流体力学向量微积分基础
向量和向量运算
\[\mathbf{v}=u \mathbf{i}+v \mathbf{j}+w \mathbf{k}
\]
\[\mathbf{v}=\left[\begin{array}{c}
u \\
v \\
w
\end{array}\right] \quad \mathbf{v}^{\mathrm{T}}=\left[\begin{array}{lll}
u & v & w
\end{array}\right]
\]
\[\|\mathbf{v}\|=\sqrt{u^{2}+v^{2}+w^{2}}
\]
两个向量的点积
\[\begin{aligned}
& \mathbf{i} \cdot \mathbf{i}=\mathbf{j} \cdot \mathbf{j}=\mathbf{k} \cdot \mathbf{k}=1 \\
& \mathbf{i} \cdot \mathbf{j}=\mathbf{i} \cdot \mathbf{k}=\mathbf{j} \cdot \mathbf{i}=\mathbf{j} \cdot \mathbf{k}=\mathbf{k} \cdot \mathbf{i}=\mathbf{k} \cdot \mathbf{j}=0
\end{aligned}
\]
\[\begin{aligned}
\mathbf{v}_{1} \cdot \mathbf{v}_{2} & =\left(u_{1} \mathbf{i}+v_{1} \mathbf{j}+w_{1} \mathbf{k}\right) \cdot\left(u_{2} \mathbf{i}+v_{2} \mathbf{j}+w_{2} \mathbf{k}\right) \\
& =u_{1} u_{2}+v_{1} v_{2}+w_{1} w_{2} \\
& =\left\|\mathbf{v}_{1}\right\|\left\|\mathbf{v}_{2}\right\| \cos \left(\mathbf{v}_{1}, \mathbf{v}_{2}\right)
\end{aligned}
\]
单位方向向量
\[\mathbf{e}_{\mathrm{v}}=\dfrac{\mathbf{v}}{\|\mathbf{v}\|}
\]
两个向量的外积
\[\begin{aligned}
& \mathbf{i} \times \mathbf{i}=\mathbf{j} \times \mathbf{j} =\mathbf{k} \times \mathbf{k}=0 \\
& \mathbf{i} \times \mathbf{j}=-\mathbf{j} \times \mathbf{i}=\mathbf{k} \\
& \mathbf{j} \times \mathbf{k}=-\mathbf{k} \times \mathbf{j}=\mathbf{i} \\
& \mathbf{k} \times \mathbf{i}=-\mathbf{i} \times \mathbf{k}=\mathbf{j}
\end{aligned}
\]
\[\mathbf{v}_{1} \times \mathbf{v}_{2}=-\mathbf{v}_{2} \times \mathbf{v}_{1}=\left|\begin{array}{ccc}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
u_{1} & v_{1} & w_{1} \\
u_{2} & v_{2} & w_{2}
\end{array}\right|=\left[\begin{array}{c}
v_{1} w_{2}-v_{2} w_{1} \\
u_{2} w_{1}-u_{1} w_{2} \\
u_{1} v_{2}-u_{2} v_{1}
\end{array}\right]
\]
\[\left\|\mathbf{v}_{1} \times \mathbf{v}_{2}\right\|=\left\|\mathbf{v}_{1}\right\|\left\|\mathbf{v}_{2}\right\|\left|\sin \left(\mathbf{v}_{1}, \mathbf{v}_{2}\right)\right|
\]
向量的外积不符合交换律，但符合分配律
标量三重积
\[\left(\mathbf{v}_{1} \cdot\left[\mathbf{v}_{2} \times \mathbf{v}_{3}\right]\right)=\left(\left[\mathbf{v}_{2} \times \mathbf{v}_{3}\right] \cdot \mathbf{v}_{1}\right)=\left|\begin{array}{lll}
u_{1} & v_{1} & w_{1} \\
u_{2} & v_{2} & w_{2} \\
u_{3} & v_{3} & w_{3}
\end{array}\right|
\]
