Initial Evaluation of an Algebraic Reynolds Stress Model for Compressible Turbulent Shear Flows

An explicit algebraic Reynolds stress model originally formulated for incompressible flows has been implemented in the WIND Navier-Stokes flow solver. Validation results are presented for an attached incompressible turbulent boundary layer and indicate good agreement with experimental data, though the model lacks the near wall terms necessary to reproduce details such as the sharp rise in turbulent kinetic energy outside of the viscous sublayer. The model is then applied to a compressible mixing layer and a supersonic elliptic jet flow. Comparisons with two-equation eddy viscosity models available in WIND indicate that the algebraic stress model appears to better replicate the shape of the mean velocity profiles in both cases. The primary deficiency of the model is that it slightly underpredicts the rate of mixing. In all of these cases, the algebraic stress model demonstrates its unique ability to predict an anisotropic distribution of the turbulence intensities. Nomenclature Aexit nozzle exit area b mixing layer thickness between transverse locations where e u = e u1 − 0.1∆e u and e u = e u2 + 0.1∆e u bij Reynolds stress anisotropy tensor, bij = ρu′′ i u ′′ j /2ρk − δij/3 Cf local skin friction coefficient Deq equivalent diameter, Deq = 2 E Aexit/π k turbulent kinetic energy, ρk = ρu′′ i u ′′ j M Mach number Mc Convective Mach number, Mc = (e u1 − e u2) / (a1 + a2) where a is the speed of sound P production of turbulent kinetic energy P pressure ∗AIAA Member Sij strain rate tensor, Sij = (ui,j + uj,i)/2 Rij rotation tensor, Rij = (ui,j − uj,i)/2 Rex Reynolds number based on x T temperature u velocity Ujet ideal jet exit velocity, Ujet = r 2γ γ−1  RgasTt  1−  Pt P∞ (1−γ)/γ uτ friction velocity, uτ = E τwall/ρwall x, y, z spatial coordinates y0 mixing layer centerline (midpoint between transverse locations where e u = e u1 − 0.1∆e u and e u = e u2 + 0.1∆e u ) y sublayer scaled wall distance, y = yuτ/νL ∆e u freestream velocity difference, ∆e u = e u1 − e u2 δ boundary layer thickness δij Kronecker delta turbulent dissipation rate κ Karman constant, κ = 0.41 μ kinematic viscosity ν dynamic viscosity Πij deviatoric pressure strain correlation tensor Πij = P ′ ∂u′′ i ∂xj + ∂u′′ j ∂xi − 23P ′ ∂u′′ k ∂xk δij ρ density τT turbulent time scale τij viscous stress tensor τwall wall shear stress ω specific dissipation rate subscripts/superscripts ∞ freestream L laminar T turbulent t total 1 high-speed stream 2 low-speed stream accents uj Reynolds-averaged mean velocity uj Reynolds fluctuating velocity e uj Favre-averaged mean velocity u′′ j Favre fluctuating velocity 1 American Institute of Aeronautics and Astronautics 41st Aerospace Sciences Meeting and Exhibit 6-9 January 2003, Reno, Nevada AIAA 2003-548 Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner. Introduction Previous work conducted at the NASA Glenn Research Center and by industry partners has indicated deficiencies of eddy viscosity turbulence models when applied to compressible, sheardominated flowfields which have flow conditions similar to those encountered in the exhaust nozzles of subsonic commercial aircraft and proposed future supersonic vehicles. Even with the inclusion of additional terms to try to account for the roundjet/plane-jet anomaly and compressibility effects, these models still fail to provide agreement with experimental data for shear layer spreading rates and jet centerline velocity decay. In addition, there is growing interest in the aeroacoustics community in predicting not only the turbulent kinetic energy, but also the individual Reynolds normal stresses which form the components of k and using these values as inputs to empirical noise prediction codes. This has lead to the pursuit of higher-order turbulence models such as algebraic Reynolds stress models. Algebraic stress models (ASMs) are derived from a simplified form of the Reynolds stress transport equation. These models provide for improved physical modeling of turbulence over traditional two-equation eddy-viscosity models and incur a relatively low cost compared to other alternatives such as full Reynolds stress models. From a computational standpoint, explicit ASMs (EASMs) are preferred over implicit formulations because: (1) explicit formulations provide direct expressions for each of the Reynolds stresses without having to sub-iterate on a system of non-linear algebraic equations, and (2) implicit formulations, when solved iteratively