extension of boussinesq turbulence constitutive relation for bridging methods

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Extension of Boussinesq turbulenceconstitutive relation for bridgingmethodsSunil Lakshmipathy a & Sharath S. Girimaji aa Aerospace Engineering Department , Texas A&M University ,College Station, TX, 77843-3141, USAPublished online: 02 Nov 2009.

To cite this article: Sunil Lakshmipathy & Sharath S. Girimaji (2007) Extension of Boussinesqturbulence constitutive relation for bridging methods, Journal of Turbulence, 8, N31, DOI:10.1080/14685240701420478

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Journal of TurbulenceVolume 8, No. 31, 2007

Extension of Boussinesq turbulence constitutive relation forbridging methods

SUNIL LAKSHMIPATHY∗ and SHARATH S. GIRIMAJI

Aerospace Engineering Department, Texas A&M University, College Station, TX 77843-3141, USA

Bridging methods for turbulence simulation are adaptive eddy-viscosity schemes based on theaccuracy-on-demand paradigm, much like hybrid approaches. The object is to resolve more scalesof motion than Reynolds-averaged Navier–Stokes (RANS) by suitably reducing the eddy viscosity asa function of grid spacing. Currently, there are two proposals for extending the RANS two-equationmodel with Boussinesq constitutive relation to bridging methods. In the first approach, it is suggestedthat the coefficient in the Boussinesq relation be reduced and the transport equations for the lengthand velocity scales left unaltered for achieving the desired level of viscosity reduction. The secondapproach suggests that the viscosity reduction is better achieved by suitably modifying the transportequations rather than altering the Boussinesq coefficient. In this paper, we compare the merits of thetwo methods in two important benchmark cases: flows past circular cylinder and backward facing step.Three types of evaluations are performed. First, we verify if the requisite viscosity reduction is indeedachieved in the calculations. Then, we compare some crucial qualitative aspects of the solutions fromthe two methods. Finally, we evaluate the two solutions against experimental data for the same levelof intended viscosity reduction.

Keywords: Partially averaged Navier–Stokes; Hybrid turbulence modeling; Boussinesq constitutive relation

1. Introduction

Bridging models [1, 2] are intended for turbulence simulations at any degree of resolutionbetween Reynolds-averaged Navier–Stokes (RANS) and direct numerical simulations (DNS).By combining the advantages of RANS and large eddy simulations (LES), bridging models –much like hybrid methods – offer an adaptive balance between computational effort andaccuracy. The computational paradigm is one of the accuracy on demand: fine resolutionwhen the complexity of flow physics or accuracy requirement demands it and coarse resolution,even RANS, at other times. In a typical bridging method computation, the intent is to resolvedynamically crucial large scales and model all the other scales of motion (including inertialrange). Thus, by resolving more scales of motion than RANS, but substantially lesser scalesthan LES, these methods potentially offer improved accuracy over RANS at a computationalcost substantially lower than standard LES. The cut-off between resolved and unresolvedscales is dictated by the local grid size and can, in principle, vary in time and space in acalculation for most efficient utility of the numerical grid.

The overall accuracy of a bridging computation depends on the combination of numericalresolution which determines the cut-off length scale and the physical fidelity of the constitutive

∗Corresponding author. E-mail: sunil [email protected]

Journal of TurbulenceISSN: 1468-5248 (online only) c© 2007 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/14685240701420478

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2 S. Lakshmipathy and S. S. Girimaji

relation, which supplies the subgrid eddy viscosity or stress for a given cut-off. If the closuremodel is poor, high numerical resolution alone cannot yield the best possible results on agiven grid. Therefore, it is imperative that we combine maximum affordable resolution withthe best possible closure model. For a given grid, a successful bridging calculation involvesthree major elements: (i) controlled and predictable reduction of eddy viscosity from RANSvalue, (ii) leading to the natural liberation and computation of more scales of motion and(iii) resulting in improved accuracy of calculation as more scales are computed directly andaccurately.

While the bridging computational paradigm is very appealing, modelling eddy viscosity as afunction of cut-off length scale can be very challenging. If the cut-off is in the dissipation range,one can invoke the equilibrium assumption – production balances dissipation locally at thecut-off wavenumber – leading to a Smagorinsky-type algebraic relation between unresolvededdy viscosity and cut-off length scale

νu ≈ �2S, (1)

where � is the grid-size and S is the resolved strain rate magnitude. The implicit assumptionhere is that the spectral cut-off length scale is equal to the grid size. In general, when the cut-offis located at large or inertial scales, a more sophisticated eddy-viscosity relation is needed,e.g.,

νu = νu(uu, lu), (2)

where uu and lu are the velocity and length scales that characterize the unresolved motion.To motivate a general closure model for the unresolved eddy viscosity valid for cut-off at anyscale of motion, we look to the tried and tested RANS two-equation model.

