On the reliability of eddy viscosity based turbulencemodels in predicting turbulent flow past a circular cylinder

using URANS approach

B. N. Rajani1, A. Kandasamy2, and Sekhar Majumdar3

1 Scientist, CTFD Division, National Aerospace Laboratories(CSIR), Bangalore, 560 017 India2 Professor, DMACS, National Institute of Technology Karnataka, Surathkal 575 025, India

3 Professor, Dept. of Mech. Engg., NITTE Meenakshi Instituteof Technology, Bangalore 560 640, India

Email: rajani@ctfd.cmmacs.ernet.in

ABSTRACT

Turbulent flow past circular cylinder at moderate to high Reynolds number has been analysed employing ansecond-order time accurate pressure-based finite volume method solving two-dimensional Unsteady ReynoldsAveraged Navier Stokes (URANS) equations for incompressible flow, coupled to eddy-viscosity based turbu-lence models. The major focus of the paper is to test the capabilities and limitations of the present turbulencemodel-based 2D URANS procedure to predict the phenomenon ofDrag Crisis, usually manifested in reliablemeasurement data, as a sharp drop in the mean drag coefficientaround a critical Reynolds number. The com-putation results are compared to corresponding measurement data for instantaneous aerodynamic coefficientsand mean surface pressure and skin friction coefficients. Turbulence model-based URANS computations arein general found to be inadequate for correct prediction of the mean drag coefficients, the Strouhal numberand also the coefficients of maximum fluctuating lift over therange of flow Reynolds number varying from104 to 107.

KEYWORDS: Turbulent flow, Circular cylinder, Drag Crisis, Unsteady RANS procedure, Eddy viscositybased Turbulence models, Implicit Finite Volume Method

NOMENCLATURE

D cylinder diameterRe flow Reynolds number based on DCP non-dimensional pressure coefficientβ skin friction coefficientCl lift coefficientCd drag coefficientf frequencySt Strouhal number〈Ui〉 phase-averaged velocity components

alongi direction〈P〉 phase-averaged pressurek turbulence kinetic energy

ε turbulence energy dissipation∆t time step size∆v cell volumey+ non-dimensional wall normal distanceµ fluid viscosityµt eddy viscosityT transformation matrixJ Jacobian of transformation matrixTηi

k metric coefficients of transformationbi

k projection area of the cell faceφ any flow variableSU andSP linearised source terms

1 INTRODUCTION

Flow around a circular cylinder is a challengingkaleidoscopic phenomenon. Cross flow normal tothe axis of a circular cylinder and the associatedproblems of heat and mass transport are encounteredin a wide variety of engineering applications. Bothmeasurements and computations for flow past cir-

cular cylinder over a wide range of Reynolds num-ber, have revealed distinct flow patterns which havelater been classified (Zdravkovich 1997; Williamson1996; Lienhard 1966; Roshko 1954) under differentflow regimes. The flow remains steady and lami-nar for Reynolds number between 5 and 40 and thewake starts becoming unstable at a critical Reynoldsnumber of around 47, leading finally to the shed-

ding of alternate vortices from the cylinder surfaceat definite frequencies, well known in literature asthe Von Karman vortex street. The laminar vortexshedding is observed to be continuing up to a valueof Reof about 190, beyond which the two dimen-sional flow becomes unstable and three dimensional,eventually leading to the simultaneous formation ofspanwise and streamwise vortex structures. Thesethree-dimensional disturbances propagate furtherdownstream and the flow in the thin free shear layerbordering the far wake zone undergoes slow transi-tion from laminar to turbulent state. As the Reynoldsnumber is further increased, the transition inceptionlocation moves further upstream and alters even thestructure and evolution of the vortex shedding inthe near wake zone. At aroundRe= 2× 105, theboundary layer on the cylinder surface is observedto becomes unstable and the transition point reachesclose to the laminar separation point on the cylin-der surface. Eventually the transition takes placein the boundary layer which slowly turns to be tur-bulent (Achenbach 1968; Roshko 1961). At thecritical Reynolds number, when the laminar to tur-bulent transition takes place in the boundary layer ata location somewhere before the laminar separationpoint, the turbulent boundary layer is able to sustainthe adverse pressure gradient for a longer distanceand hence the separation point is shifted furtherdownstream. As a consequence of such downstreamshift of the separation point, the base pressure onthe rear part of the cylinder is increased and hencethe drag is observed to be drastically reduced. TheReynolds number at which this phenomenon of theso-called ’Drag Crisis’ occurs is known as the crit-ical Reynolds number, which however depends toa great extent on the other disturbances of the ex-perimental situation. Measurement data reported byvarious researchers (Zdravkovich 1997; Achenbach1968; Roshko 1961) have shown that the value ofthe drag coefficient reduces from approximately 1.2at the Subcritical Reynolds number range to about0.3 in the Supercritical flow regime. It is also worth-while to note from the measurement data that as theReynolds number is increased further to the Trans-critical flow regime(Re> 3.4×106), the drag coef-ficient once again increases slowly to about 0.7. Asthe Reynolds number reaches the range ofRe= 107,the transition inception point is shifted further up-stream to a location very close to the front stagnationpoint and the flow is fully turbulent almost every-where around the cylinder and in the wake as well.

