Explicit Algebraic Reynolds Stress Models for Anisotropic
F. R. Menter1, A. V. Garbaruk2 and Y. Egorov1
1 ANSYS Germany GmbH, Germany
Current industrial and aeronautics CFD simulations are largely based on linear eddyviscosity turbulence models (LEVM). There are several reasons why higher order models like Reynolds Stress Models (RSM) have rarely made their way into mainstream industrial CFD applications, even though they are available in most general-purpose codes. The two main reasons are the increased computational requirements, often caused by a lack of robustness and the observation that such models have in many cases not resulted in a
systematic improvement of results. Still there are numerous areas where EV models are known to fail systematically and where further model improvements are required. The goal should be to allow the inclusion of specific additional effects without a large penalty on speed and robustness.
Explicit Algebraic RSM (EARSM) (Pope, 1975, Rodi, 1976, Gatski and Speziale,1993, Wallin and Johansson, 2000) offer an attractive framework for such enhancements. These models can be considered as a subset of nonlinear constitutive relations in which apart of the higher-order description of physical processes on the RSM-level is transferred into the two-equation modeling level. As a result, they are much less demanding than RSM from the computational standpoint and, at the same time, are capable of reproducing some important features of turbulence (e.g. its anisotropy in the normal stresses), which are
beyond the capabilities of LEVM. The emphasis in the development of EARSM has largely been on the correct mathematical formulation of the stress-strain relationship and not so much in the formulation of an industrial CFD model (see however Hellsten and Laine, 2000).
When increasing the complexity in the formulation of a turbulence model, one should first answer the question, which types of flows will benefit from it? One of the weaknesses of EARSMs is that they do not naturally account for swirling and rotating flows. This being one of the major practical arguments for RSM, one has to find other areas where EARSM could be beneficial (or add appropriate rotation terms). One such area is the prediction of flows parallel to corners formed by intersecting walls, as observed in wing-body junctions or hub-blade regions. There is a strong indication that LEVM predict much too early separation from such corners when the flow is exposed to adverse pressure gradients. This can have a severe impact on the computed performance characteristics of these technical devices. The goal of the current paper is to explore model enhancements which allow the inclusion of such effects at the lowest level of complexity. For this purpose, two EARSM formulations will be investigated. The underlying idea being that the secondary flow of the second kind which is observed near corners is the main mechanism for obtaining delayed separation in these regions. In other words, it is anticipated that the secondary flow caused by the differences in the normal stresses drives additional momentum into the corner, thereby delaying separation. The goal is the identification of the most appropriate and simplest EARSM formulation which allows the inclusion of anisotropic effects into the formulation. There are two areas which will be investigated.
The first is the formulation of a scale-equation used in combination with the EARSM. Experience over the last decade has shown that ω-equation based models offer significant advantages over ε-equation based models, especially if integration through the viscous sublayer is desired (Menter, 2009). Unfortunately, the standard ω-equation of Wilcox in its different forms is not suitable due to the persistent freestream sensitivity of this model, even in its latest version (Menter, 2004, 2009). In addition, the Shear Stress Transport (SST) model (Menter, 1994) has been widely used in aerodynamic and other industrial flow simulations. It is therefore desirable to formulate the EARSM on the basis of a scaleequation very similar to the SST model in order to isolate the impact of the stress-strain relationship on the solution. The starting point is therefore the Baseline (BSL) model, which underlies the SST model (Menter, 1994). It will be shown that an EARSM can be formulated without a need for re-calibrating the BSL model. This allows the direct comparison of the EARSM stress-strain relationship with the SST model formulation, without additional changes of other parts of the model.
The second area of interest is the formulation of the non-linear stress-strain relationship. Here, numerous options are available in the literature and can be used as a starting point. Starting point in this paper is the stress-strain relation proposed by Wallin and Johansson (1997, 2000) (WJ) which already provides a relatively simple EARSM closure. In addition, the question is posed: “what is the simplest EARSM which would still account for the anisotropy in the normal stresses in wall-bounded flows?” and would thereby allow the prediction of secondary flow into the corner. A model variant is explored which is linear in the implicit formulation, resulting in a simplification of the WJ stressstrain relation.
The models are tested for a set of flows with an emphasis of corner flow behavior. In addition, a flat plate boundary layer will be shown to demonstrate that the model gives the correct wall shear stress and log. boundary layer profile. In some of the testcases, a comparison with the SST model is given. It should be emphasized that the superior performance for some of the corner flow cases does not imply a general recommendation for replacing the SST with the EARSM at current point. For such a recommendation, significantly more testing and model optimization (especially in terms of robustness) is required.