The importance of turbulent equilibrium for Reynolds-stress modeling

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.

Evolutionary Algorithm applied to Differential Reynolds Stress Model for Turbulent Boundary Layer subjected to an Adverse Pressure Gradient

“…Note that, although the original SSG model [27] is based on the length scale variable 𝜖, Menter's baseline 𝜔-equation, which is based on the specific dissipation rate 𝜔, is used in this work to supply the length scale variable. As mentioned by Eisfeld in [37], this helps avoid the need for any additional near-wall treatment.…”

“…( 6) exceeds the number of Reynolds stress anisotropies in two-dimensional flow at turbulent equilibrium. As a result, different sets of coefficients can yield various similar solutions [37]. Hence, there is no unique solution and the problem is over-determined.…”

“…The test case contains several local flow types which cannot be predicted correctly with one set of coefficients for the pressure-strain term of the SSG model [37]. Although the area of interest is the separated flow region, an improvement there should not deteriorate the prediction accuracy of other flow areas or other basic test cases.…”

Data-driven RANS closures for improving mean field calculation of separated flows

“…Although this approach amplifies generality and robustness, it manifests notable constraints in complex flows, including flow separation and reattachment [6,7]. The third category comprises the Reynolds Stress Model (RSM), which formulates transport equations for each component of the Reynolds stress, frequently termed as the sevenequation model [8]. Although RSM offers higher accuracy, it is computationally demanding and performs poorly in flows with substantial curvature.…”

Modeling of Two-Phase Flow Parameters of a Multi-Channel Cylindrical Cyclone

“…This model is based on the transfer equations for all components of the Reynolds stress tensor and the rate of dissipation. The RSM presents anisotropic turbulence for flows, and, according to the turbulent viscosity hypothesis, isotropic turbulence [24][25][26]. In the first case, the Reynolds stress transfer equations for the individual stress components are solved [27,28].…”