RANS closure. Two-equation turbulence closure model with the Boussinesq constitutive re-lation has long been used for RANS turbulence computations. The Boussinesq eddy-viscosityhypothesis relates the turbulent stresses to the mean strain:

〈ui u j 〉 = 2

3kδi j − 2νT Si j , (3)

where 〈ui u j 〉 is the turbulent stress, k is the turbulent kinetic energy, δi j is the Kronecker delta,νT is the turbulent eddy viscosity and Si j is the mean strain field. The eddy viscosity is givenby

νT = Cµ

k2

ε, (4)

where ε is dissipation and Cµ is a model coefficient calibrated to give reasonable agreementover a wide range of flows. The RANS velocity and length scales are k1/2 and k

32 /ε, respec-

tively. Modelled transport equations are solved for the kinetic energy and dissipation. To thisday, Boussinesq-based two-equation models continue to be the most popular RANS approachfor practical computations due to their robustness, low computational burden and reasonablelevel of accuracy. Therefore, the two-equation model with Boussinesq constitutive closure isa natural candidate for bridging method as well.

Bridging closures. There is no consensus yet on how to extend a Boussinesq two-equationmodel to bridging closure. Two major proposals are currently under consideration. In oneapproach (very large eddy simulations (VLES) [2]; limited numerical method(LNM) [3], andflow simulation methodology (FSM) [4]), the viscosity reduction is attempted by lowering thevalue of Cµ by a factor which depends on the desired cut-off length scale. In the second method(partially averaged Navier–Stokes (PANS) [5], partially integrated turbulence method (PITM)[6]), it is proposed that the requisite viscosity reduction is best achieved by modifying the

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Extension of Boussinesq turbulence constitutive relation for bridging methods 3

transport equations for the unresolved kinetic energy and dissipation rather than changing theBoussinesq model coefficient. Clearly, the numerical resolution must be compatible with thedesired viscosity reduction in both cases. In practice, the reduction factor in both methods isspecified based on the local grid spacing. While these two proposals have been independentlyused to compute various flows, the relative merit of one over the other has not yet beendemonstrated.

The objective of this paper is to compare the two proposals in two important benchmarktest cases: flow past circular cylinder and flow over a backward-facing step. Comparison willbe performed in three categories. First, we will examine how well the two models achievethe prescribed level of viscosity reduction. Second, we will compare qualitative features ofthe computed results from the two methods to verify if the flow characteristics are physicallyplausible. Finally, we will evaluate the accuracy of the model computations against availabledata for different levels of viscosity reduction prescription.

In section 2, we present more details on the proposals to extend the Boussinesq constitutiverelations to bridging turbulence models and formulate the basis of comparison. The computa-tional details of the chosen test cases are presented in section 3. In section 4, the findings fromthe simulations are presented and discussed. Section 5 contains summary and discussion.

2. Model equations and comparison criteria

In this section, we will first describe the two-equation Boussinesq-based bridging models andthen establish the ground rules and criteria for comparing the two proposals.

2.1 The bridging models

For the bridging approach, a closure relationship between the cut-off length and subgrideddy viscosity is sought. In the two methods under consideration, the relation between thegrid spacing and subgrid eddy viscosity is modelled in terms of an intermediary parametercalled the viscosity reduction factor R. The physical significance of R is very important. Itrepresents the desired reduction in viscosity from a hypothetical RANS calculation of thesame flow on the same grid. The relationship between R and grid spacing in each approachis given in the original papers [3, 7]. Clearly, the smaller the value of R, more scales will beresolved resulting higher degree of accuracy. The minimum value of R that can be used in acomputation is restricted by the grid spacing [7]. When numerical resolution is very fine, Rcan be small and when resolution is coarse R will be closer to unity. Between the two limits,R is a smooth function of grid spacing in both approaches. It is, however, important to notethat larger values of R are permissible – reflecting the fact that the cut-off length scale can belarger than the grid spacing.

We will begin with the description of the so-called unsteady Reynolds-averaged NavierStokes (URANS) [8] method which is a forerunner of the bridging methods and then presentthe bridging models.

2.1.1 URANS method. In this approach, unmodified RANS closure is used to modelsubgrid-scale (SGS) stress. The calculation is performed in a time-accurate manner on veryfine grids with the expectation of capturing unsteady scales of motion that cannot be resolvedin steady RANS. Thus, the governing equations are identical to those of RANS and the onlydifference is in the grid resolution which is finer than in the case of a steady RANS compu-tation. We include this model for the sake of comparison with the bridging methods whenappropriate.

