Implementation of a ln(ω)-based SSG/LRR Reynolds Stress Model into the DLR-TAU Code

Kurzfassung

In contemporary aircraft design processes, the used Computational fluid dynamics (CFD) methods rely on the Reynold-averaged Navier-Stokes-equations and some type of turbulence modelling in most cases. However, the accuracy of widespread eddy-viscosity models (EVM) like the Spalart-Allmaras (SA) model [26] or the shear-stress transport model (SST) published by Menter [21] seem to be insufficient at the borders of the flight envelope where massive flow separation dominates flight physics. On the contrary, dropping the Boussinesq hypothesis used in EVMs and modelling the transport equations for the Reynolds stresses directly allows for a better description of flow physics. Though, these so called Reynolds stress models (RSM) are often reported to be hard to solve and to degenerate the robustness and stability of the solution process in comparison to state of the art EVMs. Nevertheless, the SSG/LRR-ω-RSM developed by Eisfeld [11] has been demonstrated to be applicable to a wide range of aeronautical flow problems [14, 7, 8, 6] and is now routinely used within DLR and industry for fairly complex cases. However, stability and robustness issues remain a problem especially in more complex cases like high-lift configurations for example. In order to improve the robustness, Togiti and Eisfeld [27] developed the so called SSG/LRR-g model, where the transport equation of the specific dissipation rate ω is replaced by a transport equation for g = 1/√ω. This new variant of the model no longer suffers from the singular behavior of ω and the lack of a natural boundary condition on viscous walls. Furthermore, users of the DLR-TAU code reportedthat the g-formulation of the SSG/LRR-RSM is far more stable than the original model in many cases. Due to these advantages, the SSG/LRR-g has become the "standard"-RSM in the TAU code for most applications. In this report, an additional formulation of the SSG/LRR-RSM is presented in which the specific dissipation rate ω is replaced by its natural logarithm. This transformation based on ideas of Bassi et al. [2] that were developed in order to make the Wilcox-k-ω model [29] manageable for solvers based on a Discontinuous-Galerkin (DG) discretization. Using the logarithmically transformed equation solves two main problems that arise in the context of DG discretization: 1. The extremely large slope of ω near viscous walls is alleviated. Therefore, the length-scale determining variable can be much better represented with the polynomial ansatz functions DG methods inherently rely on. 2. Within the transformed equation, the new length-scale providing variable only appears in the exponent of the Euler number. This removes the necessity of a numerical limitation of the variable that is required in the original formulation in order to fulfill energy stability requirements and cannot be enforced in a DG method directly. In flow solvers based on a finite volume approach like the DLR-TAU code, the problem of negative length-scales is fixed in most cases by using a numerical or physical limiter that enforces positive values subsequent of each iteration step and guarantees that the energy stability requirements are fulfilled in the next iteration step. However, it is assumed that these inevitable supplementary changes of values also introduce instabilities into the solution procedure. Therefore, the idea of a logarithmic transport variable is transferred to the length-scale determining transport equation of the SSG/LRR-ω-RSM in order to improve the robustness of the solution procedure in TAU. It has to be mentioned, that this modification only removes the necessity of limiting the length-scale determining variable. The realizability of the Reynolds-stress tensor still has to be enforced by a limiter. Furthermore, a logarithmically transformed ω-equation still suffers from the lack of a natural boundary condition on viscous walls and the necessity of extrapolating values from the flow eld onto the wall. However, the magnitude of the extrapolated values is extremely reduced. In computations that use the SSG/LRR-ω model, the specific dissipation rate increases from values near zero outside of the boundary layer to values that are very often bigger than several trillions4 onto a viscous wall. After logarithmizing the equation, the values on a viscous wall remain always smaller than three orders of magnitude. This may improve the robustness of the implicit backward Euler solver used in TAU where large linear systems are solved using a LUSGS-scheme that may become stiff in cases where the magnitude of the entries in the matrices varies to widely. In the subsequent chapter 2, the formulation of the new SSG/LRR-ln(ω) model version will be presented. In chapter 3, the implementation of the model in the flow solver TAU will be verified and the model will be validated using several test cases. The results of stability and performance tests with the new model can be found in chapter 4 and a short summary will be given in chapter 5.