Towards an Accurate and Robust Algebraic Structure Based Model

Center for Turbulence Research Proceedings of the Summer Program 22 283 Towards an Accurate and Robust Algebraic Structure Based Model By R. Pecnik, J. P. O. Sullivan, K. Panagiotou, H. Radhakrishnan, S. Kassinos, K. Duraisamy AND G. Iaccarino The algebraic structure-based turbulence model (ASBM) incorporates information about the structure of turbulence to provide closure to the Reynolds-averaged Navier- Stokes equations. In the past, the model has been applied to several well documented Computational Fluid Dynamic (CFD) benchmark test cases, showing reasonable improvement compared to standard turbulence models, but obtaining converged solutions is not always straightforward. We therefore revisit the functional form of the ASBM and evaluate the model over a range of mean deformations of homogeneous turbulence and compare the result with reference solutions using the Particle Representation Model (PRM). The performance of the model is assessed in detail and an improved ASBM is proposed. For a test case involving separated flow over a wall mounted hump the new modifications are noted to yield better agreement when compared to experimental measurements.. Introduction Large-scale energy-containing eddies tend to organize spatially the fluctuating motion in their vicinity, and in so doing, eliminate gradients of fluctuation fields in some directions (those in which the spatial extent of the structure is significant) and to enhance gradients in other directions (those in which the extent of the structure is small). Thus, associated with each eddy are local axes of dependence and independence that determine the dimensionality of the local turbulence (Reynolds & Kassinos 995; Kassinos et al. 2). In non-equilibrium flows (strong rotation, separation and reattachment, etc.), the dimensionality of the turbulence structures dictates how the turbulence will respond to external deformation. Thus, the anisotropy of the turbulent stresses is to a large extent determined by dimensionality of the local turbulent structure, i.e. by the morphology of the turbulence eddies. This realization led to the development of the structure-based approach to turbulence modeling (Reynolds & Kassinos 995; Kassinos et al. 2), and the engineering outcome of this effort is the Algebraic Structure Based Model (ASBM). The ASBM (Langer & Reynolds 23; Kassinos & Langer 25; Kalitzin et al. 24; Kassinos et al. 26; Radhakrishnan et al. 28; Aupoix et al. 29; O Sullivan et al. 2) has the potential to lead to significant improvements in the performance of CFD codes used in aerospace design (Benton 2) and offers hope for a step change in turbulence modeling because it provides a new route to turbulence closure that is anchored on the relation between the dimensionality of the turbulent eddies and the anisotropy of the turbulent stresses. The objective of this work is to critically analyze the ASBM and identify areas where Delft University of Technology, The Netherlands University of Auckland, Auckland, New Zealand University of Cyprus, Nicosia, Cyprus

284 R. Pecnik et al. the modeling accuracy and robustness must be improved, especially with regard to application in complex flows. The analysis was carried out using a number of new techniques. A parameter space map was developed that allows the state of the turbulence to be clearly visualized with respect to the rapid distortion theory (RDT) limit and isotropic turbulence. This enabled a careful analysis of the ASBM s representation of flows transitioning from a locally hyperbolic regime to an elliptic one and vice-versa. This helped to identify inconsistencies in the parameter blending in the ASBM. Finally a number of reference solutions of homogeneous turbulence using the Particle Representation Model (PRM) (Kassinos & Reynolds 996) and Interactive Particle Representation Model (iprm) (Kassinos & Akylas 22) were carried out to obtain exact solutions for the Reynolds stress tensor for various states of turbulence. These solutions were then compared with the predictions of the ASBM and areas of improvement identified. Preliminary tests on a configuration involving separated flow over a wall mounted hump are then attempted. 