relating the fundamental frequen-cies of the oscillatory flow to the Rayleigh number and aspect ratio.Erenburg et al. [4] numerically studied the multiplicity, stability,and bifurcations stable natural convection in a two-dimensionalrectangular cavity with partially and symmetrically heated verticalwalls; the observed phenomena also occurred at larger Prandtlnumbers, which was illustrated for Pr = 10. Gelfgat [5] studiedthe oscillatory regime of natural convection of air in an 8:1 two-dimensional rectangular cavity with a global Galerkin method.

Nomenclature

c particle speedcs speed of sound in lattice scaleea discrete lattice velocityfa velocity distribution functionFy force term of vertical direction (N/kg)g gravitational acceleration (m/s2)ga temperature distribution functionkeq equivalent thermal conductivityri inner radiusro outer radiusSf slotted ratioTi, To dimensionless temperature at inner and outer cylinderu, v velocity components at horizontal and vertical direction

(m/s)u velocity vector

Greek SymbolsH dimensionless temperatured thickness of slotted annulusda2, da4 Kronecker functionc slotted angleq density (kg/m3)s dimensionless timesf, sT lattice relaxation timeDt lattice time step

Subscripti inner cylindera discretization directiono outer cylinder

Superscripteq equilibrium distribution function

96 M. Zhao et al. / International Journal of Heat and Mass Transfer 126 (2018) 95–108

Chen [6] performed a numerical study of external electrical andmagnetic effects on the thermal instability in natural convectionflow over a heated horizontal plate. Cheng [7] systematically inves-tigated the flow and heat transfer from laminar to chaos in a 2-Dsquare cavity where the flow is induced by a shear force resultingfrom the motion of the upper lid combined with buoyancy forcedue to bottom heating. Sheu and Lin [8] numerically studied therich and complex buoyancy-driven flow field due to natural con-vection over a wide range of Rayleigh numbers in a cubic cavityby virtue of the simulated bifurcation diagram, limit cycle, powerspectrum, and phase portrait. Saury et al. [9] carried out an exper-imental study on the natural convection unsteadiness occurring inan air-filled cavity having two opposite walls respectively heatedand cooled at constant and uniform temperature. Oscillation andchaos in combined heat transfer by natural convection, conduction,and surface radiation in an open cavity was solved numerically byWang et al. [10]. Louisos et al. [11] computationally investigatedthe nonlinear dynamics of unstable natural convection in a 2Dthermal convection loop with heat flux boundary conditions.

Fig. 1. Physical domain.

Most recently, Mercader et al. [12] and Sánchez et al. [13] stud-ied the thermal convection in a laterally heated horizontal cylinderrotating about its axis, and analyzed the linear stability; severalsecondary flows originated from the instabilities were computed.Dou and Jiang [14] numerically investigated the physical mecha-nism of flow instability and heat transfer of natural convection ina cavity with thin fin(s). Cimarelli and Angeli [15] analyzed thetransition to turbulence of natural convection flows between twoinfinite vertical plates by Direct Numerical Simulations (DNS).The first bifurcation from the laminar conduction regime to stableconvection and then chaotic flow regime were captured. Naghibet al. [16] investigated the unstable natural convection in awater-filled, open top tank with a black bottom subjected to radia-tive heating from a halogen theatre spotlight by laboratory-scaleexperimental study. Cho et al. [17] numerically investigated thetwo-dimensional natural convection in a square enclosure withdifferent arrays of two inner cylinders. They found that the flowand temperature fields eventually reached stable or unstablestates, depending on the distance between the cylinders.

Because of its wide engineering applications, such as solar col-lectors, thermal energy storage systems and large-current busbar.The busbar consists of two metal cylinders: an inner hollow cylin-der (current busbar: the cross-section can be circular, hexagon, oroctagon) and an outer metal cylinder. The heat generated in the

Fig. 2. Comparison of radial dimensionless temperature profiles at Ra = 5 � 104, Pr= 0.7.

(a)

(b)

Ra = 1×104, Pr = 0.7 Ra = 1×105, Pr = 5.0

Fig. 3. Comparisons of distribution of streamlines (left half) and isotherms (right half). (a) Ref. [35], (b) present study.

