European Community on Computational Methods in Applied Sciences ECCOMAS Computational Fluid Dynamics Conference 2001 Swansea, Wales, UK, 4-7 September 2001 ECCOMAS
INVESTIGATION OF ANISOTROPY-RESOLVING TURBULENCE MODELS BY REFERENCE TO HIGHLY-RESOLVED LES DATA FOR SEPARATED FLOW Y.J. Jang, L. Temmerman and M.A. Leschziner Department of Aeronautics Imperial College of Science, Technology and Medicine, Prince Consort Rd. London SW7 2BY, U.K. Email:
[email protected],
[email protected],
[email protected]
Key words: Large Eddy Simulation, Turbulence, Separation, Non-linear eddy-viscosity modelling, Reynolds-stress modelling. Abstract. The predictive properties of anisotropy-resolving non-linear eddy-viscosity models are investigated by reference to highly-resolved LES data obtained by the authors. To enhance contrast and to aid interpretation, the range of models is extended, at both ends, through the inclusion of two linear eddy-viscosity models (k-ε and k-ω) and one second-moment closure. The test geometry is a periodic segment of a channel constricted by 2D ‘hills’ on the lower wall. The Reynolds number is 10590. Periodic boundary conditions are applied in streamwise and spanwise directions. This makes the statistical properties of the simulated flow genuinely twodimensional and independent from boundary conditions, except at the walls. The simulation was performed on a high-quality, 5M-node grid. The focus of this study is on the exploitation of the data for a-priori and a-posteriori investigation of the selected models, with particular emphasis placed on four non-linear eddyviscosity models. A feature observed especially in computations with ε-based models is an underestimation of the mixing in the separated shear layer and a seriously insufficient rate of post-reattachment recovery, both leading to a general misrepresentation of the flow. Models based on the ω-equation fare better. The a-priori studies reported herein provide pointers to the origins of the defects, and current efforts are in progress to identify the specific model fragments responsible, prior to corrections being devised.
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Y.J. Jang, L.Temmerman and M.A. Leschziner
1
INTRODUCTION
Predictive inadequacies of linear eddy-viscosity models, observed over many years in computations for complex flows, have motivated the development and implementation of second-moment closure and non-linear eddy-viscosity models. Both model types are able, in principle, to resolve Reynolds-stress anisotropy, the effects of flow curvature on turbulence and the correct sensitivity of turbulence to irrotational straining. While second-moment closure offers the stronger fundamental modelling foundation, the quest for model simplicity, computational economy and numerical robustness has led to recent research being focused, principally, on the formulation of improved non-linear eddy-viscosity and explicit algebraic Reynolds-stress models. Thus, several new model forms have been derived by Gatski & Speziale1, Craft et al.2, Apsley & Leschziner3 and Wallin & Johansson4, among others. The reliable assessment of the above models by reference to experimental data poses a number of problems unrelated to the models themselves and their numerical implementation. Issues of principal concern are (i) the compatibility of the computational implementation with the experimental conditions, (ii) the completeness and accuracy of the boundary conditions and (iii) the accuracy of the data used to assess the predictions obtained with the model. The first item is concerned with flow dimensionality, directional homogeneity and symmetry. For example, there are often inescapable differences between statistically 2D computations, usually undertaken in efforts to elucidate fundamental characteristics of turbulence models, and data for nominally 2D experimental configurations, used to validate the computations. Unless the spanwise extent of the flow in question approaches infinity or the flow is perfectly axisymmetric (in a statistical sense), there will always be some contamination by 3D straining, which may have disproportionately large effects on the nominally 2D flow quantities. Incompleteness and errors in respect of boundary conditions, especially in relation to turbulence quantities (eg. Reynolds stresses and dissipation rate) is another important source of uncertainty and may be sufficiently serious to invalidate conclusions derived from a validation study. This danger is especially acute when the flow involves separation from a curved surface and is extremely sensitive to spanwise distortions and perturbations in inflow conditions. A potentially far more reliable route to investigating turbulence models, especially as a precursor to their improvement, is to use highly resolved LES (or DNS) data at conditions which hardly pose any uncertainties in respect of 3D features and boundary conditions. Appropriate simulations, if carefully undertaken and verifiably accurate, can yield data far superior and far more detailed than those derived from experiments. In particular, such simulations yield data on budgets of turbulent stresses, which differentiate the various transportive, diffusive and generative mechanisms. While DNS data have been used extensively, for over a decade, as an aid to turbulence closure, the scope and usefulness of this input has been constrained by the low Reynolds numbers, dictated by resource limitations, and the simplicity of the flows simulated (mainly homogenous strain, channel flow, boundary layers and free shear layers). Increasingly, however, Large Eddy Simulation, undertaken for complex geometries over sufficiently fine, highquality grids at moderately high Reynolds numbers, are able to provide high-quality data for reliable model assessment in challenging flow conditions. The present paper is concerned with investigating, principally, anisotropy-resolving non-linear eddy-viscosity by reference to highly resolved LES data, obtained for the geometry shown in Fig.
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Y.J. Jang, L.Temmerman and M.A. Leschziner
1. This is a streamwise-periodic segment of a channel constrained by 2D ‘hills’ which provoke massive separation on their leeward sides. The flow, at Reynolds number 10590 (based on hill height and mass-mean velocity above the hill crest), is free from transitional features and, effectively, fully turbulent over approximately 90% of its transverse (wall-to-wall) extent. At the same time, the Reynolds number is sufficiently low to allow the semi-viscous lower near-wall layer to be resolved with a high-quality mesh down to y+=O(1). Apart from offering a comprehensive database, this test case has a number of important advantages for the present purpose of model investigation. First, separation takes place from a curved surface – an especially challenging feature for turbulence models, and one that has a strong effect on the reattachment position. Second, because of streamwise periodicity, the inlet/exit conditions are two-waycoupled to the inner flow, a feature that increases the sensitivity of the predictions to the quality of the turbulence model. Third, the flow is truly statistically 2D, without any contamination associated with secondary motion arising from spanwise confinement. Finally, the geometry allows for a significant post-reattachment recovery region on the flat plate separating successive hill. The importance of this feature lies in the fact that most turbulence models which return the correct recirculation-zone dimensions in separated flow also give rise to an insufficient recovery in the post-reattachment wake.
