流动方程

速度导数

速度梯度

速度微分 (differential of velocity) 可以表示为矢径微分 (differential of position vector) 与速度梯度 (gradient of velocity) 的点乘之积： $\mathrm{d}\overrightarrow{\mathit{u}}=\mathrm{d}\overrightarrow{\mathit{r}}\mathbf{\cdot }\mathbf{\nabla }\overrightarrow{\mathit{u}}$

其中速度梯度是一个二阶张量：

$$\mathbf{\nabla }\overrightarrow{\mathit{u}}:=\left({\overrightarrow{\mathit{e}}}_i\frac{\partial }{\partial x_i}\right)\left({\overrightarrow{\mathit{e}}}_k{u}_k\right)=\left({\overrightarrow{\mathit{e}}}_i{\overrightarrow{\mathit{e}}}_k\right)\frac{\partial u_k}{\partial x_i}=:{\overrightarrow{\mathit{e}}}_i{\overrightarrow{\mathit{e}}}_k{\partial }_i{u}_k$$

它可以分解为应变率张量 (tensor of strain rate) 与角速度张量 (tensor of angular velocity) 之和：

$$\boxed{{\displaystyle \mathbf{\nabla }\overrightarrow{\mathit{u}}=\tilde{\mathit{E}}+\tilde{\mathit{\Omega }}}}⟸\{\begin{array}{ll}\tilde{\mathit{E}}:={\overrightarrow{\mathit{e}}}_i{\overrightarrow{\mathit{e}}}_k{E}_{ik}& E_{ik}:=({\partial }_i{u}_k+{\partial }_k{u}_i)/2\\ \tilde{\mathit{\Omega }}:={\overrightarrow{\mathit{e}}}_i{\overrightarrow{\mathit{e}}}_k{\Omega }_{ik}& {\Omega }_{ik}:=({\partial }_i{u}_k-{\partial }_k{u}_i)/2\end{array}$$

角速度

反对称的角速度张量只有三个独立分量：

$$\tilde{\mathit{\Omega }}=\left[\begin{array}{ccc}{\overrightarrow{\mathit{e}}}_1& {\overrightarrow{\mathit{e}}}_2& {\overrightarrow{\mathit{e}}}_3\end{array}\right]\left[\begin{array}{ccc}0& +{\Omega }_{12}& -{\Omega }_{31}\\ -{\Omega }_{12}& 0& +{\Omega }_{23}\\ +{\Omega }_{31}& -{\Omega }_{23}& 0\end{array}\right]\left[\begin{array}{c}{\overrightarrow{\mathit{e}}}_1\\ {\overrightarrow{\mathit{e}}}_2\\ {\overrightarrow{\mathit{e}}}_3\end{array}\right]$$

利用这三个独立分量，可以人为构造一个三维角速度矢量 (vector of angular velocity)：

$$\left[\begin{array}{c}{\Omega }_1\\ {\Omega }_2\\ {\Omega }_3\end{array}\right]=\left[\begin{array}{c}{\Omega }_{23}\\ {\Omega }_{31}\\ {\Omega }_{12}\end{array}\right]⟺{\Omega }_{jk}={\Omega }_i{ϵ}_{ijk}\text{其中}{ϵ}_{ijk}=\{\begin{array}{ll}+1& ijk=123,231,312\\ -1& ijk=321,132,213\\ 0& ijk=\cdots \end{array}$$

可以证明：

$$\boxed{{\displaystyle \mathrm{d}\overrightarrow{\mathit{r}}\mathbf{\cdot }\tilde{\mathit{\Omega }}=\overrightarrow{\mathit{\Omega }}\times \mathrm{d}\overrightarrow{\mathit{r}}}}⟸\{\begin{array}{l}\overrightarrow{\mathit{\Omega }}:={\overrightarrow{\mathit{e}}}_i{\Omega }_i\\ \tilde{\mathit{\Omega }}:={\overrightarrow{\mathit{e}}}_j{\overrightarrow{\mathit{e}}}_k{\Omega }_{jk}={\Omega }_i{\overrightarrow{\mathit{e}}}_j{\overrightarrow{\mathit{e}}}_k{ϵ}_{ijk}\end{array}$$