The challenging flow past circular cylinders havealways been a very interesting and exciting researchtopic for the experimental fluid dynamicists, leadingto a large volume of measurement data on flow pastcylinders. On the other hand,in the area of numeri-cal simulation of time-dependent three-dimensional

flow around circular cylinder, very few research re-sults (Rajani et al. 2009; Kakuda et al. 2007; Mittal2001; Mittal and Balachandra 1997; Karniadakisand Triantafyllou 1992; Tamura et al. 1990) havebeen reported in the recent years. Moreover mostof these detailed three-dimensional flow computa-tions are limited only to laminar range at very lowReynolds number of the order of 200-300, wherethe objective is only to demonstrate the appear-ance of three-dimensional vortical flow structuresand the chaotic state of flow in the cylinder wakeeven at a Reynolds number as low as 300 or so.As the Reynolds number increases, the phenomenaof transition and turbulence need to be appropri-ately simulated for a realistic simulation of the flowsituation. In spite of being the most accurate simu-lation methodology, Direct Numerical Simulationapproach, under the present constraints of com-putation resources, is prohibitively expensive forhigher Reynolds number flows consisting of verywide range of length and time scales. The litera-ture on the simulation of turbulent flow past circularcylinder at high Reynolds number using URANSmethodology coupled to turbulence models or us-ing LES methodology are also quite limited. LEScomputations are reported using either Smagorinsky(Singh and Mittal 2005; Selvam 1997; Song andYuan 1990) type sub grid scale models or using noturbulence models as such (Singh and Mittal 2005;Kashimaya et al. 1998; Kakuda and Tosaka 1992;Tamura and Kuwahara 1989) to simulate the tur-bulent interactions at the sub grid level. Deng andhis research group (Deng et al. 1993) have usedthe algebraic Baldwin-Lomax model as well as thek− ε turbulence models to study the phenomenonof drag crisis, whereas advanced turbulence modelslike SST and SA have been used by Cox (Cox 1997)for similar two-dimensional flow simulations. Mostof the computational results available in literaturehave usually over predicted the mean drag in thesubcritical regime compared to the correspondingmeasurement data.

The objective of the present URANS computa-tion is mainly to assess the limitation and ac-curacy level of the existing popular and widelyused turbulence models for computation of aero-dynamic drag and specially in predicting the phe-nomenon of Drag Crisis for flow past a circularcylinder over a certain range of flow Reynolds num-bers. However the present computations are re-stricted to two-dimensional flow only since measure-ment data and other computation results reportedin literature (Singh and Mittal 2005; Williamson1996; Braza et al. 1990) confirm that the freeshear layer instability in the wake which happensto be the prime source of laminar to turbulenttransition and eventually leads to drag crisis, is a

two-dimensional phenomenon only. Computation-intensive three-dimensional flow calculation usingturbulence model-based URANS methodology willbe meaningful only if the turbulence models, evenfor two-dimensional flow computation, can mimicthe physics of the transitional flow with reasonableaccuracy. In the present two dimensional numeri-cal simulation, computations have been carried outusing different turbulence models for a range ofReynolds number covering the subcritical to the su-percritical regime (Re ranging from 104 to 107 ).All the computations have been carried out usingan implicit finite volume type Navier Stokes codeRANS3D, developed at the CTFD Division, NAL,Bangalore, India (Majumdar et al. 1992; Majumdar1998; Majumdar et al. 2003). The computation re-sults have been validated against measurement andother computation data reported in unclassified liter-ature.

2 MATHEMATICAL M ODELLING OF

FLOW PHYSICS

The analysis of flow behind a circular cylinder overa wide range of flow Reynolds number has beencarried out in the present work through numericalsolution of the relevant Unsteady Reynolds Aver-aged Navier Stokes (URANS) equation system, cou-pled to four different eddy viscosity based turbu-lence models. The basic equations to be solved area set of non-linear, strongly coupled partial differen-tial equations representing the conservation of mass,conservation of three mean momentum componentsalong the cartesian directions and the conservationof relevant turbulence scalars for evaluation of theReynolds stresses.