Fig. 1: Periodic hill geometry under consideration, including LES realizations of pressure contours (left) and streamwise velocity (right)
2
THE SIMULATION
The simulation of the flow shown in Fig. 1 is not the central aspect of this paper. Hence, only some key facts are conveyed here. Details may be found in Temmerman & Leschziner5. The simulation was performed over a grid of approximately 5M nodes, covering a spanwise slab of 4.5 hill heights. The mesh was close to orthogonal, of low aspect ratio and with meshexpansion ratio below 1.05. The y+-value at the nodes closest to the lower wall was around 0.5, allowing the no-slip condition to be used directly. Sub-grid-scale effects were approximated with the WALE model (Nicoud & Ducros6), giving viscosity values below the fluid viscosity over most of the flow domain. Statistical data were assembled over a period of 55 flow-through times, at a cost of approximately 50000 processor hours on the Manchester CSAR Cray T3E computer. A similar simulation, for the same geometry and precisely the same grid, but with a different sub-grid-scale model, was performed by Mellen et al.7, and this gave results very close to those obtained by Temmerman & Leschziner5. Separation is predicted at x=0.22h and reattachment at 4.72h, and both are in close agreement with the data
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Y.J. Jang, L.Temmerman and M.A. Leschziner
by Mellen et al.7. Adherence to realisability constraints has been verified. Stress-transport budgets have been extracted from the simulation, and these are being used elsewhere for apriori studies of second-moment-closure models. In what follows, the LES data are used for comparison of turbulence model solutions as well as a-priori studies on non-linear eddy-viscosity models. In the latter, the mean strain field is inserted into the turbulence model, and the model is made to return turbulence quantities and the value of selected model fragments. 3
TURBULENCE MODELLING
3.1 Non-linear eddy-viscosity models (NLEVM) In what follows, Cartesian tensors of rank 2 are identified by bold face (e.g. a) and their components are italicized (e.g. aij). Contracted products are written as for matrix multiplication: (1) ab=aikbkj, abc=aikbklclj , etc The trace, akk of a tensor is denoted by {a}, and the second and higher invariants by a2={a2}=aijaji, a3={a3}=aijajkaki , etc
(2)
Components of mean velocity are denoted by Ui and turbulent fluctuation by ui. The turbulent kinetic energy is k= 1 uiui and I= δ ij . The anisotropy tensor is defined by 2
ui u j
2 − δ ij 3 k and the dimensionless mean strain s and vorticity w tensors are, respectively, aij =
sij =
∂U j 1 ∂U i ∂U j 2 ∂U k 1 ∂U ) τ( δ ij ) , wij = τ ( i − + − 2 ∂x j 3 ∂x k 2 ∂x j ∂xi ∂xi
For incompressible flow,
∂U k =0. ∂x k
(3)
(4)
The turbulent time scale τ is k ε τ = 1* β ω
(k −ε
models) (5)
(k −ω
models)
where β * is either a constant or a function of the turbulent Reynolds number Rw = k / νω (see Wilcox8). The general constitutive relationship between the Reynolds stresses and mean strain in incompressible flow was proposed by Pope9. Considering at most a cubic constitutive relationship, the Reynolds stresses can be written in the following canonical form:
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Y.J. Jang, L.Temmerman and M.A. Leschziner
1 3
1 3
a = -2Cµs + q1(s2- s2I) + q2(ws-sw) + q3(w2- w2I) 2 3
- γ 1 s 2 s - γ 2 w2 s - γ 3 (w2s + sw2- w2s - {wsw}I) - γ 4 (ws2-s2w) 2
(6)
2
Note that s2 = s / 2 , w2= - w / 2 where s = 2 s ij s ij , w = 2wij wij
(7)
are the dimensionless strain and vorticity invariants, respectively. The turbulence-transport equations (k and ε) used with above constitutive equation can be written as:
ν Dk = ∇ ⋅ [(ν + t )∇k ] + ( P − ε~ − D) σk Dt
(8)
ν ε~ Dε~ = ∇ ⋅ [(ν + t )∇ε~ ] + (Cε 1 f 1 P − Cε 2 f 2 ε~ ) + ( S l + S ε ) σε Dt k
(9)
where k2 ν t = Cµ f µ ~ ε
(10)
In the above, ε~ is the homogeneous dissipation rate which is related to the actual dissipation ∂k 1 / 2 2 ε through ε~ = ε − D , where D = 2ν ( ) . Additional source term S and S are used to l
∂x i
ε
control the growth of the turbulent length scale and provide the correct behaviour of the nearwall viscous sublayer, respectively. Models that use the turbulence vorticity ω make use of the following transport equations (Wilcox8):
ν Dk = ∇ ⋅ [(ν + t )∇k ] + ( P − β *ωk ) σk Dt
(11)
ν α Dω = ∇ ⋅ [(ν + t )∇ω ] + P − βω 2 σω νt Dt
(12)
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Y.J. Jang, L.Temmerman and M.A. Leschziner
where
νt = α*
k and α * is a constant or a function of the turbulent Reynolds number Rw. ω
The Gatski and Speziale1 model (GS) This is a quadratic, high-Re k-ε model, originally proposed by Gatski and Speziale1 and later “regularized” by Speziale and Xu10 to circumvent the singular behaviour observed in certain complex strain conditions. The model was derived from the equilibrium (zerotransport) form of the Reynolds-stress equations and then transformed into an explicit set of relations for the Reynolds stresses. In this model, therefore, γ1 to γ4=0. The remaining coefficients are functions of various powers (up to 6) of the strain and vorticity invariants. A seemingly minor but important feature of the model is that the separation between the εequation constants, Cε1 = 1.44, Cε2 = 1.83, is smaller than the standard, which favours separation in adverse pressure gradient. The Craft, Launder and Suga2 model (CLS) This is the first low-Re cubic k-ε model proposed, and it was designed with particular attention paid to the correct representation of near-wall viscous processes. Hence its constants are functions of the turbulent Reynolds number, in addition to strain and vorticity invariants. The functional dependence Cµ ( s , w ) was calibrated by reference to DNS data for homogeneously strained flow. There are