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12.5.2 Shear-Stress Transport (SST) $k$- $\omega$ Model



Overview


The shear-stress transport (SST) $k$- $\omega$ model was developed by Menter [ 237] to effectively blend the robust and accurate formulation of the $k$- $\omega$ model in the near-wall region with the free-stream independence of the $k$- $\epsilon$ model in the far field. To achieve this, the $k$- $\epsilon$ model is converted into a $k$- $\omega$ formulation. The SST $k$- $\omega$ model is similar to the standard $k$- $\omega$ model, but includes the following refinements:

These features make the SST $k$- $\omega$ model more accurate and reliable for a wider class of flows (e.g., adverse pressure gradient flows, airfoils, transonic shock waves) than the standard $k$- $\omega$ model. Other modifications include the addition of a cross-diffusion term in the $\omega$ equation and a blending function to ensure that the model equations behave appropriately in both the near-wall and far-field zones.



Transport Equations for the SST $k$- $\omega$ Model


The SST $k$- $\omega$ model has a similar form to the standard $k$- $\omega$ model:

 \frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\par... ...\partial k}{\partial x_j}\right) + \tilde{G}_k - Y_k + S_k (12.5-32)

and
 \frac{\partial}{\partial t} (\rho \omega) + \frac{\partial}... ..._j}\right) + G_\omega - Y_\omega + D_{\omega} + S_{\omega} (12.5-33)

In these equations, $\tilde{G}_k$ represents the generation of turbulence kinetic energy due to mean velocity gradients, calculated as described in Section  12.5.1. $G_\omega$ represents the generation of $\omega$, calculated as described in Section  12.5.1. ${\Gamma}_k$ and ${\Gamma}_\omega$ represent the effective diffusivity of $k$ and $\omega$, respectively, which are calculated as described below. $Y_k$ and $Y_\omega$ represent the dissipation of $k$ and $\omega$ due to turbulence, calculated as described in Section  12.5.1. $D_{\omega}$ represents the cross-diffusion term, calculated as described below. $S_k$ and $S_{\omega}$ are user-defined source terms.



Modeling the Effective Diffusivity


The effective diffusivities for the SST $k$- $\omega$ model are given by

$\displaystyle {\Gamma}_k$ $\textstyle =$ $\displaystyle \mu + \frac{\mu_t}{\sigma_k}$ (12.5-34)
$\displaystyle {\Gamma}_\omega$ $\textstyle =$ $\displaystyle \mu + \frac{\mu_t}{\sigma_\omega}$ (12.5-35)

where $\sigma_k$ and $\sigma_\omega$ are the turbulent Prandtl numbers for $k$ and $\omega$, respectively. The turbulent viscosity, $\mu_t$, is computed as follows:
 \mu_t = \frac{\rho k}{\omega} \frac{1} {\max \left[\frac{1}{\alpha^*}, \frac{S F_2}{a_1 \omega} \right] } (12.5-36)

where $S$ is the strain rate magnitude and
$\displaystyle \sigma_{k}$ $\textstyle =$ $\displaystyle \frac{1}{F_1/\sigma_{k,1} + (1 - F_1)/\sigma_{k,2}}$ (12.5-37)
$\displaystyle \sigma_{\omega}$ $\textstyle =$ $\displaystyle \frac{1}{F_1/\sigma_{\omega,1} + (1 - F_1)/\sigma_{\omega,2}}$ (12.5-38)

$\alpha^*$ is defined in Equation  12.5-6. The blending functions, $F_1$ and $F_2$, are given by
$\displaystyle F_1$ $\textstyle =$ $\displaystyle \tanh \left( \Phi_1^4 \right)$ (12.5-39)
$\displaystyle {\Phi}_1$ $\textstyle =$ $\displaystyle \min \left[\max \left (\frac{\sqrt{k}}{0.09 \omega y}, \frac{500 ... ... \omega} \right ), \frac{4 \rho k} {\sigma_{\omega,2} D_{\omega}^+ y^2} \right]$ (12.5-40)
$\displaystyle D_{\omega}^+$ $\textstyle =$ $\displaystyle \max \left[2 \rho \frac{1}{\sigma_{\omega,2}} \frac{1}{\omega} \f... ...\partial k}{\partial x_j}\frac{\partial \omega}{\partial x_j}, 10^{-10} \right]$ (12.5-41)


$\displaystyle F_2$ $\textstyle =$ $\displaystyle \tanh \left(\Phi_2^2 \right)$ (12.5-42)
$\displaystyle \Phi_2$ $\textstyle =$ $\displaystyle \max \left [ 2 \frac{\sqrt{k}}{0.09 \omega y}, \frac{500 \mu}{\rho y^2 \omega} \right]$ (12.5-43)

where $y$ is the distance to the next surface and $D_{\omega}^+$ is the positive portion of the cross-diffusion term (see Equation  12.5-52).



