large eddy simulation study of turbulent flow around smooth and rough domes.pdf

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Original Article

Large eddy simulation study of turbulentflow around smooth and rough domes

N Kharoua and L Khezzar

Date received: 4 September 2012; accepted: 17 December 2012

Abstract

Large eddy simulation of turbulent flow around smooth and rough hemispherical domes was conducted. The roughness

of the rough dome was generated by a special approach using quadrilateral solid blocks placed alternately on the dome

surface. It was shown that this approach is capable of generating the roughness effect with a relative success. The

subgrid-scale model based on the transport of the subgrid turbulent kinetic energy was used to account for the small

scales effect not resolved by large eddy simulation. The turbulent flow was simulated at a subcritical Reynolds number

based on the approach free stream velocity, air properties, and dome diameter of 1.4 105. Profiles of mean pressurecoefficient, mean velocity, and its root mean square were predicted with good accuracy. The comparison between the

two domes showed different flow behavior around them. A flattened horseshoe vortex was observed to develop around

the rough dome at larger distance compared with the smooth dome. The separation phenomenon occurs before the

apex of the rough dome while for the smooth dome it is shifted forward. The turbulence-affected region in the wake was

larger for the rough dome.

Keywords

Computational fluid dynamics, large eddy simulation, domes, wind load

Introduction

Flow around hemispherical domed structures is rele-vant to a variety of practical engineering applicationsnotably domed buildings, roofs, and sub-ocean struc-tures. Domes were also used in hydraulic channels tostudy the shedding of hairpin vortices in their wake ina vein to understand the development of the near-wallregion of turbulent boundary layers.1 For robustengineering design of such structures, detailed under-standing and knowledge of the flow structure arounddomes is necessary.

The flow around domes is three-dimensional andcontains large scale highly unsteady motions with sep-aration. Complex vortical structures with sheddingand multiple reattachment and separation areas char-acterize this flow. Several parameters affect the flowbehavior around domes; they include the dome shape,the Reynolds number (based on the approaching freestream velocity and dome diameter), the inflow con-ditions such as the approaching boundary layer shapeand turbulence content, the upstream-floor and domesurface, and the surroundings topology.

A number of relevant experimental studies wereconducted on domes. Taniguchi et al.2 conductedan experiment to identify the effects of the domesize and the characteristics of the approaching bound-ary layer on the pressure coefficient and integral

properties such as drag and lift coefficient. The pres-sure distributions along the symmetry plane of thehemisphere were found to be highly similar in theregion of 050, but show a marked dependency onthe Reynolds number in the region from 50 to 120.Savory and Toy3 conducted experiments to study theeffect of three different approaching boundary layersand dome surface roughness on the mean pressuredistribution and on the critical Reynolds numberbeyond which the pressure distribution becomesinvariable. For the artificially roughened dome sub-jected to a thin boundary layer, the reattachmentlength of the downstream recirculation zone wasequal to 1.25, the dome diameter from its axis.Tamai et al.4 explored the formation and sheddingof vortices from a dome in a water tunnel and deter-mined the frequencies characterizing each phenom-enon for a rather low Reynolds number of 104.Approaching the front side, successive recirculationzones, increasing in size and forming the well-known

Department of Mechanical Engineering, Petroleum Institute,

United Arab Emirates

Corresponding author:

L Khezzar, Department of Mechanical Engineering, Petroleum Institute,

PO Box 2533, Abu Dhabi, United Arab Emirates.

Email: [email protected]

Proc IMechE Part C:

J Mechanical Engineering Science

0(0) 115

! IMechE 2013

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horseshoe vortices, were observed. The number ofthese structures increases with increasing Reynoldsnumber until a critical value of about 3000 beyondwhich only one single recirculation zone persistswith a constant size. Cheng and Fu5 studied theeffect of Reynolds number on pressure distribution.This study identified three modes of flow separationin the apex region: for low Reynolds below 1.8 105,the free shear layer which develops in the wake regionseparates without reattachment on the dome surfacethis corresponded to one single peak of the fluctuatingpressure coefficient. At a Reynolds number of1.8 1053 105, a second peak was observed corres-ponding, to the reattachment point caused by a sep-aration bubble which has formed on the domesurface. Beyond a value of 3 105 of the Reynoldsnumber, only one peak was observed meaning thatthe separation bubble has disappeared.