two versions of this model, one using the k and ε equations and the other a third transport equation for the second invariant of the Reynoldsstress tensor, aijaij. Here, the former variant has been used, but this is known (eg. Loyau et al.11) not to give the correct separation between the normal stresses at the wall. It is significantly simpler, however, and widely used as a representative variant of the category of cubic models. An influential feature of the model is its use of the length-scale correction ε~ 2 k 3/ 2 ,0] , γ = ~ , cl=2.5 (13) S l = max[0.83(γ − 1)γ 2 k cl ε yn in the ε-equation. This term, going back to Yap12, is designed to depress the length scale returned by the ε-equation towards its equilibrium variation clyn. By doing so, it reduces the eddy viscosity near the wall and favours separation when the boundary layer is subjected to adverse pressure gradient. The Apsley and Leschziner3 model (AL) This cubic low-Re k-ε model was derived from a simplification of the algebraic Reynoldsstress model (Rodi13). Thus, the explicit cubic stress-strain/vorticity relationship was obtained through a two-step iterative substitution of the implicit algebraic relationships implied by the parent Reynolds-stress model. The model coefficients were then determined as function of y* ( = y k /ν ) by reference to DNS data for a number of near-wall flows, so as to ensure that the correct normal-stress separation is returned in the near-wall. Finally, functional corrections are
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Y.J. Jang, L.Temmerman and M.A. Leschziner
introduced to account for non-equilibrium conditions Pk /ε ≠1. In common with the model of Gatski & Speziale1, the present model also uses Cε1 = 1.44 and Cε2 = 1.83 in the ε-equation. As noted earlier, this enhances the tendency towards separation in the presence of adverse pressure gradient. The Wallin and Johansson4 model (WJ) This is a quadratic model that was also derived from an algebraic form of the Reynoldsstress equations. It differs from the models of Gatski & Speziale1 and Speziale & Xu10 in respect of the treatment of the production-to-dissipation ratio (Pk /ε). Wallin and Johansson have shown that the non-linear system of equations in the stresses can be solved in the form of a linear system of equations, complemented by a non-linear scalar equation for (Pk /ε), whereas other models have adopted Pk /ε (= -{as}) as a simplification, allowing the inversion of the implicit algebraic stress equations. In contrast to the previous models, that by Wallin & Johansson uses the k-ω system of equations, although a k-ε variant has also been proposed. 3.2 Reynolds-stress-transport model (RSTM) Although the primary objective of this study is to investigate non-linear eddy-viscosity models, one Reynolds-stress-transport model has been included, along with two linear eddyviscosity models, in an effort to identify the existence of any fundamental predictive differences between the different model categories. The model chosen is that of Speziale, Sarkar and Gatski14 (SSG). In its original form, the model is applicable to high Reynolds numbers only. Here, a low-Re extension by Chen et al.15 has been adopted to avoid uncertainties associated with wall functions in separated flow. The model solves equations of the form ∂ ∂ ∂ ( ρ ui u j ) + ( ρU k u i u j ) = d ijk + ρ ( Pij + Φ ij − ε ij ) ∂t ∂x k ∂x k
(14)
. Appropriate closure approximations must be provided for the pressure-strain term ( Φ ij ), diffusion term (d ijk ) and dissipation term (ε ij ) , while the production terms ( Pij ) and the advection terms are exact. Diffusion is approximated by the “generalized gradient diffusion hypothesis” (GGDH):
ρk ∂ (u i u j ) (15) u k ul ) ε ∂xl Dissipation is approximated by ε ij = 23 εδ ij . Closure then requires the solution of the εd ijk = ( µδ kl + C s
equation, which is essentially a variation on the equation used in k-ε eddy-viscosity models. The most important difference between second-moment-closure models lies in the approximation of the pressure-strain term ( Φ ij ) and the form of the transport equation for ε.
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Y.J. Jang, L.Temmerman and M.A. Leschziner
The pressure-strain correlation is typically decomposed into “slow” (return-to-isotropy), “rapid” and “wall-reflection” parts: Φ ij = Φ ij
(1)
+ Φ ij
( 2)
+ Φ ij
( w)
(16)
The original high-Re SSG14 model uses a quadratic pressure-strain model without wallreflection term and without sensitization to viscous effects via the turbulent Reynolds number. This version was then modified by Chen et al.15, so as to make it applicable to the near-wall region. This included the introduction of the damping function f w = exp[−(0.0184
ky 4 ) ] into ν
the pressure-strain model, the addition of a wall-related fragment to the pressure-strain model, also weighted by fw so as to make it vanish at the wall, and a form of the ε-equation made applicable to low-Re regions by the inclusion of fw and a second damping function fε = 1 −
2 exp[−( Rt / 6) 2 ] 9
in which Rt =
k2 νε
represents viscous effects.
Grid Independence study (X/H=2.0)
Grid Independence study (X/H=2.0) k - ε (Launder and Sharma, 1974)
3 k - ε (Launder and Sharma, 1974)
2.5 57*33 114*66 228*66 228*133
57*33 114*66 228*66 228*133
Y/H
2 1.5 1 0.5 0
-0.2
0
0.2
0.4 0.6 U/Ub
0.8
1
0
0.02
0.04 0.06 uu/U2b
0.08
0.1
Fig. 2: Grid-independence study – velocity and normal-stress profiles at x/h=2.0
4
NUMERICAL FRAMEWORK