2.1 Unsteady Reynolds AveragedNavier Stokes (URANS) Equations

In the URANS approach, the Reynolds averagingconcept, is directly used to replace the instantaneousflow variables by the so-called time-averaged vari-ables or phase-averaged variables for the steady andtime-dependent mean flow situations respectively.The only assumption in this representation is thatthe time scale of the mean flow variation has to bequite large compared to the time scale of the tur-bulent fluctuations. The effect of turbulent fluctu-ation on the mean flow is therefore represented inthe mean momentum equations in the form of a sec-ond moment correlation between the unknown fluc-tuating velocity components. The cartesian coordi-nate system(y1,y2,y3) are transformed to the non-orthogonal coordinate system(x1,x2,x3) using the

chain rules of differentiation as follows:[

∂∂y1

,∂

∂y2,

∂∂y3

]t

= T[

∂∂x1

,∂

∂x2,

∂∂x3

]t

whereT is the transformation matrix andTi j =∂y j

∂xi.

The URANS equations for turbulent incompressibleflow in non-orthogonal curvilinear coordinates withcartesian velocities as dependent variables may bewritten in a compact form as follows:

Momentum transport for the cartesian velocitycomponent〈Ui〉:

∂∂t

(ρ〈Ui〉)+1J

∂∂x j

[

ρ〈Ui〉〈Uk〉η jk + 〈P〉η j

i

−µJ

(

∂〈Ui〉∂xm

B jm+

∂〈Uk〉∂xm

ηmi η j

k

)

−

ρ〈uiuk〉η jk

]

= SUi (1)

where,ui is the fluctuating velocity component,SUi

is any momentum source other than the pressuregradient,η j

k are the metric coefficients (cofactor of

the term∂y j

∂xkin the transformation matrixT) and

B jm = η j

nηmn . η j

k is expressed in terms of the pro-jection area of the faces as follows :

η jk = b j

k∆x j/(∆x1∆x2∆x3)

where the areasbij for the control volume aroundP

are shown in Fig. 1(b) for a two-dimensional situa-tion on y1 − y2 plane. These momentum equationsare further supplemented by the mass conservationor the so-called continuity equation.

Mass conservation (Continuity):

∂∂x j

(

ρ〈Uk〉η jk

)

= 0 (2)

However Eq. 1 and Eq. 2 do not form a closed sys-tem due to the presence of the unknown turbulentstress term−ρ〈uiu j〉.

2.2 Turbulence Modelling

2.2.1 Eddy Viscosity Hypothesis

In Eddy Viscosity based turbulence models, the tur-bulent stress appearing in the Reynolds-Averagedequations is expressed in terms of the mean veloc-ity gradients as following:

−ρ〈uiuk〉 = µt

(

∂〈Ui〉∂xk

+∂〈Uk〉

∂xi

)

− 13

ρδik〈umum〉 (3)

where,δik is the Kronecker Delta andk is the sum-ming index overk = 1,2,3. The eddy viscosityµt

is assumed to be an isotropic scalar quantity whosevalue depends on the local state of turbulence. Sub-stituting the turbulent stress term in Eq. 1 andcarrying out some algebraic manipulation one mayrewrite the mean momentum equation as following :

∂(ρ〈Ui〉)∂t

+1J

∂∂x j

[

ρ〈Ui〉〈Uk〉η jk + 〈P〉η j

i

− (µ+µt)

J

(

∂〈Ui〉∂xm

B jm+

∂〈Uk〉∂xm

ηmi η j

k

)]

= SUi (4)

The algebraic or zero equation turbulence models(Baldwin and Lomax 1978; Cebeci and Smith 1974),employed very successfully for attached boundarylayer type flows, compute the eddy viscosity(µt) atany field point as an algebraic function of the meanvelocity gradients and the normal distance from thesolid surface. These models are computationallycheap but sometimes call for complicated interpo-lations to determine the normal distance from wallfor highly skewed grids near the body surface andcannot, in general, simulate separated flows. On theother hand for eddy viscosity based turbulence mod-els, transport equation are solved for one or moreturbulence scalars. Four different turbulence mod-els used in the present work are the low Reynoldsnumber version ofk− ε model proposed by Chien(Chien 1982), Shear Stress Transport (SST) modelproposed by Menter (Menter 1994), one equationmodel of Spalart-Allamaras (SA) (Spalart and Alla-maras 1992) and thek− ε− v2− f model, popularlyknown as V2F model, proposed by Durbin (Durbin1995).