涡量

$$\overrightarrow{\mathit{\omega }}:=\mathbf{\nabla }\times \overrightarrow{\mathit{u}}=\left|\begin{array}{ccc}{\overrightarrow{\mathit{e}}}_1& {\overrightarrow{\mathit{e}}}_2& {\overrightarrow{\mathit{e}}}_3\\ {\partial }_1& {\partial }_2& {\partial }_3\\ u_1& u_2& u_3\end{array}\right|=2\overrightarrow{\mathit{\Omega }}$$

物质导数

Reynolds 输运定理

$$\boxed{{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}{\int }_V\varphi ={\int }_V\frac{\partial \varphi }{\partial t}+{\oint }_{\partial V}\overrightarrow{\mathit{n}}\mathbf{\cdot }{\overrightarrow{\mathit{u}}}_{\mathrm{C}.\mathrm{S}.}\varphi }}$$

控制体上的物质导数

$$\boxed{{\displaystyle \begin{array}{rl}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{V_{\mathrm{M}.\mathrm{B}.}}\varphi =:\frac{\mathrm{D}}{\mathrm{D}t}{\int }_V\varphi & ={\int }_V\frac{\partial \varphi }{\partial t}+{\oint }_{\partial V}\overrightarrow{\mathit{n}}\mathbf{\cdot }\overrightarrow{\mathit{u}}\varphi \\ & =\frac{\mathrm{d}}{\mathrm{d}t}{\int }_V\varphi +{\oint }_{\partial V}\overrightarrow{\mathit{n}}\mathbf{\cdot }(\overrightarrow{\mathit{u}}-{\overrightarrow{\mathit{u}}}_{\mathrm{C}.\mathrm{S}.})\varphi \end{array}}}$$

物质点上的物质导数

$$\boxed{{\displaystyle \frac{\mathrm{D}\varphi }{\mathrm{D}t}\equiv {\mathrm{D}}_t\varphi :={\partial }_t\varphi +\overrightarrow{\mathit{u}}\mathbf{\cdot }\mathbf{\nabla }\varphi }}$$

守恒定律

质量守恒

$$\boxed{{\displaystyle \frac{\mathrm{D}}{\mathrm{D}t}{\int }_V\rho =0}}$$

	积分型	微分型
守恒型	${\int }_V\frac{\partial \rho }{\partial t}+{\oint }_{\partial V}\overrightarrow{\mathit{n}}\mathbf{\cdot }\overrightarrow{\mathit{u}}\rho =0$	$\frac{\partial \rho }{\partial t}+\mathbf{\nabla }\mathbf{\cdot }(\overrightarrow{\mathit{u}}\rho )=0$
非守恒型	$\frac{\mathrm{d}}{\mathrm{d}t}{\int }_V\rho +{\oint }_{\partial V}\overrightarrow{\mathit{n}}\mathbf{\cdot }(\overrightarrow{\mathit{u}}-{\overrightarrow{\mathit{u}}}_{\mathrm{C}.\mathrm{S}.})\rho =0$	$\frac{\mathrm{D}\rho }{\mathrm{D}t}+\rho \mathbf{\nabla }\mathbf{\cdot }\overrightarrow{\mathit{u}}=0$

动量守恒

$$\boxed{{\displaystyle \frac{\mathrm{D}}{\mathrm{D}t}{\int }_V\rho \overrightarrow{\mathit{u}}={\int }_V\overrightarrow{\mathit{b}}+{\oint }_{\partial V}\overrightarrow{\mathit{n}}\mathbf{\cdot }\tilde{\mathit{\sigma }}}}$$

其中 $\tilde{\mathit{\sigma }}$ 为应力张量 (stress tensor)，它与单位法向量 $\overrightarrow{\mathit{n}}$ 的点乘 $\overrightarrow{\mathit{n}}\mathbf{\cdot }\tilde{\mathit{\sigma }}$ 是一个向量，表示作用在物质体边界 $\partial V_{\mathrm{M}.\mathrm{B}.}$ 上的外力的面密度。在流体力学中，应力张量 $\tilde{\mathit{\sigma }}$ 总是被分解为静水压力张量 (hydrostatic pressure tensor) $-p\tilde{\mathbf{1}}$ 与黏性应力张量 (viscous stress tensor) $\tilde{\mathit{\tau }}$ 之和：

$$\tilde{\mathit{\sigma }}=-p\tilde{\mathbf{1}}+\tilde{\mathit{\tau }}⟺{\sigma }_{ik}=-p{\delta }_{ik}+{\tau }_{ik}$$