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2.1.2 PANS/PITS bridging method. In both the partially averaged Navier–Stokes (PANS)[5] and partially integrated turbulence method (PITM) [6], the turbulent velocity field decom-position is based on the desired level of kinetic energy to be resolved. While PANS model isderived in physical space from RANS equations, the final closure is strikingly similar to thespectral closure development of PITM. In both instances, the Boussinesq constitutive relationitself is left unmodified and the entire onus of rendering the model sensitive to cut-off falls onthe velocity and length scale transport equations. The closure models in both cases are derivedrigorously from the parent RANS equations, and we refer the readers to the original papersfor the details. Here we just present the final model equations.

The Boussinesq constitutive relation for partially averaged fields is as given in equation (3)and the eddy viscosity is given by

νu = Cµ

k2u

εu, (5)

where ku and εu are the unresolved kinetic energy and dissipation rate, respectively. Thereduction in the turbulent eddy viscosity is achieved by modifying the transport equations in amanner consistent with physics to yield lower values of ku and εu such that the ratio of k2

u/εu

is smaller than its RANS counterpart. The transport equations for unresolved kinetic energyand dissipation are

dku

dt= Pu − εu + ∂

∂x j

((νu

σku+ ν

)∂ku

∂x j

)(6)

dεu

dt= Ce1

Puεu

ku− C∗

e2ε2

u

ku+ ∂

∂x j

((νu

σεu+ ν

)∂εu

∂x j

), (7)

where Pu is the PANS production. The model coefficients are Ce1 = 1.44; C∗e2 ≡ Ce1 +

fk/ fε(Ce2 − Ce1); σku ≡ σk f 2k / fε and σεu ≡ σε f 2

k / fε; Ce2 = 1.92; σk = 1.0; σε = 1.3. InPANS, the eddy viscosity is controlled by suitably specifying fk and fε as functions of gridspacing and flow Reynolds number: fk(= ku/k) is the desired fraction of the turbulent kineticenergy to be resolved and fε(= εu/ε) gives the fraction of total dissipation rate to be resolved.In all the cases presented in this paper fε is taken to be unity as the Reynolds number is highenough [5].

2.1.3 VLES/LNS/FSM methods. As mentioned elsewhere [2–4], in these methods it issuggested that the reduction in eddy viscosity can be achieved by reducing the coefficient inthe Boussinesq relation by a factor f :

νu = f Cµ

k2u

εu. (8)

In general, f is specified according to the local grid resolution. The model transport equationsfor the VLES, LNS and FSM remain unchanged from their RANS counterpart except theynow represent unresolved kinetic energy and dissipation:

dku

dt= Pu − εu + ∂

∂x j

((νu

σk+ ν

)∂ku

∂x j

)(9)

dεu

dt= Ce1

Puεu

ku− Ce2

ε2u

ku+ ∂

∂x j

((νu

σε

+ ν

)∂εu

∂x j

), (10)

where Pu is the turbulent production. The model parameters are as given before.

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2.1.4 Viscosity reduction factor. As mentioned earlier, the viscosity reduction factor isdefined as

R =νu (Bridging model)

νT (RANS). (11)

From the PANS model closure, the desired viscosity reduction factor can be easily surmised.

R(PANS) = f 2k (�, λ). (12)

The PANS relationship between the grid spacing and viscosity reduction factor is given by [7]

R(PANS) ≥ 1

Cµ

(�

λ

)4/3

, (13)

where � is the smallest grid size and λ is the Taylor microscale. As the grid spacing decreaseswith respect to the local Taylor microscale, the viscosity reduction factor becomes smallerand more and more scales of motion can be resolved. It is important to note that equation (13)is an inequality and hence R values larger than the minimum value permitted by the grid sizecan be used.

Clearly, the LNS/FSM viscosity reduction factor is

R(LNS, FSM) = f (�). (14)

There are several proposals for specifying f as a function of grid spacing �, and we referthe readers to the original publications [2–4]. Thus, from the grid size and the relation betweengrid size and R, the bridging model parameter can be specified for both models.

2.1.5 Important caveat. The bridging closure proposals are yet to be validated compre-hensively. It is very important to note that while R is the desired viscosity reduction factor,the computed a posteriori value may be different.

2.2 Model comparison criteria

Our model comparison criteria are based on the examination of the three crucial elementslisted in the introduction as keys to bridging model success: predictable viscosity reduction,liberation of more scales of motion and, ultimately, improved model accuracy. Clearly, theaccuracy must improve with an increase in the number of resolved scales.

2.2.1 Controlled viscosity reduction. First, and foremost, the successful adaptation of theBoussinesq two-equation approach to the bridging method requires that the desired level ofviscosity reduction should be achieved in the calculation. To evaluate if the two proposedmodels posses this important quality, we devise the following test. We perform separateRANS and bridging computations of specified R (a priori viscosity reduction ratio) valueson the same numerical grid. In each calculation, we obtain the time-averaged eddy viscosityas a function of space. Then we take the ratio of the RANS and bridging viscosities at eachlocation yielding the computed (a posteriori) viscosity ratio. Comparison of the specified (apriori) and computed (a posteriori) viscosity ratios provides an important basis for evaluatingthe fidelity of the closure models. If there is a large difference between the specified andcalculated eddy-viscosity ratios, then the cut-off and the closure model will be inconsistentwith one another leading to unpredictable and unreliable results. Thus, for a reliable bridgingcalculation, the specified and computed eddy-viscosity ratios must be close.

