最近正在学习偏微分方程数值解法这门课程，发现其中很多内容在《计算流体力学入门》中存在较多的重叠部分，知识点有点多，故对所学习的进行一个大概的整理。
流体力学控制方程
黏性流动方程（N-S方程）
对于非定常、三维可压缩黏性流动，其控制方程为 连续方程 非守恒型 \[\frac{D\rho }{Dt}+ \rho \boldsymbol{\nabla} \cdot \boldsymbol{V}= 0\] 守恒型 \[\frac{\partial \rho }{\partial t}+\boldsymbol{\nabla}\cdot \left ( \rho \boldsymbol{V} \right )= 0\]
动量方程 非守恒型 x方向表达式： \[\begin{align} \rho \frac{Du}{Dt}= -\frac{\partial p}{\partial x}+\frac{\partial \tau _{xx}}{\partial x}+\frac{\partial \tau _{yx}}{\partial y}+\frac{\partial \tau \_{zx}}{\partial z}+\rho f\_{x} \end{align}\]
y方向表达式： \[\begin{align} \rho \frac{D\upsilon }{Dt}= -\frac{\partial p}{\partial y}+\frac{\partial \tau _{xy}}{\partial x}+\frac{\partial \tau _{yy}}{\partial y}+\frac{\partial \tau \_{zy}}{\partial z}+\rho f\_{y} \end{align}\]
z方向表达式： \[\begin{align} \rho \frac{D\omega }{Dt}= -\frac{\partial p}{\partial z}+\frac{\partial \tau _{xz}}{\partial x}+\frac{\partial \tau _{yz}}{\partial y}+\frac{\partial \tau \_{zz}}{\partial z}+\rho f\_{z} \end{align}\]
守恒型 x方向表达式： \[\begin{align} \frac{\partial\left ( \rho u\right ) }{\partial t}+\boldsymbol{\nabla}\cdot \left ( \rho u \boldsymbol{V}\right )= -\frac{\partial p}{\partial x}+\frac{\partial \tau _{xx}}{\partial x}+\frac{\partial \tau _{yx}}{\partial y}+\frac{\partial \tau \_{zx}}{\partial z}+\rho f\_{x} \end{align}\]
y方向表达式： \[\begin{align} \frac{\partial \left ( \rho \upsilon\right ) }{\partial t}+\boldsymbol{\nabla}\cdot \left ( \rho \upsilon \boldsymbol{V}\right )= -\frac{\partial p}{\partial y}+\frac{\partial \tau _{xy}}{\partial x}+\frac{\partial \tau _{yy}}{\partial y}+\frac{\partial \tau \_{zy}}{\partial z}+\rho f\_{y} \end{align}\]
z方向表达式： \[\begin{align} \frac{\partial \left ( \rho \omega \right ) }{\partial t}+\boldsymbol{\nabla}\cdot \left ( \rho \omega \boldsymbol{V}\right )= -\frac{\partial p}{\partial z}+\frac{\partial \tau _{xz}}{\partial x}+\frac{\partial \tau _{yz}}{\partial y}+\frac{\partial \tau \_{zz}}{\partial z}+\rho f\_{z} \end{align}\]
能量方程 非守恒型 \[ \begin{align} \rho \frac{D}{Dt}\left ( e+\frac{V^{2}}{2} \right )&=\rho \dot{q}+\frac{\partial }{\partial x}\left ( K\frac{\partial T}{\partial x} \right )+\frac{\partial }{\partial y}\left ( K\frac{\partial T}{\partial y} \right )+\frac{\partial }{\partial z}\left ( K\frac{\partial T}{\partial z} \right )-\frac{\partial \left ( up \right )}{\partial x}-\frac{\partial \left ( \upsilon p \right )}{\partial y}-\frac{\partial \left ( \omega p \right )}{\partial z}\\\\ &+\frac{\partial \left ( u\tau _{xx} \right )}{\partial x}+\frac{\partial \left ( u\tau _{yx} \right )}{\partial y}+\frac{\partial \left ( u\tau _{zx} \right )}{\partial z}+\frac{\partial \left ( \upsilon \tau _{xy} \right )}{\partial x}+\frac{\partial \left ( \upsilon \tau _{yy} \right )}{\partial y}+\frac{\partial \left ( \upsilon \tau _{zy} \right )}{\partial z}+\frac{\partial \left ( \omega \tau _{xz} \right )}{\partial x}\\\\ &+\frac{\partial \left ( \omega \tau _{yz} \right )}{\partial y}+\frac{\partial \left ( \omega \tau _{zz} \right )}{\partial z}+\rho \boldsymbol{f}\cdot \boldsymbol{V} \end{align} \]
守恒型 \[ \begin{align} \frac{\partial }{\partial t}\left [ \rho \left ( e+\frac{V^{2}}{2}\right ) \right ]+\boldsymbol{\nabla}\cdot \left [ \rho \left ( e+\frac{V^{2}}{2}\right )\boldsymbol{V} \right ]&=\rho \dot{q}+\frac{\partial }{\partial x}\left ( K\frac{\partial T}{\partial x} \right )+\frac{\partial }{\partial y}\left ( K\frac{\partial T}{\partial y} \right )+\frac{\partial }{\partial z}\left ( K\frac{\partial T}{\partial z} \right )-\frac{\partial \left ( up \right )}{\partial x}\\\\ &-\frac{\partial \left ( \upsilon p \right )}{\partial y}-\frac{\partial \left ( \omega p \right )}{\partial z}+\frac{\partial \left ( u\tau _{xx} \right )}{\partial x}+\frac{\partial \left ( u\tau _{yx} \right )}{\partial y}+\frac{\partial \left ( u\tau _{zx} \right )}{\partial z}+\frac{\partial \left ( \upsilon \tau _{xy} \right )}{\partial x}\\\\ &+\frac{\partial \left ( \upsilon \tau _{yy} \right )}{\partial y}+\frac{\partial \left ( \upsilon \tau _{zy} \right )}{\partial z}+\frac{\partial \left ( \omega \tau _{xz} \right )}{\partial x}+\frac{\partial \left ( \omega \tau _{yz} \right )}{\partial y}+\frac{\partial \left ( \omega \tau _{zz} \right )}{\partial z}+\rho \boldsymbol{f}\cdot \boldsymbol{V} \end{align} \]
无黏流动方程（欧拉方程）
对于非定常、三维可压缩无黏流动，其控制方程为 连续方程 非守恒型 \[\frac{D\rho }{Dt}+ \rho \boldsymbol{\nabla} \cdot \boldsymbol{V}= 0\] 守恒型 \[\frac{\partial \rho }{\partial t}+\boldsymbol{\nabla}\cdot \left ( \rho \boldsymbol{V} \right )= 0\]
动量方程 非守恒型 x方向表达式：\(\rho \frac{Du}{Dt}= -\frac{\partial p}{\partial x}+\rho f_{x}\)
y方向表达式：\(\rho \frac{D\upsilon }{Dt}= -\frac{\partial p}{\partial y}+\rho f_{y}\)
z方向表达式：\(\rho \frac{D\omega }{Dt}= -\frac{\partial p}{\partial z}+\rho f_{z}\)
守恒型 x方向表达式：\(\frac{\partial\left ( \rho u\right ) }{\partial t}+\boldsymbol{\nabla}\cdot \left ( \rho u \boldsymbol{V}\right )=-\frac{\partial p}{\partial x}+\rho f_{x}\)
y方向表达式：\(\frac{\partial \left ( \rho \upsilon\right ) }{\partial t}+\boldsymbol{\nabla}\cdot \left ( \rho \upsilon \boldsymbol{V}\right )= -\frac{\partial p}{\partial y}+\rho f_{y}\)
z方向表达式：\(\frac{\partial \left ( \rho \omega \right ) }{\partial t}+\boldsymbol{\nabla}\cdot \left ( \rho \omega \boldsymbol{V}\right )= -\frac{\partial p}{\partial z}+\rho f_{z}\)
