7.2.3 SST Two-Equation Model of Menter Menter 的 SST 双方程模型[cfd-7-2-3]

Menter的K- $\omega$ 剪切应力输运（SST）湍流模型[68]、[69]把Wilcox的K- $\omega$ 模型[70]、[17]与一个高雷诺数K- $\varepsilon$ 模型（变换为K- $\omega$ 形式）合并在一起。SST模型力求结合两种模型的优点。因此，在边界层的底层内采用K- $\omega$ 方法，原因是K- $\omega$ 模型不需要阻尼函数，这使得在精度相近的情况下，数值稳定性比K- $\varepsilon$ 模型显著提高。此外，在边界层的对数区内也使用K- $\omega$ 模型，因为在逆压流动和可压缩流动中它优于K- $\varepsilon$ 方法。另一方面，在边界层的尾迹区采用K- $\varepsilon$ 模型，因为K- $\omega$ 模型对 $\omega$ 的自由来流值非常敏感[71]。自由剪切层中也使用K- $\varepsilon$ 方法，因为它对尾流、射流和混合层的精度是一个较好的折中。en

The K- $\omega$ Shear Stress Transport (SST) turbulence model of Menter [68], [69] merges the K- $\omega$ model of Wilcox [70], [17] with a high Reynolds number K- $\varepsilon$ model (transformed into the K- $\omega$ formulation). The SST model seeks to combine the positive features of both models. Therefore, the K- $\omega$ approach is employed in the sublayer of the boundary layer. The reason is that the K- $\omega$ model needs no damping function. This leads, for similar accuracy, to significantly higher numerical stability in comparison to the K- $\varepsilon$ model. Furthermore, the K- $\omega$ model is also utilised in logarithmic part of the boundary layer, where it is superior to the K- $\varepsilon$ approach in adverse pressure flows and in compressible flows. On the other hand, the K- $\varepsilon$ model is employed in the wake region of the boundary layer because the K- $\omega$ model is strongly sensitive to the freestream value of $\omega$ [71]. The K- $\varepsilon$ approach is also used in free shear layers since it represents a fair compromise in accuracy for wakes, jets, and mixing layers.

SST湍流模型的一个独特之处是修改了的湍流涡黏性函数，其目的是提高对强逆压梯度流动和压力诱导边界层分离的预测精度。这一修改考虑了湍流剪切应力的输运，其依据是Bradshaw的观察：主剪切应力与湍动能成正比。en

One distinct feature of the SST turbulence model is the modified turbulent eddy-viscosity function. The purpose is to improve the accuracy of prediction of flows with strong adverse pressure gradients and of pressure-induced boundary layer separation. The modification accounts for the transport of the turbulent shear stress. It is based on the observation of Bradshaw that the principal shear stress is proportional to the turbulent kinetic energy.

SST模型的一个缺点是必须显式地知道到最近壁面的距离，这在多块结构网格或非结构网格上需要特殊处理。壁面距离的计算参见例如文献[39]。SST湍流模型的应用实例可见[72]-[74]。en

A certain disadvantage of the SST model is that distances to the nearest wall have to be known explicitly. This requires special provisions on multiblock structured or on unstructured grids. See, e.g., Ref. [39] for the computation of wall distances. Examples for applications of the SST turbulence model can be found in [72]-[74].

Differential Form 微分形式

湍动能与湍流比耗散率的输运方程的微分形式为[68]en

The transport equations for the turbulent kinetic energy and the specific dissipation of turbulence read in differential form [68]

\begin{aligned} \frac{\partial\rho K}{\partial t} + \frac{\partial}{\partial x_j}\left(\rho v_j K\right) &= \frac{\partial}{\partial x_j}\left[\left(\mu_L + \sigma_K\mu_T\right)\frac{\partial K}{\partial x_j}\right] + \tau_{ij}^F S_{ij} - \beta^{*}\rho\omega K \\ \frac{\partial\rho\omega}{\partial t} + \frac{\partial}{\partial x_j}\left(\rho v_j\omega\right) &= \frac{\partial}{\partial x_j}\left[\left(\mu_L + \sigma_{\omega}\mu_T\right)\frac{\partial\omega}{\partial x_j}\right] + \frac{C_{\omega}\rho}{\mu_T}\tau_{ij}^F S_{ij} \\ &- \beta\rho\omega^2 + 2\left(1 - f_1\right)\frac{\rho\sigma_{\omega2}}{\omega}\frac{\partial K}{\partial x_j}\frac{\partial\omega}{\partial x_j}. \end{aligned} \tag{7.65}