于是动量守恒定律可以改写为

$$\frac{\mathrm{D}}{\mathrm{D}t}{\int }_V\rho \overrightarrow{\mathit{u}}+{\oint }_{\partial V}\overrightarrow{\mathit{n}}p={\int }_V\overrightarrow{\mathit{b}}+{\oint }_{\partial V}\overrightarrow{\mathit{n}}\mathbf{\cdot }\tilde{\mathit{\tau }}$$

角动量守恒

在（不考虑分布力偶的）流体力学中，它等价于应力张量或黏性应力张量的对称性：

$$\boxed{{\displaystyle \underset{{\sigma }_{ik}}{\underbrace{-p{\delta }_{ik}+{\tau }_{ik}}}=\underset{{\sigma }_{ki}}{\underbrace{-p{\delta }_{ki}+{\tau }_{ki}}}}}$$

能量守恒

$$\boxed{{\displaystyle \frac{\mathrm{D}}{\mathrm{D}t}{\int }_V\rho e_0={\int }_V(\overrightarrow{\mathit{b}}\mathbf{\cdot }\overrightarrow{\mathit{u}}+j)+{\oint }_{\partial V}\overrightarrow{\mathit{n}}\mathbf{\cdot }(\tilde{\mathit{\sigma }}\mathbf{\cdot }\overrightarrow{\mathit{u}}-\overrightarrow{\mathit{q}})}}$$

其中

• 标量 $e_0:=e+\overrightarrow{\mathit{u}}\mathbf{\cdot }\overrightarrow{\mathit{u}}/2$ 为比总能 (specific total energy)，表示单位质量流体的内能与动能之和。
• 标量 $j$ 为热源密度 (density of heat source)，表示单位时间内单位体积热源的热生成量 (heat generation)。
• 向量 $\overrightarrow{\mathit{q}}$ 为热通量密度 (density of heat flux)，表示单位时间内穿过单位面积的热流量 (heat flow)。

Navier–Stokes 方程组

将守恒定律整理成矩阵（方程组）形式，即得 Navier–Stokes 方程组：

$$\boxed{{\displaystyle \frac{\mathrm{D}}{\mathrm{D}t}{\int }_V\underset{―}{U}+{\oint }_{\partial V}\overrightarrow{\mathit{n}}\mathbf{\cdot }\underset{―}{\overrightarrow{\mathit{P}}}={\oint }_{\partial V}\overrightarrow{\mathit{n}}\mathbf{\cdot }\underset{―}{\overrightarrow{\mathit{G}}}+{\int }_V\underset{―}{H}}}$$

其中

$$\underset{―}{U}=\left[\begin{array}{c}\rho \\ \rho \overrightarrow{\mathit{u}}\\ \rho e_0\end{array}\right]\underset{―}{\overrightarrow{\mathit{P}}}=\left[\begin{array}{c}\overrightarrow{\mathbf{0}}\\ p\tilde{\mathbf{1}}\\ p\overrightarrow{\mathit{u}}\end{array}\right]\underset{―}{\overrightarrow{\mathit{G}}}=\left[\begin{array}{c}\overrightarrow{\mathbf{0}}\\ \tilde{\mathit{\tau }}\\ \tilde{\mathit{\tau }}\mathbf{\cdot }\overrightarrow{\mathit{u}}-\overrightarrow{\mathit{q}}\end{array}\right]\underset{―}{H}=\left[\begin{array}{c}0\\ \overrightarrow{\mathit{b}}\\ \overrightarrow{\mathit{b}}\mathbf{\cdot }\overrightarrow{\mathit{u}}+j\end{array}\right]$$

