如果把时间平均（式(7.8)）或系综平均（式(7.10)）应用于不可压缩Navier-Stokes方程(7.6)，便得到质量守恒和动量守恒的如下关系en
If we apply either the time averaging Eq. (7.8) or the ensemble averaging Eq. (7.10) to the incompressible Navier-Stokes equations (7.6), we obtain the following relations for the mass and momentum conservation
\[\begin{aligned}
&\frac{\partial\bar{v}_i}{\partial x_i}=0\\
&\rho\frac{\partial\bar{v}_i}{\partial t}+\rho\bar{v}_j\frac{\partial\bar{v}_i}{\partial x_j}=-\frac{\partial\bar{p}}{\partial x_i}+\frac{\partial}{\partial x_j}\left(\tau_{ij}-\rho\overline{v'_iv'_j}\right).
\end{aligned} \tag{7.14}\]
这就是著名的雷诺平均Navier-Stokes方程（Reynolds-Averaged Navier-Stokes equations，RANS）。方程(7.14)在形式上与Navier-Stokes方程(7.6)完全相同，只是多出如下一项en
These are known as the Reynolds-Averaged Navier-Stokes equations (RANS). The equations (7.14) are formally identical to the Navier-Stokes equations (7.6) with the exception of the additional term
\[\tau^R_{ij}=-\rho\overline{v'_iv'_j}=-\rho\left(\overline{v_iv_j}-\bar{v}_i\bar{v}_j\right), \tag{7.15}\]
该项构成所谓的雷诺应力张量（Reynolds-stress tensor），它表示由湍流脉动引起的动量输运。层流黏性应力仍按式(7.2)和(7.3)用雷诺平均速度分量计算，即en
which constitutes the so-called Reynolds-stress tensor. It represents the transfer of momentum due to turbulent fluctuations. The laminar viscous stresses are evaluated according to Eqs. (7.2) and (7.3) using Reynolds-averaged velocity components, i.e.,
\[\bar{\tau}_{ij}=2\mu\,\overline{S}_{ij}=\mu\left(\frac{\partial\bar{v}_i}{\partial x_j}+\frac{\partial\bar{v}_j}{\partial x_i}\right) \tag{7.16}\]
雷诺应力张量在三维情形下由九个分量组成en
The Reynolds-stress tensor consists in 3D of the nine components
\[\rho\overline{v'_iv'_j}=\begin{bmatrix}
\rho\overline{(v'_1)^2} & \rho\overline{v'_1v'_2} & \rho\overline{v'_1v'_3}\\
\rho\overline{v'_2v'_1} & \rho\overline{(v'_2)^2} & \rho\overline{v'_2v'_3}\\
\rho\overline{v'_3v'_1} & \rho\overline{v'_3v'_2} & \rho\overline{(v'_3)^2}
\end{bmatrix}. \tag{7.17}\]
不过，由于关联中的\(v'_i\)与\(v'_j\)可以互换，雷诺应力张量只包含六个独立分量。各法向应力之和除以密度即定义为湍动能（turbulent kinetic energy），即en
However, since \(v'_i\) and \(v'_j\) in the correlations can be interchanged, the Reynolds-stress tensor contains only six independent components. The sum of the normal stresses divided by density defines the turbulent kinetic energy, i.e.,
\[K=\frac{1}{2}\overline{v'_iv'_i}=\frac{1}{2}\left[\overline{(v'_1)^2}+\overline{(v'_2)^2}+\overline{(v'_3)^2}\right]. \tag{7.18}\]
如我们所见，基于雷诺平均Navier-Stokes方程的湍流建模，其根本问题在于找出六个附加关系式，以使方程(7.14)封闭。我们将在7.1.5–7.1.7小节中介绍基本方法。en
As we can see, the fundamental problem of turbulence modelling based on the Reynolds-averaged Navier-Stokes equations is to find six additional relations in order to close the equations (7.14). We shall introduce the basic methodologies in the Subsections 7.1.5--7.1.7.