(a) Experimental result (b) numerical result

Fig. 4. Comparisons of distribution of isotherm with experimental results [36].

M. Zhao et al. / International Journal of Heat and Mass Transfer 126 (2018) 95–108 97

busbar from the Joule heating is transferred to the outer envelopeby radiation and natural convection. The flow and thermal fieldsare well studied for the natural convection induced by internalheating in horizontal concentric cylindrical annuli. There also havebeen many reports for the transition of the natural convection in ahorizontal annulus between two concentric cylinders. Most ofthem are based on numerical simulations, and their results haveshown exchanges of flow patterns and multiple stable state solu-tions; a partial review of the relevant works can be found in Dyko

et al. [18], Desrayaud et al. [19], and Yoo [20]. The others are linearstability analyses based on the parallel flow assumption whichmay be found in Mizushima et al. [21], Petrone et al. [22], Adachiand Imai [23]. Most recently, Hu et al. [24] simulated the multiplesteady solutions of 2D natural convection in a concentric horizon-tal annulus with a constant flux wall using a novel thermalimmersed boundary-lattice Boltzmann method. Kahveci [25]numerically investigated the stability of unsteady mixed convec-tion in a horizontal annulus between two concentric cylinders.

t=0s

t=20s

t=40s

(a) Experimental result (b) numerical result Fig. 5. Comparisons of isotherm with time variation at Ra = 6.46 � 104 [36].


The surfaces of the cylinders were considered to be at fixed tem-peratures and the hot inner cylinder was rotating at a constantangular velocity. Ma et al. [26] studied the instability of the staticdiffusive state of Rayleigh-Bénard convection of cold water near itsdensity maximum in a vertical annular container heated frombelow and cooled from the above by linear stability analysis. Com-paratively, little works have been reported on unsteady naturalconvection heat transfer in more relevant complex domain. Liangand Jiang [27] conducted a numerical investigation of natural con-vection within horizontal annulus with a heated protrusion. Mou-kalled and Acharyaf [28] studied natural convection in an annulusbetween concentric horizontal circular and square cylinders.

Zhang et al. [29] applied a numerical study on the nonlinearcharacteristics of natural convection in a cylindrical enclosure withan internal concentric slotted hollow. The governing equationswere discretized using the finite volume method based on stag-gered grid formulation, and solved using the SIMPLE algorithmwith QUICK scheme. Yang et al. [30] employed Lattice Boltzmannmethod (LBM) to simulate the natural convection for the same con-figuration. However, only limited parameters were chosen toobserve the characters of different solutions, and only the influenceof Rayleigh number on the oscillation are presented. The objectiveof this article is to numerically simulate natural convection in ahorizontal annulus with an internal slotted circle. Examples of such

(a) Ra = 1×105 (b) Ra = 5×105

(c) Ra = 8×105 (d) Ra = 1×106

Fig. 6. Isotherms (left) and streamlines (right) for different Ra at Sf = 0.1.

(a) U - V Phase portrait (b) Time variation

Fig. 7. Phase portrait and time variation of dimensionless velocity at Sf = 0.1, Ra = 5 � 105.


flows and transport phenomena can be found in large-currentbusbar, which is used for transmitting large electric current inpower plants. In this problem, the buoyancy driven flow (Rayleighnumber is a measure) generated from the difference intemperatures between the cold and the heated wall leads to thepossible complex flows that are substantially unstable and mayexhibit self-sustained oscillations and bifurcations phenomena.Thus, the flows will easily undergo transition from laminar flowto chaos and will show very complex non-linear characteristicsat large Rayleigh numbers. The effects of both Rayleigh numbers,slot degree and slot numbers on route to chaos, and the study ofthe periodic, the quasi-periodic, and the chaotic regimes will beinvestigated.

2. Physical and mathematical model

2.1. Physical model

Fig. 1 shows the schematic diagram of the problem under con-sideration. A two-dimensional natural convection in cylindricalenclosure with internal concentric slotted hollow with an innerradius ri and an outer radius ro will be simulated. The angle cdenotes the range of slotted zone and the slotted ratio Sf is definedas 2c/p, where d is the thickness of solid hot area. The surfaces ofinner and outer cylinders are maintained at different uniform tem-peratures Ti and To, respectively (Ti = 1, and To = 0). ro/ri denotesradial aspect ratio and in this paper it is set as 2.6, and d=ri ¼ 0:18.