守恒型

利用控制体上的物质导数的展开式，并引入通量 (flux) 矩阵

$$\underset{―}{\overrightarrow{\mathit{F}}}:=\underset{―}{U}\overrightarrow{\mathit{u}}+\underset{―}{\overrightarrow{\mathit{P}}}=\left[\begin{array}{c}\rho \\ \rho \overrightarrow{\mathit{u}}\\ \rho e_0\end{array}\right]\overrightarrow{\mathit{u}}+\left[\begin{array}{c}\overrightarrow{\mathbf{0}}\\ p\tilde{\mathbf{1}}\\ p\overrightarrow{\mathit{u}}\end{array}\right]=\left[\begin{array}{c}\rho \overrightarrow{\mathit{u}}\\ \rho \overrightarrow{\mathit{u}}\overrightarrow{\mathit{u}}+p\tilde{\mathbf{1}}\\ \rho h_0\overrightarrow{\mathit{u}}\end{array}\right]$$

即得守恒型积分方程组：

$${\int }_V{\partial }_t\underset{―}{U}+{\oint }_{\partial V}\overrightarrow{\mathit{n}}\mathbf{\cdot }\underset{―}{\overrightarrow{\mathit{F}}}={\oint }_{\partial V}\overrightarrow{\mathit{n}}\mathbf{\cdot }\underset{―}{\overrightarrow{\mathit{G}}}+{\int }_V\underset{―}{H}$$

对其中的面积分应用 Gauss 散度定理，即得守恒型微分方程组：

$${\partial }_t\underset{―}{U}+\mathbf{\nabla }\mathbf{\cdot }\underset{―}{\overrightarrow{\mathit{F}}}=\mathbf{\nabla }\mathbf{\cdot }\underset{―}{\overrightarrow{\mathit{G}}}+\underset{―}{H}$$

它在三维直角坐标系中的分量形式为

$${\partial }_t\left[\begin{array}{c}\rho \\ \rho u_1\\ \rho u_2\\ \rho u_3\\ \rho e_0\end{array}\right]+{\partial }_{\alpha }\left[\begin{array}{c}\rho u_{\alpha }\\ \rho u_1{u}_{\alpha }+p{\delta }_{1\alpha }\\ \rho u_2{u}_{\alpha }+p{\delta }_{2\alpha }\\ \rho u_3{u}_{\alpha }+p{\delta }_{3\alpha }\\ \rho h_0{u}_{\alpha }\end{array}\right]={\partial }_{\alpha }\left[\begin{array}{c}0\\ {\tau }_{\alpha 1}\\ {\tau }_{\alpha 2}\\ {\tau }_{\alpha 3}\\ {\tau }_{\alpha \beta }{u}_{\beta }-q_{\alpha }\end{array}\right]+\left[\begin{array}{c}0\\ b_1\\ b_2\\ b_3\\ b_{\alpha }{u}_{\alpha }+j\end{array}\right]$$

非守恒型

注意到守恒项 $\underset{―}{U}$ 中的 $\rho$ 可以被提出：

$$\underset{―}{U}=\rho \underset{―}{W}⟸\underset{―}{W}:=\left[\begin{array}{c}1\\ \overrightarrow{\mathit{u}}\\ e_0\end{array}\right]$$

故守恒型积分方程组中的物质导数可移入积分号内，由此即得非守恒型积分方程组：

$${\int }_V\rho \stackrel{{\mathrm{D}}_t}{\overbrace{({\partial }_t+\overrightarrow{\mathit{u}}\mathbf{\cdot }\mathbf{\nabla })}}\underset{―}{W}+{\oint }_{\partial V}\overrightarrow{n}\mathbf{\cdot }\underset{―}{\overrightarrow{\mathit{P}}}={\oint }_{\partial V}\overrightarrow{n}\mathbf{\cdot }\underset{―}{\overrightarrow{\mathit{G}}}+{\int }_V\underset{―}{H}$$

相应地有非守恒型微分方程组：

$$(\rho {\partial }_t+\rho \overrightarrow{\mathit{u}}\mathbf{\cdot }\mathbf{\nabla })\underset{―}{W}+\mathbf{\nabla }\mathbf{\cdot }\underset{―}{\overrightarrow{\mathit{P}}}=\mathbf{\nabla }\mathbf{\cdot }\underset{―}{\overrightarrow{\mathit{G}}}+\underset{―}{H}$$