2.3 Finite Volume Formulation

The present flow solution algorithm (RANS3D) is aniterative two-step predictor-corrector procedure sim-ilar to the SIMPLE algorithm of Patankar (Patankar1980). In the predictor step, the momentum equa-tions are solved to advance the velocity field par-tially in time for a guessed pressure field. The con-tinuity is then ensured by solving a Poisson equa-tion for pressure correction, followed by relevantcorrection of the pressure and velocity field in thecorrector step. Integration of the momentum trans-port equations (Eq. 4) over each control volumetransforms the relevant pde’s in the form of dis-crete algebraic equations representing a balance be-tween the convective and diffusive fluxes through

the cell faces and the other remaining terms as vol-ume sources. Second order accurate central dif-ference scheme is used for spatial discretisation ofthe convective and diffusive fluxes at the cell faces.The temporal derivatives are also discretised usingthe second order accurate three-level fully implicitscheme. The numerical stability is ensured by meansof a deferred correction procedure (Khosla and Ru-bin 1974) where a suitable weighting function isused to blend the flux from the desired scheme withupwind fluxes which allow some small numericaldiffusion for stability of the solution. Using therelevant geometric factors with second order spatialand temporal discretisation schemes for the cell facefluxes and linearisation of the source terms, the fluxbalance equation is expressed in an implicit mannerin the following quasi-linear form:

(

1.5φn+1P +0.5φn−1

P −2φnP

)

∆v/∆t +APφn+1P =

∑nb

Anbφn+1nb +SU (5)

whereAP = ∑Anb−SP; the coefficientAnb repre-sents the combined effect of convection and diffu-sion from the neighbouring nodes at the cell faces;SU and SP are the components of the linearisedsource term,∆v is the cell volume and∆t is the timestep size. The superscripts ofφ represent the respec-tive time step. The detailed derivation of Eq. 5 interms of the flow variables and the cell face pro-jection areas is given elsewhere (Majumdar et al.1992; Majumdar et al. 2003). The transformationof continuity equation to an equation of pressure-correction, described in the following subsectionis not that straightforward. A segregated iterativemethod, coupled to the strongly implicit procedureof Stone (Stone 1968), is used to solve the systemof linear equations (Eq. 5) corresponding to eachtransport equation in a sequential manner. At eachtime step, the normalized residue for all the equa-tions solved are brought down to a convergence cri-terion of 10−5.

2.4 Computation of Pressure Field

Algorithms based on pressure-velocity solutionstrategy transform the continuity equation into afield equation for pressure correction using momen-tum equations as link between pressure and velocitycorrections. For three dimensional flow analysis us-ing a collocated variable arrangement, the problemof the so-called checkerboard spitting is avoided byemploying the principle of Momentum Interpolation(Majumdar 1988; Majumdar et al. 1992). This givesthe cell-face velocity components,Uiw (as an illus-tration, at the cell west face‘w’ ) and hence the link

between pressure and velocity correctionU ′iw as:

Uiw = αv

[

H iP +Di

2(Ps−Pn)+Di3(Pb−Pt)+

Di1(Pw−Pp)

]

+(1−αv)U0iw (6)

U ′iw = αvDi

1(p′w− p′p) (7)

where H ip =

(

∑AinbUinb +SU

)

/Aip and

Dik = bi

k/Aip; Ai

nb represents the coefficients forneighbouring nodes, the superscript′0′ is the valueat the previous iteration,Ai

P is the value ofAP forUi at the nodeP, bi

k is the relevant projection areaof the cell face (Fig.1),αv is an under-relaxation pa-rameter and the expressions with overbar representthe linear average of the same quantities evaluatedat the cell-centersP andW adjacent to the face’w’ .Now the conservation of mass in a control volumemay be expressed as:

Ce−Cw +Cn−Cs+Ct −Cb = 0 (8)

where the face mass fluxCw, for example, may bewritten as follows using’i’ as the summation indexfor three directions.

Cw = ρwb1iw(Uiw +U ′

iw) (9)

SubstitutingUiw andU ′iw for all ‘i’ from equations

similar to Eqs. 6 and 7 respectively in mass flux ex-pression similar to Eq. 9 for each of the six faces inEq. 8, one obtains the equation of pressure correc-tion in a form very similar to Eq. 5. Once the fieldof pressure correction is solved, corrections are ap-plied to the cell-centre pressure and velocities, to allthe cell-face velocities and finally to the convectivemass-fluxes to satisfy the cell wise continuity.

3 RESULTS AND DISCUSSION

3.1 Computational Details

The computational domain consists of the annularregion between two circles of diameterD and 40DwhereD is the diameter of the cylinder and the num-ber of nodes used is 144 along the radial and 121along the circumferential direction divided into twoblocks overlapped with one control volume on ei-ther side of the block interface. In order to handle atwo-dimensional problem by the three-dimensionalflow solver RANS3D, only two parallel planes areconsidered along the axial or spanwise direction(z)with the same 2D curvilinear grid stacked on each ofthe twoz planes. At the spanwise end planes Sym-metry conditions are applied to ensure two dimen-sionality of the flow. The two block radial polar grid(Fig. 2(a)) is used on the two-dimensional plane of