The present RANS calculations were performed with a non-orthogonal, collocated, cellcentered finite-volume approach ‘STREAM’ (Lien and Leschziner16, Lien et al.17). Convection of mean flow as well as turbulence quantities is approximated by the ‘UMIST’ scheme (Lien and Leschziner18) - a second-order TVD approximation of QUICK scheme of Leonard19. Mass continuity is enforced indirectly by way of a pressure-correction algorithm. Within this scheme, the transport and the pressure-correction equations are solved sequentially and iterated to convergence. The computational domain employed a finite-volume mesh of 114 × 66, based on gridindependence studies, one result of which (with the LS model20) is shown in Fig. 2. To provide adequate resolution of the viscous sublayer and buffer layer adjacent to a solid wall, the minimum grid spacing in the near-wall region is maintained at 10-3 of hill height, which
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Y.J. Jang, L.Temmerman and M.A. Leschziner
corresponds to a wall coordinate y+ of order of 0.1 for low-Re calculation. For the high-Re GS model, y+ value was kept above 11. LES solution (Temmerman & Leschziner, 2001) (Separation : X/H = 0.22, Reattachment : X/H = 4.72)
Y/H
3
3
2
2
1
1 X/H
0 0
1
2
3
4
5
6
7
8
9
0
3
2
2
1
1 X/H
0 1
2
3
4
5
6
7
8
9
3
2
2
1
1 X/H
1
2
3
4
5
6
7
8
4
5
6
7
8
1
2
3
4
5
6
7
8
9
Quadratic k - ω (Wallin & Johansson, 2000) (WJ) (Separation : X/H = 0.22, Reattachment : X/H = 4.75)
X/H
0
9
9
X/H
0
3
0
3
Cubic k - ε (Apsley & Leschziner, 1998) (AL) (Separation : X/H = 0.26, Reattachment : X/H = 5.3)
Y/H
0
2
0
k - ω (Wilcox, 1994) (WX) (Separation : X/H = 0.22, Reattachment : X/H = 4.9)
Y/H
1
Y/H
3
0
X/H
0
k - ε (Launder & Sharma, 1974) (LS) (Separation : X/H = 0.35, Reattachment : X/H = 3.42)
Y/H
RSM (SSG+Chen, 2000) (Separation : X/H = 0.21, Reattachment : X/H = 4.85)
Y/H
0
1
2
3
4
5
6
7
8
9
Fig. 3: Streamfunction contours compared to LES
5
RANS-LES COMPARISONS
This section presents a conventional set of comparisons between model predictions and the LES data, which simply contrast the predictive performance of the various models examined. An overall view is conveyed in Fig. 3 by way of streamfunction contours for five models in comparison with the LES solution. It is already at this global level that some substantial differences between different models are observed. This sensitivity to modelling is due, in part, to the streamwise periodicity and the fact that the reattachment point depends sensitively on the separation point and the angle of the separation streamline at that point. The LES solution gives the reattachment point at x=4.7h. The linear k-ε returns, much as expected, a premature reattachment at x=3.4h, while the majority of the more elaborate models overestimate the recirculation length, providing a first indication that turbulent mixing in the separated shear layer is insufficient. This applies to both non-linear ε-based eddy-viscosity models (NLEVM) and the Reynolds-stress-transport model (RSTM). The latter gives, in
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Y.J. Jang, L.Temmerman and M.A. Leschziner
addition, an unrealistic bulge in the separation streamline close to the reattachment point - a feature often observed in second-moment calculations of separated flows and attributed to an erroneous representation of the near-wall variation of the stresses and/or length scale. The only models which come close to the LES solution are the linear Wilcox k-ω model and the quadratic Wallin-Johansson form of this model. Both give the correct reattachment point, but the latter gives a better representation of the shape of the recirculation zone and the velocity distribution within it. This observation suggests that the precise form of the equation governing the length-scale surrogate (ε,ω) is extremely influential, perhaps more so than the nature of the stress/strain-vorticity relationship. 3 (X/H=0.05)
(X/H=2.0)
(X/H=6.0)
(X/H=8.0)
2.5 LS AL CLS GS LES
Y/H
2
LS AL CLS GS LES
LS AL CLS GS LES
LS AL CLS GS LES
1.5 1
0.5 0
-0.2
0
0.2
0.4 0.6 U/Ub
0.8
1
-0.2
0
0.2
0.4 0.6 U/Ub
0.8
1
-0.2
0
0.2
0.4 0.6 U/Ub
0.8
1
-0.2
0
0.2
0.4 0.6 U/Ub
0.8
1
0.8
1
3 (X/H=0.05)
(X/H=2.0)
(X/H=6.0)
(X/H=8.0)
2.5 WX WJ SSG(Chen) LES
WX WJ SSG(Chen) LES
Y/H
2
WX WJ SSG(Chen) LES
WX WJ SSG(Chen) LES
1.5 1
0.5 0
-0.2
0
0.2
0.4 0.6 U/Ub
0.8
1
-0.2
0
0.2
0.4 0.6 U/Ub
0.8
1
-0.2
0
0.2
0.4 0.6 U/Ub
0.8
1
-0.2
0
0.2
0.4 0.6 U/Ub
Fig. 4: Mean-velocity distributions
A more differentiated view of the mean-flow field is offered by Fig. 4 which shows velocity profiles at four streamwise stations: x=0.05h, 2h, 6h and 8h – respectively, just downstream of the ‘inlet’, in the middle of the recirculation zone, in the post-reattachment recovery zone and in the accelerating windward side of the hill. The first station is of considerable interest here because it indicates the degree to which the ‘inlet’ conditions are affected by the inner solution. Normally, the flow at this position is ‘fixed’ by prescribed inlet conditions. Here, however, these conditions ‘float’ and depend on the quality of the solution between the inlet and outlet planes. The comparisons in Fig. 4 demonstrate that the excessive recirculation length returned by ε-based NLEVMs and the RSTM, observed in Fig. 3, does indeed seem to be linked to insufficient turbulent diffusion in the separated shear layer – signified by excessive shear strain. This leads to a delayed reattachment. In addition, there are clear indications that the rate of post-reattachment recovery predicted by the ε-based
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Y.J. Jang, L.Temmerman and M.A. Leschziner
