7.1.4 Favre- and Reynolds-Averaged Navier-Stokes Equations Favre平均与雷诺平均Navier-Stokes方程[cfd-7-1-4]

在湍流建模中，通常假设Morkovin假设[16]成立。该假设指出：若 $\rho'\ll\bar{\rho}$ ，边界层的湍流结构不会受到密度脉动的显著影响。对于壁面约束流动，直到马赫数约为5这一结论一般都成立。然而，对于高超声速流动或可压缩自由剪切层，则必须计及密度脉动。对于有燃烧或有显著传热的流动，情况也是如此。en

In turbulence modelling, it is quite common to assume that Morkovin's hypothesis [16] is valid. It states that the turbulent structure of a boundary layer is not notably influenced by density fluctuations if $\rho'\ll\bar{\rho}$ . This is generally true for wall-bounded flows up to a Mach number of about five. However, in the case of hypersonic flows or for compressible free shear layers, density fluctuations have to be taken into account. The same holds also for flows with combustion or with significant heat transfer.

对可压缩Navier-Stokes方程(7.1)中的密度和压力应用雷诺平均（式(7.8)或(7.10)），对其余流动变量应用Favre平均（式(7.11)），便得到[17]en

Application of the Reynolds averaging (Eq. (7.8) or (7.10)) to density and pressure, and of the Favre averaging Eq. (7.11) to the remaining flow variables in the compressible Navier-Stokes equations (7.1) yields [17]

\begin{aligned} &\frac{\partial\bar{\rho}}{\partial t}+\frac{\partial}{\partial x_i}\left(\bar{\rho}\tilde{v}_i\right)=0\\ &\frac{\partial}{\partial t}\left(\bar{\rho}\tilde{v}_i\right)+\frac{\partial}{\partial x_j}\left(\bar{\rho}\tilde{v}_j\tilde{v}_i\right)=-\frac{\partial\bar{p}}{\partial x_i}+\frac{\partial}{\partial x_j}\left(\bar{\tau}_{ij}-\bar{\rho}\widetilde{v''_iv''_j}\right)\\ &\frac{\partial}{\partial t}\left(\bar{\rho}\tilde{E}\right)+\frac{\partial}{\partial x_j}\left(\bar{\rho}\tilde{v}_j\tilde{H}\right)=\frac{\partial}{\partial x_j}\left(k\frac{\partial\tilde{T}}{\partial x_j}-\bar{\rho}\widetilde{v''_jh''}+\widetilde{\tau_{ij}v''_i}-\bar{\rho}\widetilde{v''_jK}\right)\\ &\qquad+\frac{\partial}{\partial x_j}\left[\tilde{v}_i\left(\bar{\tau}_{ij}-\bar{\rho}\widetilde{v''_iv''_j}\right)\right]. \end{aligned} \tag{7.19}

这就是Favre平均与雷诺平均Navier-Stokes方程。与雷诺平均类似，动量（和能量）方程中的黏性应力张量要再加上Favre平均雷诺应力张量，即en

These are the Favre- and Reynolds-Averaged Navier-Stokes equations. Similarly to the Reynolds averaging, the viscous stress tensor in the momentum (and energy) equation is extended by the Favre-averaged Reynolds-stress tensor, i.e.,

\tau^F_{ij}=-\bar{\rho}\,\widetilde{v''_iv''_j}. \tag{7.20}

其形式与式(7.17)类似，只是把雷诺平均换成了Favre平均。层流（分子）黏性应力张量 $\bar{\tau}_{ij}$ 的分量按式(7.2)用Favre平均速度分量计算。en

Its form is similar to Eq. (7.17) with Favre instead of Reynolds averaging. The components of the laminar (molecular) viscous stress tensor $\bar{\tau}_{ij}$ are evaluated by Eq. (7.2) using Favre-averaged velocity components.