interest, between the cylinder of unit diameter(D)and the far field of radius twenty units. The grid linesare stretched radially near the wall to resolve thesharp local gradients of the flow variables. Fig. 2(b)shows the boundary conditions used for the turbu-lent flow computation. Block 1 consists of an inflowboundary, a wall boundary for the half cylinder andtwo cut boundaries separating this block from theneighbouring Block 2 as shown in the figure. Simi-larly, Block 2 consists of an outflow, a wall boundaryfor the rest half of the cylinder and the two cuts. Animpulsive start of the cylinder is simulated by speci-fying uniform inflow velocity (U = U∞ andV = 0)at all nodes with no slip conditions (U = 0 andV = 0),imposed on the cylinder wall as initial con-ditions (t = 0) and maintained thereafter at all timeinstants (t > 0). For the turbulence scalars, appro-priate near wall boundary conditions have been useddepending on the turbulence model chosen. Thelevel of free stream turbulence energy(k) is main-tained at 1% of the mean kinetic energy of the freestream whereas the value of the turbulence energydissipation(ε) at the inflow boundary nodes is pre-scribed assuming the local eddy viscosity to be ap-proximately equal to the laminar viscosity. Effect ofthe spatial and temporal discretisation schemes andthe effect of far field location have earlier been stud-ied in detail for laminar flows. In the present studyextensive parametric analysis have been carried outfor turbulent flows atRe= 105 andRe= 3.6×106 inorder to study the effect of grid-size, time step sizeand local eddy viscosity level at the inflow boundary.Based on the previous and present studies, centraldifference scheme for spatial discretisation coupledto second order time discretisation with∆t = 0.05have been used for all the turbulent flow computationat different Reynolds number. For very high valuesof Re, the number of grid lines and the near wallgrid spacing are so adjusted that the value ofy+ atthe first near wall point is maintained to be less thanunity for all the turbulence models used.

3.2 Temporal Evolution of Lift andDrag Coefficients

The typical temporal evolution of the lift(Cl ) anddrag(Cd) coefficients for the cylinder computed forReynolds number of 105, using different turbulencemodels are shown in Fig. 3. The variation of theaerodynamic coefficients are plotted against non-dimensional time(t = U∞t

D ). Irrespective of the tur-bulence model used, the figures clearly demonstratethe phenomenon of periodic fluctuation of the liftand drag coefficient due to turbulent vortex shed-ding. In each computation however, large number oftime steps are required for the initial transients to dieout and finally when a statistically stationary state is

reached, the number of iterations required per timestep is reduced. The temporal variation of the lift(Cl ) coefficient shows a periodic fluctuation aboutthe zero mean as expected physically for a symmet-ric mean flow and the corresponding Fourier spec-tra (not shown in any figure) indicates a dominantfrequency(f ), expressed in non-dimensional form asthe Strouhal number(St= f D

U∞) . In case of drag(Cd)

also similar periodic behaviour is demonstrated butthe Strouhal number forCd is observed to be almostdouble the corresponding value for theCl . Such dif-ference in the frequency of the periodic mean motionbetween the longitudinal and the transverse directionhas also been observed and explained in other mea-surement and computation data reported in unclas-sified literature (Mittal 2001; Song and Yuan 1990;Drescher 1956). For the same flow Reynolds num-ber of 105, the mean value of the drag coefficient andthe maximum or RMS amplitude of the lift coeffi-cient, obtained by different turbulence models, indi-cated in the same Fig. 3, are observed to be different.The Chien’s lowRe k− ε and the SST model appearto be predicting nearly the same value of maximumCl amplitude as 1.0 at the statistically stationary statewhereas the SA model consistently under predictsand the V2F model over predicts the same quan-tity by about 15 to 20%. Comparison to measure-ment data on mean drag coefficient(Cd ≈ 1.2) re-ported in literature (Achenbach 1968) demonstratesthat the predictions by V2F model agree reason-ably well; Chien and SST models mildly under pre-dict the Cd value whereas SA model grossly un-der predicts this value. AtRe= 105, the experi-mentally observed laminar flow separation is consis-tently over predicted by all the four models. How-ever, the predicted location of separation is at around90◦ whereas the corresponding measured value re-ported by Aachenbach (Achenbach 1968) is around78◦. Further the post separation behaviour of the tur-bulent flow, specially the combined effect of the tur-bulent free shear layer bordering the wake and thelarge vortex structure of the near wake itself in thevicinity of the rear surface of the cylinder, is ob-served to be quite different for the different turbu-lence models used. Under prediction of mean dragcoefficients for flow past circular cylinder, observedin the present computation with some of the turbu-lence models, has also been reported by other re-searchers (Cox 1997; Selvam 1997). Fig. 4 showsthe temporal evolution ofCl andCd at four differentReynolds number using the Chien’s lowReversionof k− ε model. The sharp drop in the level of meandrag coefficient is clearly seen in the figure when theReis increased from 105 to 8.5×105, whereas whentheRe is further increased to 3.6×106 and 107 thelevel of mean drag further decreases but at a muchslower rate. The increase of the drag observed inmeasurement data (Zdravkovich 1997; Achenbach

1968; Schlichting 1968) when the Reynolds num-ber in increased to 107 is not captured by the presentcomputation with any of the four turbulence mod-els, even using reasonably high spatial and temporalresolution (Section 3.3). The maximum value ofCl

is drastically reduced from about 1.0 atRe= 105 toapproximately 0.3 atRe= 8.5×105, but the level ofmaximumCl remains almost unchanged as theReisincreased further up to 107.