(a) U - V Phase portrait (b) Time variation

Fig. 8. Phase portrait and time variation of dimensionless velocity at Sf = 0.1, Ra = 8 � 105.

(a) Ra = 8×104 (b) Ra = 1×105

Fig. 9. Isotherms and streamlines for different Ra at Sf = 0.2.


2.2. Mathematical model

Lattice Boltzmann Method is a rapid developing numericalmethod for simulating viscous compressible or incompressibleflows in the subsonic regime. Instead of solving the usual contin-uum hydrodynamic equations for the conserved fields, the LBMtries to model the fluid flow by tracking the evolution of the distri-bution functions of the microscopic fluid particles. This kinetic nat-ure of the LBM introduces some important features thatdistinguish it from other numerical methods, such as finite volumemethod (FVM).

The D2G9 model, which is an improved isothermal D2Q9 LBGKmodel will be employed in this work [31–33]. It is based on incom-pressible model for thermal fluid flows by introducing an addi-tional LBGK equation using a temperature distribution functionto describe the evolution of the temperature field. The temperaturedistribution is then coupled to the velocity distribution functionbased on the Boussinesq assumption. The main feature of the LBGKmethod that is different from the LGA method is to replace theBoolean variables by the single-particle distribution function, andreplace the complicated collision operator of LGA by the simpleBhatnagar-Gross-Krook (BGK) collision operator. Li et al. [34] alsoemployed the LBM model to simulate the natural convection oflow Prandtl number fluids.

The thermal lattice Boltzmann model utilizes two distributionfunctions, f and g, for the flow and temperature fields, respectively.It is used with modeling of movement of fluid particles to capturemacroscopic fluid quantities such as velocity, pressure, andtemperature. In this approach, the fluid domain is discretized touniform Cartesian cells. Each cell holds a fixed number of distribu-

tion functions, which represent the number of fluid particles mov-ing in discrete directions.

The velocity and temperature distribution functions, f and T, arecalculated by solving the Lattice Boltzmann equation (LBE), whichis a special discretization of the kinetic Boltzmann equation. Afterintroducing BGK approximation, the general form of lattice Boltz-mann equation with external force (the model D2G9) is:

For the flow field:

f aðr þ eaDt; t þ DtÞ � f aðr; tÞ ¼ �1sf

½f aðr; tÞ � f eqa ðr; tÞ� þ DtFa ð1Þ

For the temperature field:

Taðr þ eaDt; t þ DtÞ � Taðr; tÞ ¼ � 1sT ½Taðr; tÞ � Teqa ðr; tÞ� ð2Þ

where Dt denotes lattice time step, ea is the discrete lattice velocityin the direction a, Fa is the external force in the direction of latticevelocity, sf and sT respectively denote the dimensionless latticerelaxation times of the flow and temperature distribution functiontowards the local equilibrium feq.

In order to incorporate buoyancy force into this model, the forceterm in Eq. (1) needs to be calculated as following in the verticaldirection (y):

Fa ¼ � 12c ðda2 þ da4Þea � bðT � T0Þg ð3Þ

where c is the particle speed, da2 and da4 are Kronecker function, b isvolumetric coefficient of expansion, T0 is reference temperature, g isgravitational acceleration.

s14103.0s81692.0

0.26374 s 0.26164 s

0.25641 s 0.25955 s

0.28362 s 0.27839 s

0.31188 s 0.31397 s

Fig. 10. Variation of isotherms and streamlines with time at Ra = 2 � 105 and Sf = 0.2.