方程(7.65)右端各项分别代表守恒性扩散、涡黏性生成和耗散；此外， $\omega$ 方程中的最后一项描述交叉扩散（cross diffusion）。Favre平均湍流应力 $\tau_{ij}^F$ 由式(7.25)给出，应变率张量 $S_{ij}$ 由式(7.3)得到。式(7.28)和(7.29)中的湍流涡黏性由下式得到[68]en

The terms on the right-hand side of Eq. (7.65) represent conservative diffusion, eddy-viscosity production and dissipation, respectively. Furthermore, the last term in the $\omega$ -equation describes the cross diffusion. The Favre-averaged turbulent stresses $\tau_{ij}^F$ are given by Eq. (7.25) and the strain-rate tensor $S_{ij}$ follows from Eq. (7.3). The turbulent eddy viscosity in Eq. (7.28) and (7.29) is obtained from [68]

\mu_T = \frac{a_1\rho K}{\max\left(a_1\omega,\ f_2\|\text{curl}\,\vec{v}\|_2\right)}. \tag{7.66}

湍流黏性的这一定义保证了在逆压梯度边界层内——那里 $$K$$ 的生成大于其耗散 $\omega$ （因而 $a_1\omega < \|\text{curl}\,\vec{v}\|_2$ ）——Bradshaw假设，即 $\tau = a_1\rho K$ （剪切应力正比于湍动能）得到满足。en

This definition of the turbulent viscosity guarantees that in an adverse pressure gradient boundary layer, where the production of $$K$$ is larger than its dissipation $\omega$ (hence $a_1\omega < \|\text{curl}\,\vec{v}\|_2$ ), Bradshaw's assumption, i.e., $\tau = a_1\rho K$ (shear stress proportional to turbulent kinetic energy) is satisfied.

方程(7.65)中的函数 $$f_1$$ 用于把边界层内K- $\omega$ 模型的模型系数与自由剪切层及自由来流区中变换后的K- $\varepsilon$ 模型的系数混合起来，其定义为en

The function $$f_1$$ in Eq. (7.65), which blends the model coefficients of the K- $\omega$ model in boundary layers with the transformed K- $\varepsilon$ model in free-shear layers and freestream zones, is defined as

\begin{aligned} f_1 &= \tanh\left(arg_1^4\right) \\ arg_1 &= \min\left[\max\left(\frac{\sqrt{K}}{0.09\,\omega d},\ \frac{500\,\mu_L}{\rho\omega d^2}\right),\ \frac{4\rho\sigma_{\omega2}K}{CD_{K\omega}d^2}\right], \end{aligned} \tag{7.67}

其中 $$d$$ 表示到最近壁面的距离， $CD_{K\omega}$ 是方程(7.65)中交叉扩散项的正部，即en

where $$d$$ stands for the distance to the nearest wall and $CD_{K\omega}$ is the positive part of the cross-diffusion term in Eq. (7.65), i.e.,

CD_{K\omega} = \max\left(2\,\frac{\rho\sigma_{\omega2}}{\omega}\frac{\partial K}{\partial x_j}\frac{\partial\omega}{\partial x_j},\ 10^{-20}\right). \tag{7.68}

方程(7.66)中的辅助函数 $$f_2$$ 由下式给出en

The auxiliary function $$f_2$$ in Eq. (7.66) is given by

\begin{aligned} f_2 &= \tanh\left(arg_2^2\right) \\ arg_2 &= \max\left(\frac{2\sqrt{K}}{0.09\,\omega d},\ \frac{500\,\mu_L}{\rho\omega d^2}\right). \end{aligned} \tag{7.69}

