7.2.1 Spalart-Allmaras One-Equation Model Spalart-Allmaras 单方程模型[cfd-7-2-1]

Spalart-Allmaras单方程湍流模型[38]对涡黏性变量 $\tilde{\nu}$ 求解输运方程。它是在经验、量纲分析和伽利略不变性的基础上发展起来的，并利用二维混合层、尾流和平板边界层的结果进行了标定。Spalart-Allmaras模型还能对具有逆压梯度的湍流流动给出相当准确的预测。此外，它能够在用户指定的位置实现从层流到湍流的平滑转捩。Spalart-Allmaras模型具有若干良好的数值特性：它是“局部的”，即某一点处的方程不依赖于其他点上的解，因此可以方便地在结构多块网格或非结构网格上实现；它还稳健、向定态收敛快，并且近壁区域只需适中的网格分辨率。en

The Spalart-Allmaras one-equation turbulence model [38] employs transport equation for an eddy-viscosity variable $\tilde{\nu}$ . It was developed based on empiricism, dimensional analysis and Galilean invariance. It was calibrated using results for 2-D mixing layers, wakes and flat-plate boundary layers. The Spalart-Allmaras model also allows for reasonably accurate predictions of turbulent flows with adverse pressure gradients. Furthermore, it is capable of smooth transition from laminar to turbulent flow at user specified locations. The Spalart-Allmaras model has several favourable numerical features. It is "local" which means that the equation at one point does not depend on the solution at other points. Therefore, it can be readily implemented on structured multi-block or on unstructured grids. It is also robust, converges fast to steady-state and requires only moderate grid resolution in the near-wall region.

Differential Form 微分形式

Spalart-Allmaras湍流模型用张量记号可以写成如下形式[38]en

The Spalart-Allmaras turbulence model can be written in tensor notation as follows [38]

\begin{aligned} \frac{\partial \tilde{\nu}}{\partial t} + \frac{\partial}{\partial x_j}\left(\tilde{\nu}\, v_j\right) &= C_{b1}\left(1 - f_{t2}\right)\tilde{S}\,\tilde{\nu} \\ &+ \frac{1}{\sigma}\left\{ \frac{\partial}{\partial x_j}\left[\left(\nu_L + \tilde{\nu}\right)\frac{\partial\tilde{\nu}}{\partial x_j}\right] + C_{b2}\frac{\partial\tilde{\nu}}{\partial x_j}\frac{\partial\tilde{\nu}}{\partial x_j} \right\} \\ &- \left[C_{w1} f_w - \frac{C_{b1}}{\kappa^2} f_{t2}\right]\left(\frac{\tilde{\nu}}{d}\right)^2 + f_{t1}\|\Delta\vec{v}\|_2^2. \end{aligned} \tag{7.36}

右端各项分别代表涡黏性生成、守恒性扩散、非守恒性扩散、近壁湍流破坏、生成的转捩阻尼以及湍流的转捩源。此外， $\nu_L = \mu_L/\rho$ 表示层流运动黏度， $$d$$ 为到最近壁面的距离（壁面距离的计算见文献[39]）。式(7.28)和(7.29)中的湍流涡黏性由下式得到en

The terms on the right-hand side represent eddy-viscosity production, conservative diffusion, non-conservative diffusion, near-wall turbulence destruction, transition damping of production, and transition source of turbulence. Furthermore, $\nu_L = \mu_L/\rho$ denotes the laminar kinematic viscosity and $$d$$ is the distance to the closest wall (for the computation of wall distances see Ref. [39]). The turbulent eddy viscosity in Eq. (7.28) and (7.29) is obtained from

\mu_T = f_{v1}\,\rho\,\tilde{\nu}. \tag{7.37}

生成项用下列公式求值en

The production term is evaluated with the following formulae

\begin{aligned} \tilde{S} &= f_{v3}\, S + \frac{\tilde{\nu}}{\kappa^2 d^2}\, f_{v2}, \\ f_{v1} &= \frac{\chi^3}{\chi^3 + C_{v1}^3}, \qquad f_{v2} = \left(1 + \frac{\chi}{C_{v2}}\right)^{-3}, \\ f_{v3} &= \frac{\left(1 + \chi f_{v1}\right)\left(1 - f_{v2}\right)}{\max(\chi,\, 0.001)}, \qquad \chi = \frac{\tilde{\nu}}{\nu_L} \end{aligned} \tag{7.38}

