动量方程中viscous term的离散，书里写的有点问题？
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《The finite volume in computational fluid dynamics》中，$\nabla\mathbf v$写成
\begin{equation}
\nabla \mathbf{v} = \left[
\begin{matrix}
\frac{\partial u}{\partial x} & \frac{\partial v}{\partial x} & \frac{\partial w}{\partial x}\\
\frac{\partial u}{\partial y} & \frac{\partial v}{\partial y} & \frac{\partial w}{\partial y} \\
\frac{\partial u}{\partial z} & \frac{\partial v}{\partial z} & \frac{\partial w}{\partial z}\\
\end{matrix}
\right]
\end{equation}
在书中15.5.1节对SIMPLE的推导中，把$\nabla\cdot\{\mu\nabla\mathbf v\}$作为扩散项离散，把$\nabla\cdot\{\mu(\nabla\mathbf v)^\top\}$作为源项处理，是不是弄反了？ -
@李东岳
问题可能在这儿，Moukalled书p589这两个式子有问题，应该像下面这样写：\begin{equation}
\int_\limits{V_C}\nabla\cdot\{\mu(\nabla\mathbf{v})^\top\}dV=\sum_\limits{f\sim nb(C)}\mathbf{S}_f\cdot\{\mu(\nabla\mathbf{v})_f^\top\}
\end{equation}\begin{equation}
\mathbf{S}_f\cdot\{\mu(\nabla\mathbf{v})^\top_f\} = \left[
\begin{matrix}
\frac{\partial u}{\partial x}S_f^x + \frac{\partial v}{\partial x}S_f^y + \frac{\partial w}{\partial x}S_f^z\\
\frac{\partial u}{\partial y}S_f^x + \frac{\partial v}{\partial y}S_f^y + \frac{\partial w}{\partial y}S_f^z \\
\frac{\partial u}{\partial z}S_f^x + \frac{\partial v}{\partial z}S_f^y + \frac{\partial w}{\partial z}S_f^z\\
\end{matrix}
\right]
\end{equation}就是说，把$\nabla\cdot\{\mu\nabla\mathbf v\}$作为diffusion，把$\nabla\cdot\{\mu(\nabla\mathbf v)^\top\}$作为source处理是对的。diffusion term用高斯定理积分后应该是
\begin{equation}
\int_\limits{V_C}\nabla\cdot\{\mu(\nabla\mathbf{v})\}dV=\sum_\limits{f\sim nb(C)}\mathbf{S}_f\cdot\{\mu(\nabla\mathbf{v})\}
\end{equation}\begin{equation}
\mathbf{S}_f\cdot\{\mu(\nabla\mathbf{v})\} = \left[
\begin{matrix}
\frac{\partial u}{\partial x}S_f^x + \frac{\partial u}{\partial y}S_f^y + \frac{\partial u}{\partial z}S_f^z\\
\frac{\partial v}{\partial x}S_f^x + \frac{\partial v}{\partial y}S_f^y + \frac{\partial v}{\partial z}S_f^z \\
\frac{\partial w}{\partial x}S_f^x + \frac{\partial w}{\partial y}S_f^y + \frac{\partial w}{\partial z}S_f^z\\
\end{matrix}
\right]
\end{equation}$\nabla\cdot\{\mu\nabla\mathbf v\}$和(5)的第$i$分量分别和Ferziger的Computational Method for Fluid Dynamics p185式(7.8)的微分形式和积分形式一致。
其中
\begin{equation}
\nabla u_i\cdot\mathbf ndS=\frac{\partial u_i}{\partial x}S_f^x + \frac{\partial u_i}{\partial y}S_f^y + \frac{\partial u_i}{\partial z}S_f^z
\end{equation}
总结，$\nabla\cdot\{\mu\nabla\mathbf v\}$是扩散项没问题，可以应用中心差分等数值格式计算。Moukalled书p589的式(15.65)张量形式的高斯定理写错了，$\mathbf{a}\cdot\{\tau\}$写成了$\{\tau\}\cdot\mathbf{a}$，式(15.66)展开成分量也就跟着错了。麻烦大佬看看我理解的对不对？ -
@李东岳 感谢大佬回复，我说的可能有点乱，核心问题就是，$\nabla\cdot\{\tau\}$拆成两部分：
\begin{equation}
\begin{aligned}
&\int_\limits{V_C}\nabla\cdot\{\tau\}dV\\
=&\int_\limits{V_C}\nabla\cdot\{\mu\nabla\mathbf v\}dV+\int_\limits{V_C}\nabla\cdot\{\mu(\nabla\mathbf v)^\top\}dV\\
=&\sum\limits_f\Box\cdot\mathbf{S}_f+\sum\limits_f\Box\cdot\mathbf{S}_f
\end{aligned}
\end{equation}不管$\Box$里面是分子布局还是分母布局，反正可以得到下面两项：
\begin{equation}
\sum\limits_f\Box\cdot\mathbf{S}_f=\sum\limits_f\left[
\begin{matrix}
\frac{\partial u}{\partial x}S_f^x + \frac{\partial u}{\partial y}S_f^y + \frac{\partial u}{\partial z}S_f^z\\
\frac{\partial v}{\partial x}S_f^x + \frac{\partial v}{\partial y}S_f^y + \frac{\partial v}{\partial z}S_f^z \\
\frac{\partial w}{\partial x}S_f^x + \frac{\partial w}{\partial y}S_f^y + \frac{\partial w}{\partial z}S_f^z\\
\end{matrix}
\right]
\end{equation}\begin{equation}
\sum\limits_f\Box\cdot\mathbf{S}_f=\sum\limits_f\left[
\begin{matrix}
\frac{\partial u}{\partial x}S_f^x + \frac{\partial v}{\partial x}S_f^y + \frac{\partial w}{\partial x}S_f^z\\
\frac{\partial u}{\partial y}S_f^x + \frac{\partial v}{\partial y}S_f^y + \frac{\partial w}{\partial y}S_f^z \\
\frac{\partial u}{\partial z}S_f^x + \frac{\partial v}{\partial z}S_f^y + \frac{\partial w}{\partial z}S_f^z\\
\end{matrix}
\right]
\end{equation}应该是前面一个，式(8)作为diffusion处理，对每个分量都有$\nabla\phi\cdot\mathbf S_f=\frac{\phi_F-\phi_C}{d_{CF}}S_f$（正交网格）。是这样吗？