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2.2.2 Liberation of more turbulent scales. Direct computation of more unsteady scalesof motion is crucial to the success of bridging methods. Bridging model viscosity reduction isintended to liberate scales of motion that would be suppressed in a RANS calculation. Withprogressive viscosity reduction, more and more scales of motion should be liberated. We willqualitatively examine the increase in unsteadiness with viscosity reduction.

2.2.3 Comparison with experiments. In the final step of our study, we will comparecalculations from each model at different R values against experimental and LES data.

Both PANS and FSM methods are intended for spatially changing resolution. However,we will perform the investigation for spatially invariant specification of viscosity reductionfactor due to two important factors. (i) The main reason for the spatially invariant filter is theminimization of commutation error. When the filter size (related to viscosity reduction factor)varies with time or space in a manner that filtering and differentiation are non-commutative,there will be additional terms in the resolved and unresolved flow evolution equations. Theseextra terms cannot be modelled resulting in an error termed as the commutation error. (ii) Ithas been found that grid insensitive results are very difficult to obtain if the viscosity reductionfactor is a function of grid spacing. Therefore, we will restrict ourselves to constant viscosityreduction specification – spatially invariant R.

For a given flow configuration, all URANS and bridging model calculations (for all R values)are performed on the same grid. From detailed sensitivity study, the grid used is shown to bein the grid-insensitive range for all R cases. As mentioned elsewhere, larger R values can becomputed on the same grid as the model cut-off length scale is larger than the grid resolution.Thus, we ensure that any differences in the observed results are due only to the closure modeland not grid related issues.

3. Test case description

The two benchmark flows chosen for this comparison study are flow past a circular cylinder atReD 1.4 × 105 and flow past a backward facing step at ReH 3.75 × 104. These flows exhibitlarge scale unsteadiness and other flow features that are not easily captured with simple RANSmodels. Simulations are performed for various values of R with PANS and FSM to comparethe two bridging model proposals.

3.1 Cylinder flow computational details

The domain chosen for the cylinder flow simulations is box-shaped with dimensions as shownin figure 1. The spanwise width of the cylinder is in accordance with the LES of Breuer [9].Structured O-type grid is employed with 240 nodes along the wake centreline with the firstgrid point in the wall normal direction placed at y+ ≤ 3. In the circumferential direction, 320grid nodes are uniformly distributed. The spanwise length of the domain is divided into 32equal parts. Therefore, the grid has the same resolution as the fine grid cases of Breuer [9] inthe circumferential and spanwise direction to accommodate the smallest R value case. Theresolution along the wake centreline is less fine and the grid nodes are spaced at an expansionratio of 1.2. At the inlet, zero-turbulence constant velocity field is specified such that the flowReynolds number based on the cylinder diameter is 1.4 × 105. Outflow boundary conditionsare specified at the outlet. The domain boundaries in the crosswise direction are specified asslip walls. Periodic boundary condition is imposed in the spanwise direction. The turbulence

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Figure 1. Computational domain for the cylinder flow simulations.

model is active at all times during the simulations. The turbulence model parameters arespecified depending upon the closure used and the desired viscosity reduction factor R.

3.2 Backward facing step computational details

The computational domain used for the backward facing step simulations is shown in figure 2.The grid resolution in the inlet section is 50 × 115 × 36. In the step section, there are 280nodes along the flow direction. The grid is clustered near the step so that the y+ near the stepwall is less than 1. The grid is gradually stretched in this direction with an expansion ratio of1.05. Along the normal direction, there are 215 nodes with 100 nodes placed within the stepheight and the first grid point placed in the viscous sub-layer (average y+ ∼ 0.6). The spanwiseresolution in this section is same as the inlet section with 36 nodes distributed uniformly. Theinlet conditions for the velocity, turbulent kinetic energy and dissipation are generated from aPANS simulation of flat plate boundary layer performed with a maximum mean inlet velocity

Figure 2. Computational domain for the backward facing step simulations.

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of Uo and with a turbulent intensity of 2%. The outlet is modelled as pressure outlet with zerogauge pressure. In the wall-normal direction, no-slip boundary condition is imposed at thebottom wall and slip condition at the top wall. In the z-direction, periodic boundary conditionis imposed.