如果采用Favre平均湍动能的定义，即en

If we employ the definition of the Favre-averaged turbulent kinetic energy, i.e.,

\bar{\rho}\tilde{K}=\frac{1}{2}\,\bar{\rho}\,\widetilde{v''_iv''_i}, \tag{7.21}

就可以把方程(7.19)中的总能量表示为en

we can express the total energy in Eq. (7.19) as

\bar{\rho}\tilde{E}=\bar{\rho}\tilde{e}+\frac{1}{2}\bar{\rho}\tilde{v}_i\tilde{v}_i+\frac{1}{2}\bar{\rho}\widetilde{v''_iv''_i}=\bar{\rho}\tilde{e}+\frac{1}{2}\bar{\rho}\tilde{v}_i\tilde{v}_i+\bar{\rho}\tilde{K}. \tag{7.22}

总焓定义为en

The total enthalpy is defined as

\bar{\rho}\tilde{H}=\bar{\rho}\tilde{h}+\frac{1}{2}\bar{\rho}\tilde{v}_i\tilde{v}_i+\frac{1}{2}\bar{\rho}\widetilde{v''_iv''_i}=\bar{\rho}\tilde{h}+\frac{1}{2}\bar{\rho}\tilde{v}_i\tilde{v}_i+\bar{\rho}\tilde{K}. \tag{7.23}

Favre平均与雷诺平均Navier-Stokes方程(7.19)的各个部分具有如下物理意义[17]：

$\dfrac{\partial}{\partial x_j}\left(k\dfrac{\partial\tilde{T}}{\partial x_j}\right)$ ——热的分子扩散
$\dfrac{\partial}{\partial x_j}\left(\bar{\rho}\widetilde{v''_jh''}\right)$ ——热的湍流输运
$\dfrac{\partial}{\partial x_j}\left(\widetilde{\tau_{ij}v''_i}\right)$ —— $\tilde{K}$ 的分子扩散
$\dfrac{\partial}{\partial x_j}\left(\bar{\rho}\widetilde{v''_jK}\right)$ —— $\tilde{K}$ 的湍流输运
$\dfrac{\partial}{\partial x_j}\left(\tilde{v}_i\bar{\tau}_{ij}\right)$ ——分子应力做的功
$\dfrac{\partial}{\partial x_j}\left(\tilde{v}_i\tau^F_{ij}\right)$ ——Favre平均雷诺应力做的功

The individual parts of the Favre- and Reynolds-averaged Navier-Stokes equations (7.19) have the following physical meaning [17]:

$\dfrac{\partial}{\partial x_j}\left(k\dfrac{\partial\tilde{T}}{\partial x_j}\right)$ - molecular diffusion of heat
$\dfrac{\partial}{\partial x_j}\left(\bar{\rho}\widetilde{v''_jh''}\right)$ - turbulent transport of heat
$\dfrac{\partial}{\partial x_j}\left(\widetilde{\tau_{ij}v''_i}\right)$ - molecular diffusion of $\tilde{K}$
$\dfrac{\partial}{\partial x_j}\left(\bar{\rho}\widetilde{v''_jK}\right)$ - turbulent transport of $\tilde{K}$
$\dfrac{\partial}{\partial x_j}\left(\tilde{v}_i\bar{\tau}_{ij}\right)$ - work done by the molecular stresses
$\dfrac{\partial}{\partial x_j}\left(\tilde{v}_i\tau^F_{ij}\right)$ - work done by the Favre-averaged Reynolds stresses

$\tilde{K}$ 的分子扩散和湍流输运在很多时候都可以忽略。对于跨声速和超声速流动，这是有效的近似。为使Favre平均与雷诺平均方程(7.19)封闭，还必须提供Favre平均雷诺应力张量（式(7.20)）的六个分量以及湍流热通量向量的三个分量。我们将在后面几个小节中讨论这三种基本方法。en

The molecular diffusion and turbulent transport of $\tilde{K}$ are very often neglected. This is a valid approximation for transonic and supersonic flows. In order to close the Favre- and Reynolds-averaged equations (7.19), we also have to supply six components of the Favre-averaged Reynolds-stress tensor (Eq. (7.20)) and three components of the turbulent heat-flux vector. We shall discuss the three basic approaches in the next subsections.