\[\left(\mathbf{v}_{1} \cdot\left[\mathbf{v}_{2} \times \mathbf{v}_{3}\right]\right)=\left(\mathbf{v}_{2} \cdot\left[\mathbf{v}_{3} \times \mathbf{v}_{1}\right]\right)=\left(\mathbf{v}_{3} \cdot\left[\mathbf{v}_{1} \times \mathbf{v}_{2}\right]\right)
\]
矢量三重积
\[\mathbf{v}_{1} \times \left(\mathbf{v}_{2} \times \mathbf{v}_{3}\right)=(\mathbf{v}_{1} \cdot \mathbf{v}_{3})\mathbf{v}_{2}-(\mathbf{v}_{1} \cdot \mathbf{v}_{2})\mathbf{v}_{3}
\]
哈密顿算符
\[\nabla=\dfrac{\partial}{\partial x} \mathbf{i}+\dfrac{\partial}{\partial y} \mathbf{j}+\dfrac{\partial}{\partial z} \mathbf{k}
\]
\[\nabla s=\dfrac{\partial s}{\partial x} \mathbf{i}+\dfrac{\partial s}{\partial y} \mathbf{j}+\dfrac{\partial s}{\partial z} \mathbf{k}
\]
\[\dfrac{\mathrm{d} s}{\mathrm{d} l}=\nabla s \cdot \mathbf{e}_{l}=\|\nabla s\| \cos \left(\nabla s, \mathbf{e}_{l}\right)
\]
\[\nabla \cdot \mathbf{v}=\dfrac{\partial u}{\partial x}+\dfrac{\partial v}{\partial y}+\dfrac{\partial w}{\partial z}
\]
\[\nabla \cdot(\nabla s)=\nabla^{2} s=\dfrac{\partial^{2} s}{\partial x^{2}}+\dfrac{\partial^{2} s}{\partial y^{2}}+\dfrac{\partial^{2} s}{\partial z^{2}}
\]
\[\nabla^{2} \mathbf{v}=\left(\nabla^{2} u\right) \mathbf{i}+\left(\nabla^{2} v\right) \mathbf{j}+\left(\nabla^{2} w\right) \mathbf{k}
\]
\[\begin{aligned}
\nabla \times \mathbf{v} & =\left(\dfrac{\partial}{\partial x} \mathbf{i}+\dfrac{\partial}{\partial y} \mathbf{j}+\dfrac{\partial}{\partial z} \mathbf{k}\right) \times(u \mathbf{i}+v \mathbf{j}+w \mathbf{k}) \\
& =\left|\begin{array}{ccc}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
u & v & w
\end{array}\right| \\
& =\left(\dfrac{\partial w}{\partial y}-\dfrac{\partial v}{\partial z}\right) \mathbf{i}+\left(\dfrac{\partial u}{\partial z}-\dfrac{\partial w}{\partial x}\right) \mathbf{j}+\left(\dfrac{\partial v}{\partial x}-\dfrac{\partial u}{\partial y}\right) \mathbf{k}
\end{aligned}
\]
\[\begin{aligned}
{[(\mathbf{v} \cdot \nabla) \mathbf{v}] } & =(u \mathbf{i}+v \mathbf{j}+w \mathbf{k}) \cdot\left(\dfrac{\partial}{\partial x} \mathbf{i}+\dfrac{\partial}{\partial y} \mathbf{j}+\dfrac{\partial}{\partial z} \mathbf{k}\right)(u \mathbf{i}+v \mathbf{j}+w \mathbf{k}) \\
& =\left(u \dfrac{\partial}{\partial x}+v \dfrac{\partial}{\partial y}+w \dfrac{\partial}{\partial z}\right)(u \mathbf{i}+v \mathbf{j}+w \mathbf{k}) \\