模型常数如下en

The model constants are as follows

a_1 = 0.31, \qquad \beta^{*} = 0.09, \qquad \kappa = 0.41. \tag{7.70}

最后，SST湍流模型的系数 $\beta$ 、 $C_{\omega}$ 、 $\sigma_K$ 和 $\sigma_{\omega}$ 由K- $\omega$ 模型的系数（记作 $\phi_1$ ）与变换后的K- $\varepsilon$ 模型的系数（ $\phi_2$ ）混合得到。相应的关系式为en

Finally, the coefficients of the SST turbulence model $\beta$ , $C_{\omega}$ , $\sigma_K$ , and $\sigma_{\omega}$ are obtained by blending the coefficients of the K- $\omega$ model, denoted as $\phi_1$ , with those of the transformed K- $\varepsilon$ model ( $\phi_2$ ). The corresponding relation reads

\phi = f_1\phi_1 + \left(1 - f_1\right)\phi_2. \tag{7.71}

内层模型（K- $\omega$ ）的系数为en

The coefficients of the inner model (K- $\omega$ ) are given by

\begin{aligned} &\sigma_{K1} = 0.85, \qquad \sigma_{\omega1} = 0.5, \qquad \beta_1 = 0.075, \\ &C_{\omega1} = \beta_1/\beta^{*} - \sigma_{\omega1}\kappa^2/\sqrt{\beta^{*}} = 0.533. \end{aligned} \tag{7.72}

外层模型（K- $\varepsilon$ ）的系数定义为en

The coefficients of the outer model (K- $\varepsilon$ ) are defined as

\begin{aligned} &\sigma_{K2} = 1.0, \qquad \sigma_{\omega2} = 0.856, \qquad \beta_2 = 0.0828, \\ &C_{\omega2} = \beta_2/\beta^{*} - \sigma_{\omega2}\kappa^2/\sqrt{\beta^{*}} = 0.440. \end{aligned} \tag{7.73}

SST湍流模型的积分形式原则上与小节7.2.2中K- $\varepsilon$ 模型的积分形式相同，故此处不再重复。en

The integral formulation of the SST turbulence model corresponds, in principle, to that of the K- $\varepsilon$ model from Subsection 7.2.2. Therefore, it is not repeated here.

Boundary Conditions 边界条件

固体壁面上湍动能与比耗散率的边界条件为en

The boundary conditions for the kinetic turbulent energy and the specific dissipation at solid walls are

K = 0 \qquad \text{and} \qquad \omega = 10\,\frac{6\mu_L}{\rho\beta_1\left(d_1\right)^2} \tag{7.74}

其中 $$d_1$$ 为第一个节点（单元质心）到壁面的距离。网格须加密至 $y^{+} < 3$ 。en

with $$d_1$$ being the distance of the first node (cell centroid) from the wall. The grid has to be refined such that $y^{+} < 3$ .

对入口边界，建议采用如下自由来流值en

For the inflow boundaries, the following freestream values are recommended

\omega_{\infty} = C_1\frac{\|\vec{v}_{\infty}\|_2}{L}, \qquad \left(\mu_T\right)_{\infty} = \left(\mu_L\right)_{\infty}10^{-C_2}, \qquad K_{\infty} = \frac{\left(\mu_T\right)_{\infty}}{\rho_{\infty}}\,\omega_{\infty}, \tag{7.75}

其中 $$L$$ 表示计算域的长度， $1 \le C_1 \le 10$ ， $2 \le C_2 \le 5$ 。在出口边界上， $$K$$ 和 $\omega$ 的值由计算域内部外推。en

where $$L$$ denotes the length of the computational domain, $1 \le C_1 \le 10$ and $2 \le C_2 \le 5$ , respectively. The values of $$K$$ and $\omega$ are extrapolated from the interior at outflow boundaries.