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Numerical Heat Transfer, Part A:ApplicationsAn International Journal of Computation andMethodologyPublication details, including instructions for authors and subscription information:http://www-intra.informaworld.com/smpp/title~content=t713657973

Three-Dimensional Simulation of Flow Past a CircularCylinder by Nonlinear Turbulence ModelT. Ayyappan a; S. Vengadesan aa Fluid Mechanics Laboratory, Department of Applied Mechanics, Indian Institute ofTechnology Madras, Chennai, India

Online Publication Date: 01 January 2008

To cite this Article: Ayyappan, T. and Vengadesan, S. (2008) 'Three-Dimensional Simulation of Flow Past a CircularCylinder by Nonlinear Turbulence Model', Numerical Heat Transfer, Part A: Applications, 54:2, 221 — 234

To link to this article: DOI: 10.1080/10407780802084694URL: http://dx.doi.org/10.1080/10407780802084694

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THREE-DIMENSIONAL SIMULATION OF FLOWPAST A CIRCULAR CYLINDER BY NONLINEARTURBULENCE MODEL

T. Ayyappan and S. VengadesanFluid Mechanics Laboratory, Department of Applied Mechanics,Indian Institute of Technology Madras, Chennai, India

Numerical simulation of flow past a circular cylinder at sub-critical Reynolds number

Re ¼ 3,900 is performed using three-dimensional, unsteady, Reynolds–Averaged Navier-

Stokes (URANS) equations. A nonlinear turbulence model based on the k–e formulation

is used to achieve the turbulent closure. The results obtained by the simulations are com-

pared with experimental and previously reported numerical results. The grid used for the

present simulation is reasonable, and the accuracy obtained is good considering the compu-

tational cost involved in carrying out large-eddy simulations (LES) for the same test case.

The test flow is also simulated using standard k–e model, and the results obtained by the

nonlinear k–e model are found to be better.

1. INTRODUCTION

The flow past a cylinder of circular cross section has been the subject of interestfor industrial researchers as well as scientists, because of its wide range of applica-tions. To cite a few examples, flow in bridge piers, chimney stacks, and tower struc-tures in civil engineering; electrodes in chemical engineering; nuclear fuel rods in theatomic field and heat exchanger tubes in thermal engineering, etc., fall under thissubject of study. Although the geometry is simple, the flow has complicated featuressuch as stagnation points, laminar boundary-layer separation, turbulent shear layers,periodic vortex shedding, and wakes. Even though there is much literature availableon numerical simulation of laminar flow past a two-dimensional circular cylinder atlow Reynolds number, a focus on practical high Reynolds numbers is less. Thiscould be due to the complexity of formulating Reynolds stresses in turbulent flows.The majority of turbulent flow calculations carried out in earlier days used two-equation models such as the standard k–e model (hereafter referred as SKE) andthe k-x model, because of their robustness, computational efficiency, and complete-ness. In the classical SKE model, the turbulent kinetic energy (k) and the turbulentkinetic energy dissipation rate (e) were calculated using modeled transport equationsseparately for k and e along with the Boussinesq eddy viscosity approximation. Sub-sequently, nonlinear models were proposed by many researchers such as Gatski and

Received 27 September 2007; accepted 7 March 2008.

Address correspondence to S. Vengadesan, Fluid Mechanics Laboratory, Department of Applied

Mechanics, IIT Madras, Chennai 36, India. E-mail: [email protected]

221

Numerical Heat Transfer, Part A, 54: 221–234, 2008

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7782 print=1521-0634 online

DOI: 10.1080/10407780802084694

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Speziale [1], Craft et al. [2], and Shih et al. [3]. Some researchers have tested theappropriateness of nonlinear turbulence models for simple flows such as homo-geneous shear flow, separated flow over a backward-facing step, and flow in aconfined jet. They obtained encouraging results in the prediction of mean andturbulent quantities.