3.3 Flow solver

The simulations were performed using the commercially available Fluent CFD package, whichis a finite volume code. For all the simulations, a double-precision solver with second-orderupwind discretization for the momentum, turbulent kinetic energy and dissipation equationsis used. Third-order MUSCL scheme was also used in the selected cases. As the differencebetween the second-order and third-order schemes was small for the quantities investigatedin this paper, we present results only from the second-order scheme. It must be, however,pointed out that if unsteady velocity-gradient flow details must be captured accurately, thethird-order MUSCL scheme must be employed. Continuity of the incompressible flow wasensured through SIMPLE algorithm.

The near-wall regions are treated with enhanced wall treatment. Since the near-wall meshgenerated for the test cases is sufficiently fine to resolve the laminar sub-layer, Fluent uses atwo-layer zonal model in the wall region [10]. In this approach the whole domain is dividedinto a viscous affected region and a fully turbulent region based on the turbulent Reynoldsnumber, Rey (≡ y

√k/ν). In the fully turbulent region (Rey ≥ 200), the ku and εu equations

are solved. In the viscous affected wall region, Fluent employs a one-equation model [10]. Themomentum equations and the ku equation are retained. The turbulent viscosity is computedfrom

νTu,two-layer = ρCµlu

√ku, (15)

where lµ = yCl(1−e− ReyAµ ). The two-layer definition and the high Reynolds number definition

are smoothly blended using

νTu,enh = λενTu(1 − λε)νTu,two-layer. (16)

The blending function λε is chosen such that it is equal to unity far away from the walls andzero very near to the walls and is given in [10]. The εu field is computed from

εu = k3/2u

yCl(1 − e− Rey

Aε

) (17)

The constants used in the length scale formulas are Cl = kC−3/4µ ; Aµ = 70; Aε = 2Cl . The

PANS formulation extends to the first grid point of the wall.

4. Results and discussion

The test cases chosen in this study are long-standing benchmark flows widely used for vali-dating turbulence models and verifying numerical schemes. The geometry is very simple toset-up but the flow physics exhibits complex features involving separation, large-scale coherentstructures and reattachment. The inflow and boundary conditions in each case is described inthe previous section. To demonstrate the versatility of the bridging methods, k-ε two-equationformulation is used for the cylinder flow and k-ω model is used in the backward-facing stepcase.

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Each simulation (unless otherwise mentioned) is started from a steady-state initial field andallow to develop unsteadiness. The statistics are gathered after many flow-through times whenthe statistical stationarity of the flow field in clearly established. The number of flow throughtimes for achieving statistical stationarity is a strong function of R. For fine-resolution runs(low R), the time taken for stationarity is significantly larger than that of low-resolution (highR) cases.

4.1 Results from flow past a circular cylinder at ReD 1.4 × 105

4.1.1 Controlled eddy-viscosity reduction. First, RANS calculation is performed torecord the eddy viscosity as a function of space. Then, PANS simulations are performedwith specified R values of 0.6, 0.49, 0.36, 0.25 and 0.16. In the case of FSM, computations areperformed with R values of 0.8, 0.6, 0.49 and 0.36. Data from the first 30 shedding cycles arediscarded to allow for vortex shedding to be established. Once statistically steady vortex shed-ding is established, flow data from subsequent 50 shedding cycles are gathered to compute theneeded statistics. The bridging model eddy viscosity is then obtained as a function of space.At each location, bridging model to RANS eddy-viscosity ratio is constructed. Locations inwhich the background velocity is unaffected or modified by less than 2% of the backgroundflow are not considered, as the closure model does not play any role at these grid points. Allthe other locations constitute the statistical ensemble investigated.

The probability density function (PDF) of the computed viscosity ratio is plotted in figure3(a) for each PANS simulation. The specified R value is also shown for each case. While there issome spread in the computed ratio distribution, the PDF peaks very close to the specified valuein all cases. For reference, uniform distribution of eddy-viscosity ratio will yield a flat PDF ata value of about 1.0. The computed peak PDF value is about 10, clearly indicating the strongpropensity of the computed viscosity to be close to the specified value. The eddy-viscosityratio at over 75% of the flow field is within ±0.05 of the specified ratio in all simulations.Closer examination of the flow field reveals that the departure from the specified value isseen mostly in areas where turbulent transport effects are significant. The reader should bereminded that the PANS formulation is based on homogeneous turbulence assumptions and,hence, can be expected to be somewhat inaccurate when transport is the leading cause ofturbulence. The PANS (and PITM) paradigm is most valid when the local effects of productionand dissipation are most dominant. In figure 3(b), the most probable a posteriori PANSviscosity ratio is plotted as a function of specified (a priori) value. Clearly, the computationsindicate that PANS/PITM bridging model successfully produce the desired level of viscosityreduction.