Using Reynolds averaged NavierStokes (RANS)based turbulence models, Meroney et al.6 found thatthe three turbulence models, SpalartAllmaras, k",and Reynolds stress model, gave similar pressure dis-tribution results with minor variations. Tavakol et al.7

used the RNG k" turbulence model to investigate theeffects of thin and thick approaching boundary layerson the flow aerodynamics. The reattachment pointbehind the obstacle which is of the order of1.0851.17 the dome diameter was located at a shorterdistance for the thick boundary layer. All these studieshave simulated the flow using the steady-state equa-tions of motion. In general steady and to some extentunsteady RANS simulate such flows with acceptablequalitative accuracy. Manhart8 used the large eddysimulation (LES) turbulence model based on theSmagorinsky subgrid-stress model with the law ofthe wall. In this study, the dome roughness was gen-erated by a stepwise approximation of the curvedwall. The study was based on the case reported inSavory and Toy,3 of an artificially roughened domeat a Reynolds number of 1.4 105 and two grid sizesof 650,000 and 1,863,680 computational cells wereused with an integration time step of 0.00250.005 s.Two mechanisms of vortex generation are mentioned,one from the top face due to the separation of thefree shear layer from the region surrounding thedome apex causing symmetric vortex shedding andan another one from the side faces engendering asym-metric vortex shedding. Tavakol et al.9 conducted acomparison between the RNG k" and severalLES subgrid turbulence models over smooth domes.Although their comparison was limited to few meanstreamwise velocity profiles, the RNG k" model wasoverestimating the streamwise velocity over the hemi-sphere. The results showed that LES simulation usingthe single equation transport model for the subgridkinetic energy turbulence model (KETM) withproper grid resolution greater than 4 million cells,are able to capture the mean velocity profiles accur-ately in the apex and wake regions.

From the review of previous work, it is clear thatRANS-based approaches to simulate flows over hemi-spherical domes performed with relative success andthat relatively very few LES-based numerical simula-tions were conducted on such flows. LES-based simu-lations offer the advantage of simulating explicitly thelarge scales while modeling the small subgrid scales.This approach is hence appropriate for aerodynamicflows around domes. In this study, a numerical simu-lation based on the LES model using the KETM sub-grid-scale model is used to calculate and benchmarkthe flow over two hemispherical domes of Savory andToy3,10,11 and examine in detail the features of the sur-rounding flow field. Available, experimental mean sur-face pressure coefficient and velocity profiles withcorresponding root mean square (RMS) are used inthe validation of the numerical results. This particulartest case consists of turbulent flow over an (a) artifi-cially roughened surface which can exist in practicalapplications and, hence, presents an additional levelof complexity in modeling the surface details in con-trast to flows over smooth surfaces for the LES frame-work and (b) over a smooth dome of the same size.Taking advantage of the visualization of the flowthat can be provided by LES the effect of the surfaceroughness is examined in detail and contrasted to thesmooth dome case. In particular, this article presents anovel methodology of how to model and account forsurface roughness in structures such as domes withinthe LES approach for industrial applications. Thestandard ke model was used to generate the startingflow for the LES simulation over the rough surface andis also used in the discussion of the results.

Mathematical model

LES is used in this study. The fluid is assumed incom-pressible and the filtered continuity and momentumtime dependent equations solved are given by

@ Ui@xi

0 1

@ Ui@t

@Ui Uj@xj

@ij@xj

@ p@xj

@ij@xj

2

where the variables with an overbar, Ui and p, repre-sent the filtered (locally averaged) values of the vel-ocity and pressure, respectively. The laminar stresstensor is given by

ij @ Ui@xj

@Uj

@xi

3

is the molecular viscosity.According to Tavakol et al.,9 where three subgrid-

scale models were contrasted for flows around domes,the subgrid-scale model based on the transport of thesubgrid-scale turbulent kinetic energy performs betterfor flows of this type and it is hence used in this study.

2 Proc IMechE Part C: J Mechanical Engineering Science 0(0)




This model developed independently by a number ofauthors, is not based on the local equilibrium hypoth-esis and belongs to the class of model based onsubgrid scales in contrast to the dynamicSmagorinskyLilly model which is based on theresolved scales properties.12 The grid scale cutoff canbe in the inertial range and thus allowing the usage ofrelatively coarse grids.13

The subgrid stress accounting for the unresolvedscales contribution defined by

ij UiUj Ui Uj 4

is modeled according to the following equation

ij 2

3ksgsij 2Ckk

12sgsf Sij 5

where ksgs is the subgrid-scale turbulent kineticenergy, f V1=3 the filter size or characteristiclength, based on the average dimension of a compu-tational finite volume, and Sij the usual resolved meanstrain rate tensor given by

Sij 1

2

@ Ui@xj

@Uj

@xi

6

and the kinetic energy of the subgrid modes is given by

ksgs 1

2U2k U

2k

7

From equation (5), the subgrid-scale turbulenteddy viscosity is then implicitly defined from kineticenergy of the subgrid modes by

t Ckk12sgsf 8

The subgrid-scale kinetic energy adds anotherunknown to the problem and is evaluated by solvinga modeled transport equation which takes the formgiven by equation (9).14

@ksgs@t

@Ujksgs@xj

ij@ Ui@xj

C"k

32sgs

f

Ck@

@xj

k12sgsf

k

@ksgs@xj

! 9

The constants Ck and C" are computed dynamic-ally, based on the least-square method proposed byLilly,15 usually applied to the standard Smagorinskymodel to compute its constant dynamically, while thePrandtl number k is equal to unity.