The two-dimensional, unsteady, Reynolds-averaged Navier-Stokes (URANS)simulation of subcritical flow past a circular cylinder at Re ¼ 1.4� 105 was carriedout by Tutar and Holdo [4] using the SKE, nonlinear k�e formulation and large-eddysimulation (LES) technique. The wake centerline velocity recovery predicted by the non-linear k�e turbulence model showed better agreement with the LES and experimentalresults than those predicted by the SKE model. Saghafian et al. [5] simulated the two-dimensional flow past a circular cylinder using a nonlinear eddy viscosity model andtested the mean drag coefficient (Cd,mean), root-mean-square (rms) lift coefficient (Cl,rms),and Strouhal number value (St ¼ fD=U1) for the range of Reynolds numbers from sub-critical laminar separation to supercritical turbulent separation. The authors used amodified form of the Craft et al. [2] model and predicted the drag crisis at Re ¼ 2� 105,and they proved the suitability of nonlinear turbulence models. The standard and renor-malized group versions of the k�e model were examined by Jennifer [6] for two-dimen-sional flow past a circular cylinder at subcritical Reynolds number Re ¼ 5,232.

With this increasing interest in nonlinear turbulence models, Kimura andHosoda [7] proposed a cubic nonlinear k�e model by accounting for the effect of ani-sotropy. They tested the model for two-dimensional flow around a square cylinderand a surface-mounted cubic obstacle. The model, which included the realizabilitycondition, performed better than linear models when compared with experimentalresults. Recently, the above model was used by Ramesh et al. [8] for simulating three-dimensional flow around a square cylinder, and they found it performs better thanresults by the standard k�e and RNG k�e models.

In the recent past, LES and direct numerical simulation (DNS) techniquesemerged as tools for predicting turbulent flows and are being tested for cases suchas homogeneous shear flows, open-channel flows, and flow past cylinders. Theabove-mentioned two tools give better prediction of both mean and turbulentquantities, as they solve for instantaneous flow. However, the computational costinvolved in these methods is very high, as the mesh resolution requirement isstringent. This limits the Reynolds number to be studied by these methods. Beaudanand Moin [9] (hereafter referred to as BM) were the first to carry out a detailed LESstudy for the subcritical flow past a circular cylinder at Re ¼ 3,900.

NOMENCLATURE

Cd mean mean drag coefficient

Cl rms rms lift coefficient

D diameter of cylinder

f frequency of vortex shedding

K turbulent kinetic energy

rmin location of minimum velocity

Re Reynolds number

St Strouhal number

U1 streamwise velocity

U2 cross-streamwise velocity

U1 free-stream velocity

e turbulent kinetic energy dissipation

rate

222 T. AYYAPPAN AND S. VENGADESAN

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Later, Ong and Wallace [10] conducted detailed experiments using an X-arrayprobe hotwire and reported data to verify the LES simulation of BM. The mean velo-city profile at different downstream locations showed the validity of prior LES results,but the turbulent Reynolds stress by the experiment deviated considerably from that ofLES. This factor was later explained by Mittal and Moin [11]. They attributed it to thedissipative property associated with the upwind difference scheme, which suppressesthe small-scale turbulences in the downstream, where the grid was coarse as well.

The same test case was simulated using LES technique by Breuer [12] and byKrevechenko and Moin [13]. DNS study by Ma et al. [14] with a spectral finite-element method explained the extent of recirculation length (Lr) for two domain sizesin the spanwise directions. Large recirculation length was formed for the domain sizeof pD and short recirculation length for the domain size of 2pD. The spanwiseresolution was maintained in both cases.

Quantitative and qualitative comparison of three-dimensional LES and two-dimensional Reynolds-averaged Navier-Stokes (RANS) simulation based on thek�x formulation for the test case of a circular cylinder at Re ¼ 3,900 was carriedout by Lubcke et al. [15] (hereafter referred as LU01). In RANS, both linear andnonlinear equations are used for the Reynolds stress formulation. Simulation bythe nonlinear RANS with curvature correction showed better performance thanthe simulation without it. The realizability of the modeled subgrid-scale stressesand the computed Reynolds stresses were analyzed by Franke and Frank [16] (here-after referred as FF02) by simulating the same flow by LES. By taking large timesequences for unsteady averaging, the accuracy was reported to have been improved.