它在三维直角坐标系中的分量形式为

$$(\rho {\partial }_t+\rho \overrightarrow{\mathit{u}}\mathbf{\cdot }\mathbf{\nabla })\left[\begin{array}{c}\rho \\ u_1\\ u_2\\ u_3\\ e_0\end{array}\right]+\left[\begin{array}{c}{\rho }^2{\partial }_{\alpha }{u}_{\alpha }\\ {\partial }_{1}p\\ {\partial }_{2}p\\ {\partial }_{3}p\\ {\partial }_{\alpha }(p{u}_{\alpha })\end{array}\right]={\partial }_{\alpha }\left[\begin{array}{c}0\\ {\tau }_{\alpha 1}\\ {\tau }_{\alpha 2}\\ {\tau }_{\alpha 3}\\ {\tau }_{\alpha \beta }{u}_{\beta }-q_{\alpha }\end{array}\right]+\left[\begin{array}{c}0\\ b_1\\ b_2\\ b_3\\ b_{\alpha }{u}_{\alpha }+j\end{array}\right]$$

有限体积

间断有限元

局部弱形式

守恒律方程（组）的微分形式：

$$\boxed{{\displaystyle {\partial }_{t}U(\overrightarrow{\mathit{r}},t)+\mathbf{\nabla }\mathbf{\cdot }\overrightarrow{\mathit{F}}(\overrightarrow{\mathit{r}},t)=0}}$$

可借助于测函数 $V(\overrightarrow{\mathit{r}})$ 改写积分形式：

$${\int }_{\Omega }({\partial }_{t}U(\overrightarrow{\mathit{r}},t)+\mathbf{\nabla }\mathbf{\cdot }\overrightarrow{\mathit{F}}(\overrightarrow{\mathit{r}},t))V(\overrightarrow{\mathit{r}})=0\mathrm{\forall }V(\overrightarrow{\mathit{r}}),\mathrm{\forall }\Omega$$

分部积分，即得守恒律方程（组）的弱形式 (weak form)：

$${\int }_{\Omega }V(\overrightarrow{\mathit{r}}){\partial }_{t}U(\overrightarrow{\mathit{r}},t)={\int }_{\Omega }\overrightarrow{\mathit{F}}(\overrightarrow{\mathit{r}},t)\mathbf{\cdot }\mathbf{\nabla }V(\overrightarrow{\mathit{r}})-{\oint }_{\partial \Omega }\overrightarrow{\mathit{\nu }}(\overrightarrow{\mathit{r}})\mathbf{\cdot }\overrightarrow{\mathit{F}}(\overrightarrow{\mathit{r}},t)V(\overrightarrow{\mathit{r}})\mathrm{\forall }V(\overrightarrow{\mathit{r}}),\mathrm{\forall }\Omega$$

正交基函数

选定两组线性独立的基函数 (basis functions)：

$$\underset{―}{\varphi }(\overrightarrow{\mathit{r}})=\left[\begin{array}{ccc}{\varphi }_1(\overrightarrow{\mathit{r}})& \cdots & {\varphi }_K(\overrightarrow{\mathit{r}})\end{array}\right]\underset{―}{\psi }(\overrightarrow{\mathit{r}})=\left[\begin{array}{ccc}{\psi }_1(\overrightarrow{\mathit{r}})& \cdots & {\psi }_L(\overrightarrow{\mathit{r}})\end{array}\right]$$

在由其张成在试函数空间 (space of trial functions) 与测函数空间 (space of test functions) 中，分别寻找未知函数 $U(\overrightarrow{\mathit{r}},t)$ 与测函数 $V(\overrightarrow{\mathit{r}})$ 的最优逼近：

$$U(\overrightarrow{\mathit{r}},t)\approx U^h(\overrightarrow{\mathit{r}},t):=\underset{―}{\varphi }(\overrightarrow{\mathit{r}})\cdot \underset{―}{\hat{U}}(t)V(\overrightarrow{\mathit{r}})\approx V^h(\overrightarrow{\mathit{r}}):=\underset{―}{\psi }(\overrightarrow{\mathit{r}})\cdot \underset{―}{\hat{V}}$$

将上述近似代入弱形式，并利用 $\underset{―}{V}$ 的任意性，即得一组常微分方程

$$\boxed{{\displaystyle {\underset{―}{A}}_{\Omega }\cdot \frac{\mathrm{d}}{\mathrm{d}t}\underset{―}{\hat{U}}(t)={\underset{―}{B}}_{\Omega }(\underset{―}{\hat{U}}(t))-{\underset{―}{B}}_{\partial \Omega }(\underset{―}{\hat{U}}(t))}}$$