3.3 Surface Pressure and Skin Frictionon Cylinder Surface

The time averaged flow quantities at any field nodeare computed from arithmetic averaging of the rel-evant flow variable over at least 50 vortex shed-ding cycles after the statistically stationary state isreached. Both the surface pressure and the wallshear stress on the cylinder surface are appropriatelynon-dimensionalised for pressure asCp = (p−p∞)

1/2ρU2∞

and for skin friction asβ = τwall1/2ρU2

∞

√Re. Fig. 5 shows

the circumferential variation ofCp and β at fourdifferent Reynolds number (Re= 105, 8.5× 105,3.6× 106 and 107) using two different grid sizes -one with 121 nodes (∆θ = 3◦) along the circum-ferential direction and 144 points in the radial di-rection (∆r = 1× 10−4D for Re= 105 ) and theother with 241 nodes (∆θ = 1.5◦) along the cir-cumferential direction and 305 points in the radialdirection (∆r = 1×10−4D for Re= 105). Howeverinsignificant effect of grid refinement on the circum-ferential variation ofCp andβ, observed in Fig. 5,indicates the results with 121×144 nodes to be moreor less grid-independent. Similarly for time step sizealso, the results obtained using∆t = 0.05 and 0.005shown in Fig. 5(a) and (c) justify the independenceof the solution on the time step size for∆t = 0.05.

The effect of the four different turbulence modelson circumferential variation ofCp andβ at four dif-ferent Reynolds number are shown in Fig. 6. Ir-respective of the turbulence model used, the com-puted pressure distribution and the skin friction coef-ficient agree reasonably well with the measurementdata (Jones et al. 1969; Achenbach 1968) for thefront segment of the cylinder covering almost onethird of the periphery (about 60◦ on either side of thefront stagnation point) where the flow mostly con-sists of the development of a laminar boundary layerunder a favourable pressure gradient. But beyondthe subcritical regime ofRe= 105, discrepancies areobserved between the computed and measured val-ues of the suction peak attained by this acceleratingflow. The corresponding peak of the skin friction co-efficient are largely over predicted at almost all theflow Reynolds number beyond 105. The other major

disagreement lies here in the prediction of the sepa-ration point location which, compared to the corre-sponding measurement data, is invariably predictedfurther downstream for all the flow situations stud-ied. The prominent double peak in skin frictionβobserved in the measurement data atRe= 8.5×105,possibly due to the transition taking place well be-fore the separation, has not at all been captured byany of the turbulence models used. However themeasurement data for flow atRe= 3.6×106 do notexhibit such double peaks ofβ and it is not clearwhether flow at thisReis fully turbulent and the tran-sition point falls very close to the stagnation point.However the large over prediction of the single peakmagnitude ofβ at the same Reynolds number is notin conformity with the argument of fully turbulentflow everywhere in the boundary layer on the cylin-der surface. AtRe= 107 when the boundary layeris physically expected to be fully turbulent for mostpart of the cylinder, the base pressure at the rearsurface predicted by all the models is found to beclose to the measurement data but the magnitudeof the suction peak is over predicted with a down-stream displacement of the peak location as well.This disagreement which also explains the discrep-ancy in the value of the aerodynamic coefficientsmay be attributed to the inadequacy of the eddy vis-cosity based turbulence models for flow in the pres-ence of laminar to turbulent transition and also tothe strong adverse pressure gradient. Perhaps LESor DNS simulations might be more reasonable ap-proaches for resolving such flows with large vorticalstructures and often with more than one characteris-tic frequencies.

3.4 Phenomenon of Drag Crisis

In order to understand whether the Unsteady 2DRANS computation employing eddy viscosity basedturbulence models, is capable of predicting the inter-esting phenomenon of Drag Crisis for circular cylin-ders, observed in measurement data forCd againstRe, 2D computations have been carried out for awide range of flow Reynolds number spanning fromthe laminar regime to as high as 107. Fig. 7 showsthe available measurement data (Zdravkovich 1997;Achenbach 1968; Schlichting 1968) on the meandrag coefficient collected from different sources andalso the present computation results for laminar flowas well as for turbulent flow situations using varietyof turbulence models. Excellent agreement withthe compilation data of Zdravkovich (Zdravkovich1997) in the laminar regime up to aboutRe= 200confirms the adequacy of the space and time res-olution and also the accuracy of the discretisationschemes used in the flow solution algorithm ofRANS3D. BeyondRe= 200, the important effects