能量方程 非守恒型 \[\rho \frac{D}{Dt}\left ( e+\frac{V^{2}}{2} \right )=\rho \dot{q}-\frac{\partial \left ( up \right )}{\partial x}-\frac{\partial \left ( \upsilon p \right )}{\partial y}-\frac{\partial \left ( \omega p \right )}{\partial z}+\rho \boldsymbol{f}\cdot \boldsymbol{V}\]
守恒型 \[\frac{\partial }{\partial t}\left [ \rho \left ( e+\frac{V^{2}}{2}\right ) \right ]+\boldsymbol{\nabla}\cdot \left [ \rho \left ( e+\frac{V^{2}}{2}\right )\boldsymbol{V} \right ]=\rho \dot{q}-\frac{\partial \left ( up \right )}{\partial x}-\frac{\partial \left ( \upsilon p \right )}{\partial y}-\frac{\partial \left ( \omega p \right )}{\partial z}+\rho \boldsymbol{f}\cdot \boldsymbol{V}\]
适用于CFD的控制方程形式
守恒型的连续、动量和能量方程均可以表示成统一的形式。由下式给出： \[\frac{\partial \boldsymbol{U}}{\partial t}+\frac{\partial \boldsymbol{F}}{\partial x}+\frac{\partial \boldsymbol{G}}{\partial y}+\frac{\partial \boldsymbol{H}}{\partial z}=\boldsymbol{J}\]
其中 \[ \boldsymbol{U}=\begin{pmatrix} \rho \\\\ \rho u\\\\ \rho \upsilon \\\\ \rho \omega \\\\ \rho \left ( e+\frac{V^{2}}{2} \right ) \end{pmatrix} \]
\[ \boldsymbol{F}=\begin{pmatrix} \rho u\\\\ \rho u^{2}+p-\tau _{xx}\\\\ \rho \upsilon u-\tau _{xy} \\\\ \rho \omega u-\tau _{xz} \\\\ \rho \left ( e+\frac{V^{2}}{2} \right )u+pu-K\frac{\partial T}{\partial x}-u\tau _{xx}-\upsilon \tau _{xy}-\omega \tau _{xz} \end{pmatrix} \]
\[ \boldsymbol{G}=\begin{pmatrix} \rho \upsilon \\\\ \rho u\upsilon -\tau _{yx}\\\\ \rho \upsilon^{2} +p-\tau _{yy} \\\\ \rho \omega \upsilon -\tau _{yz} \\\\ \rho \left ( e+\frac{V^{2}}{2} \right )\omega +p\upsilon -K\frac{\partial T}{\partial y}-u\tau _{yx}-\upsilon \tau _{yy}-\omega \tau _{yz} \end{pmatrix} \]
\[ \boldsymbol{H}=\begin{pmatrix} \rho \omega \\\\ \rho u\omega -\tau _{zx}\\\\ \rho \upsilon\omega -\tau _{zy} \\\\ \rho \omega^{2} +p -\tau _{zz} \\\\ \rho \left ( e+\frac{V^{2}}{2} \right )\omega +p\omega -K\frac{\partial T}{\partial z}-u\tau _{zx}-\upsilon \tau _{zy}-\omega \tau _{zz} \end{pmatrix} \]
\[ \boldsymbol{J}=\begin{pmatrix} 0 \\\\ \rho f_{x}\\\\ \rho f_{y} \\\\ \rho f_{z} \\\\ \rho\left ( uf_{x}+\upsilon f_{y}+\omega f_{z} \right )+\rho \dot{q} \end{pmatrix} \]
二阶线性偏微分方程的分类
二阶线性偏微分方程的一般形式为 \[\begin{align} a_{11}u_{xx}+2a_{12}u_{xy}+a_{22}u_{yy}+b_{1}u_{x}+b_{2}u_{y}+cu=f \end{align}\]
其中\(a_{11}\)，\(a_{12}\)，\(a_{22}\)，\(b_{1}\)，\(b_{2}\)，\(c\)，\(f\)均为区域\(\Omega\)上的实值函数且连续可微。
若在区域\(\Omega\)上某点\(\left ( x_{0} ,y_{0}\right )\) \(\Delta =a_{12}^{2}-a_{11}a_{22}>0\)，则称该偏微分方程为双曲型； \(\Delta =a_{12}^{2}-a_{11}a_{22}=0\)，则称该偏微分方程为抛物型； \(\Delta =a_{12}^{2}-a_{11}a_{22}<0\)，则称该偏微分方程为椭圆型；
参考链接
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