NLEVMs and the second-moment model is too low, which further increases the near-wall momentum deficit caused by the late reattachment. This last defect is displayed especially clearly by the linear k-ε model which, despite returning a seriously premature reattachment, gives just about the correct near-wall profile at x=6h. The consistency of this defect across all ε-based model forms provides a pointer to the ε-equation being the source. It is tempting to conclude that the momentum deficit in the recovery region is responsible for the excessively slow and thick boundary layer close to the hill crest at x=0.05h. This may, in turn, be sensibly argued to lead to a misrepresentation of the separated shear layer, which then causes the aforementioned defects in the recirculation zone and the field further downstream. However, this cause-and-effect argument is not given support by solutions of the ω-based models which yield remarkably better solutions than the ε-based models, with the quadratic Wallin-Johansson form performing especially well. Both also return a broadly correct rate of recovery, yet the behaviour at x=0.05h is not improved. As will be argued later, the near-wall behaviour at this position appears to depend greatly on the response of the turbulence to the straining associated with the strong acceleration upstream of this location. 3
(X/H=0.05)
(X/H=2.0)
LS AL CLS GS LES
2.5
(X/H=8.0)
LS AL CLS GS LES
LS AL CLS GS LES
Y/H
2
(X/H=6.0)
LS AL CLS GS LES
1.5 1
0.5 0
3
-0.04
-0.02 uv/U2b
0
(X/H=0.05)
0.02 -0.04
0
0.02 -0.04
(X/H=2.0)
WX WJ SSG(Chen) LES
2.5
-0.02 uv/U2b
-0.02 uv/U2b
0
(X/H=6.0)
WX WJ SSG(Chen) LES
0.02 -0.04
-0.02 uv/U2b
0
0.02
(X/H=8.0)
WX WJ SSG(Chen) LES
WX WJ SSG(Chen) LES
Y/H
2 1.5 1 0.5 0
-0.04
-0.02 uv/U2b
0
0.02 -0.04
-0.02 uv/U2b
0
0.02 -0.04
-0.02 uv/U2b
0
0.02 -0.04
-0.02 uv/U2b
0
0.02
Fig. 5. Shear stress distributions
The mean-flow behaviour is evidently most strongly linked to the shear-stress level, and Fig. 5 provides related comparisons. The profiles at x=2h confirm the earlier supposition that most models underestimate the mixing in the separated shear layer: those models predicting the lowest level tend also to return the longest recirculation zones. The ε-based NLEVMs as well as the RSTM give far too low levels of shear stress within the recirculation zone, especially in the near-wall reverse-flow layer, and this also applies to streamwise traverses
11
Y.J. Jang, L.Temmerman and M.A. Leschziner
within that zone. The ω-based models also underestimate the shear stress in the separated shear layer, but to lesser extent, especially in the near-wall region. The Wallin-Johansson model gives the closest agreement with the LES data and hence also the best representation of the mean flow. All models return reasonable levels of shear stress after reattachment, although the ε-based forms return the maximum too close to the wall, reflecting defects in the upstream separated shear layer - in particular, the insufficient level of turbulence therein. Beyond the above defects, neither the comparisons at x=6h nor those further downstream (before the next hill is reached) provide obvious clues as to why the rate of flow recovery of εbased models is too low. It is interesting to examine the shear-stress behaviour at x=8h, a location in which the flow is strongly accelerating on the wind-ward side of the hill. Here, all models but the secondmoment closure and, to some degree, the quadratic Gatski-Speziale model, give far too high levels of shear stress, and it is the this defect which appears primarily responsible for the misrepresentation of the mean-velocity profile close to the wall as x=0.05h. Thus, only the RSTM returns a velocity peak near the wall at x=0.05h. The large acceleration around x=8h is accompanied by significant stress transport. In effect, the stress field tends to be ‘frozen’. In contrast, the steep, erroneous increase predicted by the eddy-viscosity models reflects the absence of stress transport in the models and the overly rigid linkage they provide between stress and strain. The wrong response during the acceleration phase gives then rise to wrong shear-stress distributions at x=0.05h. Again, the only exception is the RSTM which accounts for stress transport. It is finally of interest to compare the ability of the models to return the anisotropy in the normal stresses. This is done in Fig. 6. Here, the linear models (LS, WX) may be ignored, for these are intrinsically unable to represent anisotropy. At x=2h – a representative location within the separated shear layer, the LES solution gives u 2 / v 2 ratios of order 1.7. For the NLEVMs, the separation between the normal stresses is dictated, primarily, by the quadratic fragments in the stress/strain-vorticity relations. As can be seen, all indeed return a significant level of anisotropy, but quantitative agreement is generally poor. The predicted u 2 / v 2 ratio is the range 1.4-3.5, with all models returning far too low wall-normal stress and also too low streamwise stress. In contrast to earlier comparisons, the performance of Wallin & Johansson’s model is here unremarkable. In fact, it gives especially low levels of wall-normal intensity in the middle of the flow. Consistently, the intensity the model returns in the acceleration region is broadly correct, whereas other models tend to give excessive level. The above defects combine to an insufficient level of turbulence energy, which is consistent with the low level of shear stress in the separated shear layer. Some of the most serious errors occur in the near-wall reverse-flow region. The message sent out by these comparisons is, therefore, that some fundamental element(s) of all models is (are) seriously wrong, especially in the ε-based variants. The RSTM fares better in some respects, especially in the acceleration region, but it also gives insufficient shear stress and wall-normal stress in the shear layer, and hence delayed reattachment.
12
Y.J. Jang, L.Temmerman and M.A. Leschziner
3
(X/H=0.05)
(X/H=2.0)
LS AL CLS GS LES
2.5
LS AL CLS GS LES
LS AL CLS GS LES
Y/H
2
(X/H=8.0)
(X/H=6.0)
LS AL CLS GS LES
1.5 1 0.5 0 0
0.02
3
0.04 0.06 uu/U2b
0.08
0
0.02
(X/H=0.05)
0.04 0.06 uu/U2b
0
0.02
(X/H=2.0)
WX WJ SSG(Chen) LES
2.5
0.08
0.04 0.06 uu/U2b
0.08
0
0.02
0.04 0.06 uu/U2b
0.1
(X/H=8.0)
(X/H=6.0) WX WJ SSG(Chen) LES