其中

$$\begin{array}{c}\underset{―}{\hat{U}}(t):=\left[\begin{array}{ccc}{\hat{U}}_1(t)& \cdots & {\hat{U}}_K(t)\end{array}\right]{\underset{―}{A}}_{\Omega }:=\left[\begin{array}{ccc}⟨{\varphi }_1|{\psi }_1⟩& \cdots & ⟨{\varphi }_1|{\psi }_L⟩\\ ⋮& \ddots & ⋮\\ ⟨{\varphi }_K|{\varphi }_1⟩& \cdots & ⟨{\varphi }_K|{\psi }_L⟩\end{array}\right]\\ {\underset{―}{B}}_{\Omega }(\underset{―}{\hat{U}}(t)):=⟨\overrightarrow{\mathit{F}}|\mathbf{\nabla }\underset{―}{\psi }{⟩}_{\Omega }:={\int }_{\Omega }\overrightarrow{\mathit{F}}\mathbf{\cdot }\left[\begin{array}{ccc}\mathbf{\nabla }{\psi }_1& \cdots & \mathbf{\nabla }{\psi }_L\end{array}\right]\\ {\underset{―}{B}}_{\partial \Omega }(\underset{―}{\hat{U}}(t)):=⟨F_{\nu }|\underset{―}{\psi }{⟩}_{\partial \Omega }:={\oint }_{\partial \Omega }{F}_{\nu }\left[\begin{array}{ccc}{\psi }_1& \cdots & {\psi }_L\end{array}\right]\end{array}$$

将积分区域取为单元及其边界，即得有限单元法 (finite element methed, FEM)。

• 若 $(K=L)$ 且 $(\mathrm{\forall }i)({\varphi }_i={\psi }_i)$，则称相应的 FEM 为 Galerkin 型，否则称其为 Petrov–Galerkin 型。
• 若为每个单元独立地选择基函数，并且不要求在单元边界上保证连续性，则称相应的 FEM 为间断的 (discontinuous)。
• 计算流体力学中较为常用的是间断的 Galerkin 型有限单元法 (DG-FEM)。

为避免频繁对 $\underset{―}{A}$ 求逆（解线性方程组），可对所选的基函数作正交化 (orthogonalization)，使 $\underset{―}{A}$ 为对角阵。

以上讨论可推广到方程组：

$${\partial }_t\underset{―}{U}+{\partial }_1\underset{―}{F_1}+{\partial }_2\underset{―}{F_2}+{\partial }_3\underset{―}{F_3}=\underset{―}{0}⟹\frac{\mathrm{d}}{\mathrm{d}t}{\underset{―}{\hat{U}}}_{K\times N}={\underset{―}{B}}_{K\times N}{\underset{―}{A}}_{N\times N}^{-1}$$

其中

$$\begin{array}{c}{\underset{―}{U}}_{K\times 1}^h(\overrightarrow{x},t)=\underset{{\underset{―}{\hat{U}}}_{K\times N}(t)}{\underbrace{\left[\begin{array}{ccc}⟨U_1|{\varphi }_1⟩& \cdots & ⟨U_1|{\varphi }_N⟩\\ ⋮& \ddots & ⋮\\ ⟨U_K|{\varphi }_1⟩& \cdots & ⟨U_K|{\varphi }_N⟩\end{array}\right]}}\underset{{\underset{―}{\varphi }}_{N\times 1}(\overrightarrow{x})}{\underbrace{\left[\begin{array}{c}{\varphi }_1(\overrightarrow{x})\\ ⋮\\ {\varphi }_N(\overrightarrow{x})\end{array}\right]}}\\ {\underset{―}{B}}_{K\times N}:=\sum _{i=1}^3{\int }_{E_i}{\underset{―}{F_i}}_{K\times 1}{\left({\partial }_i{\underset{―}{\varphi }}^{†}\right)}_{1\times N}-{\oint }_{\partial E_i}{\underset{―}{F_{\nu }}}_{K\times 1}{\left({\underset{―}{\varphi }}^{†}\right)}_{1\times N}\end{array}$$