Geometry and computational approach

Geometrical configuration

Figure 1 shows the geometry used for this study andtaken from Savory and Toy.3,10,11 The hemispherical

dome had a diameter of d 190mm and wasimmersed in a thin turbulent boundary layer of86mm thickness as in the experiment. In the experi-ments, two dome models were considered one with asmooth surface and the other with an artificiallyroughened surface using a random coat of sphericalbeads to give an equivalent roughness ratio ofk0/d 0.01, where k0 is the diameter of the sphericalbeads. The size of the computational domain, in thespanwise y-direction, was equal to 4d. The experi-ments were conducted at a subcritical Reynoldsnumber based on the approach free stream velocity,air properties, and dome diameter of 1.4 105.However, the surface roughness induced flow separ-ation resulted in an effectively supercritical flow.

Computational approach

In the present LES approach, the near-wall flow fieldis resolved rather than modeled. To simulate the sur-face roughness of the rough dome of Savory andToy,11 Manhart and Wengle16 adopted a techniqueof blocking out the body-filled grid cells of thedome within a Cartesian grid. In this study, a ratherdifferent and more elaborate approach is adopted.Alternate rectangular solid blocks (Figure 2) wereextruded from the curved wall of the dome. Thesehad a perpendicular depth to the dome surfaceequal to 1.5mm and edge sides less than 6mm. Inthis way, the artificial roughness introduced bySavory and Toy11 which corresponds to an averagebead size of 1.5mm was modeled. The ideal represen-tation of the dome roughness would use surface pro-trusions based on solid cubic cells with an average sizeof 1.5 mm or even smaller. However, the generationof the corresponding mesh, and the implementation ofthe boundary conditions would, in this case, be highlyprohibitive. The approach used represents an accept-able compromise between a reasonable representationof the surface details and time resources. The distri-bution of the solid blocks in Figure 2 is uniform in theazimuthal direction but is in some places non-uniformin the latitude direction. These would have a minorimpact since real surface roughness non-uniformity israndom. This simple modeling strategy will be shownto account for the main flow features.

A hybrid mesh (Figure 3) of hexahedral and tetra-hedral cells was generated. The computationaldomain was decomposed into several blocks. A firstblock of clustered grids having the shape of a hemi-sphere surrounding the dome was created to facilitatethe generation of a structured hexahedral mesh in thezones where most of the relevant flow phenomenawere expected to occur namely; the front side, theapex region, and the wake. Further out, this hemi-spherical mesh was surrounded by a cube of tetrahe-dral cells due to the difficulty of generating ahexahedral mesh of good quality within the outercube truncated by the hemisphere surrounding

Kharoua and Khezzar 3




the dome. Finally, a relatively coarse mesh was gen-erated in the rest of the domain. The refined-mesh sizein the hemispherical block surrounding the dome is1.5 mm and approximately the same size was kept forthe tetrahedral mesh. The computational domain wasdivided into about 9 million cells for the two casesstudied. Studies on optimization of computationalresources based on LES requirements, such asAddad et al.17 for example, showed that an efficientgrid should have a cell size around the Taylor

microscales. Based on an a priori RANS simulation,the corresponding scale for this problem, was,approximately, equal to 0.2mm. The cell size in thisproblem was chosen, due to the computationalresources limitations, to be 0.4 mm inside the regionsurrounding the dome where smaller scales areexpected to reside. The values of y near the wall ofthe dome was in the range 114. This mesh resolutionnecessitated the use of a near-wall treatment modelwhich is discussed below.

Air is then working fluid at ambient temperature,having a density of 1.225 kg/m3 and dynamic viscosityof 1.7894 105kg/ms.

The boundary conditions were imposed in suchway to mimic the conditions under which the experi-ments of Savory and Toy3,10,11 were conducted. At theinlet, measured profiles of the mean streamwise vel-ocity component in addition to the turbulent param-eters required (k, ", and Reynolds stresses) wereimposed. The turbulent parameters were calculatedbased on the profile of the streamwise turbulenceintensity measured in Savory and Toy.3 The spectralsynthesizer technique18,19 was employed for the gen-eration of the fluctuating velocity components.

In this method, fluctuating velocity componentsare computed by synthesizing a divergence-free velo-city-vector field from the summation of Fourier har-monics. In ANSYS FLUENT, the number of Fourierharmonics is fixed to 100.

The turbulent kinetic energy and its dissipationrate profiles were calculated from the measured tur-bulence intensity profiles.

k 32

UavgIt 2 10

" C3=4k3=2

l11

where It is the turbulence intensity, C the constantequal to 0.09, and l the turbulence length scale, and isequal to 0.07 the hydraulic diameter based on fullydeveloped duct flow empirical results.

