ASBM-BSL: An Easy Access to the StructureBased Model Technology

Bertrand Aupoix, Stavros C. Kassinos,and Carlos A. Langer

Abstract The Algebraic Structure Based Model (ASBM) offers unique featuresto represent the Reynolds stress tensor from the underlying turbulent structures. Itis usually coupled with a non standard length scale equation. A way of coupling itwith the more popular BSL ω equation is proposed here. Only a minor modificationof the ω equation is required to obtain a realistic turbulent kinetic energy profile andthus achieve fair predictions.

1 Introduction

Present industrial design heavily relies upon CFD, using mainly eddy viscosity mod-els, the two most popular ones being the Spalart–Allmaras one equation model andthe Menter SST k–ω model. However, they are known to fail to capture many flowissues, first of all because of the eddy viscosity assumption. Two routes are generallyconsidered to get rid of this assumption.

The use of transport equations for the Reynolds stress tensor (DRSM models)allows one to circumvent most of the failures of the eddy viscosity assumption, byaccounting for turbulence memory effects, most of the rotation and curvature effects.

Explicit Algebraic Reynolds Stress Models (EARSM) only require two transportequations and replace the eddy viscosity assumption by an equilibrium assump-

B. Aupoix (�)Aerodynamics and Energetics Department, ONERA, BP 74025, 2 Avenue E. Belin, 31055Toulouse Cedex 4, Francee-mail: Bertrand.Aupoix@onera.fr

S.C. Kassinos · C.A. LangerComputational Sciences Laboratory — UCY-CompSci, Department of Mechanical &Manufacturing Engineering, University of Cyprus, 75 Kallipoleos, Nicosia 1678, Cypruse-mail: kassinos@ucy.ac.cyC.A. Langere-mail: langer@ucy.ac.cy

M. Stanislas et al. (eds.), Progress in Wall Turbulence: Understanding and Modeling,ERCOFTAC Series 14,DOI 10.1007/978-90-481-9603-6_29, © Springer Science+Business Media B.V. 2011

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278 B. Aupoix et al.

tion for the anisotropy tensor. They so inherit many properties of the underlyingReynolds stress model.

A third modelling route was proposed by Reynolds and Kassinos [2–4, 9] tocircumvent most of the failures of DRSM models. While DRSM models assume thatall the information about turbulence is in the turbulence length and velocity scalesand in the anisotropy tensor, they pointed out the importance of the dimensionalityof turbulence. This led them to Structure Based Models (SBM), which are able tobetter represent rapid distortion, rotation effects and many subtle non-equilibriumeffects than current DRSM models. But these models use extra transport equations,compared to DRSM models and are thus less prone to a prompt industrial use.

A simplified version, using the SBM approach to directly represent the anisotropyand dimensionality tensors was derived [5, 6]. This Algebraic Structure BasedModel (ASBM) only requires two transport equations for turbulence velocity andlength scales, and brings some similarities with EARSM models. This approach isvery powerful as it allows to sensitize the length scale equation to the dimension-ality of turbulence. But the length scale equation is unusual [10] and may requiresome changes to be implemented in a commercial code.

Therefore, a simpler version, coupling the ASBM with a more standard set oflength scale equations has been developed, to give an easy access to the ASBMtechnology.

2 ASBM Modelling

Structure Based Models, and especially ASBM models, can be envisioned in twocomplementary ways.

A first way is the use of turbulence structure tensors to characterize the turbulentmotion. While DRSM models only deal with the anisotropy tensor, other tensorscan be derived from the turbulent stream function vector, the curl of which is theturbulent velocity field. The dimensionality tensor characterizes the changes of theturbulence structure along the various directions; the circulicity tensor characterizesthe vorticity field associated with the energy bearing structures, the inhomogeneitytensor the degree of inhomogeneity of the turbulent field. These three tensors and theReynolds stress tensor are related and, in SBM complete models, transport equationsare solved for these tensors to fully describe the turbulent field.