of the three dimensional spanwise instabilities onthe flow have already been identified by many re-searchers including the present author (Rajani et al.2009; Mittal 2001; Mittal and Balachandra 1997;Williamson 1996; Zhang et al. 1995; Williamsonand Roshko 1990; Roshko 1955; Roshko 1954). Asstated earlier, this complex flow always consists oflaminar, turbulent and transitional patches and thediscrepancies obviously are expected to be morewhile using either purely laminar flow equations orfully turbulent flow-based turbulence models for sit-uations with large laminar or transitional patches.Accordingly the zone ofRebetween 400 and 3000has been simply omitted for the time being and theunsteady RANS computation has been carried outbetweenRe= 3900 and 107. Up toRe= 6.9×104,the turbulence models appear to have insignificanteffect on the predicted mean drag coefficient. How-ever beyondRe= 6.9×104, a sharp fall in the dragcoefficient is observed - but at different rates withdifferent turbulence models. The computation withSA model shows a very sharp fall atRe= 1×105 re-sulting in a strong under prediction ofCd comparedto the measurement data. The Chienk− ε modelmore or less follows the data from other source upto Re= 8.5×105, with a sudden drop inCd occur-ring atRe= 1.4×105 which is slightly earlier thanthat observed in the measurement data. The SSTand V2F models also follow a similar trend of thedrag variation where the phenomenon of drag crisisoccurs at a relatively high value of flow Reynoldsnumber(Re= 3.6× 105) and hence the mean dragcoefficient is over predicted forRe> 105. Howeveras theRe increases further, results from all the fourmodels more or less follow the same trend of dropin Cd till Re reaches about 106. Therefore quali-tatively a sharp drop inCd in the critical regime ismanifested by all the turbulence models, but a widescatter is observed in the computation results ob-tained by different models. Further, the falling trendof Cd is continued in the prediction even when theReynolds number is increased to higher value of theorder of 107. None of the 2D URANS computationis able to capture the rise inCd observed in the mea-surement data beyond a Reynolds number of 106.Fig. 8 shows the variation of the mean drag coef-ficient with flow Reynolds number using a singleturbulence model coupled to different grid and timestep sizes. The overlapping of the data points clearlyshows the solution obtained using 121× 144 gridand∆t = 0.05 may be accepted as the grid indepen-dent solution and further refinement does not bringany significant effect on the solution. The discrepan-cies between the measurement data and computationresults may mostly be attributed to the inadequacyof the eddy viscosity based turbulence models tocapture the transition specially beyond the criticalregime when the transition takes place somewhere

in the cylinder surface boundary layers and that tooat a point upstream of the separation point.

Fig. 9 compares the computed variation of theStrouhal number (St) with the flow Reynolds num-ber against measurement data available from differ-ent sources (Williamson 1992; Cantwell and Coles1983; Roshko 1961). Similar to the mean drag varia-tion, the prediction matches reasonably well to mea-surement data in the pure laminar regime (Re< 250).However for turbulent flow regime, the predictedStrouhal number is observed to be almost constantapproximately at a level of 0.25 tillRe= 105 fol-lowed by a sharp rise as the value ofRe increasestowards 107. The measurement data for Strouhalnumber shows a similar trend of variation in a qual-itative sense but the approximately constant levelof Strouhal number for 103 < Re< 3× 105 is ataround 0.2 whereas the predicted level is found tobe at the order of 0.25. Further the peak value ofSt= 0.45 atRe= 2×106, followed by a sharp drop,as observed in measurement data of Roshko (Roshko1961), could not also be captured by the presentcomputation where instead of the drastic oscillatorybehaviour, a slowly increasing trend is clearly ob-served tillRe= 107, for all the turbulence modelsused. The major discrepancies indicate gross inade-quacy of the two dimensional flow computation us-ing eddy viscosity based turbulence models for sim-ulation of the present flow situation where the tran-sition taking place in the free-shear layer borderingthe near wake zone plays a very important role andeventually the transition is observed to occur alongthe spanwise direction within the separation bubbleitself.

4 CONCLUDING REMARKS

Two dimensional turbulent flow past a circular cylin-der has been computed for flow Reynolds numbervarying from 104 to 107 using the Unsteady RANScode RANS3D coupled to four eddy viscosity-basedturbulence models. In the subcritical range up toRe= 105, all the four turbulence models predict rea-sonable flow patterns with turbulence vortex shed-ding at definite frequency (Strouhal number) for agiven Reynolds number. The circumferential varia-tion of the time averaged pressure and skin frictionaround the cylinder for different values ofReshowthat compared to measurement data, the flow sepa-ration location on the cylinder surface is always pre-dicted further downstream, irrespective of whetherthe separation is laminar or turbulent. As a con-sequence of that the width of the wake and hencethe base pressure at the cylinder rear surface whichdecides the value ofCd is changed. The skin fric-tion coefficient is, in general, largely over predicted