A zero gradient boundary condition was imposedat the outlet of the domain relying on the fact that it

x y z

Inlet Wall

Symmetry

Outlet

d8d4

3d

Figure 1. Boundary conditions and dimensions of the domain (flow from left to right).

Figure 2. Roughened dome model (the blue or dark blocks

represent the solid protrusions).

Figure 3. A view of the computational mesh.





was sufficiently far from the dome. At the top of thedomain, a symmetry condition was used at a locationwhere no effect of the dome on the flow is noticeable.Manhart8 invoked the fact that, with such boundaryconditions, the mass flux arising from the growth ofthe approaching boundary layer is omitted and mightresult in some discrepancies compared with theexperiments which were conducted inside a wall-bounded test section. In this study, the computationaldomain was truncated in the free stream, at z/d 3,relying on preliminary simulations conducted with thereal test section dimensions and the truncateddomain. The two lateral sides were taken as periodicboundaries, placed at 2d from the central verticalplane of symmetry. No-slip wall boundary conditionswere imposed at solid surfaces. The Werner andWengle20 wall function was used in regions wherethe mesh was not fine enough to resolve the viscoussub-layer. Fluent documentation states that there areno restrictions on the grid size near the wall for LESsince the wall boundary conditions were implementedusing the law of the wall approach although it isadvisable to refine the mesh near the wall to a valueof y 1 for a better accuracy of the results.

The finite volume technique implemented in thefluent code version 12.1 is used to integrate the con-servation equations. The computations were per-formed using the unsteady segregated solver. Abounded central differencing scheme was used to dis-cretize the convective term in the filtered NavierStokes and subgrid-scale turbulent kinetic energyequations. It is a composition of the pure second-order central differencing scheme, the second-orderupwind scheme and the first-order upwind scheme,see the documentation of Fluent version 12.121 refer-ring to the Normalized Variable Diagram approachproposed by Leonard.22 The pressure and velocitywere decoupled using the SIMPLE algorithm. Thepressure interpolation was performed by thePRESTO scheme.

The implicit temporal discretization was of second-order accuracy. The steady mean flow was computedusing the k" model to provide reasonable initial con-ditions for the LES simulation. Subsequently, an LESsimulation was conducted during about 1 s corres-ponding roughly to five times the residence time ofthe flow based on an average velocity of 10.76m/sand the domain length. When the flow developed,the simulation was run during another 1 s for the col-lection of statistics. The local time-averaged variableswere calculated using

; 1N

XNi1

; x, y, z, ti 12

where ; x, y, z, ti is the local and instantaneous vari-able at the position (x, y, z) of the computationaldomain at time ti and N the number of samples

collected to compute the statistical averages, the sam-pling interval time was chosen equal to the integrationtime step.

The integration time step was 104 s and insured acell Courant number in the domain less than 2. Recentstudies on optimization of LES computational effort(e.g. Kornhaas et al.23) showed that a Courantnumber equal to 2 would be sufficient for an accept-able solution, using an implicit discretization scheme,in terms of accuracy and stability for shear flows. Thecalculations were conducted with fluent running inparallel mode using 32 processors of a high perform-ance cluster HPC during about 60 days of continuousrun characterized by a total wall-clock time/time stepequal to 296 s and a total CPU time per time stepequal to 9480 s with a total of 19,597 time steps.

Results

The results of the simulations are presented in thissection for the smooth and rough domes of Savoryand Toy.3,10,11 The mean flow field, represented by thepressure coefficient, the average streamwise velocitycomponent, and the dividing streamline showing thesize of the wake recirculation zone, are discussed first.A qualitative and quantitative assessment of the flowfeatures, in different regions surrounding the dome, isthen presented. Finally, the development of theReynolds stress profiles, in the wake region arediscussed.

Mean flow field

Figure 4 illustrates the distribution of the pressurecoefficient on the dome centerline, in the streamwisex-direction, for the smooth and rough domes sub-jected to an approaching thin boundary layer.Savory and Toy3 stated that for the Reynoldsnumber considered, 1.4 105, the flow regime overthe rough model was supercritical in which case thepressure distribution over the dome surface becomesindependent of the Reynolds number while it was stillsubcritical for the smooth dome. The calculated andexperimental values of the mean pressure coefficientfor the smooth dome are shown on Figure 4(a). BothLES and k"models predictions are good on the frontside until the apex region at about 85, where the peakcorresponds to the separation of the boundary layerdeveloping on the dome surface. A slight decrease isobserved between 0 and 7. On the front side, a smallrecirculation zone was generated in the corner formedby the dome curved surface and the bottom wall. Thesecond peak at about 21 corresponds to the stagna-tion point caused by the impact of the approachingflow on the dome surface. The incoming flow impactsfirst on a recirculation zone generated upstream of thedome which acts as an obstacle. The approaching flowis, then, deviated upward impacting the dome surfaceat the position of 21. Discrepancies are observed





