7.1.7 Reynolds-Stress Transport Equation 雷诺应力输运方程[cfd-7-1-7]

通过取时间平均（二阶矩），可以为雷诺应力导出精确方程en

It is possible to derive exact equations for the Reynolds stresses by taking the time average (second-order moment)

\overline{v'_i\mathcal{N}(v_j)+v'_j\mathcal{N}(v_i)}=0, \tag{7.32}

其中 $\mathcal{N}(v_i)$ 表示Navier-Stokes算子，即en

where $\mathcal{N}(v_i)$ denotes the Navier-Stokes operator, i.e.,

\mathcal{N}(v_i)=\rho\frac{\partial v_i}{\partial t}+\rho v_j\frac{\partial v_i}{\partial x_j}+\frac{\partial p}{\partial x_i}-\mu\nabla^2 v_i. \tag{7.33}

把平均式(7.32)与式(7.33)联立，便得到如下雷诺应力输运方程（Reynolds-stress transport equation）[31]en

Using the average Eq. (7.32) together with Eq. (7.33), we obtain the following Reynolds-stress transport equation [31]

\frac{\partial\tau^R_{ij}}{\partial t}+\bar{v}_k\frac{\partial\tau^R_{ij}}{\partial x_k}=P_{ij}+\Pi_{ij}-\varepsilon_{ij}-\frac{\partial C_{ijk}}{\partial x_k}+\mu\nabla^2\tau^R_{ij} \tag{7.34}

它适用于不可压缩流动。可压缩流动的表述见文献[17]第179页，或[32]、[33]。方程(7.34)中，湍动能的生成项 $P_{ij}$ 、压力—应变项 $\Pi_{ij}$ 、耗散率项 $\varepsilon_{ij}$ 以及三阶扩散项 $C_{ijk}$ 定义为en

for incompressible flow. The formulation for compressible flows can be found in Ref. [17], p. 179, or in [32], [33]. The production of the turbulent kinetic energy $P_{ij}$ , the pressure-strain term $\Pi_{ij}$ , the dissipation-rate term $\varepsilon_{ij}$ , and the third-order diffusion term $C_{ijk}$ in Eq. (7.34) are defined as

\begin{aligned} P_{ij}&=-\tau^R_{ik}\frac{\partial\bar{v}_j}{\partial x_k}-\tau^R_{jk}\frac{\partial\bar{v}_i}{\partial x_k}\\ \Pi_{ij}&=\overline{p'\left(\frac{\partial v'_i}{\partial x_j}+\frac{\partial v'_j}{\partial x_i}\right)}=2\,\overline{p'S'_{ij}}\\ \varepsilon_{ij}&=2\mu\,\overline{\frac{\partial v'_i}{\partial x_k}\frac{\partial v'_j}{\partial x_k}}\\ C_{ijk}&=\rho\overline{v'_iv'_jv'_k}+\overline{p'v'_i}\,\delta_{jk}+\overline{p'v'_j}\,\delta_{ik}. \end{aligned} \tag{7.35}

在方程(7.35)中， $S'_{ij}$ 表示应变率张量的脉动部分。 $C_{ijk}$ 的第一部分（三重速度项）表示由脉动对流驱动的输运，另外两部分则分别是压力输运项（压力—速度关联）。en

In Eq. (7.35), $S'_{ij}$ denotes the fluctuating part of the strain-rate tensor. The first part of $C_{ijk}$ , the triple velocity term, represents transport driven by fluctuating convection, the two other parts are the pressure transport terms (pressure-velocity correlations), respectively.

如我们所见，精确的雷诺应力方程包含新的未知高阶关联（例如 $\overline{v'_iv'_jv'_k}$ ）。因此，方程(7.34)只能借助经验模型来封闭。这是由Navier-Stokes方程的非线性本质造成的。二阶封闭——雷诺应力模型——为求解方程(7.34)提供了必要的框架。实施的例子可参见文献[34]–[36]。en

As we can see, the exact Reynolds-stress equation contains new unknown higher-order correlations (e.g., $\overline{v'_iv'_jv'_k}$ ). Therefore, Equation (7.34) can be closed only by using empirical models. This is caused by the non-linear nature of the Navier-Stokes equations. The second-order closures -- the Reynolds-stress models -- provide the necessary framework for solving Eq. (7.34). Examples of implementations can be found, e.g., in Refs. [34]--[36].