A second way is to consider that turbulence can be mimicked with a combinationof simplified turbulent structures, i.e. as an ensemble of hypothetical 2D eddiesthe direction of independence of which is aligned with the eddy-axis. These eddiesdiffer by their componentality and dimensionality and are:

• Jetal motions: 2D-1C fields where the motion is only along the eddy axis,• Vortical motions: 2D-2C fields where the motion is around the eddy axis,• Helical eddies: superpositions of the above two motions

A last important notion is the flattening of the eddies, which is related to thedegree of asymmetry of the turbulent kinetic energy distribution around the hypo-thetical eddies.

BSL Structure Based Model 279

A complete turbulent field is envisioned as a large ensemble of these eddies, withvarying characters and orientations. Averaging over the ensemble produces statis-tical quantities representative of the field. Decomposing the fluctuating velocity incomponents aligned and normal to the eddy-axis direction, taking the product u′

iu′j ,

and then averaging, results in the algebraic constitutive relation for the normalizedReynolds stress tensor (related to statistics of the ensemble)

rij = u′iu

′j

2k= 1

2(1 − φ)(δij − aij ) + φaij

+ (1 − φ)χ

[1

2(1 − anmbnm)δij − 1

2(1 + anmbnm)aij − bij

+ ainbnj + ajnbni

]− γ

Ωk

Ω(εiprapj + εjprapi)

{1

2

[1 − χ(1 − anmbnm)

]δkr

+ χ(bkr − aknbnr)

}(1)

where the different symbols are defined below. The eddy-axis tensor aij representsthe energy-weighted average direction cosine of the eddy-axis vector. Eddies, likematerial lines, tend to align with the direction of positive mean strain rate, and arerotated kinematically by the mean rotation rate. To ensure the algebraic model forthe eddy-axis tensor is realizable, it is computed via a two-step procedure. Initiallya strained eddy axis, as

ij , is evaluated based upon the mean strain rate tensor, Sij .

asij = δij

3+ τ

Sikaskj + Sjka

ski − 2

3 |Sas |δij

a0 + 2√

a21 + τ 2SkpSkqas

pq

, (2)

where τ is a turbulent time scale, |Sas | = Spqasqp , and {a0, a1} are “slow” param-

eters. Next a rotation operation is applied on asij , so that the final (strained, then

rotated) eddy-axis tensor, aij , is obtained as

aij = HikHjmasij . (3)

The rotation operator Hij is modeled as

Hij = δij + h1Ωij√|Ω2| + h2

ΩikΩkj

|Ω2| , (4)

which satisfies the orthonormal conditions HikHjk = δij for h1 =√

2h2 − h22/2.

h2 is chosen to satisfy theoretical rapid distortion limits for combined homogeneousplane strain and rotation.

h2 =⎧⎨⎩

2 − 2√

(1 + √1 − r )/2 if r ≤ 1

2 − 2√

(1 − √1 − 1/r )/2 if r ≥ 1

, (5)

where r = (apqΩqrSrp)/(SknSnmamk).bij , the eddy-flattening tensor, is based upon an ensemble average of the flat-

tening direction of the hypothetical eddies. This flattening direction is found to be

280 B. Aupoix et al.

dependent on the vorticity vector, so that the model for bij takes the simple formbij = ΩiΩj/|Ω2| in a fixed reference frame.