WX WJ SSG(Chen) LES
0.08
WX WJ SSG(Chen) LES
Y/H
2 1.5 1 0.5 0 0
0.02
3
0.04 0.06 uu/U2b
0.08
0
0.02
(X/H=0.05)
0.04 0.06 uu/U2b
0.02
0.04 0.06 uu/U2b
0.08
0
0.02
(X/H=6.0)
LS AL CLS GS LES
0.04 0.06 uu/U2b
0.08
0.1
(X/H=8.0)
LS AL CLS GS LES
LS AL CLS GS LES
Y/H
2
0
(X/H=2.0)
LS AL CLS GS LES
2.5
0.08
1.5 1 0.5 0 0
0.02
3
0.04 0.06 vv/U2b
0.08
0
0.02
(X/H=0.05)
0.04 0.06 vv/U2b
0
0.02
(X/H=2.0)
WX WJ SSG(Chen) LES
2.5
0.08
0.04 0.06 vv/U2b
0.08
0
0.02
(X/H=6.0)
WX WJ SSG(Chen) LES
0.04 0.06 vv/U2b
0.08
0.1
(X/H=8.0)
WX WJ SSG(Chen) LES
WX WJ SSG(Chen) LES
Y/H
2 1.5 1 0.5 0 0
0.02
0.04 0.06 vv/U2b
0.08
0
0.02
0.04 0.06 vv/U2b
0.08
0
0.02
0.04 0.06 vv/U2b
0.08
0
0.02
0.04 0.06 vv/U2b
0.08
0.1
Fig. 6: Normal-stress distributions
6
A-PRIORI ANALYSIS
The outstanding advantage of well-resolved simulation data, in contrast to experimental data, is that the former essentially offer a complete description of the flow evolution and can
13
Y.J. Jang, L.Temmerman and M.A. Leschziner
therefore be post-processed to yield whole-field distributions of most statistical correlations of interest. Such distributions may then be used for calibrating and/or a-priori testing of model fragments. In the case of LES data, some limitations are imposed by the imperfect resolution. This applies, in particular, to correlations of small-scale fluctuations, notably dissipation rate. When attention focuses on Reynolds-stress models, all terms but stress dissipation in the exact second-moment-transport equations are dominated by large-scale turbulent motion and can be extracted with fair accuracy. The dissipation can then be obtained as an imbalance of all other terms contributing to the budget. The above range and detail of data allow the realism of closure proposals to be assessed in some depth. In the case of non-linear eddy-viscosity models, the stresses are linked to the mean-velocity field by the constitutive equations (6), and the scope for a-priori testing is more limited. What can be done, however, is to insert the LES mean-flow fields into (6), as well as into the associated transport equations, and then examine the stresses, the fragments of (6), the turbulence energy and (less reliably) the length-scale surrogate that are returned by the model in comparison with the corresponding LES data. Clearly, differences between corresponding data for any one quantity are responsible for driving the modelled mean flow (that which arises from a full RANS solution) away from the simulated solution. Model improvements must strive to minimize the differences, and this is the ultimate purpose of performing the apriori analysis. The distributions for the stresses, the fragments of (6), etc. that arise from the full RANS solutions can also be compared with the corresponding data arising from the apriori evaluation, and this can give additional information of which model elements respond especially sensitively to changes in the mean flow, thus identifying these elements as requiring special attention. The above process is conveyed in Figs. 7-10 for four models, respectively: the linear LS and WX models and the cubic AL and quadratic WJ models. Figs. 7 and 8 give, on the left, distributions of 2/3k, the deviatoric component of the streamwise normal stress (L≡) u 2 − 23 k and the total stress u 2 for the locations x=2h (upper row) and 6h (lower row). For each variable, the symbols denote the a-priori analysis, while the corresponding lines come from the complete (a-posteriori) RANS calculation. The adjacent central plots give, again, a-priori and a-posteriori results for the deviatoric normal-stress component, this time compared with the actual LES solution (the line formed with the symbol L). Finally, the right-hand-side plots show corresponding distributions for the shear stress. The linear WX and LS models display similar features, but also show important differences, the former doing considerably better. The fact that the a-priori distributions for the WX model are close to those from the complete (a-posteriori) RANS computations is consistent with the observation, Fig. 3, that the model predicts a mean flow which is close to that arising from the LES simulation. This applies, in particular, to the post-reattachment zone. In the separated shear layer, the model gives a shear stress which is well below the LES level, despite the closeness of the a-priori and a-posteriori distributions. This raises the intriguing question as to whether the shear-stress returned by the LES in the separated shear layer is excessive, perhaps due to the presence of a periodic (flapping-like) flow component. What is unambiguously wrong with both linear stress-strain models is the representation of the
14
Y.J. Jang, L.Temmerman and M.A. Leschziner
Symbols : a-priori analysis Lines : LS model
3 2.5
Y/H
2 1.5 1 0.5 0
(X/H=2.0) L* L L* L* L* * L * 〉 2/3k L * * L * L * L * 〉 L L * L * L * L ** 〉 uu(=2/3k+L) L L ** L L L uu-2/3k (actual ** L L * L ** L L * L ** L L L ** L ** L L ** L L L ** L L ** L * L * L L ** L L ** L L **
LES)
L L L L L L L L L L L L L L L L L L L L L L L
L L
0 0.02 0.04 0.06 0.08 0.1 2/3k, mean strain and normal stress
L
(X/H=2.0)
(X/H=2.0) LS model a-priori analysis uu-2/3k (actual LES)
L
L
L
L
L
L
L
L
L
L
L
L L
L
L
L L
L
0 0.01 0.02 (L=)Linear term of uu/U2b
L
L L L L
L L L L L L L L L L L L L L L L L L L L L L L
LS model a-priori analysis Actual LES
L
L
L
L
L
L
L
L
L
L
L
L
L
L
L
-0.04 -0.03 -0.02 -0.01 uv/U2b
L
0
0.01
0.02
0.01
0.02
0.01
0.02
0.01
0.02
Symbols : a-priori analysis Lines : LS model
3 2.5
Y/H
2 1.5 1 0.5 0
(X/H=6.0) *L* L *L L L* * L 〉 2/3k L* * L* L* L* 〉 L L ** L* L* L* 〉 uu(=2/3k+L) L* L* L* L* L uu-2/3k (actula LES) L* L* L* L* L* L* L* L* L * L L * L ** L ** L L * L L ** L * L L ** L * L L ** L L **
0 0.02 0.04 0.06 0.08 0.1 2/3k, mean strain and normal stress
L
L L L L L L L L L L L L L L L L L L L