starting from 93 for the LES model and 100 for thek" model. The LES exhibits a local jump in the pres-sure coefficient profile due probably to a reattachmentof an existing separation bubble causing the secondnegative peak at about 97 but during recoverymatches the experimental values. In the wakeregion, the LES model slightly under-predicts thepressure coefficient, whereas the k" model showsbetter agreement with experiments. For the roughdome and in the front stagnation region, the max-imum value of the pressure coefficient is slightlyover-predicted by the ke and LES calculations until40. This behavior might be due to a lack of resolutionof the flow in this region. This was also observed byManhart8 who attributed it to the different shapes ofthe oncoming experimental and calculated boundarylayers. The decrease from the maximum value is cap-tured correctly by the calculations until the peak suc-tion value around 80. Both the LES and the kemodels exhibit a first local negative maximum at 65

followed by a decreasing trend until 75, where asecond sharp increase is observed. The roughnesseffect has moved the separation point backward

compared with the smooth dome, i.e. there is an ear-lier separation. The separation and reattachment phe-nomena are more pronounced on the surface of theroughened dome due to the nature of the surface. Therecovery of the ke model appears to be more rapidthan the LES results.

Savory and Toy3 referred to the local centerlinedrag coefficient as a useful parameter to predict thetrend of the pressure forces acting on the dome sur-face for different Reynolds numbers. The local dragcoefficient exhibits a minimum which designates thecritical flow regime (critical Reynolds number). In thisstudy, the local drag centerline drag coefficient wasequal to 0.5 for the rough dome and 0.63 for thesmooth dome with a deviation from the experimentalresults equal to 4.6% and 40%, respectively.

The fluctuating pressure coefficient is presented inFigure 5 along the centerline of the dome in the sym-metry plane y/d 0. The distribution along the roughdome is characterized by three peaks at 10, 90, and109, respectively, while, for the smooth dome, thepeaks are observed at 2, 14, and 112, respectively.Cheng and Fu5 explained that the fluctuating pressure

1.0

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

0 20

Mea

n pr

essu

re c

oeff

icie

ntM

ean

pres

sure

coe

ffic

ient

Mea

n pr

essu

re c

oeff

icie

nt

40 60 80 100 120 140 160 180

Angle ()

(a)

[3]

LES

k

1.0

0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

0 20 40 60 80 100 120 140 160 180

Angle ()

(b)

[3]

LES

k

Figure 4. Mean pressure coefficient distribution on the dome centerline: (a) smooth dome and (b) rough dome.





coefficient peaks are related to transition of separationbubbles and reattachment phenomena. ForRe< 1.8 105, as in the present case, they observeda first peak near the apex region (8090) and abroader lump at 140. In the present simulation witha smooth dome, only one peak, near the separationregion, is observed at about 112, the second peak isnot detected probably because the separation bubbleis not resolved. Two other peaks are generated in thefront side under the effect of the impingement of theincoming flow on the dome surface at 14 and thereattachment of a small recirculation zone, generatedin the corner formed by the dome and the floor, at 2.The peak at 112 corresponds approximately to theseparation of the boundary layer developing along theupper surface of the dome. The related flow behavioris illustrated in Figure 6(a) on which was superim-posed the surface contours of the RMS of the pressurecoefficient, and the mean velocity vectors plotted inthe plane of symmetry. The position of the RMS

Cp peak is indicated by the thick horizontal line.The abovementioned small recirculation zone andimpact point are clearly seen. The situation is differentfor the rough dome case where only one peak isobserved on the front side due to the absence of thesmall recirculation zone (Figure 6(b)). Near the apex,a first peak is observed at 90 and second one at 109

which, according to Cheng and Fu,5 are due to sep-aration and reattachment phenomena, respectively.

Figure 7 shows the dividing streamline in the wakeregion. Taking the axis of the dome as a reference inthe streamwise x-direction, the position of the meanreattachment is 0.8d and 1d for the smooth and roughdomes, respectively. An underestimation of thereattachment distance is clearly seen compared withthe values measured by Savory and Toy3 for the samecase or the value of Tavakol et al.7 which was 1.17 fora Reynolds number equal to 6.4 104, based on a freestream velocity of 8.5m/s and a diameter of the domeof 0.12m.

0.04

0

0.04

0.08

0.12

0.16

0 20 40 60 80 100 120 140 160 180

rms

pres

sure

coe

ffic

ient

Angle ()

LES rough

LES smooth

Figure 5. RMS surface pressure coefficient distribution on the dome centerline.

RMS: root mean square.

(a) (b)

Figure 6. Mean velocity vectors (plane of symmetry) superimposed with the contours of RMS of Cp (dome surface).

Thick lines indicate the maximum RMS of Cp. (a) Smooth and (b) rough.

RMS: root mean square.





