Abstract
Turbulence equilibrium state is analyzed for the modeled Reynolds-stress transport equation, assuming the most general formulation of pressure–strain correlation. In a two-dimensional mean flow at a high-Reynolds number, an algebraic equation system is obtained, providing Reynolds-stress anisotropies as functions of pressure–strain model coefficients. Conversely, the equations provide calibration conditions for the model coefficients to predict specified equilibrium anisotropies. The predicted von-Kármán constant depends on the predicted equilibrium anisotropies and, hence, the pressure–strain model coefficients. Identical equilibrium anisotropies can be obtained with different sets of model coefficients. Identical equilibrium values of invariants of the Reynolds-stress anisotropy tensor can be achieved, despite the differing anisotropy components. Numerical simulations with the Speziale–Sarkar–Gatski (SSG) model, using different sets of model coefficients, confirm the results of the theoretical analysis. They show that the predicted equilibrium value of the Reynolds-shear stress anisotropy determines the predicted skin friction of a boundary layer as well as the predicted spreading rate of a plane mixing layer. However, different values and, hence, different sets of model coefficients are required for achieving good agreement with experimental data for both flows. Therefore, for general improvement of turbulence models, the set of model coefficients probably needs to be adapted to the local type of flow. The required classification is supposed to be suitable for machine learning methods.
Highlights
Turbulent equilibrium, in the sense of balance between production and dissipation of specific kinetic turbulence energy, is a key assumption in turbulence modeling
Identical equilibrium anisotropies can be obtained with different sets of model coefficients
As will be demonstrated, the turbulent equilibrium state is of utmost importance for Reynolds-stress models for predicting canonical turbulent flows
Summary
In the sense of balance between production and dissipation of specific kinetic turbulence energy, is a key assumption in turbulence modeling. The so-called rapid term is proportional to the mean-flow velocity gradient. Both terms are assumed to depend on the Reynolds-stress anisotropies. The most general model formulations of the slow and rapid terms as functions of the Reynolds-stress anisotropies have been provided by Lumley and Newman[34] and by Johansson and Hallb€ack,[35] respectively. They form the basis of the subsequent analysis of the relationship between the model coefficients and the predicted turbulent equilibrium state. For general improvement of turbulence models, methods involving a classification of different regions of the flow field, e.g., Ref. 45, could be promising
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