L
L
L
LS model a-priori analysis uu-2/3k (actual LES)
L
L
L
L
L
L
L
L
L
L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L
(X/H=6.0)
(X/H=6.0)
LS model a-priori analysis Actual LES
L
L L L L L L L
L L L L L
0 0.01 0.02 (L=)Linear term of uu/U2b
-0.04 -0.03 -0.02 -0.01 uv/U2b
0
Fig. 7: A-priori analysis of Launder & Sharma model. Symbols : a-priori analysis Lines : WX model
3 2.5
Y/H
2 1.5 1 0.5 0
(X/H=2.0) L* L L* L* L* L ** 〉 2/3k L * * L * L * L * 〉 L L * L * L ** L 〉 uu(=2/3k+L) L ** L L ** L L uu-2/3k (actual L ** L L ** L L ** L L ** L L ** L L ** L L L ** L L ** L * L * L * L L ** L L ** L *L *
LES)
0 0.02 0.04 0.06 0.08 0.1 2/3k, mean strain and normal stress
L L L L L L L L L L L L L L L L L L L L L L L
L L
L
(X/H=2.0)
(X/H=2.0) WX model a-priori analysis uu-2/3k (actual LES)
L
L
L
L
L
L
L
L
L
L
L
L
L L
L
L L
L
0 0.01 0.02 (L=)Linear term of uu/U2b
L
L L L L
L L L L L L L L L L L L L L L L L L L L L L L
WX model a-priori analysis Actual LES
L
L
L
L L
L
L
L
L
L
L
L
L
-0.04 -0.03 -0.02 -0.01 uv/U2b
L
L
L
0
Symbols : a-priori analysis Lines : WX model
3 2.5
Y/H
2 1.5 1 0.5 0
L *
(X/H=6.0)
L L* L* L* L* L* * L** L* L* L * L * L * L * L * L * L * L * L L * L * L * L * L * L L ** L ** L L ** L L ** L L * L L ** L * L L ** L * L * L L ** *LL *
L
〉 2/3k 〉 L 〉 uu(=2/3k+L) uu-2/3k (actula LES)
0 0.02 0.04 0.06 0.08 0.1 2/3k, mean strain and normal stress
L L L L L L L L L L L L L L L L L L L
L L L L L L
L
L
WX model a-priori analysis uu-2/3k (actual LES)
L
L
L
L
L
L
L
L
L
L
WX model a-priori analysis Actual LES
L L L L L L L
0 0.01 0.02 (L=)Linear term of uu/U2b
-0.04 -0.03 -0.02 -0.01 uv/U2b
Fig. 8: A-priori analysis of Wilcox model
15
L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L
(X/H=6.0)
(X/H=6.0)
0
Y.J. Jang, L.Temmerman and M.A. Leschziner
Symbols : a-priori analysis Lines : AL model
3 2.5
Y/H
2 1.5 1 0.5 0
L* L L * L * L * L * L * * L * L ** L ** L L ** L L ** L L ** L L L ** L L ** L L ** L L ** L L ** L L ** L L L ** L L ** L * L * L L ** L L ** L * L * L L **
0
(X/H=2.0)
(X/H=2.0)
(X/H=2.0)
(X/H=2.0)
〉 2/3k
AL model a-priori analysis
〉 L+Q+C
AL model a-priori analysis
AL model a-priori analysis
〉 uu(=2/3k+L+Q+C) uu-2/3k(actual LES)
0.02 0.04 0.06 0.08 0.1 2/3k, mean strain and normal stress
0 0.01 0.02 (L=)Linear term of uu/U2b
0.03
0 0.01 0.02 (Q=)Quadratic term of uu/U2b
0.03
0 0.01 0.02 (C=)Cubic term of uu/U2b
0.03
Symbols : a-priori analysis Lines : AL model
2.5
Y/H
2 1.5 1 0.5 0
0
L L* L * L * L * L * L * L * L * L * L * L ** L * L * L * L * L * L * L * L* L* L* L* L* L* L* L L * L * L ** L * L L ** L L ** L * L L ** L L ** L L **
*
(X/H=2.0)
L
Y/H
0.5
LL
LL
0
-0.04 -0.03 -0.02 -0.01 uv/U2b
2.5
2
1.5
1
0.5
0
LL
LL
0
L
0.01
0.02
0
-0.03 -0.02 -0.01 0 Linear term of uv/U2b
0.02
-0.03 -0.02 -0.01 0 Linear term of uv/U2b
0.03
0.01
0.02
-0.03 -0.02 -0.01 0 0.01 Quadratic term of uv/U2b
0.02
-0.03 -0.02 -0.01 0 0.01 Quadratic term of uv/U2b
0.01
0.02
0.01
0.02
AL model a-priori analysis
0.02
Fig. 9: A-priori analysis of Apsley and Leschziner model
16
-0.03 -0.02 -0.01 0 Cubic term of uv/U2b (X/H=6.0)
AL model a-priori analysis
0.02
0.03
AL model a-priori analysis
(X/H=6.0)
0.01
0 0.01 0.02 (C=)Cubic term of uu/U2b
(X/H=2.0)
AL model a-priori analysis
AL model a-priori analysis
0.01
0 0.01 0.02 (Q=)Quadratic term of uu/U2b
(X/H=2.0)
(X/H=6.0)
AL model a-priori analysis Actual LES
-0.04 -0.03 -0.02 -0.01 uv/U2b
0.03
AL model a-priori analysis
(X/H=6.0)
L L L LL LL LL L L LL LL LL LL L L LL LL LL LL L L LL LL LL LL L LL LL L L L LL LL LL LL LL LL LL LL LL
0 0.01 0.02 (L=)Linear term of uu/U2b
(X/H=2.0)
L L L LL LL L L L L LL LL LL LL L LL LL L L L L L L L LL LL LL
L L L L L LL L LL LL LL LL LL LL
3
AL model a-priori analysis
AL model a-priori analysis
〉 uu(=2/3k+L+Q+C) uu-2/3k(actual LES)
AL model a-priori analysis Actual LES
1.5
(X/H=6.0)
(X/H=6.0) AL model a-priori analysis
〉 L+Q+C
L
2
1
(X/H=6.0)
〉 2/3k
0.02 0.04 0.06 0.08 0.1 2/3k, mean strain and normal stress
3
2.5
(X/H=6.0)
L*
3
-0.03 -0.02 -0.01 0 Cubic term of uv/U2b
Y.J. Jang, L.Temmerman and M.A. Leschziner
Symbols : a-priori analysis Lines : WJ model
3 2.5
Y/H
2 1.5 1 0.5 0
(X/H=2.0)
L* L L * L* L* L* * L* * L* L* L* L * L * L * L * L * L * L ** L L L ** L L ** L L ** L L ** L L ** L L * L L ** L L ** L * L L ** L L ** L L ** L L ** *L
0
(X/H=2.0) (X/H=2.0)
〉 2/3k
WJ model a-priori analysis
〉 L+Q
WJ model a-priori analysis
〉 uu(=2/3k+L+Q) uu-2/3k (actual LES)
0.02 0.04 0.06 0.08 0.1 2/3k, mean strain and normal stress
0 0.01 0.02 (L=)Linear term of uu/U2b
0.03
0 0.01 0.02 (Q=)Quadratic term of uu/U2b
0.03
Symbols : a-priori analysis Lines : WJ model
2.5
Y/H
2 1.5 1 0.5 0
0
L L* L* L* L** L* L* L* L* L* L* L* L* L* L* L* L* L* L* L* L* L* L* L* L* L * L L * L ** L * L L ** L L ** L * L * L L ** L L ** *LL *
(X/H=2.0)
Y/H
L L L L
0.5
L
L
L
L
L
L
L
L
L
L
L
L
L
0
-0.04 -0.03 -0.02 -0.01 uv/U2b
Y/H
1.5
1
0.5
0
L
L
0
L
L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L
0.01
0.02
0
0.02
0.03
-0.03 -0.02 -0.01 0 Linear term of uv/U2b
WJ model a-priori analysis
0.01
0.02
-0.03 -0.02 -0.01 0 0.01 Quadratic term of uv/U2b
0.02
(X/H=6.0)
WJ model a-priori analysis
0.01
0 0.01 0.02 (Q=)Quadratic term of uu/U2b
(X/H=2.0)
(X/H=6.0)
WJ model a-priori analysis Actual LES
-0.04 -0.03 -0.02 -0.01 uv/U2b
0.03
WJ model a-priori analysis
(X/H=6.0)
3
2
L
0 0.01 0.02 (L=)Linear term of uu/U2b (X/H=2.0)
L L L L L L L L L L L L L L L L L L L L L L L
WJ model a-priori analysis Actual LES
L
WJ model a-priori analysis
〉 uu(=2/3k+L+Q) uu-2/3k (actual LES)
L
1.5
2.5
(X/H=6.0) WJ model a-priori analysis
〉 L+Q
2
1
(X/H=6.0)
〉 2/3k
*
0.02 0.04 0.06 0.08 0.1 2/3k, mean strain and normal stress
3
2.5
(X/H=6.0)
L *
3
-0.03 -0.02 -0.01 0 Linear term of uv/U2b
WJ model a-priori analysis
0.01
0.02
-0.03 -0.02 -0.01 0 0.01 Quadratic term of uv/U2b
Fig. 10: A-priori analysis of Wallin & Johansson model.