2. Algebraic Structure Based Model In conventional RANS closures, the Reynolds stress tensor components are either derived from gradients of the mean velocity field using the eddy-viscosity concept or using partial differential or algebraic equations for the Reynolds stress tensors. In the ASBM, the Reynolds stress tensor is calculated in terms of one-point turbulence structure tensors that sensitize the model not only to the anisotropy of the turbulence componentality, but also to the structural anisotropy. 2.. Structure tensors The turbulence tensors, as defined in Kassinos & Reynolds (994) and Kassinos et al. (2), are R ij, the Reynolds stress tensor, D ij, the dimensionality structure tensor, and F ij, the circulicity structure tensor. D ij and F ij contain information about the largescale, energy-bearing structures that is not conveyed by R ij alone. In the case of homogeneous turbulence, the contractions of the structure tensors are all twice the turbulence kinetic energy, i.e. R ii = D ii = F ii = q 2 = 2k. Thus, normalized structure tensors can be defined as r ij R ij /q 2, d ij D ij /q 2, f ij F ij /q 2, with q 2 being twice the turbulent kinetic energy. An exact constitutive relation between the three normalized tensors can be derived and it shows that in homogeneous turbulence only two of them are linearly independent: r ij + d ij + f ij = δ ij. (2.) The eddy-axis concept (Kassinos & Reynolds 994) is used to relate the Reynolds stress r ij and structure tensors to parameters of a hypothetical turbulent eddy field. Each eddy represents a two-dimensional turbulence field, and is characterized by an eddy-axis vector a i. The turbulent motion and its energy content along this axis can be decomposed into a jettal component along the eddy axis and a vortical component perpendicular to the eddy axis. Additionally, if the motion around the eddy axis is not axisymmetric the eddy is flattened, as it is in the case of turbulence subjected to rotation. Averaging over an ensemble of such eddies gives statistical quantities representative of the eddy field. The eddy axis tensor a ij = q V 2 a 2 i a j (where V is the velocity) is the energy-weighted average direction-cosine of the ensemble that characterizes its structure. Exact constitutive equations are then derived to relate the Reynolds stresses r ij and the

eddy axis tensor a ij as follows: Algebraic structure based model 285 r ij = u i u j 2k = ( φ) 2 (δ ij a ij ) + φa ij + ( φ)χ[ 2 ( a nmb mn )δ ij 2 ( + a nmb mn )a ij b ij + a in b nj + a jn b ni ] (2.2) γωt k Ω T (ǫ ipra pj + ǫ jpr a pi ){ 2 [ χ( a nmb mn )]δ kr + χb kr χa kn b nr }, with δ ij the Kronecker delta, ǫ ijk the Levi-Civita tensor and Ω T the total rotation vector (sum of mean rotation and frame rotation). φ, χ, γ are three additional structure parameters and b ij is the flattening tensor (normalized direction-cosine of flattening vector). The flattening tensor is modeled in terms of mean rotation Ω i and frame rotation vector Ω f i as, b ij = (Ω i + C b Ω f i )(Ω j + C b Ω f j ) (Ω k + C b Ω f k )(Ω k + C b Ω f k ), C b =.. (2.3) The three structure parameters φ, χ, γ are discussed in more detail below. 