The PDF of the various FSM calculations is shown in figure 4(a) and the specified R valuesare also indicated. It is immediately clear that the computed viscosity ratio values are farfrom the specified ones in this case. The peaks of the PDF are located far from the specifiedvalues. In figure 4(b), we plot the most probable a posteriori viscosity ratio against the apriori one. The difference between the specified and computed ratio gets progressively worsewith decreasing R value. In fact for the R = 0.36 case, the PDF is not too far from a uniformdistribution implying very little correlation between specified and computed viscosity ratios.

To understand the reasons for the big difference between the two methods, we now plot theunresolved kinetic energy computed from the two approaches. In figure 5(a), the PDF of theratio of PANS to RANS unresolved kinetic energy is plotted for the various R calculations.A similar plot for FSM is shown in figure 5(b). The fundamental difference between the twomodels is immediately evident. In the case of PANS, the level of unresolved kinetic energygoes down progressively with R as specified. The PDF peaks fairly close to the specified value

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(a)

(b)

Figure 3. (a) PDF of computed viscosity ratio in PANS calculation: specified value (dashed line). (b) Curve-fit forthe computed viscosity ratio peaks for PANS.

of fk = √R. Lower levels of unresolved kinetic energy lead to reduced eddy viscosity. In the

case of FSM, there is no discernible decrease in unresolved kinetic energy as a function of R.Rather surprisingly, there is a high probability of the FSM unresolved kinetic energy exceedingthat of RANS kinetic energy with decreasing R, as indicated by the long and heavy tails ofthe PDF. Thus, the FSM kinetic energy behaviour appears to be contrary to expectations.

4.1.2 Liberation of unsteady scales. The performance of the bridging models in resolvingthe turbulent flow structures is presented next. Figures 6–8 show the instantaneous contourplots obtained from the two bridging methods studied for the case of R = 0.16. Although theresults presented here are qualitative in nature, they help to identify important features of the

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(a)

(b)

Figure 4. (a) PDF of computed viscosity ratio for FSM calculation: specified value (dashed line). (b) Curve-fit forthe computed viscosity ratio peaks for FSM.

two bridging proposals. Figures 6(a) and (b) show the instantaneous iso-z-vorticity surfacesobtained from the PANS and FSM, respectively. The difference between the two results isstriking. In the case of PANS (figure 6(a)), the results go from laminar two-dimensionalflow over the cylinder to highly unsteady, strongly three-dimensional flow with energeticsmall scales in the wake. Many of the flow details (striated rollers in the wake region) arequalitatively consistent with experimental observations. The FSM results (figure 6(b)) on theother hand yield a two-dimensional flow pattern throughout the entire flow domain. In figures7(a) and (b), the z-vorticity contours are compared. While FSM flow field exhibits regularlaminar-like vorticity pattern, the PANS contours are irregular composed of a wider range of

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(a)

(b)

Figure 5. (a) PDF of computed ratio of unresolved kinetic energy ratios for PANS computations. (b) PDF ofcomputed ratio of unresolved kinetic energy ratios for FSM computations.

scales characteristic of turbulence. The reasons for the observed PANS and FSM behaviourcan be surmised from the instantaneous contour plots of turbulent eddy viscosity shown infigures 8(a) and (b). The contour colour coding used in both cases is identical: the level ofviscosity increases from blue to green to yellow to red. The PANS eddy-viscosity levels aresignificantly lower that the FSM values as quantified in figures 3(a) and (b). Therefore, in FSMmuch of the small-scale motions, including all relatively weak three-dimensional structures,are suppressed leading to a laminar-type regular two-dimensional flow field. As R = 0.16 isthe finest resolution examined, it can be expected that the FSM calculations of higher R values

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Figure 6. (a) Iso-vorticity surfaces coloured by x-velocity for (a) PANS and (b) FSM.

will be laminar-like and two dimensional, a fact that is confirmed in our calculations (resultsnot shown).

4.1.3 Comparison against data. We now compare PANS and FSM results against exper-imental data [11] and LES computations [9].

Figure 9(a) shows the mean streamwise velocity along the wake centreline for the variousPANS simulations. Experimental data [11], LES results [9] and URANS calculation are alsoshown for comparison. It is useful to note that URANS corresponds to R = 1 case. It is easy tosee that as R decreases, the PANS results go monotonically from the URANS to LES results.

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Figure 7. (a) z-vorticity contours for PANS. (b) z-vorticity contours for FSM.

The circulation bubble size is well predicted by R = 0.36 case. The out-flow recovery is notvery well predicted and probably can be improved by introducing a longer buffer zone betweenthe fully developed outflow condition and the flow region of interest. Notwithstanding this, itis very clear that the PANS model performance improves substantially with decreasing R: atR = 0.36 good agreement with data is obtained. For a given increment in R, the differencein PANS result is larger at large R values and very small at lower R values. For smaller Rvalues, the mean velocity plot is only slightly different from the R = 0.36 case.