& =\left(u \dfrac{\partial u}{\partial x}+v \dfrac{\partial u}{\partial y}+w \dfrac{\partial u}{\partial z}\right) \mathbf{i}+\left(u \dfrac{\partial v}{\partial x}+v \dfrac{\partial v}{\partial y}+w \dfrac{\partial v}{\partial z}\right) \mathbf{j}+\left(u \dfrac{\partial w}{\partial x}+v \dfrac{\partial w}{\partial y}+w \dfrac{\partial w}{\partial z}\right) \mathbf{k}
\end{aligned}
\]
附加向量运算
\[\nabla \cdot(\nabla \times \mathbf{v})=0
\]
\[\nabla \times(\nabla s)=0
\]
\[\nabla \cdot(s \mathbf{v})=s \nabla \cdot \mathbf{v}+\mathbf{v} \cdot \nabla s
\]
\[\nabla \times(s \mathbf{v})=s \nabla \times \mathbf{v}+\nabla s \times \mathbf{v}
\]
\[\nabla\left(\mathbf{v}_{1} \cdot \mathbf{v}_{2}\right)=\mathbf{v}_{1} \times\left(\nabla \times \mathbf{v}_{2}\right)+\mathbf{v}_{2} \times\left(\nabla \times \mathbf{v}_{1}\right)+\left(\mathbf{v}_{1} \cdot \nabla\right) \mathbf{v}_{2}+\left(\mathbf{v}_{2} \cdot \nabla\right) \mathbf{v}_{1}
\]
\[\nabla \cdot\left(\mathbf{v}_{1} \times \mathbf{v}_{2}\right)=\mathbf{v}_{2} \cdot\left(\nabla \times \mathbf{v}_{1}\right)-\mathbf{v}_{1} \cdot\left(\nabla \times \mathbf{v}_{2}\right)
\]
\[\nabla \times\left(\mathbf{v}_{1} \times \mathbf{v}_{2}\right)=\mathbf{v}_{1}\left(\nabla \cdot \mathbf{v}_{2}\right)-\mathbf{v}_{2}\left(\nabla \cdot \mathbf{v}_{1}\right)+\left(\mathbf{v}_{2} \cdot \nabla\right) \mathbf{v}_{1}-\left(\mathbf{v}_{1} \cdot \nabla\right) \mathbf{v}_{2}
\]
\[\nabla \times(\nabla \times \mathbf{v})=\nabla(\nabla \cdot \mathbf{v})-\nabla^{2} \mathbf{v}
\]
\[(\nabla \times \mathbf{v}) \times \mathbf{v}=\mathbf{v} \cdot(\nabla \mathbf{v})-\nabla(\mathbf{v} \cdot \mathbf{v})
\]
张量和张量运算
\[\boldsymbol{\tau}=\left[\begin{array}{lll}
\tau_{x x} & \tau_{x y} & \tau_{x z} \\
\tau_{y x} & \tau_{y y} & \tau_{y z} \\
\tau_{z x} & \tau_{z y} & \tau_{z z}
\end{array}\right] \qquad \boldsymbol{\tau}^{\mathrm{T}}=\left[\begin{array}{lll}
\tau_{x x} & \tau_{y x} & \tau_{z x} \\
\tau_{x y} & \tau_{y y} & \tau_{z y} \\
\tau_{x z} & \tau_{y z} & \tau_{z z}
\end{array}\right]
\]
\[\boldsymbol{\tau}=\mathbf{i} \mathbf{i} \tau_{x x}+\mathbf{i j} \tau_{x y}+\mathbf{i k} \tau_{x z}+\mathbf{j i} \tau_{y x}+\mathbf{j} \mathbf{j} \tau_{y y}+\mathbf{j} \mathbf{k} \tau_{y z}+\mathbf{k} \mathbf{i} \tau_{z x}+\mathbf{k} \mathbf{j} \tau_{z y}+\mathbf{k} \mathbf{k} \tau_{z z}
\]
下标记号法
\[x_{1}, x_{2}, x_{3} \rightarrow x_{i}(i=1,2,3)
\]