在方程(7.38)中， $$S$$ 表示平均旋转率的大小，即en

In Equation (7.38), $$S$$ stands for the magnitude of the mean rotation rate, i.e.,

S = \sqrt{2\,\Omega_{ij}\Omega_{ij}}, \tag{4}

其中 $\Omega_{ij}$ 由式(7.4)给出。注意 $\tilde{S}$ 与文献[38]中的原始定义不同，这一修改是Spalart提出的，目的是防止 $\tilde{S}$ 变为零（参见文献[40]，第155页）。en

where $\Omega_{ij}$ is given by Eq. (7.4). Note that $\tilde{S}$ differs from its original definition in [38]. The modification was suggested by Spalart in order to prevent $\tilde{S}$ from reaching zero (cf. Ref. [40], p. 155).

控制涡黏性破坏的项为en

The terms controlling the destruction of the eddy viscosity read

\begin{aligned} f_w &= g\left(\frac{1 + C_{w3}^6}{g^6 + C_{w3}^6}\right)^{1/6}, \\ g &= r + C_{w2}\left(r^6 - r\right), \qquad r = \frac{\tilde{\nu}}{\tilde{S}\,\kappa^2 d^2}. \end{aligned} \tag{7.39}

用于模拟层流-湍流转捩的函数为en

Functions used for modelling the laminar-turbulent transition are given by

\begin{aligned} f_{t1} &= g_t\, C_{t1}\exp\left(-C_{t2}\,\frac{\omega_t^2}{\Delta U^2}\left(d^2 + g_t^2 d_t^2\right)\right), \\ f_{t2} &= C_{t3}\exp\left(-C_{t4}\,\chi^2\right), \qquad g_t = \min\left[0.1,\ \|\Delta\vec{v}\|_2/(\omega_t\,\Delta x_t)\right], \end{aligned} \tag{7.40}

其中 $\omega_t$ 表示转捩触发点（trip point，位置须由用户指定）处壁面上的涡量， $\|\Delta\vec{v}\|_2$ 表示触发点速度与当前流场点速度之差的2-范数， $$d_t$$ 为到最近触发点的距离， $\Delta x_t$ 表示触发点处沿壁面的网格间距。en

where $\omega_t$ represents the vorticity at the wall at the trip point (position has to be specified by the user), $\|\Delta\vec{v}\|_2$ denotes the 2-norm of the difference between the velocity at the trip point and the current field point, $$d_t$$ is the distance to the nearest trip point, and $\Delta x_t$ stands for the spacing along the wall at the trip point.

最后，式(7.36)-(7.40)中的各个常数定义为en

Finally, the various constants in Eqs. (7.36)-(7.40) are defined as

\begin{aligned} &C_{b1} = 0.1355, \qquad C_{b2} = 0.622, \\ &C_{v1} = 7.1, \qquad C_{v2} = 5, \qquad \sigma = 2/3, \qquad \kappa = 0.41, \\ &C_{w1} = C_{b1}/\kappa^2 + (1 + C_{b2})/\sigma, \qquad C_{w2} = 0.3, \qquad C_{w3} = 2, \\ &C_{t1} = 1, \qquad C_{t2} = 2, \qquad C_{t3} = 1.3, \qquad C_{t4} = 0.5. \end{aligned} \tag{7.41}

为了考虑非平衡效应对生成项的影响，最近有文献提出把 $C_{b1}$ 表示为应变率的函数[41]。en

In order to account for non-equilibrium effects on the production term, it was recently proposed to express $C_{b1}$ as a function of the strain rate [41].

正如文献[38]所指出的，把方程(7.36)中的非守恒性扩散项，即en

As pointed out in [38], it is convenient to substitute the non-conservative diffusion term in Eq. (7.36), i.e.,

\frac{1}{\sigma}\left\{ \frac{\partial}{\partial x_j}\left[\left(\nu_L + \tilde{\nu}\right)\frac{\partial\tilde{\nu}}{\partial x_j}\right] + C_{b2}\frac{\partial\tilde{\nu}}{\partial x_j}\frac{\partial\tilde{\nu}}{\partial x_j} \right\} \tag{8}

用下面的表达式来代替en

by the following expression

\frac{1 + C_{b2}}{\sigma}\frac{\partial}{\partial x_j}\left[\left(\nu_L + \tilde{\nu}\right)\frac{\partial\tilde{\nu}}{\partial x_j}\right] - \frac{C_{b2}}{\sigma}\left(\nu_L + \tilde{\nu}\right)\nabla^2\tilde{\nu}. \tag{7.42}

这样就可以避开对项 $\left(\partial\tilde{\nu}/\partial x_j\right)^2$ 做离散化的困难。en

In this way, difficulties with the discretisation of the term $\left(\partial\tilde{\nu}/\partial x_j\right)^2$ are circumvented.

