Reynolds Stress Transport Equation

The transport equation for the Reynolds stress tensor $R$ is:

Figure 1. EQUATION_DISPLAY

\begin{matrix} \frac{\partial}{\partial t} (ρ R) + \nabla\cdot (ρ R \bar{v}) = \end{matrix} \begin{matrix} \nabla\cdot D + P + G - \frac{2}{3} I ϒ_{M} + \overset{ϕ}{_} - ρ \underset{̲}{ε} + S_{R} \end{matrix}

(1309)

where:

$ρ$ is the density.
$\bar{v}$ is the mean velocity.
$D$ is the Reynolds Stress Diffusion.
$P$ is the Turbulent Production.
$G$ is the Buoyancy Production.
$I$ is the identity tensor.
$γ_{M}$ is the Dilatation Dissipation.
$\underset{̲}{ϕ}$ is the pressure strain tensor.
$\underset{̲}{ε}$ is the turbulent dissipation rate tensor.
$S_{R}$ is the user-specified source.

For the Linear Pressure Strain and Quadratic Pressure Strain models, the dissipation is simply:

\underset{̲}{ε} = \frac{2}{3} ε I

(1310)

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 $ε$ (see Eqn. (1169) (SKE)).

Reynolds Stress Diffusion

Two different models for the Reynolds-stress diffusion are available. By default, following [343], a simple isotropic form of the turbulent diffusion is adopted, such that:

Figure 2. EQUATION_DISPLAY

D = (μ + \frac{μ_{t}}{σ_{k}}) \nabla R

(1311)

where:

$μ$ is the dynamic viscosity.
$σ_{k}$ is a Model Coefficient.

The turbulent viscosity $μ_{t}$ is computed as:

Figure 3. EQUATION_DISPLAY

μ_{t} = ρ C_{μ} \frac{k^{2}}{ε}

(1312)

where $C_{μ}$ is a Model Coefficient. The turbulent kinetic energy $k$ is defined as:

Figure 4. EQUATION_DISPLAY

k = \frac{1}{2} t r (R)

(1313)

where $t r (R)$ is the trace of the Reynolds stress tensor (see Eqn. (1146)).

Alternatively, the Reynolds-stress diffusion can be written according to Daly and Harlow [338] in the form:

Figure 5. EQUATION_DISPLAY

D = μ \nabla R + C_{s} \frac{k}{ε} (R \cdot \nabla) R

(1314)

where $C_{s}$ is a Model Coefficient.

This model is also known as the Generalized Gradient Diffusion Hypothesis, GGDH.

Turbulent Production

The turbulent production is obtained directly, without recourse to modeling as follows:

Figure 6. EQUATION_DISPLAY

P = - ρ (R \cdot \nabla {\bar{v}}^{T} + \nabla \bar{v} \cdot R^{T}) = - ρ (R \cdot \nabla {\bar{v}}^{T} + \nabla \bar{v} \cdot R)

(1315)

Buoyancy Production

For constant-density flows using the Boussinesq approximation, the buoyancy production is modeled as:

Figure 7. EQUATION_DISPLAY

G = β \frac{μ_{t}}{\Pr_{t}} (\nabla \bar{T} g + g \nabla \bar{T})

(1316)

where:

$β$ is the coefficient of thermal expansion.
$g$ is the gravitational vector.
$\Pr_{t}$ is the turbulent Prandtl number.
$\bar{T}$ is the mean temperature.

For flows with varying density:

Figure 8. EQUATION_DISPLAY

G = \frac{μ_{t}}{{ρ \Pr}_{t}} (\nabla ρ g + g \nabla ρ)

(1317)

Turbulent Dissipation Rate

The isotropic turbulent dissipation rate is obtained from a transport equation analogous to the K-Epsilon model (and with identical boundary conditions):

Figure 9. EQUATION_DISPLAY

\begin{matrix} \frac{\partial}{\partial t} (ρ ε) + \nabla\cdot (ρ ε \bar{v}) = \\ \nabla\cdot [(μ + \frac{μ_{t}}{σ_{ε}}) \nabla ε] + \frac{ε}{k} [C_{ε 1} (\frac{1}{2} tr (P) + \frac{1}{2} C_{ε 3} tr (G)) - C_{ε 2} ρ ε] \end{matrix}

(1318)

where:

$C_{ε 1}$ and $C_{ε 2}$ are Model Coefficients.
$C_{ε 3}$ is determined as in the Standard K-Epsilon model (see K-Epsilon Model Coefficients).

Dilatation Dissipation Rate

The dilatation dissipation $ϒ_{M}$ is modeled as in the K-Epsilon model, using the model of Sarkar et al. [344]:

Figure 10. EQUATION_DISPLAY

ϒ_{M} = \frac{ρ C_{M} k ε}{c^{2}}

(1319)

where:

$C_{M}$ is a Model Coefficient.
$c$ is the speed of sound.

Model Coefficients


Coefficient	Linear Pressure Strain	Linear Pressure Strain Two Layer	Quadratic Pressure Strain	Elliptic Blending
$σ_{k}$	1	0.82	1	1
$σ_{ε}$	1.22	1	1.22	1.15
$C_{M}$ (Sarkar)	2	2	2	2
$C_{s}$	0.2	0.2	0.2	0.21
$C_{ε 1}$	1.44	1.44	1.44	1.44
$C_{ε 2}$	1.92	1.92	1.83	1.83
$C_{μ}$	0.065536	0.09	0.098596	0.07