2.2. Structure parameters Modeling the structure parameters φ, χ and γ is a crucial part in the construction of the model. The eddy jetting parameter φ represents the amount of energy in the jetal mode, with φ = purely vortical and φ = purely jettal eddies, as in the case for turbulence subjected to mean shear in the limit of RDT. The eddy helix vector γ correlates the jettal and the vortical mode and is the key parameter in setting the shear stress level in the turbulent field. The eddy flattening parameter, together with the direction-cosine of the flattening vector, defines the deformation of the eddy due to rotation. These three parameters are defined in terms of the ratio of mean rotation to mean strain, the ratio of frame rotation to mean strain η f and a 2 the second invariant of the eddy axis tensor as a measure of anisotropy of turbulence. Since in this work only cases without frame rotation have been considered, the discussion of the parameterization is restricted to two parameters. They are, ˆΩ = 2 Ŝ, 2 a2 = a mn a mn, (2.4) with: ˆΩ 2 = a ij Ω ik Ω kj, Ŝ 2 = a ij S ik S kj. (2.5) Figure (a) shows the two-dimensional parameter plane used to parameterize the structure parameters. It is limited on the left at = by plane strain, at the bottom with a 2 = /3 by isotropic turbulence and at the top with a 2 = by RDT. The vertical line at = corresponds to shear flows, which separates hyperbolic and elliptic streamline flows. For > the flow is dominated by rotation. In Figure (b) the computed Reynolds stress r is plotted as a function of and a 2 as obtained using the ASBM model from Langer & Reynolds (23). Along the plane strain line = and for a 2 = /3 the Reynolds stress tensor is isotropic. At the RDT limit of pure shear, for which an analytic solution exists, the turbulence consists of purely jettal eddies with r =. A discontinuity is clearly visible at the boundary between elliptic and hyperbolic flow. Since the asymptotic state in the elliptic region is oscillatory, prior modeling

286 R. Pecnik et al. rapid distortion limit.8 plane strain.8.4.6 a2 shear a2.6.4 isotropic turbulence.2.2 hyperbolic.5 elliptic streamline flow.5 2 R:.3.4.5.6.7.8.9 2.5.5 ηm.5 2 ηm (a) ηm -a2 parameter plane. (b) Normalized Reynolds stress r plotted as a function of ηm and a2. Figure. Parameter plane used to construct the structure parameters φ, χ and γ..8 a2.6.4.2.5.5 2 2.5 ηm Figure 2. Post-processed results from the periodic channel flow with separation (Fro hlich et al. 25) plotted in the parameter map. Dots represent the whole flow field, while the tringles are extracted within the flow separation indicated by the line in the inset below. of the ASBM was limited by simply making the Reynolds stress tensor isotropic. The discontinuity also typically leads to convergence issues in CFD solvers and thus motivates the present improvements of the ASBM at the shear line. in Figure 2 the results of a large eddy simulation (Fro hlich et al. 25) are postprocessed and plotted in the ηm -a2 parameter plane. The black dots correspond to points in the entire flow domain, while the red dots indicate the points within the separation bubble. It is clearly seen that many points are located near the shear line, where the discontinuity of the Reynolds stresses computed by the ASBM can potentially lead to numerical instabilities in CFD simulations.

Algebraic structure based model 287 3. Assessment of the ASBM in homogeneous turbulence In the original formulation as detailed in Langer & Reynolds (23), the equations of the structure parameters are found by comparing target turbulent states obtained at the RDT limit, given a local mean deformation rate, and then these results are extrapolated for slower mean deformations (away from RDT). During the summer program, we revisited the ASBM functional forms and evaluated them over a range of mean deformations. To establish reference solutions, we have made extensive use of the Particle Representation Model (PRM) and the Interactive Particle Representation Model (iprm) (Kassinos & Reynolds 996; Kassinos & Akylas 22). The PRM is essentially a reduced Fourier representation retaining the minimum information beyond one-point that allows an exact closure of the Rapid Distortion Theory (RDT) governing equations without using a model. From a different point of view, the PRM formalism can be thought of as a real space Monte Carlo type of