The discrete velocity directions of velocity distribution func-tions for D2G9 are given by:

ea ¼ð0;0Þ a ¼ 0cðcos½ða� 1Þ p2�; sin½ða� 1Þ p2�Þ a ¼ 1;2;3;4ffiffiffi2

pcðcos½ð2a� 1Þ p4�; sin½ð2a� 1Þ p4�Þ a ¼ 5;6;7;8

8><>: ð4Þ

The discrete velocity directions of temperature distributionfunctions for D2G9 are given by:

ea ¼ cðcos½ða� 1Þp2�; sin½ða� 1Þp2�Þ a ¼ 1;2;3;4 ð5Þ

The equilibrium distribution functions of D2G9 are given by:


f eqa ¼q0 � 4d0 pc2 þ q0s0ðuÞ a ¼ 0d1

pc2 þ q0saðuÞ a ¼ 1;2;3;4

d2pc2 þ q0saðuÞ a ¼ 5;6;7;8

8><>: ð6Þ

saðuÞ ¼ xa ea � uc2sþ ðea � uÞ

2

2c2s� u

2

2c2s

" #ð7Þ

xa ¼49 a ¼ 019 a ¼ 1;2;3;4136 a ¼ 5;6;7;8

8><>: ð8Þ

d0 ¼ 512 d1 ¼13d2 ¼ 112 ð9Þ

Teqa ¼T4ð1þ 2 ea � u

c2Þ;a ¼ 1;2;3;4 ð10Þ

where cs is the speed of sound and defined by cs ¼ c=ffiffiffi3

p, the macro-

scopic variables are calculated by the following formulas:

q0 ¼X8a¼0

f eqa ; q0u ¼X8a¼0

eaf a;

p ¼ q0c2

4d0

Xa–0

f a þ s0ðuÞ" #

; T ¼X4a¼1

Ta ð11Þ

A Chapman-Enskog procedure can be applied to Eq. (1) to derivethe macroscopic equations of the model. They are given by:

0.98289 s

Fig. 11. Variation of isotherms and streamlin

(a) U-V Phase portrait

Fig. 12. Phase portrait and time variation of dim

r � u ¼ 0 ð12Þ

@u@t

þr � ðuuÞ ¼ �rpq0

þ mr2u� gbðT � T0Þ ð13Þ

@T@t

þr � ðuTÞ ¼ r � ðarTÞ ð14Þ

where the kinetic viscosity, m, and the thermal diffusivity, a, aredefined in terms of their respective relaxation times:

t ¼ 13c2 sf � 12� �

Dt; a ¼ 12c2 sT � 12� �

Dt ð15Þ

For natural convection, the Boussinesq approximation isassumed and radiation heat transfer is assumed to be negligible.To ensure that the code works in a near incompressible regime,the characteristic velocity of the flow for the natural(unatural ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibgyDTH

p) regime must be low compared with the speed

of sound of the fluid. In this work, the characteristic velocity isselected as 0.1 of the speed of sound.

Boundary conditions play important roles in lattice Boltzmannmethods in that they will influence the accuracy and stability ofthe LBM. The non-equilibrium extrapolation method for boundaryconditions [20] is adopted in this work, and the basic idea of theextrapolation method is to decompose the distribution functionon boundary node into its equilibrium and non-equilibrium parts.

0.99459 s

es with time at Ra = 1 � 106 and Sf = 0.2.

(b) Time variation

ensionless velocity at Sf = 0.2, Ra = 1 � 106.


3. Grid testing and code validation

Grid refinement tests have been performed for the case Sf = 0.3using two uniform grids systems: 130 � 130 and 260 � 260. Theresults show that when we change the mesh size from a grid of130 � 130 to a grid of 260 � 260, the mean temperature of the ref-erence point undergoes an increase of 1.73%, and the 130 � 130grid may be retained after trade-off between accuracy and compu-tational cost. However, when Rayleigh number increases to Ra � 1

(a)

(c)

(e)

Sf=0.1, Ra=2×105

Sf = 0.3, Ra = 2.5×104

Sf = 0.4, Ra = 2×104

time(s)

time(s)

time(s)

Fig. 13. Time variation of dimensionl

� 105, grid of 260 � 260 is adopted because of its larger character-istic length to improve the convergence performance.