\[\sigma_{x x}, \sigma_{x y}, \sigma_{x z}, \sigma_{y z}, \sigma_{y y}, \sigma_{y z}, \sigma_{z x}, \sigma_{z y}, \sigma_{z z} \rightarrow \sigma_{i j}(i, j=x, y, z)
\]
爱因斯坦求和约定
当一个下标符号在一项中出现多次，称为哑标号，这个下标符号应理解为取变程中所有的值然后求和
\[a_{i} x_{i}=a_{1} x_{1}+a_{2} x_{2}+a_{3} x_{3}
\]
\[\sigma_{i i}=\sigma_{11}+\sigma_{22}+\sigma_{33}
\]
\[\sigma_{i j} l_{j}=\sigma_{i 1} l_{1}+\sigma_{i 2} l_{2}+\sigma_{i 3} l_{3}
\]
\[\left(\sigma_{i i}\right)^{2}=\left(\sigma_{11}+\sigma_{22}+\sigma_{33}\right)^{2}
\]
\[\begin{aligned}
\sigma_{i j} \varepsilon_{i j} = & \sigma_{11} \varepsilon_{11}+\sigma_{12} \varepsilon_{12}+\sigma_{13} \varepsilon_{13} \\
& +\sigma_{21} \varepsilon_{21}+\sigma_{22} \varepsilon_{22}+\sigma_{23} \varepsilon_{23} \\
& +\sigma_{31} \varepsilon_{31}+\sigma_{32} \varepsilon_{32}+\sigma_{33} \varepsilon_{33}
\end{aligned}
\]
柯氏符号，亦称单位张量
\[\delta_{i j}=\left\{\begin{array}{l}
1, i=j \\
0, i \neq j
\end{array}\right. \qquad \delta_{i j}=\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]
\]
- \(\delta_{i i}=\delta_{11}+\delta_{22}+\delta_{33}=3\)
- \(\delta_{i j} \delta_{i j}=\left(\delta_{11}\right)^{2}+\left(\delta_{22}\right)^{2}+\left(\delta_{33}\right)^{2}=3\)
- \(\delta_{i j} \delta_{j k}=\delta_{i 1} \delta_{1 k}+\delta_{i 2} \delta_{2 k}+\delta_{i 3} \delta_{3 k}=\delta_{i k}\)
- \(a_{i j} \delta_{i j}=a_{11} \delta_{11}+a_{22} \delta_{22}+a_{33} \delta_{33}=a_{i i}\)
- \(a_{i} \delta_{i j}=a_{1} \delta_{1 j}+a_{2} \delta_{2 j}+a_{3} \delta_{3 j}=a_{j}\)
- \(\sigma_{i j} l_{j}-\lambda l_{i}=\sigma_{i j} l_{j}-\lambda \delta_{i j} l_{j}=\left(\sigma_{i j}-\lambda \delta_{i j}\right) l_{j}\)
张量与向量的点积
\[\boldsymbol{\tau} \cdot \mathbf{v}=\tau_{ij} u_{j}=\left[\begin{array}{l}
\tau_{x x} u+\tau_{x y} v+\tau_{x z} w \\
\tau_{y x} u+\tau_{y y} v+\tau_{y z} w \\
\tau_{z x} u+\tau_{z y} v+\tau_{z z} w
\end{array}\right]
\]
\[\mathbf{v} \cdot \boldsymbol{\tau}=u_{i} \tau_{ij}=\left[\begin{array}{c}
u \tau_{xx} + v \tau_{yx} + w \tau_{zx} \\
u \tau_{xy} + v \tau_{yy} + w \tau_{zy} \\
u \tau_{xz} + v \tau_{yz} + w \tau_{zz}
\end{array}\right]
\]
\[\begin{aligned}
\nabla \cdot \boldsymbol{\tau}= & \left(\dfrac{\partial \tau_{x x}}{\partial x}+\dfrac{\partial \tau_{y x}}{\partial y}+\dfrac{\partial \tau_{z x}}{\partial z}\right) \mathbf{i}+\left(\dfrac{\partial \tau_{x y}}{\partial x}+\dfrac{\partial \tau_{y y}}{\partial y}+\dfrac{\partial \tau_{z y}}{\partial z}\right) \mathbf{j} \\