approach, and this duality offers a valuable framework for developing one-point structure-based closures. The iprm, or Interactive Particle Representation Model, is an extension of the PRM method for flows with weak deformation rates, but unlike the PRM, involves an approximate model for slower mean deformations. The iprm has been successfully applied to all standard benchmark cases for homogeneous turbulence. The iprm particularly utilizes the concept of Effective Gradients. It is postulated that the nonlinear turbulence-turbulence interactions can be represented by an effective deformation rate, which acts on each eddy or particle in addition to the mean gradients as a result of the deformation caused by the sea of all other eddies (particles). It has to be mentioned that the iprm transport equations retain the same form as the RDT (PRM) equations, with just the mean velocity gradient tensor replaced by the sum of the mean and effective gradient tensors. The effective gradient tensors are modeled in terms of the one-point structure tensors thus providing the route for closure even at the one-point level. Figure 3 compares the Reynolds stress components from the ASBM with iprm results for homogeneous turbulence subjected to planar hyperbolic mean deformation (mean strain of magnitude S 2 = S 2 = S in the 2-plane) combined with mean rotation along the 3-axis with magnitude S (i.e. Ω 3 = Ω 3 = S). The results confirm that for high Sq 2 /ǫ, i.e., near the RDT limit, the ASBM results match the iprm results extremely well as they were designed to. However, for slower deformation rates, significant deviations are seen in the ASBM results. In Figure 4 the Reynolds stresses and the structure parameters φ and γ are compared to the PRM at the RDT limit for < < 3.5. As mentioned previously, the discontinuity at = is clearly visible for the Reynolds stresses as given by the ASBM (Langer & Reynolds 23) (dashed line). New functional relations for the structure parameters in the elliptic flow regime are proposed to avoid these discontinuities and to better match the PRM results (solid line). The details of the changes to these functional forms for the structure parameters are not outlined in this paper, as they are preliminary - representing a possible progress towards a more accurate model. Furthermore the description of all the equations involved would not be possible within this paper format. However, as an example the functional forms of the equation for φ in the previous version and the new version of the ASBM are given as: φ old = 3 + 2 ( ); φ new = η 3 + m 3 + 2 ( ) ; (3.) η 3 + m 2 ( A 2 ) 5/2 2 (4/3 A 2 ) 5/2 These equations are representative of the simplified case of a non-rotating frame in the

288 R. Pecnik et al..8.8.8.6 r.6 r.6 r.4.4.4.2 r22.2 r22.2 r22 -.2 r2 -.2 r2 -.2 r2.2.4.6.8 (a) Sq 2 /ǫ =.2.4.6.8 (b) Sq 2 /ǫ =.2.4.6.8 (c) Sq 2 /ǫ = Figure 3. Comparison of ASBM (lines) and iprm (symbols) computed normalized Reynolds stress components for homogeneous turbulence undergoing hyperbolic mean deformation elliptic regime and the interpolation away from the RDT limit is not included. Notice that the only change is in the denominator of the second term. This small change ensures that the second term has a non-zero value for = at the RDT limit where A 2 =. The resulting functional forms are shown in Figure 4(e) and it can be seen that the discontinuity in φ has been removed by this change also improving the predictions of r ij in plots Figures 4(a) to 4(a). Figure 5 shows the same quantities, but for slower deformation rates of Sq 2 /ǫ =. The new ASBM (solid line) results in a better agreement with the iprm for the turbulent shear stress in the elliptic region, but still large disagreements are observed for the normal turbulent stresses. This has been identified as a key area of improvement of the ASBM in the future. 