Integral Form 积分形式

把Spalart-Allmaras湍流模型（方程(7.36)）变换到有限体积框架后，形式如下en

The Spalart-Allmaras turbulence model (Eq. (7.36)) reads after the transformation into the finite-volume framework as follows

\frac{\partial}{\partial t}\int_{\Omega}\tilde{\nu}\ d\Omega + \oint_{\partial\Omega}\left(F_{c,T} - F_{v,T}\right)dS = \int_{\Omega}Q_T\ d\Omega, \tag{7.43}

其中 $\Omega$ 表示控制体， $\partial\Omega$ 为其表面， $$dS$$ 是 $\Omega$ 的面元。对流通量定义为en

where $\Omega$ represents the control volume, $\partial\Omega$ its surface, and $$dS$$ is a surface element of $\Omega$ . The convective flux is defined as

F_{c,T} = \tilde{\nu}\, V \tag{7.44}

其中 $$V$$ 为逆变速度（见式(2.22)）。对流通量一般用一阶上风格式离散。黏性通量为en

with $$V$$ being the contravariant velocity (see Eq. (2.22)). The convective flux is in general discretised using a first-order upwind scheme. The viscous flux is given by

F_{v,T} = n_x\tau_{xx}^T + n_y\tau_{yy}^T + n_z\tau_{zz}^T, \tag{7.45}

其中 $$n_x$$ 、 $$n_y$$ 和 $$n_z$$ 是单位法向量的分量。法向黏性应力为en

where $$n_x$$ , $$n_y$$ , and $$n_z$$ are the components of the unit normal vector. The normal viscous stresses read

\begin{aligned} \tau_{xx}^T &= \frac{1}{\sigma}\left(\nu_L + \tilde{\nu}\right)\frac{\partial\tilde{\nu}}{\partial x}, \qquad \tau_{yy}^T = \frac{1}{\sigma}\left(\nu_L + \tilde{\nu}\right)\frac{\partial\tilde{\nu}}{\partial y}, \\ \tau_{zz}^T &= \frac{1}{\sigma}\left(\nu_L + \tilde{\nu}\right)\frac{\partial\tilde{\nu}}{\partial z}. \end{aligned} \tag{7.46}

最后，方程(7.43)中的源项为en

Finally, the source term in Eq. (7.43) becomes

\begin{aligned} Q_T &= C_{b1}\left(1 - f_{t2}\right)\tilde{S}\,\tilde{\nu} + \frac{C_{b2}}{\sigma}\left[\left(\frac{\partial\tilde{\nu}}{\partial x}\right)^2 + \left(\frac{\partial\tilde{\nu}}{\partial y}\right)^2 + \left(\frac{\partial\tilde{\nu}}{\partial z}\right)^2\right] \\ &- \left[C_{w1} f_w - \frac{C_{b1}}{\kappa^2} f_{t2}\right]\left(\frac{\tilde{\nu}}{d}\right)^2 + f_{t1}\|\Delta\vec{v}\|_2^2. \end{aligned} \tag{7.47}

另一种做法是按方程(7.42)所建议的方式来表述非守恒性扩散。模型常数由方程(7.41)给出。en

Alternatively, the non-conservative diffusion can be formulated as suggested by Eq. (7.42). The model constants are provided in Eq. (7.41).

Initial and Boundary Conditions 初始条件与边界条件

$\tilde{\nu}$ 的初值通常取 $\tilde{\nu} = 0.1\,\nu_L$ 。入口边界上也指定同样的值。在出口边界上， $\tilde{\nu}$ 直接由计算域内部外推。在固体壁面上，宜取 $\tilde{\nu} = 0$ ，从而 $\mu_T = 0$ 。en

The initial value of $\tilde{\nu}$ is usually taken as $\tilde{\nu} = 0.1\,\nu_L$ . The same value is also specified at inflow boundaries. At outflow boundaries, $\tilde{\nu}$ is simply extrapolated from the interior of the computational domain. At solid walls, it is appropriate to set $\tilde{\nu} = 0$ and hence $\mu_T = 0$ .