Reynolds Stress Transport (RST) Turbulence

Reynolds Stress Transport (RST) models, also known as second-moment closure models, directly calculate the components of the Reynolds stress tensor $R$ by solving their governing transport equations.

RST models have the potential to predict complex flows more accurately than eddy viscosity models because the transport equations for the Reynolds stresses naturally account for the effects of turbulence anisotropy, streamline curvature, swirl rotation and high strain rates.

The starting point for the development of an RST model is generally the exact differential transport equation for the Reynolds stresses, which is derived by multiplying the instantaneous Navier-Stokes equations by a fluctuating property and Reynolds-averaging their product (see [344] for this exact equation). In the resulting equations, the transient, convective and molecular diffusion terms do not require modeling. The terms remaining to be modeled are the turbulent diffusion term, the dissipation term and, perhaps the greatest challenge, the pressure-strain term. Appropriate models for these terms have received much attention during the past few decades, some of which are reviewed in [345].

Note

A RST model carries significant computational overhead. Seven equations must be solved (as opposed to the two equations of a K-Epsilon or a K-Omega model): six equations for the Reynolds stresses (symmetric tensor) and one equation for the isotropic turbulent dissipation

ε

. Apart from the additional memory and computational time required for these equations to be solved, there is also likely to be a penalty in the total number of iterations required to obtain a converged solution due to the numerical stiffness of the RST equations.

Pressure Strain Models

To model the pressure-strain term, the following approaches are implemented in Simcenter STAR-CCM+:

Linear Pressure Strain
Quadratic Pressure Strain

The Linear Pressure-Strain model can be used with a high- $y^{+}$ wall treatment, but it is also available with a two-layer formulation, which makes it applicable right down to the wall.

The Quadratic Pressure-Strain model and its low Reynolds number variant, the Elliptic Blending model, are based on the most recent and precise formulation of the pressure-strain term. Thus, they are likely to be the models of choice in a majority of cases, the Elliptic Blending model being preferred as it also contains specific treatments for all- $y^{+}$ meshes. The Quadratic Pressure-Strain model can only be used with a high- $y^{+}$ wall treatment (that is, using wall functions) without resolving the viscous-affected near-wall region.

Linear Pressure Strain

The linear model of Gibson and Launder [339] for the pressure-strain term comprises five terms; these are the rapid term, the slow (return-to-isotropy) term, and their respective wall-reflection terms.

Linear Pressure Strain Two-Layer

The advantage of the linear pressure-strain model is that it lends itself to being incorporated into a two-layer formulation, which can be used to resolve the viscous sublayer for low Reynolds number type applications.

In the linear pressure-strain model approach, suggested by Rodi [313], the computation is divided into two layers. In the layer adjacent to the wall, the turbulent dissipation rate $ε$ and the turbulent viscosity $μ_{t}$ are specified as functions of wall distance. The values of $ε$ specified in the near-wall layer are blended smoothly with the values computed from solving the transport equation far from the wall.

Quadratic Pressure Strain

The quadratic pressure-strain model of Sarkar, Speziale and Gatski [345] uses a higher-order truncation of the expansion for the pressure-strain term.

Elliptic Blending

The Elliptic Blending RST model of Manceau and Hanjalić [334] is a low Reynolds number model that is based on a inhomogeneous near-wall formulation of the quasi-linear quadratic pressure-strain term. The blending function is used to blend the viscous sub-layer with the log-layer formulation of the pressure-strain term. This approach requires the solution of an elliptic equation for the blending parameter $α$ . The version of the model implemented in Simcenter STAR-CCM+ was revised by Lardeau and Manceau [341].