In order to validate the numerical simulation, the radial dimen-sionless temperature is obtained for the natural convection in thismodel and the results are compared with those of Kuehn and Gold-stein [35] in Fig. 2. The capital R is the radius of the experimentalannulus, Ri and Ro is the radius of inner and outer cylinder in thehorizontal axis. The comparisons of distribution of streamlinesand isotherms are shown in Fig. 3. It can be seen that the largest

(b)

(d)

(f)

time(s)

Sf=0.2, Ra=2×105

Sf = 0.3, Ra = 4×104

Sf = 0.4, Ra = 2×105

time(s)

time(s)

ess temperatures at Ra = 2 � 105.


difference percentage is 3.9%, and all of the comparison resultsshowed good agreement. Comparisons with experimental resultsreported in [36] are also made. The experimental environmentaltemperature Te = 291.4 K, and the outer and inner cylinder temper-ature To = 294.8 K, Ti = 321.7 K, respectively. We can obtain that Pr= 0.6983 and Ra = 5.64 � 104. Comparisons of distribution of stableisotherms are shown in Fig. 4. Comparison with oscillatory iso-therm at Ra = 6.46 � 104 are also made to validate the oscillatorynumerical simulation and the results are shown in Fig. 5. All ofthe results show that it have good agreement with experimentalresult.

4. Results and discussions

4.1. Oscillations and bifurcations of nature convection

Numerical simulation is first carried out for the case of Sf = 0.1while Rayleigh number, Ra, varying from 105 to 106. The streamli-nes and temperature contours at different Ra are displayed inFig. 6. It can be observed that, at low slotted ratio, the flow is stableand the flow pattern is reflection symmetric consisting of two con-vective rolls that rotate in opposite directions. The flow pattern isregarded as the base-two-cells-flow that a part of flow raised fromthe inner cylinder goes through the slot and sinks to the bottomalong the outer cold cylinder forming a large cycle. Another partof flow rises from the outer side of hot solid areas and sinks alongthe cold outer cylinder forming a crescent shaped eddy. Despite ofthe increasing Ra, the flow inside the horizontal annulus with aninternal slotted circle are stable, because at low slotted ratio, thehot air flow that goes through the slot is limited and it producesthe two convective rolls that can retain its balance; the disturbanceinduced by the increasing buoyancy force cannot destroy thebalance.

Figs. 7 and 8 show the phase portrait and the correspondingtime variation of the dimensionless velocity. A phase portrait is atwo-dimensional projection of the phase space and it representseach of the state variables’ instantaneous state to each other. Forexample, the angular velocity position graph of a pendulum is aphase portrait. Chaotic and other motions can be distinguishedvisually from each other according to the phase portrait. It can

Sf =0.1 Sf =0.2

(a)

(b)

Fig.14. Isotherms (a) and streamlines (b) fo

be observed from Figs. 7 and 8 that the solutions trend to a fixedpoint in phase portrait for Sf = 0.1, and the flow is stable at differentRayleigh numbers.

The flow at fixed slotted ratio of Sf = 0.2 with a series of Rayleighnumbers are simulated to observe different kinds of flow patternsand their formation processes. Isotherms, streamlines, time varia-tions of dimensionless velocity at certain point are used to describethe flow characters. Figs. 9 and 10 show the temperature contoursand streamlines for various Ra at a fixed slotted ratio of Sf = 0.2. Itcan be seen that the buoyancy driven flow (measured by Rayleighnumber, Ra) is generated because the difference in temperaturesbetween the cold and the heated wall leads to the possibility ofcomplex flows. At Ra = 8 � 104 and 1 � 105 (as shown in Fig. 9),the flow domains with two stationary cells are still stable. Athigher Raleigh number of Ra = 2 � 105 (as shown in Fig. 10), onemay observe that an abrupt transition towards a multi-cellularflow with four-stable-symmetrical-secondary cells (two in eachhalf-annulus), instead of the two cells that we have found justabove the transitional Ra, are observed. The flow pattern thenevolves to four-oscillatory-secondary cells regime, and finally toa two-dissymmetrical-oscillatory cells flow pattern. By increasingRa through the threshold at which onset of multicellular flowoccurs at the annulus top, the type of bifurcation is similar to thepitchfork bifurcation found in the Rayleigh-Bénard problem. Forthe Rayleigh-Bénard problem, the fluid is initially at rest becausethe stabilizing effect of the viscous forces is stronger than thedestabilizing effect of buoyancy. For a fluid layer of infinite hori-zontal extension, this state is modified above Rac = 1708; themulti-cellular flow patterns and types of bifurcations occurringwere considered in a number of works by Yang [37].