φ, the jetal parameter, is representative of the amount of energy in the jetal modeof motion. (1 −φ) is representative of the amount of energy in the vortical mode. Inirrotational flows φ = 0, and of course 0 ≤ φ ≤ 1. Mathematically, it represents theaverage (over all eddies) of the dot product between the eddy-axis vector and thevelocity vector. In simple shear flows (this study), it reads

φ = 1 × fslow(a2), a2 = almaml

fslow(a2) = 0.35f 2.5iso (a2) + (1 − 0.35)f 0.5

iso (a2) (6)

fiso(a2) = (3/2)(a2 − 1/3)

γ , the helical parameter, is representative of the correlation between the jetal andthe vortical modes. In the general case,

γ = β

√2φ(1 − φ)

1 + χ, (7)

and for simple shear flows β = 1.χ is the flattening parameter, a representative of the average magnitude of the

lack of symmetry in the energy distribution around the eddies. In simple shear flows

χ = 0.2 × fslow(a2). (8)

The model is moreover sensitized to the wall blocking with the help of an ellipticrelaxation equation for a blocking parameter Φ , from which a partial projection ofthe eddy axis tensor towards the wall is deduced

aij = PikahklPlj , Pik = 1

Da

(δik − Bik), D2a = 1 − (2 − Bkk)a

hmnBnm, (9)

where Pik is the partial-projection operator, and D2a is such that the trace of aij

remains unity (note the superscript “h” is used to indicate the unblocked tensor).The blockage tensor Bij gives the strength and the direction of the projection. Ifthe wall-normal direction is x2, then B22 = Φ is the sole non-zero component, andvaries between 0 (no blocking) far enough from the wall, to 1 (full blocking) at thewall.

The blocking parameter, Φ , is computed by an elliptic relaxation equation

L2 ∂2Φ

∂x2l

= Φ, L = CL max

(k3/2

ε,Cν

4

√ν3

ε

)(10)

with Φ = 1 at solid boundaries, and ∂Φ∂xn

= 0 at open boundaries, where xn is thedirection normal to the boundary. Values of the constants are CL = 0.17, Cν = 80.

To retrieve the proper near-wall asymptotics for the Reynolds stress tensor, theblocking parameter Φ is also used for blending of φ and γ with their wall values;1 and 0 respectively,

φ = 1 + (φh − 1)(1 − Φ)2, γ = γ h(1 − Φ). (11)

At last, δij and εijk are the identity and Ricci tensors, as usual.

BSL Structure Based Model 281

It must be pointed out that the so obtained Reynolds stress tensor is fully realiz-able. Model details can be found in Reynolds et al. [9] and Langer and Reynolds [6].

3 Coupling with a k–ω Model

Among two-equation models, the k–ω model is known to give fair predictions ofpressure gradient effects. However, as the Wilcox [12] model is sensitive to freestream values, Menter’s BSL model [7] has been preferred. Coupling the aboveASBM representation of the Reynolds stress tensor with the BSL model howeverrequires two significant changes.

To take advantage of the fair representation of the Reynolds stress tensor pro-vided by the ASBM model, the turbulent kinetic energy production must be com-puted with the ASBM model as Pk = −u′

iu′j

∂Ui

∂xj. This is used for both k and ω

equations, i.e. the source term in the ω equation is expressed as αPkωk

.Moreover, the ASBM model has been designed to represent real near wall tur-

bulence while the basic k–ω model yields turbulent kinetic energy profiles whichare more similar to v′2 profiles in the wall region and do not exhibit a near-wallpeak. Wilcox later proposed to add three wall functions to retrieve the correct wallbehaviour. When using the ASBM model, modifications on the eddy viscosity andon the ω production rate are not needed. Only the dissipation rate is altered, using afunction adapted from Peng and Davidson [8] as

ε = βfwωk, fw = 1 − 13

18exp

(−

(Rt

10

)2), Rt = k

νω. (12)

This form ensures that k ∝ y2, ε having a plateau in the wall region (see Fig. 2). Atlast, the length scale definition in the elliptic wall blocking has been re-tuned, withCL = 0.1285.

4 Validation Results

Channel flows are first addressed. Comparisons are performed with DNS data atvarious Reynolds numbers provided by Madrid Polytechnic University [1]. Thetransport equations used here are Wilcox’ k–ω ones, which are nearly equivalentto BSL ones for channel flows. They are coupled either with eddy viscosity or withASBM constitutive relations. Results are given for the largest Reynolds numberRτ = huτ

ν= 2000, using wall scaling.