for Rebeyond the critical regime. This may be at-tributed to the gross inadequacy of all the turbulencemodels to capture the effect of laminar to turbulenttransition in the boundary layer on the cylinder sur-face. A sharp fall of the mean drag coefficientCd iscaptured by all the turbulence models, as observedin measurement; but only the low Reynolds num-ber version of the Chienk− ε turbulence model isable to predict the phenomenon of drag crisis at aRecloser to the corresponding measurement data. Awide scatter is observed in the predicted variationof the mean drag coefficient with Reynolds number.It is important to note that unlike the measurementdata, the falling trend ofCd with Re is not reversedin the prediction for the post or transcritical regimewhenReexceeds 5×105. The phenomenon of DragCrisis is captured by the eddy viscosity based tur-bulence models in a qualitative sense only. Moreaccurate prediction demands realistic simulation ofthe transition process in the free-shear layer in thesubcritical regime and also the complex transition inthe boundary layer interacting with the flow separa-tion in the critical and post-critical regime. Work isin progress on Large Eddy Simulation of the sameflow, which is likely to capture the physical processof transition through direct numerical simulation ofthe large eddies and produce more accurate resultson this complex unsteady three dimensional flow sit-uation.

Acknowledgements

The authors wish to thank Director NAL Bangalorefor his kind permission to publish this paper.

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(a) Hexahedral control volume (b) Projection areas for 2D Cell

Figure 1: Typical control volume and cell face projection area

Block 1 Block 2

(a) 2-block grid arrangement

D

40D

Cut

Cut

Inflow Outflow

Cylinder Wallin

U

(b) Boundary conditions

Figure 2: Multiblock polar grid(124×144) and boundary conditionsused for computation of flow past circular cylinder

25 50 75 100 125 150

-1

0

1

Clrms= 0.749

t

Cl

25 50 75 100 125 1500.25

0.5

0.75

1

1.25

1.5Cdavg= 1.00

Cd

t

(a) Chien LowRe k− ε Model

25 50 75 100 125 150

-1

0

1

Clrms= 0.996

t

Cl

25 50 75 100 125 1500.25

0.5

0.75

1

1.25

1.5Cdavg= 1.21

t

Cd

(b) V2F Model

25 50 75 100 125 150

-1

0

1

Clrms= 0.786

t

Cl

25 50 75 100 125 1500.25

0.5

0.75

1

1.25

1.5Cdavg= 0.96

t

Cd

(c) SST Model

25 50 75 100 125 150

-1

0

1

Clrms= 0.582

t

Cl

25 50 75 100 125 1500.25

0.5

0.75

1

1.25

1.5Cdavg= 0.74

t

Cd

(d) SA Model

Figure 3: Temporal variation of lift (Cl ) and drag (Cd) coefficientusing different turbulence models (Grid:121×144,∆t = 0.05,Re= 105)

25 50 75 100 125 150

-1

0

1

Clrms= 0.749

t

Cl

25 50 75 100 125 1500.25

0.5

0.75

1

1.25

1.5Cdavg= 1.00

Cd

t

(a) Re= 105

25 50 75 100 125 150

-1

0

1

Clrms= 0.332

Cl

t 25 50 75 100 125 1500.25

0.5Cdavg= 0.37

Cd

t

(b) Re= 8.5×105

25 50 75 100 125 150

-1

0

1

Clrms= 0.371

Cl

t 25 50 75 100 125 1500.25

0.5Cdavg= 0.31

Cd

t

(c) Re= 3.6×106

25 50 75 100 125 150

-1

0

1

Clrms= 0.391

Cl

t 25 50 75 100 125 1500.25

0.5Cdavg= 0.28

Cd

t

(d) Re= 107

Figure 4: Temporal variation of lift (Cl ) and drag (Cd) coefficient at differentReusing Chien lowRe k− ε turbulence model (Grid:121×144,∆t = 0.05)

(a) Re= 105

(b) Re= 8.5×105

(c) Re= 3.6×106

(d) Re= 107

Figure 5: Surface pressure (Cp) and skin friction coefficient (β) around the cylinder at differentReusing Chien’s lowRe k− ε turbulence model

(a) Re= 105

(b) Re= 8.5×105

(c) Re= 3.6×106

(d) Re= 107

Figure 6: Surface pressure (Cp) and skin friction coefficient (β) around the cylinder at differentReusing different turbulence models (Grid:121×144,∆t = 0.05)

Figure 7: Effect of turbulence models on the variation of mean drag coefficient withRe(Grid:121×144,∆t = 0.05)

Figure 8: Effect of grid and time step size on the variation ofmean drag coefficient withRe(Turbulence Model : Chien low Rek− ε)

Figure 9: Effect of turbulence models on the variation of Strouhal number (St) with Re(Grid:121×144,∆t = 0.05)