4. Test results The improved functional forms of the structure parameters were tested on a high Reynolds number flow over a two-dimensional bump that has been investigated as part of an ongoing NASA research project (Greenblatt et al. 26). The flow has a large separated region similar in size to the periodic channel flow discussed in Section 2.2 and a good set of experimental data for comparison. The Mach number and the Reynolds number based on the chord length of the bump were Ma=. and Re = 9.36 5, respectively. The grid used for the simulations fulfilled the standard requirement of y + <. and had 8 cells in the wall-normal direction and 28 in the streamwise direction. Symmetry conditions were applied at the top of the domain, an outlet boundary condition downstream of the bump and freestream conditions at the inlet. The domain was sufficiently long for the boundary layer to develop fully and the profiles of the mean and turbulent quantities agreed well with those measured upstream of the bump. Figure 6 shows that the modifications to the functional forms of the structure parameters significantly improves the agreement between the flow calculated using the ASBM and the experimental data. Comparing plots Figure 6(a) and Figure 6(c) it can be seen that the new ASBM no longer overestimates the size of the separated region. The profile of the streamwise velocity shown in Fig. 7(a) more clearly demonstrates the improvement in the mean flow predicted by the new ASBM. Previous studies have shown that the ASBM underestimated the turbulent shear stress in separated regions (O Sullivan et al. 2). Figure 7(b) shows that the new ASBM accurately estimates r 2 for most of the flow. Close to the turbulent shear layer, the new ASBM still underestimates r 2 but

Algebraic structure based model 289 r r33.8.6.4.2 -.2 -.4.5.5 2 2.5 3 3.5.8.6.4.2 -.2 (a) r -.4.5.5 2 2.5 3 3.5 (c) r 33 r22 r2.8.6.4.2 -.2 -.4.5.5 2 2.5 3 3.5.8.6.4.2 -.2 (b) r 22 -.4.5.5 2 2.5 3 3.5 (d) r 2 φ.8.6.4.2 -.2.5.5 2 2.5 3 3.5 (e) φ γ.8.6.4.2 -.2.5.5 2 2.5 3 3.5 Figure 4. Reynolds stresses and structure parameter at RDT limit. Symbols: PRM, solid line: new ASBM, dashed line: ASBM (Langer & Reynolds 23). (f) γ the improvement is significant. The comparison of the skin friction coefficient shown in Figure 7(c) highlights the effect of the higher turbulent shear stress seen by the better agreement reattachment point, but also the need for further research as the new ASBM improves the predictions of c f in the separated region but is still inaccurate close to separation. 5. Conclusions The Algebraic Structure Based Model has been critically evaluated and a number of problems have been identified. By analyzing flows using novel parameter space maps, specific areas have been identified where the ASBM does not agree well with DNS and LES results. In particular the ASBM performs poorly at the transition between hyperbolic and the elliptic flow regimes. Investigations show that this occurs often in many flows

29 R. Pecnik et al..6.6.4.4 r.2 r22.2 -.2 -.2.5.5 2 2.5 3 3.5 (a) r.5.5 2 2.5 3 3.5 (b) r 22.6.6.4.4 r33.2 r2.2 -.2 -.2.5.5 2 2.5 3 3.5.5.5 2 2.5 3 3.5 (c) r 33 (d) r 2.8.8.6.6 φ.4 γ.4.2.2 -.2 -.2.5.5 2 2.5 3 3.5 (e) φ.5.5 2 2.5 3 3.5 (f) γ Figure 5. Reynolds stresses and structure parameter at Sq 2 /ǫ =. Symbols: iprm, solid line: new ASBM, dashed line: ASBM (Langer & Reynolds 23). and the treatment of the functional form of the structure paramters must be improved in this region. PRM and iprm simulations have also been used to identify issues with the ASBM and more importantly they have been used to determine corrections to the functional forms of the structure parameters. The corrections have been implemented in a new version of the ASBM and preliminary results from a representative test case involving flow separation show improvements in the prediction of the mean flow and the turbulent quantities. Acknowledgements The authors would like to thank Prof. Paul Durbin for discussions during the 22 summer program.

