7.1.2 Favre (Mass) Averaging Favre(质量)平均[cfd-7-1-2]

在密度不为常数的情形下，对方程(7.1)中的某些量，宜采用密度（质量）加权即Favre分解[14]、[15]来代替雷诺平均。否则，由于出现涉及密度脉动的额外关联，平均后的控制方程会变得复杂得多。最方便的做法是：对密度和压力采用雷诺平均，对速度、内能、焓和温度等其他变量采用Favre平均。Favre平均量（例如速度分量）由如下关系求得[14]、[15]en

In cases where the density is not constant, it is advisable to apply the density (mass) weighted or Favre decomposition [14], [15] to certain quantities in Eq. (7.1) instead of Reynolds averaging. Otherwise, the averaged governing equations would become considerably more complicated due to additional correlations involving density fluctuations. The most convenient way is to employ Reynolds averaging for density and pressure, and Favre averaging for other variables such as velocity, internal energy, enthalpy and temperature. Favre averaged quantities, for example the velocity components, are obtained from the relation [14], [15]

\tilde{v}_i=\frac{1}{\bar{\rho}}\lim_{T\to\infty}\frac{1}{T}\int_t^{t+T}\rho v_i\,dt, \tag{7.11}

其中 $\bar{\rho}$ 表示雷诺平均密度。于是，Favre分解写为en

where $\bar{\rho}$ denotes the Reynolds-averaged density. Hence, the Favre decomposition reads

v_i=\tilde{v}_i+v''_i, \tag{7.12}

其中 $\tilde{v}_i$ 表示平均值， $$v''_i$$ 表示速度 $$v_i$$ 的脉动部分。同样，脉动部分的平均值为零，即 $\widetilde{v''_i}=0$ 。此外，若两个脉动量相关，则其乘积的平均值不为零。例如， $\widetilde{v''_iv''_i}\neq 0$ ，并且一般地 $\widetilde{v''_iv''_j}\neq 0$ 。en

where $\tilde{v}_i$ represents the mean value and $$v''_i$$ the fluctuating part of the velocity $$v_i$$ . Again, the average of the fluctuating part is zero, i.e., $\widetilde{v''_i}=0$ . Furthermore, the average of the product of two fluctuating quantities is not zero, if the quantities are correlated. Hence, for example, $\widetilde{v''_iv''_i}\neq 0$ and in general $\widetilde{v''_iv''_j}\neq 0$ .

对于Favre平均与雷诺平均的混合使用，可以导出如下关系en

The following relationships can be derived for a mix between Favre and Reynolds averaging

\widetilde{\rho v_i}=\bar{\rho}\tilde{v}_i\,,\qquad \overline{\rho v''_i}=0\,,\qquad \text{but }\overline{v''_i}\neq 0. \tag{7.13}

这些关系将在后面的小节中用到。en

These relations will be utilised in later subsections.