17
0.02
Y.J. Jang, L.Temmerman and M.A. Leschziner
normal stress. Indeed, the predicted deviatoric stress component even has the wrong sign. Hence, in any flow or region in which the normal stresses are dynamically influential, this model is bound to be erroneous. The LS model not only gives a similarly wrong behaviour for the normal stress, but also returns an inaccurate mean flow and considerably larger errors in the shear stress, as indicated by the large differences between the a-priori and a-posteriori distributions. Because the LES mean flow is incompatible with the turbulence field returned by the model operating in a-priori mode, the coupled RANS solution elevates the turbulence energy and shear stress, thus leading to a more diffusive behaviour, with a shorter recirculation zone and smoother gradients. This worse performance appears to be linked to the low a-priori shear stress and especially the turbulence energy this model gives, but the precise cause-andeffect relationship is unclear. The tentative message thus emerging from this first set of comparisons for the linear models is that the form of the length-scale-governing equation is more influential than the ability of the models to return the correct normal stresses. Comparisons corresponding to the above for the cubic AL and quadratic WJ models are shown in Figs. 9 and 10, respectively. These are organized in a similar manner to Figs. 7 and 8, but are more elaborate because of the need to examine different fragments of the stressstrain relationships. First, the left-most plots on the upper two rows include distributions for “L+Q” (or “L+Q+C”) ≡ u 2 − 23 k . Next, the 2 (or 3) right-hand-side plots show, separately, the contributions of the linear, quadratic and (if appropriate) cubic terms. An analogous representation is adopted for the shear stress on the lower half of Figs. 9 and 10. Again, the curves identified by L represent the LES solution. Both figures show, first, that the quadratic terms contribute greatly to the normal-stress levels. In fact, these fragments are primarily responsible for the fairly close agreement between the distributions of u 2 − 23 k predicted by the models and returned by the LES. Although the quadratic contribution in the AL model appears excessive, the normal-stress comparisons shown in Fig. 6 at x/h=2 and 6 do not support this supposition. Second, Fig. 9 shows that the cubic contributions are minor (in fact, the CLS model, not included, gives even lower cubic contributions). Third, the mean-flow field predicted by the AL model differs significantly from the LES solution (see Fig. 3), and this is the source of significant differences between a-priori and a-posteriori distributions. This is, most likely, a consequence of the far too low shear stress predicted by the model in the shear layer, possibly because the cubic term is too influential (it is of opposite sign to the linear term). The quadratic WJ model performs better, giving a mean flow that agrees closely with the LES solution and also satisfactory normal and shear stresses. Hence, one important lesson arising from the above comparisons is that the cubic terms, while perhaps important to include from a fundamental point of view, are not evidently helpful in the present flow. The ω-based formulation provides a superior representation to that returned by ε-based models of the shear stress and mean flow, while the quadratic WJ form adds a realistic (though by no means perfect) representation of normal-stress anisotropy. 7
CONCLUSIONS
An investigation has been presented of the performance of several turbulence models when applied to a separated flow in a periodically-constricted channel for which highly-resolved
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Y.J. Jang, L.Temmerman and M.A. Leschziner
LES data are available. The emphasis has been on the performance of non-linear eddyviscosity models, in contrast to linear and second-moment-closure forms. The initial (“a-posteriori”) comparisons of model predictions with the LES data illustrated the high level of sensitivity of the solutions to modelling details. The expected poor performance of the linear k-ε model was confirmed, while the k-ω form returned surprisingly good results, pointing to the outstanding importance of the length-scale-governing equation and to the relatively weak dynamic role played by the normal stresses (which are seriously in error with any linear eddy-viscosity model). The non-linear ε-based models gave, as did second-moment closure, insufficient mixing in the separated region, delayed reattachment and poor post-reattachment recovery characteristics. The only formulation found to do well was the quadratic ω-based model of Wallin & Johansson4, although even this gives some poor results for the normal-stress anisotropy in the separated shear layer. The a-priori analysis examined, for the eddy-viscosity models, the role of the linear and (if appropriate) quadratic and cubic contributions in the constitutive stress-strain/vorticity relations. The analysis highlighted the inapplicability of the linear formulation in respect of representing the normal stresses. Thus, the deviatoric part of streamwise normal stress is even predicted to have the wrong sign in the recirculation zone. The analysis also showed that the quadratic fragments are crucially important for a realistic representation of the normal stresses, while the cubic terms play a sub-ordinate role. In the case of the shear stress, both non-linear fragments are relatively small, and the linear term dominates. Thus, if a model is calibrated to give the correct shear stress, it will return a broadly correct mean flow. A defect of the ε-based cubic models, especially the variant of Apsley & Leschziner3 is that the cubic fragment appears to be too influential. The Wallin-Johansson model lacks the cubic contribution, and its quadratic fragment appears to be well constructed. This, aided by the favourable characteristics of the ω-equation, gives this model a decisive edge over other forms, at least for the flow investigated. The model returns closely similar a-priori and a-posteriori distributions of stresses and turbulence energy, which is consistent with the fidelity of the velocity field predicted by the model with the corresponding LES solution. ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial support provided by EPSRC and BAE Systems and the promotion of the research through the MSTAR DARP, especially by DERA. The LES data used in the paper were generated as part of Framework IV BRITE/EURAM project LESFOIL, supported by the CEC. REFERENCES [1] T.B. Gatski and C.G. Speziale, “On explicit algebraic stress models for complex turbulent flows”, Journal of Fluid Mech., 254, 59-78 (1993). [2] T.J. Craft, B.E. Launder and K. Suga, “Development and application of a cubic eddyviscosity model of turbulence”, Int. J. Num. Meth. In Fluids,17, 108-115 (1996). [3] D.D. Apsley and M.A. Leschziner, “A new low-Reynolds-number nonlinear two- equation turbulence model for complex flows”, Int. J. Heat and Fluid Flow, 19, 209-222 (1998).
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