It is clear that the test case at Re ¼ 3,900 has been used widely and has become abenchmark for many numerical simulations such as URANS, LES, and DNS. LES andDNS are inherently three-dimensional, whereas URANS results have been reportedonly for two-dimensional simulations. That is, URANS performed for the subcriticaland the supercritical flow used the two-dimensional approach, and the effect of three-dimensionality was not considered. The three-dimensional LES study by BM revealsthe irregularity of vortex shedding present in the flow, which cannot be captured bytwo-dimensional simulation. In the present study, we perform unsteady three-dimen-sional simulation of flow past a circular cylinder at subcritical Reynolds numberRe ¼ 3,900 by a nonlinear turbulence model. Reynolds number is defined as U1D=n,where U1 is the free-stream velocity (m=s), D is the diameter of the cylinder (m), andn is the kinematic viscosity (m2=s). The results are compared with experimental andpreviously reported numerical results and the performance of the model is discussed.

2. COMPUTATIONAL METHODS

2.1. Basic Equations

The ensemble-averaged RANS equations for an incompressible flow areContinuity equation:

qUi

qxi¼ 0 ð1Þ

SIMULATION OF FLOW PAST A CIRCULAR CYLINDER 223

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Momentum equation:

qUi

qtþ qUjUi

qxj¼ � qP

qxiþqð�u0iu

0jÞ

qxjþRe�1 q2Ui

qxj qxjð2Þ

where xi is the spatial coordinate, t is the time, Ui is the ensemble-averagedvelocity, u0i is the fluctuating velocity, and P is the averaged pressure dividedby the density. As a result of ensemble-averaging process, further unknownsare introduced into the momentum equations by means of Reynolds stressesð�u0iu

0jÞ. In engineering flows, closure approximation using two-equation models

for ð�u0iu0jÞ have gained popularity because of their simplicity. In this article

the study is confined to the k�e model, which employs additional transportequations for turbulent kinetic energy k and its dissipation rate e, and they aregiven as

qk

qtþ qkUj

qxj¼ �u0iu

0j

qUi

qxj� eþ q

qxj

nt

rkþ n

� �qk

qxj

� �ð3Þ

qeqtþ qeUj

qxj¼ �Ce1

ek

u0iu0j � Ce2

e2

kþ qqxj

nt

reþ n

� �qeqxj

� �ð4Þ

where k is the turbulent kinetic energy, e is the turbulent kinetic energy dissi-pation rate, n is the fluid kinematic viscosity, and nt is the eddy viscosity.Ce1;Ce2, rk, and re are the model constants.

2.2. The Nonlinear k�e Model

In the standard k�e model, the Reynolds stresses are calculated by the linearrelation proposed by Boussinesq,

�u0iu0j ¼ ntSij �

2

3kdij ð5Þ

where Sij is the mean strain rate and dij is the Kronecker delta. The above isotropicrelation assumes that the principal axis of the Reynolds stress tensor sij coincideswith that of the mean strain rate. The standard k�e model does not take intoaccount the anisotropic effects and fails to represent the complex interactionmechanisms between Reynolds stresses and the mean velocity field. For example,the linear model fails to mimic the effects related to streamline curvature, secondarymotion, or flow with extra strain rates. These anisotropic effects can be predicted byintroducing a nonlinear expression for the Reynolds stresses as given in the followingexpression:

�u0iu0j ¼ ntSij �

2

3kdij þ � k

ent

� �� nonlinear terms

� �ð6Þ


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Constitutive relations for the nonlinear terms in the Reynolds stress equation havebeen proposed by many (e.g., [1–3, 7]). A general expression is given as

a1ðSilXlj þ XilSljÞ þ a2 SilSlj �1

3SkmSmkdij

� �þ a3 XilXlj �

1

3XkmXmkdij

� �

þ a4k

eðSklXlj þ SkjXliÞSkl þ a5

k

eXilXlmSmj þ SilXlmXmj þ

2

3SlmXmnXnldij

� �

þ a6k

eðSilSklSklÞ þ a7

k

eðSijXklXklÞ

Coefficients ai (i ¼ 1–7) in these nonlinear terms are determined through rapiddistortion theory and the realizability principle. In the present study, coefficientsproposed by Kimura and Hosoda [7] for bluff-body flows are used, and they aregiven as