can result in non-unique solutions and singularities which inhibit numerical convergence. EASMs are generally written in terms of the turbulent kinetic energy (k), turbulent dissipation rate ( ), mean strain-rate tensor (e Sij), and mean rotation tensor ( e Rij). The resultant expression used to calculate the turbulent stresses resembles that used with typical two-equation eddy viscosity models with additional terms that are nonlinear in e Sij and e Rij . However, unlike non-linear eddy-viscosity models, which can have somewhat arbitrary formulations for the higher-order terms, the resulting expression for the stresses in an ASM follows directly from simplifications made to the Reynolds stress transport equation. Therefore while the link between the ASM and k − model may appear to be weak and involve merely replacing the Boussinesq approximation with a slightly more complicated formula, these ASMs are more closely aligned with full Reynolds stress models. In addition, these ASMs are calibrated for use with a particular underlying k− model, which means one cannot simply swap k − models as when using the Boussinesq expression. Rodi was among the first to propose an algebraic Reynolds stress model. Since the convection and diffusion terms make the Reynolds stress transport equation a differential equation, he proposed that these terms be related to the convection and diffusion of the turbulent kinetic energy. The key to this simplification is the assumption that derivatives of the anisotropy tensor bij are small and can be neglected. The transport equation for the turbulent kinetic energy equation is then used to replace the convection and diffusion of turbulent kinetic energy with the production and dissipation of k. This results in a nonlinear algebraic equation for the Reynolds stresses. Unfortunately, Rodi’s model is an implicit formulation subject to the undesirable numerical behavior previously described. Building upon Rodi’s work, Pope presented a methodology based on the use of integrity bases from linear algebra which resulted in semiexplicit relations for the turbulent stresses. Gatski and Speziale extended this technique to threedimensional flows in non-inertial frames by treating the ratio of production to dissipation of turbulent kinetic energy as a known quantity and linearizing the problem. Girimaji demonstrated that if one does not linearize about the equilibrium value of production to dissipation, it is possible to reduce the non-linear problem to the solution of a single cubic equation for one of the closure coefficients. In solving the cubic equation, Girimaji’s selection of the proper root is governed by the argument that the closure coefficient must be a continuous function. Instead of using the continuity arguments of Girimaji to provide an explicit solution to the algebraic Reynolds stress equations, Jongen and Gatski use the limiting behavior of a dynamic system to show that the correct root to be used is the one with the lowest real part. This approach has been used by Rumsey and Gatski in implementing k − and k − ω based EASMs in the CFL3D code and is the basis upon which the WIND EASM has been formulated. Because k − models have historically performed better than k−ω models for computing free shear flows, the k − ASM formulation of reference 9 is the focus of this investigation. 2 American Institute of Aeronautics and Astronautics The goal of the present study is to implement this explicit algebraic stress model in WIND and determine its level of accuracy relative to linear two-equation eddy viscosity models in predicting basic shear flows which are have flow characteristics similar to those encountered in aircraft engine exhaust nozzles. In order to provide an equivalent comparison between models, no explicit corrections for compressibility effects, such as those for pressuredilatation or dilatation dissipation, have been used. Appropriate extension of the ASM formulation for compressible flows shall be the subject of future work. ASM Formulation In the ASM formulation, the turbulent stress tensor is computed from τ ij = −ρu′′ i u′′ j (1) = − 3 ρkδij+2μT  e Sij− 13 e Skkδij +F1 e Sik e Rkj− e Rik e Skj +F2 e Sik e Skj− 13 e Skl e Sklδij (2) where the first line of Eq. (2) is identical to the expression obtained using the Boussinesq approximation. The coefficients multiplying the nonlinear terms are given by F1 = a2a4 (3) F2 = −2a3a4 (4) The “nonlinear” eddy viscosity expression used is μT = C ∗ μρkτT (5) where the turbulent time scale is given by