Figure 9(b) presents similar comparison of FSM calculations. Again, URANS case corre-sponds to R = 1 calculation. While the FSM accuracy certainly improves with decreasing R,

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Figure 8. (a) Contours of instantaneous eddy viscosity for PANS. (b) Contours of instantaneous eddy viscosity forFSM.

the rate of improvement is much slower than PANS. For a given value of R, the FSM resultsare much inferior to the PANS computations. For a given R, due to higher FSM eddy-viscositylevels, there is less mixing in the wake leading to larger recirculation bubble size that what isseen in PANS results and LES data. The change from R = 0.49 case to R = 0.36 case is moresubstantial in PANS than in FSM. For values of R < 0.36, the FSM results do not show muchimprovement over the R = 0.36 case shown in the figure. The reason for this can perhaps befound in figure 4 where the computed values of eddy viscosity are shown. It is very clear thatthe computed FSM eddy viscosity responds very slowly to decreasing R in the fine resolutionrange.

In figure 10, the mean streamwise velocity statistics at two different x-planes in the near-wake region is presented. At x/D = 1.0, the bridging models predict a V-shaped velocityprofile whereas the experiments and LES predict a U-shaped velocity profile. At the centre,the PANS model predictions are closer to the experimental observations than URANS andFSM. At x/D = 3.0, the URANS results predict the mean velocity defect to be higher than

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Figure 9. (a) Mean streamwise velocity along the wake centerline for PANS. (b) Mean streamwise velocity alongthe wake centerline for FSM.

in experimental data. The PANS and FSM data agree fairy well with experiments at thislocation.

4.2 Backward facing step at ReH = 3.75 × 104

The simulation of flow past a backward facing step is a challenging test case as it involves pre-dicting the boundary layer accurately. The backward facing step simulations were performedusing the k-ω turbulence model [12]. The PANS k-ω model has the advantage of predictingthe turbulent boundary layer at solid walls without having to apply any viscous correction toreproduce the law of the wall [13].

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Figure 10. Mean streamwise velocity at various x-planes.

4.2.1 Controlled eddy-viscosity reduction. For this test case, the results of eddy-viscosityreduction factor are presented only for the PANS simulations. Figure 11 shows the distributionof the computed viscosity factor from various PANS simulations along with the prescribedvalue. The peak of the computed ratio agrees very precisely with the specified value. Althoughthe spread in the PDF is a little larger than in the cylinder case, it is clear that the eddy-viscosityreduction over a substantial portion of the flow field is close to the specified value. Much ofthe deviation from the prescribed value occurs close to the walls underscoring the need forbetter wall treatment. Accurate wall treatment at reasonable cost continues to be a major topicof research interest in LES and hybrid methods. We will address this issue in later works.Overall, this figure clearly validates the PANS/PITM eddy-viscosity reduction rationale.

Figure 11. PDF of computed viscosity ratio in PANS k-ω calculation: specified value (dashed line).

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Figure 12. (a) Iso-vorticity surfaces coloured by x-velocity for PANS computations of backward facing step. (b)Iso-vorticity surfaces coloured by x-velocity for FSM computations of backward facing step.

4.2.2 Liberation of unsteady scales. The iso-vorticity surfaces from PANS and FSMcalculations (R = 0.25) are shown in figures 12(a) and 12(b) respectively. The shown PANSresult is at an arbitrary time after statistical steady state is established. The PANS calculationagain exhibits irregular three-dimensional flow structure with motions over a wide range ofscales typical of turbulent flows. The shown PANS profile is used as the initial condition for theFSM calculation. Thus the FSM initial field is three-dimensional with energetic small scales.During the course of the FSM evolution it is found that the small scales and three-dimensionalstructures quickly disappear leading to laminar-type two-dimensional flow field. The predictedFSM flow pattern is similar to RANS results.

4.2.3 Comparison against data. Figures 13(a) and 13(b) compare the mean velocity statis-tics obtained from the PANS and FSM model computations for R = 0.25. Figure 13(a) presentsthe results for the mean velocity statistics at x/h = 1.0 and figure 13(b) shows the same statis-tics at several planes further downstream of the step region. The symbols in the plots represent

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(b)

Figure 13. (a) Comparison between PANS and FSM for streamwise x-velocity at x /H = 1.0. (b) Mean streamwisevelocity at various x-planes.

measured data of Driver and Seegmiller [14]. Recall (from figures 11) that the flow fieldsthat yield the PANS and FSM results are fundamentally different. At x/h = 1.0, the PANSmodel captures the trend for the mean streamwise velocity quite adequately. The FSM modelpredicts a shallow velocity profile in comparison to experimental data. At x/h = 5.0, theFSM computations do not capture the reverse flow characteristics of the flow since the modelunder-predicts the reattachment length. The PANS model captures the trend and the predictedvalue for the reattachment length is in agreement with experimental data. At x-planes furtherdownstream, the FSM model consistently over-predicts the mean streamwise velocity insidethe step region.