When increasing Ra up to 1 � 106 (as shown in Fig. 11), aasymmetrical-oscillatory flow patterns appear with the maincrescent-shaped cells; the buoyancy force becomes more intense,which results in the onset of quasi-periodic oscillatory solution.These hydrodynamic instabilities take the form of longitudinalrolls often observed in a vertical slot. As a consequence, the twomain cells at the annulus start to oscillate, changing their sizesalternatively and irregularly. Low amplitude characterizes thisquasi-periodic motion so that the asymmetry of the flow structureis weak.

Sf =0.3 Sf =0.4

.

r different slotted ratio at Ra = 2 � 105.

Sf=0.3, Ra=3.5×104 Sf =0.3, Ra=3.5×104

(a) (b)

(c) (d)

Sf =0.3, Ra=8×105 Sf =0.3, Ra=8×105

(e) (f)

Sf =0.3, Ra=1×106Sf =0.3, Ra=1×106

(g) (h)

Sf =0.3, Ra=2×106 Sf =0.3, Ra=2×106

Fig. 15. Phase portrait (left) and Time variation of dimensionless velocity (right) at Sf = 0.3.


Fig. 12 shows the phase portrait and the corresponding timevariation of dimensionless velocity at S = 0.2 and Ra = 1 � 106. It

can be seen that a multi-closed curve is in the phase portrait (seeFig. 12(a)), the solution is quasi-periodic, and the velocity is


quasi-periodic oscillatory (see Fig. 12(b)). It seems that the oscilla-tory solution always correspond to non-symmetric states, whichexhibits an unexpected multiplicity.

4.2. Effect of slot ratio (Sf)

For a horizontal air-filled annulus with an internal slotted circle,the slot ratio can be considered as an external perturbation thatcontrols the secondary cell generation because the slot ratio hasa strong effect on the oscillations and bifurcations of nature con-vection. Therefore, onset of the Rayleigh-Bénard type flow is possi-ble at the annulus top while slot ratio differs. One can thus expectto observe thermal instabilities of the Rayleigh-Bénard type at thetop of the annulus provided that the radius ratio is large enoughand also hydrodynamic instabilities like in vertical slots filled withlow Prandtl number fluids [19].

Fig. 13 shows the variation of velocity with time at slotted ratioSf = 0.1, 0.2, 0.3 and 0.4. Fig. 14 shows the temperature contoursand streamlines for various slotted ratio at Ra = 2 � 105. It can be

Fig. 16. Classification of flow patterns at different slotted ratios.

a

b

No slot Bottom slot

Fig. 17. Isotherms (a) and streamlines (b) for dif

seen that the flow is weak oscillatory at Sf = 0.2, Ra = 2 � 105 (seeFig. 13(b)) while the solutions remain stable-state for Sf = 0.1 atthe same Ra. It can also be observed that the flow is stable at Sf= 0.3, Ra = 2.5 � 104 (see Fig. 13(c)) and starts to oscillate at Ra =4 � 104 (see Fig. 13(d)). On the other hand, the flow is unstableat Sf = 0.4, Ra = 2 � 104 (see Fig. 13(e)) and the solution is irregu-larly oscillatory at Sf = 0.4 and Ra = 2 � 105 (see Fig. 13(f)).

The above observed phenomena show that the effect of slottedratio on the oscillations and bifurcations of nature convection inthe system is evident: the largest slotted ratio for Sf = 0.4 seemsto be the most unstable among the four values of Sf. One of the pos-sible reasons is due to the strongest competition between the twocells at the annulus top for Sf = 0.4 (see Fig. 14). It more easilychanges the morphology of solution when Ra are varied. Anotherpossible explanation can be that at larger slotted ratio, the base-two-cells-flow will unlikely to evolve to multi-cell flow.