& +\left(\dfrac{\partial \tau_{x z}}{\partial x}+\dfrac{\partial \tau_{y z}}{\partial y}+\dfrac{\partial \tau_{z z}}{\partial z}\right) \mathbf{k}
\end{aligned}
\]
两个向量的并积
\[\mathbf{v v}=\left[\begin{array}{ccc}
u u & u v & u w \\
v u & v v & v w \\
w u & w v & w w
\end{array}\right]
\]
\[\nabla \mathbf{v}=\left[\begin{array}{lll}
\frac{\partial u}{\partial x} & \frac{\partial v}{\partial x} & \frac{\partial w}{\partial x} \\
\frac{\partial u}{\partial y} & \frac{\partial v}{\partial y} & \frac{\partial w}{\partial y} \\
\frac{\partial u}{\partial z} & \frac{\partial v}{\partial z} & \frac{\partial w}{\partial z}
\end{array}\right]
\]
两个张量的二重点积
\[\begin{aligned}
\boldsymbol{\tau}:\boldsymbol{\sigma}=\tau_{ij} \sigma_{ji}=\tau_{xx}& \sigma_{xx} + \tau_{xy} \sigma_{yx} + \tau_{xz} \sigma_{zx} + \tau_{yx} \sigma_{xy} \\
&+ \tau_{yy} \sigma_{yy} + \tau_{yz} \sigma_{zy} + \tau_{zx} \sigma_{xz} + \tau_{zy} \sigma_{yz} + \tau_{zz} \sigma_{zz}
\end{aligned}
\]
\[\begin{aligned}
\boldsymbol{\tau}: \nabla \mathbf{v}=\tau_{x x} & \dfrac{\partial u}{\partial x}+\tau_{x y} \dfrac{\partial u}{\partial y}+\tau_{x z} \dfrac{\partial u}{\partial z}+\tau_{y x} \dfrac{\partial v}{\partial x} \\
& +\tau_{y y} \dfrac{\partial v}{\partial y}+\tau_{y z} \dfrac{\partial v}{\partial z}+\tau_{z x} \dfrac{\partial w}{\partial x}+\tau_{z y} \dfrac{\partial w}{\partial y}+\tau_{z z} \dfrac{\partial w}{\partial z}
\end{aligned}
\]
向量微积分基本定理
线积分的梯度公式
\[\displaystyle\int_{C} \nabla s \cdot d \mathbf{r}=s(r(b))-s(r(a))
\]
格林公式
\[\displaystyle\oint_{C}(u d x+v d y)=\iint_{R}\left(\dfrac{\partial v}{\partial x}-\dfrac{\partial u}{\partial y}\right) d x d y
\]
\[\displaystyle\oint_{C} \mathbf{v} \cdot d \mathbf{r}=\iint_{R}[\nabla \times \mathbf{v}] \cdot d \mathbf{S}
\]
斯托克斯公式
\[\displaystyle\int_{S}[\nabla \times \mathbf{v}] \cdot d \mathbf{S}=\oint_{C} \mathbf{v} \cdot d \mathbf{r}
\]
散度定理
\[\displaystyle\int_{V}(\nabla \cdot \mathbf{v}) d V=\oint_{S} \mathbf{v} \cdot \mathbf{n} d S
\]
\[\displaystyle\int_{V}[\nabla s] d V=\oint_{S} s d \mathbf{S}
\]
\[\displaystyle\int_{V}[\nabla \cdot \boldsymbol{\tau}] d V=\oint_{S}[\boldsymbol{\tau} \cdot \mathbf{n}] d S
\]
参考资料
The Finite Volume Method in Computational Fluid Dynamics- An Advanced Introduction with OpenFOAM and Matlab, F. Moukalled