29.25.2.2 y/c y/c Algebraic structure based model.25.5.5...5.5.7.8.9..2.3 x/c (a) new ASBM.7.8.9..2.3 x/c (b) ASBM (Langer & Reynolds 23).25 y/c.2.5..5.7.8.9..2.3 x/c (c) experiment Figure 6. Streamlines for wall-mounted hump case of Greenblatt et al. (26)(no flow-control case)..25.25.2.2.5.5. skin friction cf y/c y/c 6.E-3. 4.E-3 2.E-3.E+.5.5-2.E-3 -.5.5 U/U (a) U/U -.24 -.6 -.8 r2 (b) r2.5.75.25.5.75 x/c (c) skin friction cf 2 Figure 7. Streamwise velocity U/U (a) and turbulent shear stress u v /U (b) at x/c =.9, and wall skin friction coefficient cf (c) for wall-mounted hump. Symbols: experiment, solid line: new ASBM, dashed line: ASBM (Langer & Reynolds 23). REFERENCES Aupoix, B., Kassinos, S. & Langer, C. 29 An easy access to the structure-based model technology. In 6th Int. Symp. on Turbulence and Shear Flow Phenomena, pp. 22 24. Seoul, Korea. Benton, J. J. 2 Evaluation of v2-f and ASBM turbulence models for transonic aerofoil RAE2822. In Progress in Wall Turbulence: Understanding and Modeling (ed. M. Stanislas, J. Jimenez & I. Marusic), pp. 76 8. ERCOFTAC Series, Vol. 4. Fro hlich, J., Mellen, C. P., Rodi, W., Temmerman, L. & Leschziner, M. A. 25 Highly resolved large-eddy simulation of separated flow in a channel with streamwise periodic constrictions. Journal of Fluid Mechanics 526, 9 66.

292 R. Pecnik et al. Greenblatt, D., Paschal, K. B., Yao, C.-S., Harris, J., Schaeffler, N. W. & Washburn, A. E. 26 A separation control cfd validation test case, part : Baseline and steady suction. AIAA 44 (2), 282 283. Kalitzin, G., Iaccarino, G., Langer, C. & Kassinos, S. 24 Combining eddyviscosity models and the algebraic structure-based reynolds stress closure. In Proceedings of the Summer Program, pp. 7 82. Stanford. Kassinos, S. & Akylas, E. 22 Advances in Particle Representation Modeling of homogeneous turbulence. From the linear PRM version to the interacting viscoelastic IPRM. In New Approached in Modelling Multiphase Flows and Dispersion in Turbulence, Fractal Methods and Synthetic Turbulence (ed. F. Nicolleau, C. Cambon, J.-M. Redondo, J. Vassilicos, M. Reeks & A. Nowakowski), pp. 8. ERCOFTAC Series, Vol. 8. Kassinos, S. & Langer, C. 25 A new algebraic structure-based model with proper handling of strong rotation. In Int. Congress on Computational Mechanics. Limassol, Cyprus. Kassinos, S. & Reynolds, W. C. 996 A particle representation model for the deformation of homogeneous turbulence. Tech. Rep. Annual Briefs 996. Center for Turbulence Research, Stanford/NASA-Ames Research Center. Kassinos, S. C., Langer, C. A., Kalitzin, G. & Iaccarino, G. 26 A simplified structure-based model using standard turbulence scale equations: computation of rotating wall-bounded flows. International Journal of Heat and Fluid Flow 27 (4), 653 66. Kassinos, S. C. & Reynolds, W. C. 994 A structure-based model for the rapid distortion of homogeneous turbulence. Tech. Rep. TF-6. Mech. Eng. Department, Stanford University. Kassinos, S. C., Reynolds, W. C. & Rogers, M. 2 One-point turbulence structure tensors. Journal of Fluid Mechanics 428, 23 248. Langer, C. A. & Reynolds, W. C. 23 A new algebraic structure-based turbulence model for rotating wall-bounded flows. Tech. Rep. TF-85. Mech. Eng. Department, Stanford University. O Sullivan, J. P., Pecnik, R. & Iaccarino, G. 2 Investigating turbulence in wind flow over complex terrain. In Proceedings of the Summer Program 2, pp. 29 39. Stanford University. Radhakrishnan, H., Pecnik, R., Iaccarino, G. & Kassinos, S. C. 28 Computation of separated turbulent flows using the ASBM model. In Proceedings of the Summer Program 28, pp. 365 376. Stanford University. Reynolds, W. & Kassinos, S. 995 One-point modelling of rapidly deformed homogeneous turbulence. Proceedings - Royal Society of London, A 45 (94), 87 4.