Coupling with Turbulence Models

The intermittency solved by the Gamma ReTheta transition model and the Gamma transition model is used to scale the turbulence equation of the SST (Menter) K-Omega model and the Spalart-Allmaras model.

Coupling with SST (Menter) K-Omega Model

The Gamma ReTheta transition model and the Gamma transition model enter the transport equation for the turbulent kinetic energy $k$ and the specific dissipation rate $ω$ (see SST (Menter) K-Omega Model) as follows:

图 1. EQUATION_DISPLAY

\begin{matrix} \frac{\partial}{\partial t} (ρ k) + \nabla\cdot [ρ k (\bar{v} - v_{g})] = \end{matrix} \begin{matrix} \nabla\cdot [(μ + {σ_{k} μ}_{t}) \nabla k] + P_{k} - γ^{'} ρ β^{*} f_{β^{*}} (ω k - {ω_{0} k}_{0}) + S_{k} \end{matrix}

(1576)

图 2. EQUATION_DISPLAY

\begin{matrix} \frac{\partial}{\partial t} (ρ ω) + \nabla\cdot (ρ ω \bar{v}) = \end{matrix} \begin{matrix} \nabla\cdot [((μ + {σ_{ω} μ}_{t}) \nabla ω] + P_{ω} - γ^{'} ρ β f_{β} (ω^{2} - ω_{0}^{2}) + S_{ω} \end{matrix}

(1577)

where the production term $P_{k}$ is defined as:

图 3. EQUATION_DISPLAY

P_{k} = γ_{t r a n s} (G_{k} + G_{n l} + G_{b}) + G_{k}^{\lim}

(1578)

and:

图 4. EQUATION_DISPLAY

γ' = {\begin{matrix} 0, \\ 1, \end{matrix} \begin{matrix} γ_{t r a n s} < 0.9 \\ γ_{t r a n s} \geq 0.9 \end{matrix}

(1579)

where:

图 5. EQUATION_DISPLAY

γ_{t r a n s} = {\begin{matrix} γ_{e f f} & for the Gamma ReTheta transition model \\ γ & for the Gamma transition model \end{matrix}

(1580)

$γ_{e f f}$ and $γ$ are provided by the Gamma ReTheta Transition Model and Gamma Transition Model), respectively.

$γ_{t r a n s}$ and $γ^{'}$ are unity if no transition model is activated.

The Gamma transition model introduces an additional production term $G_{k}^{\lim}$ to ensure proper generation of $k$ at transition points for low values of free-stream turbulence:

图 6. EQUATION_DISPLAY

G_{k}^{\lim} = 5 \max (γ - 0.2, 0) (1 - γ) F_{o n}^{\lim} \max (3 μ - μ_{t}, 0) S W

(1581)

where:

图 7. EQUATION_DISPLAY

F_{o n}^{\lim} = \min (\max (\frac{{Re}_{ν}}{2.2 \cdot 1100} - 1, 0), 3)

(1582)

and:

$S$ and $W$ are modulus of the mean strain rate tensor and the modulus of the mean vorticity tensor. See Eqn. (1129) and Eqn. (1131), respectively.
${Re}_{ν}$ is the strain-rate Reynolds number given by Eqn. (1139).

The additional production term is set to zero if the Gamma transition model is not activated.

Coupling with Spalart-Allmaras Model

The SA Gamma transition model enters the transport equation for the modified diffusivity $\tilde{ν}$ (see Spalart-Allmaras Model) as follows:

\frac{\partial}{\partial t} (ρ \tilde{ν}) + \nabla\cdot (ρ \tilde{ν} \bar{v}) = \begin{matrix} \frac{1}{σ_{\tilde{ν}}} \nabla\cdot [(μ + ρ \tilde{ν}) \nabla \tilde{ν}] + γ P_{\tilde{ν}} - \max (γ, 0.1) D_{\tilde{ν}} {+ S}_{\tilde{ν}} \end{matrix}

(1583)