Fig. 15 shows the change of phase portraits and time variationof dimensionless velocity for the case of Sf = 0.3, and Ra given inthe range from 3.5 � 104 to 2 � 106. It is shown that as Raincreases, the solution may exhibit a change from a periodic oscil-lation, and then to a quasi-periodic oscillatory state, and finally toan irregularly oscillatory state. Fig. 13 clearly demonstrated thatthe phase portraits show the evolution of the attractor from amulti-limited cycle, and finally, to chaos.

By further increasing Ra, the rolls within the main cells maymerge and a transition to chaos motion, and at larger Ra thenon-symmetric oscillation takes place. As a result, a non-symmetric flow pattern develops (see Fig. 14), which can also beindicated by Figs. 10 and 11. As seen from the examples of the cor-responding flow pattern (Fig. 14), the break of symmetry leads to aslight growth of one of the vortices and a slight decay of the other;the non-symmetric flow pattern continuously transforms and theflow pattern changes drastically and a chaotic motion appears.Such a transition from a stable symmetric to a non-symmetricoscillatory state was described by Desrayaud et al. [19] and Raoet al. [38]. The results show that the oscillatory flow undergoesseveral bifurcations and ultimately evolves to a chaotic flow afterthe first bifurcation. Fig. 15 also shows that, despite being chaotic,the trajectories in phase space are actually quite organized. Thesystem is far from being completely random, and there is order

Top slot Top and bottom slot

ferent slotted configuration at Ra = 2 � 105.

Fig. 18. Time variation of dimensionless temperatures at Ra = 2 � 105 for different slotted configuration.


in chaos. The trajectories remain in a localized region of phasespace forming the attractor. Numerical simulations are also per-formed for different slot ratios to identify the critical Rayleighnumbers for transition and the results are shown in Fig. 16. It fol-lows from the preceding results that a lower critical Rayleigh num-ber exists, below which stable base-two-cells flow appears.Furthermore, there exists an upper critical Rayleigh number,beyond which oscillatory asymmetrical flow occurs. As can beseen, according to different Ra and Sf, the flow inside the annulusmay be: (1) a stable base-two-cells regime, (2) a multi-cellular flowwith four-stable-symmetrical-secondary cells regime, (3) a multi-cellular flow with four-oscillatory-secondary cells regime, and (4)an asymmetrical oscillation regime. At rather small slotted ratio,like Sf = 0.1, only stable base-two-cells flow appears and it directlyturns to asymmetrical oscillatory flow as Ra � 8.0 � 105. At Sf = 0.4,oscillatory flow occurs firstly when Ra � 4 � 104.

4.3. Effect of slotted configuration

Four slotted configurations are considered here: (1) No slot; (2)Bottom slot; (3) Top slot; (4) Top and bottom slots. Fig. 17 showsthe isotherms and streamlines for different slotted configurationat Ra = 2 � 105. Fig. 18 shows the corresponding time variation ofdimensionless temperatures for four configurations. It can be seen

from Figs. 17 and 18 that the slotted configuration has a greatinfluence on the oscillations and bifurcations of nature convectionin this system, and the top and bottom slot configuration seems tobe the most unstable model in the four configurations. The reasonof the phenomena may be that it is easier to transform the multi-competitive cells in the fourth model, so it is more convenient tolose the stability and the oscillation appears.

5. Conclusions

The characteristics of transition from laminar to chaotic natureconvection inside the horizontal annulus with an internal slottedcircle is investigated in this paper. Slot ratio, slot configuration,and Rayleigh number are found to be influential to oscillation phe-nomenon in this model. As Ra increases for different slot ratio,there exist four kinds of heat transfer regimes depending on themagnitude of Ra: (1) a stable base-two-cells regime, (2) a multi-cellular flow with four-stable-symmetrical-secondary cells regime,(3) a multi-cellular flow with four-oscillatory-secondary cellsregime, and (4) an asymmetrical oscillation regime. The resultsalso show that the oscillatory flow undergoes several bifurcationsand ultimately evolves to a chaotic flow after the first bifurcation.Furthermore, with the larger slotted ratio, the flow is more unsta-


ble, and the configuration with top and bottom slot seems to be themost unstable among the given four models.

Conflict of interest

None declared.

Acknowledgments

Financial support from the China Scholarship Council and theChinese National Nature Science Foundation under grants number51306120 are gratefully acknowledged.

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