To investigate the causes of the underestimation ofthe reattachment point, profiles of the average stream-wise velocity component, obtained by LES for bothrough and smooth domes and experiments ofSavory24 for the rough dome, are plotted at differentlongitudinal positions in the vertical z-direction inFigure 8. At x/d 0.8 and approaching the floor sur-face, the reversed flow starts to become underestimatedcausing the location of zero-streamwise velocity tooccur at a lower position z/d. At x/d 1, the simulated

flow has already reattached while the experiments stillshow a small backflow in the region below z/d 0.33.At x/d 1.2, which is approximately the measuredreattachment position, the streamwise velocity compo-nent, predicted by LES, is positive everywhere in thevertical direction while it is equal to zero belowz/d 0.3. From the mean velocity profiles, the maindiscrepancy between a rough and smooth dome existwithin the separating shear layer, with the rough domeprofiles showing a higher momentum deficit.

0

0.1

0.2

0.3

0.4

0.5

z/d

0.6

0.7

0.8

0.5 0.5 1.5

x/d=0.4

0.5 0.5 1.5

x/d=0.6

0.5 0.5 1.5

x/d=0.8

0.5 0.5 1.5

x/d=1

0.5 0.5 1.5

x/d=1.2

Figure 8. Profiles of the mean streamwise velocity component in the vertical direction at different x/d positions: , Savory;24 , LESrough; and - - -, LES smooth.

LES: large eddy simulation.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

z/d

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

x/d

[3]: rough

LES rough

LES smooth

DomeSurface

Figure 7. Dividing stream line.





Flow structures and patterns

Isosurfaces of vorticity and mean path lines are usedto illustrate features of the flow. In addition, streamtraces obtained from the mean velocity field are alsoused. A comparison between the smooth and roughdomes is conducted in the light of the qualitative andquantitative results presented.

Figure 9 shows isosurface vorticity contours whichillustrate the main flow features previously described.An intense turbulent activity close to the floor isobserved in the front side of the dome and developsaround it. This turbulent behavior corresponds to thehorseshoe vortex and is more pronounced for thesmooth dome. For the rough dome, it is not clearlyshown due to the value of the vorticity, which waschosen to capture the wake turbulence rather. Theflow separation occurs earlier for the rough than forthe smooth dome as shown before with the wall pres-sure coefficient results. The turbulent structures gen-erated by the separated boundary layer from the topface of the smooth dome, tend to be entrained in adirection slopping downward, while for the roughmodel, they move parallel to the longitudinaldirection.

Figure 10 shows the streamlines projected on sev-eral horizontal planes. Close to the bottom atz/d 0.005, the horseshoe vortex is present with a lat-eral axis at about x/d0.9 for the smooth dome andx/d1.1 for the rough dome. Two counter-rotatingrecirculation zones, generated in the wake region,extend to a longitudinal position of x/d 0.8 for thesmooth dome and x/d 1 for the rough dome. Atz/d 0.05, the horseshoe vortex for the rough dome,disappears while, in the wake, the recirculation zonesare still present with a larger size for the rough dome.

Once the horseshoe, for the rough dome, disappears,the streamlines tend to converge in a more pro-nounced way in the far wake compared with thesmooth dome case. For both geometries, it seemsthat, in the positive lateral wake side, some of thestreamlines circumventing the recirculation zone areentrained by the opposite recirculation zonerather than the streamwise flow. The counter-rotatingrecirculation zones vanish at a vertical position ofz/d 0.25 for the smooth dome, while for the roughdome, they disappear at z/d 0.375.

The most important flow feature seen in the verti-cal planes perpendicular to the main flow direction(Figure 11) is the rolled up vortices in the lateralsides of the dome belonging to the horseshoe vortex.At x/d 0 (apex), trailing vortices, with a quasi-circu-lar cross section, are seen to be closer to the dome forthe smooth model while flattened vortices are locatedfarther from the rough dome, at about y/d 1.25. Inthe wake region at x/d 1, the trailing longitudinalvortices are flattened for both cases under the effectof diffusion and decay. In the vertical streamwise dir-ection (Figure 12), the main observed features are thehorseshoe vortex upstream of the dome and the recir-culation zone in the wake. In the plane y/d 0, thehorseshoe vortex axis is located at x/d0.65 andz/d 0.05 for the smooth dome. A flattened thinnerhorseshoe vortex is generated at x/d0.98 andz/d 0.025. In the wake of the smooth dome, thecenter of the recirculation zone has the coordinatesx/d 0.58 and z/d 0.27 while a larger one is gener-ated behind the rough dome at x/d 0.59 and at ahigher vertical position z/d 0.355. The center ofthe recirculation zone obtained in the experiments ofSavory and Toy,3 was at approximately x/d 0.77and z/d 0.38. The recirculation zone axis predicted

(a1) smooth (b1) rough

(a2) smooth (b2) rough

Figure 9. Isosurface vorticity contours from side and isometric views: (a) smooth (1000 s1) and (b) rough (1200 s1).





by LES is located at a closer distance to the dome andslightly lower which explains the shorter reattachmentdistance compared with the experimental results(Figure 7). At y/d 0.5, the recirculation zone in thewake has disappeared for the smooth dome while it isstill present for the rough dome.

