Reynolds Stress Transport (RST) Models

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

RST models approximate the stress tensor as:

Figure 1. EQUATION_DISPLAY

T_{RANS} = - ρ R + \frac{2}{3} tr (R) I

(1307)

where:

$ρ$ is the density.
$I$ is the identity tensor.

$R$ is the Reynolds stress tensor defined as:

Figure 2. EQUATION_DISPLAY

R = (\begin{matrix} \bar{u' u'} & \bar{u' v'} & \bar{u' w'} \\ \bar{u' v'} & \bar{v' v'} & \bar{v' w'} \\ \bar{u' w'} & \bar{v' w'} & \bar{w' w'} \end{matrix})

(1308)

RST models solve transport equations for each component of $R$ .

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].

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

Linear Pressure-Strain model
Quadratic Pressure-Strain model

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.