a1 ¼ðC3 � C1Þ

4:0a2 ¼

ðC1 þ C2 þ C3Þ4:0

a3 ¼ðC2 � C1 � C3Þ

4:0

a4 ¼ 0:02 fMðMÞ a5 ¼ 0 a6 ¼ 0 a7 ¼ 0

where C1 ¼ 0:4 fMðMÞ; C2 ¼ 0; C3 ¼ �0:13 fMðMÞ, and fMðMÞ ¼ ð1þ 0:01M2Þ�1,with

M ¼ maxðS;XÞ S ¼ k

e

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2

qUi

qxjþ qUj

qxi

� �2s

X ¼ k

e

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

2

qUi

qxj� qUj

qxi

� �2s

S is the strain parameter and X is the rotation parameter. In RANS models, the tur-bulent viscosity is given by the expression nt ¼ Cmðk2=eÞ, and in the standard k�emodel, Cm is set to a constant value of 0.09. It is known that this constant value doesnot satisfy the realizability constraint under certain circumstances. In the presentmodel, Cm is expressed as a function of S and X and is given by

Cm ¼ min 0:09;0:3

1þ 0:09M2

� �ð7Þ

2.3. Numerical Strategies

The governing equations for velocities and turbulent quantities are solvedusing the finite-volume-based commercial solver FLUENT 6.2. The equations arediscretized on a collocated grid in fully implicit form. Momentum equations aresolved using the QUICK scheme, and the SIMPLE algorithm is used for couplingthe pressure and velocity terms. The second-order upwind scheme is used to discre-tize convective terms and also the terms in equations for turbulent quantities. Thesecond-order implicit scheme is used for time integration of each equation. Thepresent nonlinear model is incorporated in FLUENT through user-defined functions(UDFs). The nonlinear stress term is added as a source term in equations for k and e.The turbulent viscosity is also made to vary according to Eq. (7) through a UDF.


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The implementation of the present nonlinear k�e model (hereafter referred to asNLKE) was validated for turbulent flow around a square cylinder [8].

A box-type computational domain (Figure 1a) with structured grid inCartesian coordinates having origin at the center of the cylinder is used. Streamwisedirection is along the x axis with x ¼ 0 at the center of the cylinder, the y axis is thevertical axis with y ¼ 0 the wake centerline, and the z axis is the spanwise directionwith z ¼ 0 being the midspan of the cylinder. The upstream boundary with uniforminlet velocity is placed at 7D from the center of the body, where D is the diameter ofthe cylinder. This extent is less than that used in the LES simulation of FF02, wherethe inlet boundary was kept at 10D from center of the body and the URANS, andthe LES simulation of LU01, who used 8D from the center of the body. At the inletboundary, 2% turbulence intensity is specified. The convective boundary conditionat the outlet is specified at 20D downstream of the body. The same domain size wasused by FF02, whereas LU01 used 24D. In the y direction, slip boundary condition isapplied at 8D from the center of the body, which is the same as the one used by pre-vious references. Most other reported LES simulations have used either an O-grid ora C-grid with larger boundary dimensions. All the LES simulations were done withspanwise domain size of pD, whereas the URANS simulations have been reportedonly for two-dimensional flow.

The three-dimensional Floquet stability analysis reported by Barkley andHenderson [17] and the experiment done by Prasad and Williamson [18] at subcriti-cal Reynolds number showed the dominant spanwise scales having wavelengths ofapproximately three to four cylinder diameters in the Reynolds number range180 < Re < 240. After this Reynolds number, the wavelength shortens to nearlyone diameter. In our simulation, we have taken a spanwise length of 4D along thez direction, and periodic boundary condition is enforced on the boundary. Thisextent is slightly larger than that used in LES and DNS calculations. A non-equilibrium wall function approach is used to capture the adverse pressure gradienteffect.

The present NLKE model was simulated with five different grids for griddependence studies. In all grids, along the circumference of the cylinder, 144 meshpoints were placed, and in the cross-stream direction, 85 mesh points were used.For all grids, the mesh points were placed uniformly in the spanwise direction. Dif-ferent grids were achieved by systematically changing the resolution in the stream-wise and spanwise directions. In Grid A, the first grid point is placed at a distanceof 0.01D from the cylinder wall. For Grid B, Grid C, and Grid E, the first grid pointis placed at 0.005D, and for Grid D, it is placed at 0.0025D. Accordingly, in thestreamwise direction, the total numbers of mesh points are 100 for Grid A, 145for Grid B, Grid C, and Grid E, and 180 for Grid D. The effect of spanwise resol-ution is studied with 20, 30, and 38 points. Figure 1b shows the typical grid used.Bulk parameters obtained by different grids are compared in Table 1. It can be seenthat Grid C gives better results considering also the computational cost. In Grid Cwe have taken 145 mesh points in the streamwise direction, 85 mesh points in thecross-streamwise direction, and 30 mesh points in the spanwise direction. It is alsoto be noted that the overall grid size is coarser than those used by LES and DNS.The final results reported here are for simulations obtained with Grid C. The samegrid is used to simulate three-dimensional flow by the standard k�e model (hereafter