The main findings through this comparison are thedifferent shapes and positions of the features charac-terizing the turbulent flow around domes especiallythe flattened shape of the horseshoe vortex for therough dome and its position away from the obstacle.Bradshaw25 explained that the pressure gradientsmight play a key role in determining the position ofthe characteristic phenomena. Longitudinal pressuregradients affect the separation position along thedome surface while lateral pressure gradients mightmove the vortex shoe away from the corner formedby the dome and the floor. Unfortunately, the litera-ture does not contain enough experimental results for

a detailed comparison between the smooth and therough domes.

Turbulent flow field prediction

Figures 13 and 14 show the profiles of the Reynoldsstresses , , and turbulent kinetic energy k inthe wake region in the plane of symmetry y/d 0 andin a horizontal plane at z/d 0.158. These results referto the rough dome for which experimental values areavailable in Savory and Toy.11

In the plane of symmetry, the calculated longitu-dinal correlation over-predicts the experimentalmaximum peaks within the shear layers betweenx/d 0.4 to 0.8. The values inside the recirculationwake zone are, however, nicely replicated.Nonetheless, the positions of the peaks, indicatingthe approximate location of the shear layer boundingthe wake region, are well captured. In view of the

Streamlines at z/d =0.005

x/d

y/d

-1 0 1 2 3

-1

-0.5

0

0.5

1

x/d

y/d

-1 0 1 2 3 4

-1

0

1

a) smooth b) rough

Streamlines at z/d = 0.05

x/d

y/d

-1 0 1 2 3

-1

-0.5

0

0.5

1

x/d

y/d

-1 0 1 2 3-1

-0.5

0

0.5

1

a) smooth b) rough

Streamlines in different planes xy

Figure 10. Time-averaged streamlines plotted in xy planes at different vertical positions z/d.




Kharoua and Khezzar

Streamlines at 1Jd = 0.1

0 x/d

a) smooth

Streamlines at d cl =0.25

0.5

~ 0

-0.5

0 2 x/d

a) smooth

Streamlines at d cl =0.375

~ 0

-0.5

x/d a) smooth

nature and structure of the wake for both rough and smooth domes, the profiles of for the smooth dome case exhibit large deviations. The region bounded by the shear layer is remarkably larger for the rough dome. The profiles for the component are, also, better predicted for the rough geometry. They show that the velocity fluctuations are Jess important at high z/d positions corresponding to the shear layer separated from the top surface of the dome. In contrast , the Reynolds stress component in the spanwise direction exhibit a highly turbulent behavior at z/d < 0.2 far from the dome at 0.8 :( x/d :( 1.2. This region corresponds to the zone of impact of the turbulent structures resulting from the separation phenomenon from the lateral sides of the dome, at low z/d positions, known as Von Karman vortices. Similarly to the Reynolds stresses, the turbu-lent kinetic energy, shown in Figure 13(c), is well pre-dicted at x /d = 1.2 while, for the remaining axial positions, discrepancies, especially for the peaks, can be observed. It is noteworthy to mention that the

2

11

x/d b) rough

0.5

~ o

-0.5

x/d b) rough

0.5

" ->. 0 -0.5

x/d

b) rough

RNG k- e model, also, predicted, correctly, the loca-tion of the peaks of turbulent kinetic energy although with a clear underestimation. This is expected since such model is unable to simulate correctly flows with strong mean streamline curvature and anisotropy as exists in the wake of the dome.

Figure 14(a) shows profiles of the component in a horizontal plane at z/d= 0.158 and shows that the profiles are, relatively, well predicted for the positions x/d = I and x/d = 1.2 while remarkable overestimated peaks are observed close to the obstacle at x /d = 0.6 and x /d = 0.8. This is, probably, due to the strong turbulent effect induced by the technique used to rep-resent the surface roughness of the dome which is not entirely perfect. However, contrary to the predictions on the plane of symmetry, the location of the peaks is slightly shifted downward compared with the experi-mental results which means that a shorter shear-Jayer-bounded region is being predicted by the numerical simulation and which might , also, explain the shorter reattachment length observed from Figure 8.


Figure 11. Time-averaged streamlines plotted in yz planes at different longitudinal positions x/d.

x/d

z/d

-1 -0.5 0 0.5 1 1.5 20

0.5

x/d

z/d

-1 -0.5 0 0.5 1 1.5 20

0.5

a) smooth b) rough

x/d

z/d

-1 -0.5 0 0.5 1 1.5 20

0.5

x/d

z/d

-1 -0.5 0 0.5 1 1.5 20

0.5

a) smooth b) rough

Figure 12. Time-averaged streamlines plotted in xz planes at different lateral positions y/d.