Figure 1 shows that the model inherits many characteristics from the underlyingk–ω model and yields very similar predictions for the mean velocity profile, withsome discrepancies with respect to DNS in the buffer and wake regions. The erroron the friction coefficient, w.r.t. Prandtl’s correlation, is about −4% for a wide rangeof Reynolds numbers, while it is about −6% with the BSL model.

282 B. Aupoix et al.

Fig. 1 Channel flowRτ = 2000 — Mean velocitypredictions

Fig. 2 Channel flowRτ = 2000 — Turbulentkinetic energy and dissipationrate predictions

Figure 2 points out that the model, as any other, is unable to reproduce the growthin the turbulent kinetic energy peak as it does not account for the inactive motions.The improvement brought about by the use of the damping function (12) is apparenton the dissipation profile, which is now in better agreement with DNS data, andalso on the appearance of the near-wall turbulent kinetic energy peak, which is notreproduced by Wilcox’ model.

The three diagonal stresses are plotted in Fig. 3. “Wilcox” values are obtainedassuming u′2 = k, v′2 = 0.4k, w′2 = 0.6k. v′2 profile is well reproduced, w′2 is de-pleted in the buffer region and of course u′2 and w′2 are underestimated as inactivemotions are not accounted for. Profiles of all turbulent quantities are very similar tothe ones provided by the k–ω model in the logarithmic region and above.

BSL Structure Based Model 283

Fig. 3 Channel flowRτ = 2000 — Diagonalstresses predictions

Fig. 4 Relative error on skinfriction predictions versusReynolds number for zeropressure gradient boundarylayers (in per cent)

For zero pressure gradient boundary layers, errors on the skin friction coefficientprediction w.r.t. Fernholz’ correlation are plotted in Fig. 4. The error is less thanone percent over a wide range of Reynolds numbers Rθ based upon the momentumthickness.

Comparisons were also performed with experimental data for high Reynoldsnumber boundary layers, performed in the framework of WALLTURB in Labora-toire de Mécanique de Lille (LML). Nice agreement is obtained on the mean veloc-ity profiles but the turbulent quantities evidence that the model, as others, yields thesame solution in the wall region whatever the Reynolds number and hence does notaccount for inactive motions.

At last, several adverse pressure gradient test cases have been considered. Onlythe most difficult case, i.e. the nearly equilibrium flow subjected to a strong adversepressure gradient investigated by Skåre and Krogstad [11] is presented. Figures 5

284 B. Aupoix et al.

Fig. 5 Skin frictionpredictions for Skåre andKrogstad experiment

Fig. 6 Mean velocity profilepredictions for Skåre andKrogstad experiment — Fromright to left stations X = 3,4.2, 4.6 and 5 m

and 6 show that the model is able to reproduce the very low skin friction levelsencountered in the experiment and even does better than the SST model.

5 Conclusions and Perspectives

An easy way to use the Algebraic Structure Based Model, coupling it with thewidespread BSL model, has been proposed. It only requires slight modifications ofthe BSL part, besides of course adding an ASBM routine to compute the Reynoldsstress tensor. It thus gives access to the much better representation of the Reynoldsstress tensor, e.g. for flows with rotation, provided by the ASBM model.

The model has been shown to give fair predictions and to rather nicely reproducethe near wall turbulence, what the basic BSL model is not able of.

BSL Structure Based Model 285

Another way to determine the wall function fw , from known near-wall physics,will be investigated and reported later.

Author names appear in alphabetical order.

Acknowledgements This work has been performed under the WALLTURB project. WALL-TURB (A European synergy for the assessment of wall turbulence) is funded by the EC under the6th framework program (CONTRACT N: AST4-CT-2005-516008).

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