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Figure 1. (a) Schematic representation of the computational domain. (b) Typical grid used in the

computation.


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referred to as 3DSKE) and two-dimensional flow by the nonlinear k�e model (here-after referred to as 2DNLKE) for comparison purposes.

3. RESULTS AND DISCUSSION

The solution is initiated and allowed to march in time with a nondimensionalincrement of dt( ¼ Dt U1=D) ¼ 5� 10�3 until the vortex shedding becomes per-iodic. The time variation of the lift coefficient is shown in Figure 2, and periodicityis observed. The simulation is continued for five more vortex shedding cycles toadvect all the numerical errors to downstream. In Breuer’s LES work [12], in the timehistory of the lift coefficient, there was a low-frequency component over a regularperiodic component. Hence, in order to achieve reproducible statistics, averagingwas done over 22 vortex shedding cycles. In the present work, as there are no suchfeatures, the time averaging is done over 10 vortex shedding cycles to obtain bothbulk and mean field quantities. The streamline plots at different instances for onevortex shedding cycle are plotted in Figure 3. Here T (¼ tU1=D) is nondimensional

Figure 2. Variation of lift coefficient with time.

Table 1. Mesh sensitivity study

Grid Mesh size Cd mean St U1 min=U1 rmin=D Lr=D

Exp — 0.98� 0.05 0.215� 0.005 �0.24� 0.1 [21] 0.72� 0.1 [21] 1.33� 0.2 [20]

Grid A 100� 85� 30 0.83 0.179 �0.17 0.61 1.20

Grid B 145� 85� 20 0.81 0.181 �0.19 0.68 1.35

Grid C 145� 85� 30 0.94 0.211 20.22 0.74 1.39

Grid D 180� 85� 30 0.95 0.212 �0.22 0.78 1.38

Grid E 145� 85� 38 0.97 0.212 �0.21 0.74 1.34


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time for one vortex shedding cycle. The bulk parameters by the present simulationand the experimental and previous numerical results are compared in Table 2. Thepresent three-dimensional simulation with the NLKE model (hereafter referred toas 3DNLKE) shows better agreement with the experimental results. The present3DNLKE simulation predicted the mean drag coefficient (Cd mean) and Strouhalnumber (St) within the experimental uncertainty, but the two-dimensionalsimulation (2DNLKE) underpredicts these values.

The recirculation length (Lr=D) and negative velocity (U1 min=U1) in the wakepredicted by the present 3DNLKE simulation agreed well with those of experimentaland LES results. Simulation by 3DSKE predicts longer recirculation length (Lr=D)and less negative velocity in the wake. Location of the minimum velocity (rmin=D)by the present two-dimensional simulation with the NLKE model (2DNLKE) andthe two-dimensional URANS simulation result from LU01 showed an underpredicting

Figure 3. Streamline patterns at four different time intervals for one vortex shedding cycle.

Table 2. Comparison of bulk parameters for flow over circular cylinder at Re ¼ 3,900

Contribution Model Cd mean St U1 min=U1 rmin=D Lr=D

Present 3DNLKE 0.94 0.211 20.22 0.74 1.39

Present 2DNLKE 0.89 0.195 �0.2 0.61 1.25

Present 3DSKE 0.97 0.182 �0.18 0.81 1.75

Exp — 0.98� 0.05 [19] 0.215� 0.005 [20] �0.24� 0.1 [21] 0.72� 0.1 [21] 1.33� 0.2 [20]

Beaudan and

Moin [9]

LES 1.00 0.203 �0.32 0.88 1.36

Breuer [12] LES 1.016 — �0.234 0.91 1.372

Lubcke

et al. [15]

2-D

URANS

0.98 0.203 �0.360 0.66 1.2


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trend. However, the present 3DNLKE results match well with experiment. The meanstreamwise velocity recovery along the wake centerline is shown in Figure 4. It canalso be observed that the experimental data showed some scatter. Except for theselocations, the present 3DNLKE results are in good agreement with experiment.However, slower wake velocity recovery is observed with 3DSKE model.