0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1

y/d

x/d=0.6

0 0.12

x/d=0.8

0 0.1

x/d=1

0 0.1

x/d=1.2(c)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1

y/d

x/d=0.6

0 0.12

x/d=0.8

0 0.1

x/d=1

0 0.1

x/d=1.2(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.05

y/d

x/d=0.6

0 0.052

x/d=0.8

0 0.05

x/d=1

0 0.05

x/d=1.2(b)

Figure 14. Profiles of the Reynolds stresses and the turbulent kinetic energy in the horizontal plane z/d 0.158 at different lon-gitudinal positions: , Savory and Toy;11 , LES rough; - - -, LES smooth; and , RNG k".LES: large eddy simulation.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1

z/d

x/d=0.4

0 0.1

x/d=0.6

0 0.1

x/d=0.8

0 0.1

x/d=1

0 0.1

x/d=1.2(a)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1

z/d

x/d=0.4

0 0.1

x/d=0.6

0 0.1

x/d=0.8

0 0.1

x/d=1

0 0.1

x/d=1.2(b)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1

z/d

x/d=0.4

0 0.1

x/d=0.6

0 0.1

x/d=0.8

0 0.1

x/d=1

0 0.1

x/d=1.2(c)

Figure 13. Profiles of the Reynolds stresses and the turbulent kinetic energy in the plane of symmetry y/d 0 at different longi-tudinal positions: , Savory and Toy;11 , LES rough; - - -, LES smooth; and , RNG k".LES: large eddy simulation.





The peaks are much lower for the smooth model lead-ing to a smaller shear-layer-bounded region in thewake and an even shorter reattachment longitudinallocation. Figure 14(b) illustrates profiles of the lateralReynolds stress component in the same hori-zontal plane close to the bottom surface. It can beobserved that, close to the dome at x/d 0.6, aslight peak exists at approximately y/d 0.47 whichis not well predicted by LES. Beyond x/d 0.6, thelateral stress component is overestimated andthe trend is a monotonic decay of the lateral Reynoldsstress component from the symmetry plane towardthe outer region until, approximately, y/d 0.5.Profiles of the turbulent kinetic energy are shown inFigure 14(c). At x/d 0.6 and x/d 0.8, the peaks aty/d 0.38 and y/d 0.32, respectively, represent theeffect of the longitudinal component while the rela-tively good prediction of k beyond x/d 1 is consist-ent with the good prediction of the stress componentsat the same positions.

Conclusions

A LES of a turbulent flow around smooth and roughdomes was conducted. The subgrid-scale model basedon the transport of the subgrid-scale turbulent kineticenergy was used. The rough dome surface modelingpresented a challenge to capture its geometricaldetails. The surface roughness of the rough dome sur-face was incorporated into the geometry using solidblocks extruded with an average size of the glassbeads diameter used in the experiment and representsa reasonable approach to model surface details.

The pressure coefficient distribution along the cen-terline of the dome was predicted with very goodaccuracy although with slight discrepancies in therecovery region. The LES under-predicted the pres-sure coefficient slightly for the smooth dome butover-predicted it for the rough dome although thedifferences were slight. The predicted rear reattach-ment point downstream of the dome was locatedcloser to the dome compared to the experiments.

For the rough dome, the Reynolds stresses werewell predicted in the vertical plane of symmetryalthough the peaks were overestimated. In the hori-zontal plane close to the floor, however, the peakswere underestimated announcing a smaller turbu-lence-affected region in the wake compared with theexperiments.

The LES model allowed the visualization of fea-tures of the flow that are difficult to observe experi-mentally at high Reynolds numbers. The flow aroundthe rough dome was characterized by a flattenedhorseshoe vortex shifted away from the obstacle com-pared to the smooth dome. The horseshoe vortexdevelops around and behind the dome in the formof trailing vortices, becoming progressively parallelto the flow direction with a separating lateral distancebetween either sides being larger for the rough dome.

In the wake, a larger vorticity content region boundedby the separated shear layers was observed for therough dome.

Funding

This research received no specific grant from anyfunding agency in the public, commercial, or not-for-profit

sectors.

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Appendix

Notation

d dome diameterIt turbulence intensityk turbulent kinetic energyp pressureSij strain rate tensort timeu, v, w fluctuating velocity componentsU, V, W velocity componentsUi filtered velocity component in the xi

directionx, y, z Cartesian coordinates

f LES filter size laminar dynamic viscosityt turbulent dynamic viscosityij laminar stress tensorij subgrid-scale stress tensor




large eddy simulation study of turbulent flow around smooth and rough domes.pdf

Documents

smooth dome

dome diameter

aroundthe rough dome

dome shape

rough hemispherical

thetwo domes

mean velocity

different flow behavior