Modeling the Turbulence Production


Production of $k$

The term $\tilde{G}_k$ represents the production of turbulence kinetic energy, and is defined as:

 \tilde{G}_k = \min (G_k, 10 \rho \beta^* k \omega) (12.5-44)

where $G_k$ is defined in the same manner as in the standard $k$- $\omega$ model. See Section  12.5.1 for details.

Production of $\omega$

The term $G_\omega$ represents the production of $\omega$ and is given by

 G_\omega = \frac{\alpha}{\nu_t} G_k (12.5-45)

Note that this formulation differs from the standard $k$- $\omega$ model. The difference between the two models also exists in the way the term $\alpha_{\infty}$ is evaluated. In the standard $k$- $\omega$ model, $\alpha_{\infty}$ is defined as a constant (0.52). For the SST $k$- $\omega$ model, $\alpha_{\infty}$ is given by
 \alpha_\infty = F_1 \alpha_{\infty,1} + (1 - F_1) \alpha_{\infty,2} (12.5-46)

where
$\displaystyle \alpha_{\infty,1}$ $\textstyle =$ $\displaystyle \frac{\beta_{i,1}}{\beta^*_\infty} - \frac{\kappa^2}{\sigma_{w,1} \sqrt{\beta^*_\infty}}$ (12.5-47)
$\displaystyle \alpha_{\infty,2}$ $\textstyle =$ $\displaystyle \frac{\beta_{i,2}}{\beta^*_\infty} - \frac{\kappa^2}{\sigma_{w,2} \sqrt{\beta^*_\infty}}$ (12.5-48)

where $\kappa$ is 0.41.



Modeling the Turbulence Dissipation


Dissipation of $k$

The term $Y_k$ represents the dissipation of turbulence kinetic energy, and is defined in a similar manner as in the standard $k$- $\omega$ model (see Section  12.5.1). The difference is in the way the term $f_{\beta^*}$ is evaluated. In the standard $k$- $\omega$ model, $f_{\beta^*}$ is defined as a piecewise function. For the SST $k$- $\omega$ model, $f_{\beta^*}$ is a constant equal to 1. Thus,

 Y_k = \rho \beta^* k \omega (12.5-49)

Dissipation of $\omega$

The term $Y_\omega$ represents the dissipation of $\omega$, and is defined in a similar manner as in the standard $k$- $\omega$ model (see Section  12.5.1). The difference is in the way the terms $\beta_i$ and $f_\beta$ are evaluated. In the standard $k$- $\omega$ model, $\beta_i$ is defined as a constant (0.072) and $f_\beta$ is defined in Equation  12.5-24. For the SST $k$- $\omega$ model, $f_{\beta}$ is a constant equal to 1. Thus,

 Y_k = \rho \beta \omega^2 (12.5-50)

Instead of a having a constant value, $\beta_i$ is given by
 \beta_i =F_1 \beta_{i,1} + (1 - F_1) \beta_{i,2} (12.5-51)

and $F_1$ is obtained from Equation  12.5-39.



Cross-Diffusion Modification


The SST $k$- $\omega$ model is based on both the standard $k$- $\omega$ model and the standard $k$- $\epsilon$ model. To blend these two models together, the standard $k$- $\epsilon$ model has been transformed into equations based on $k$ and $\omega$, which leads to the introduction of a cross-diffusion term ( $D_\omega$ in Equation  12.5-33). $D_\omega$ is defined as

 D_\omega = 2 \left(1 - F_1 \right) \rho \sigma_{\omega,2}... ...\partial k}{\partial x_j}\frac{\partial \omega}{\partial x_j} (12.5-52)

For details about the various $k$- $\epsilon$ models, see Section  12.4.



Model Constants



\sigma_{k,1} = 1.176, \;\; \sigma_{\omega,1} = 2.0, \;\; \sigma_{k,2} = 1.0, \;\; \sigma_{\omega,2} = 1.168


a_1 = 0.31, \;\; \beta_{i,1} = 0.075 \;\; \beta_{i,2} = 0.0828

All additional model constants ( $\alpha_{\infty}^*$, $\alpha_{\infty}$, $\alpha_0$, $\beta_{\infty}^*$, $R_\beta$, $R_k$, $R_\omega$, $\zeta^*$, and M $_{t0}$) have the same values as for the standard $k$- $\omega$ model (see Section  12.5.1).


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