The 2-D URANS simulation by LU01 predicted well the negative velocity justbehind the body and the recovery in the wake. In the present 2DNLKE simulation,we were able to get vortex shedding and negative velocity behind the body, but therecirculation length is shorter than that reported for the experiment. This aspect canbe clearly understood by observing the contours of instantaneous streamwise velo-city (Figure 5a) and cross-streamwise velocity (Figure 5b) in the x–z plane. Contoursexhibit the presence of three-dimensionality in the flow. This makes the two-dimensional simulation at this Reynolds number and beyond ineffective, asvariations in the spanwise direction are neglected.

Figure 6 provides a comparison of the wall pressure coefficient withexperimental and different numerical simulations. The LES results agree well withthe experimental results, but all our three simulations (3DSKE, 2DNLKE, and3DNLKE) show different trends. The overprediction of stagnation-point valueby the SKE model is a well-known fact and is attributed to the steep velocitygradient on the upstream side. The 2DNLKE and 3DNLKE models predict thestagnation point correctly, but the maximum negative pressure and base pressurecoefficient differ from the experimental results. The mean streamwise velocity pro-files at three different locations in the wake are shown in Figures 7a–c. These com-parisons demonstrate the appropriateness of the present nonlinear model. Both the2-D and 3-D simulations are in good agreement with experimental results; in some

Figure 4. Mean streamwise velocity along the wake centerline.


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places, the present 3-D simulation performs better than LES results. The improve-ment in prediction of these values by the present simulation is good when com-pared to the 2-D URANS simulation of LU01. The cross-streamwise velocityprofile at x=D ¼ 1.54 plotted in Figure 8 shows that the standard k�e modelunderpredicts the trend. The results by almost all LES simulations showed

Figure 5. (a) Instantaneous streamwise velocity contours in the x–z plane. (b) Instantaneous cross-

streamwise velocity contours in the x–z plane.

Figure 6. Comparison of wall pressure coefficient.


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Figure 7. Mean streamwise velocity at (a) x=D ¼ 1.06, (b) x=D ¼ 1.54, and (c) x=D ¼ 2.02.

Figure 8. Mean cross-streamwise velocity at x=D ¼ 1.54.


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overprediction of the profile, but the present simulation results match well with theexperiment.

4. CONCLUSIONS

In the present work, 3-D unsteady computation of flow past a circular cylin-der at subcritical Reynolds number has been performed using a nonlinear k�emodel to evaluate its applicability. The same test case was simulated with2DNLKE and 3DSKE to understand the effectiveness of the present model. Thebulk parameters and the wake velocity recovery match well with experimental dataand LES results. Since the grid requirement is not as severe as in LES and thenumber of cycles required to do averaging is also less, computational cost associa-ted with the present model is very much less. For high-Re flows and flows encoun-tered in practical engineering applications, there is a restriction on the mesh sizeand the LES technique is prohibitively expensive. Encouraging performance ofthe present NLKE model suggests that it could be used as an alternative tool inthese situations. Further improvement in the prediction may be possible by makingthe model fully cubic form.

REFERENCES

1. T. B. Gatski and C. G. Speziale, On Explicit Algebraic Stress Models for ComplexTurbulent Flows, J. Fluid Mech., vol. 254, pp. 59–78, 1993.

2. T. J. Craft, B. E. Launder, and K. Suga, Development and Applications of a CubicEddy-Viscosity Model of Turbulence, Int. J. Heat Fluid flow, vol. 17, pp. 108–115, 1996.

3. T. H. Shih, J. Zhu, and J. L. Lumley, A New Reynolds Stress Algebraic Equation Model,Comput. Meth. Appl. Mech. Eng., vol. 125, pp. 287–302, 1997.

4. M. Tutar and A. E. Holdo, Computational Modeling of Flow around a Circular Cylinderin Subcritical Flow Regime with Various Turbulence Models, Int. J. Numer. Meth. Fluids,vol. 35, pp. 763–784, 2000.

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