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5. Conclusion

The main objective of the present study is to evaluate the effectiveness of two bridging philoso-phies: PANS and PITM on one side and VLES, FSM and LNS on the other. Each approachemploys a Boussinesq constitutive relation and solves transport equations for length and ve-locity scales. The rationale for achieving viscosity reduction (over the base RANS model)is, however, fundamentally different. In the case of VLES, FSM and LNS it is proposed thatthe viscosity reduction can be achieved by lowering the value of the model co-efficient in theBoussinesq relationship. The FSM length and velocity scale equations are unaltered fromthe RANS form. On the contrary, in the PANS/PITM paradigm, the reduction of viscosity isattempted by modifying coefficients in the transport equations for length and velocity scales.The coefficient in the PANS Boussinessq relation is not modified.

For this evaluation, reasonably high Reynolds number simulations of flow past a circularcylinder and flow past a backward facing step are chosen as test cases as they are simpleto set-up and experimental and other numerical data are readily available. Simulations usingboth the PANS and the FSM bridging models are performed for different viscosity reductionfactors.

From our study, we conclude that PANS closure yields to a reliable and predictable reductionin subgrid viscosity. The reduction in viscosity leads to the liberation of scales of motion thatare typically suppressed in a RANS calculation. Due to the direct computation of more scalesof motion, PANS accuracy increases substantially with decreasing viscosity reduction factor.Overall, PANS appears to be a reasonable bridging model that performs as intended.

On the other hand, the FSM closure does not lead to the specified level of viscosity reduction.More importantly, there is clear evidence that unsteady scales of motion are not present in thecalculations, although the grid spacing is fine enough to resolve these scales of motion. Whilesome improvement is observed in the comparison against experimental data, this clearly isnot due to the direct computation of more scales of motion.

The PANS and FSM observations are well in line with the fixed-point analysis of Girimajiet al. [15]. It is demonstrated in that paper that modifying the transport equations for thelength and velocity scales offer the best approach to resolving more scales of motion. It is alsoargued that modifying the Cµ value alone is tantamount to a different RANS model ratherthan a bridging model.

References

[1] Germano, M., 1999, From RANS to DNS: towards a bridging model, Direct and Large Eddy Simulation III(Dordrecht: Kluwer), 225–236.

[2] Speziale, C.G., 1996, Computing non-equilibrium flows with time dependent RANS and VLES. 15th ICNMFD.[3] Batten, P., Goldberg, U. and Chakravarthy, S., 2002, LNS – an approach towards embedded LES. AIAA 2002-

0427.[4] Von Terzi, D.A. and Fasel, H.F., 2002, A new flow simulation methodology applied to the turbulent backward

facing step. AIAA 2002-0429.[5] Girimaji, S.S., 2006, Partially-averaged Navier–Stokes model for turbulence: a Reynolds-averaged Navier–

Stokes to direct numerical simulations bridging method. Journal of Applied Mechanics, Transactions ASME,73(3), 413–421.

[6] Schiestel, R., 1987, Multiple time-scale modeling of turbulent flows in one-point closures. Physics of fluids,30, 722.

[7] Girimaji, S.S. and Abdol-Hamid, K.S., 2005, Partially-averaged Navier–Stokes model for turbulence: imple-mentation and validation. AIAA 2005-0502.

[8] Khorrami, M.R., Singer, B. and Berkman, M.E., 2002, Time accurate simulations and acoustic analysis of slatfree shear layer. AIAA Journal, 40(7), 1284–1291.

[9] Breuer, M., 2000, A challenging test case for large eddy simulation: high Reynolds number circular cylinderflow. International Journal of Heat Fluid Flow, 21, 648–654.

[10] Fluent Inc. Fluent 6.2 user’s guide 2005.

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[11] Cantwell, B. and Coles, D., 1983, An experimental study of entrainment and transport in the turbulent nearwake of a circular cylinder. Journal of Fluid Mechanics, 136, 321–374.

[12] Wilcox, D.C., 1988, Multi-scale model for turbulence flows. AIAA Journal, 26(11), 1311–1320.[13] Lakshmipathy, S. and Girimaji, S.S., 2006, Partially-averaged Navier–Stokes method for turbulent flows: k-ω

model implementation. AIAA 2006-119.[14] Driver, D.M., Seegmiller, H.L. and Marvin, J.G., 1987, Time dependent behavior of a reattachment shear layer.

AIAA Journal, 27(7), 914–919.[15] Girimaji, S.S., Jeong, E. and Srinivasan, R., 2006, Partially-averaged Navier Stokes method for turbulence:

fixed point analysis and comparison with partially-averaged Navier Stokes. Journal of Applied Mechanics,Transactions ASME, 73(3), 422–429.

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extension of boussinesq turbulence constitutive relation for bridging methods

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