CFD中的张量公式

本网页中\(\tau\)tensor二阶张量,\(\bfU,\bfV\)vector矢量,\(a,b\)scalar标量。

\[\begin{split} \notag \nabla p = \left[\begin{matrix} \frac{\partial p}{\partial x} \\ \frac{\partial p}{\partial y} \\ \frac{\partial p}{\partial z} \end{matrix} \right] \end{split}\]
\[ \notag \nabla \cdot(\nabla p)=\nabla ^2p=\frac{\partial^2p}{\partial x^2}+\frac{\partial^2p}{\partial y^2}+\frac{\partial^2p}{\partial z^2} \]
\[\begin{split} \notag \mathbf{U} \cdot \mathbf{V} = [u_1, u_2, u_3] \left[\begin{matrix} v_1 \\ v_2 \\ v_3 \end{matrix} \right]=u_1v_1+u_2v_2+u_3v_3 \end{split}\]
\[\begin{split} \notag \mathbf{U}\mathbf{V}=\mathbf{U}\otimes\mathbf{V}=\left[ \begin{matrix} u_1 v_1 & u_1 v_2 & u_1 v_3\\ u_2 v_1 & u_2 v_2 & u_2 v_3\\ u_3 v_1 & u_3 v_2 & u_3 v_3 \end{matrix} \right] \end{split}\]
\[\begin{split} \notag \mathbf{U} \times \mathbf{V}=\left[ \begin{matrix} u_2v_3-u_3v_2\\ u_3v_1-u_1v_3\\ u_1v_2-u_2v_1\\ \end{matrix} \right] \end{split}\]
\[ \notag \nabla \cdot \mathbf{U} = \frac{\partial u_1}{\partial x}+\frac{\partial u_2}{\partial y}+\frac{\partial u_3}{\partial z} \]
\[\begin{split} \notag \nabla \mathbf{U} = \left[ \begin{matrix} \frac{\partial u_1}{\partial x} & \frac{\partial u_2}{\partial x} & \frac{\partial u_3}{\partial x}\\ \frac{\partial u_1}{\partial y} & \frac{\partial u_2}{\partial y} & \frac{\partial u_3}{\partial y} \\ \frac{\partial u_1}{\partial z} & \frac{\partial u_2}{\partial z} & \frac{\partial u_3}{\partial z}\\ \end{matrix} \right] \end{split}\]
\[\begin{split} \notag \nabla \cdot(\nabla \mathbf{U})= \left[ \begin{matrix} \frac{\partial}{\partial x}\left(\frac{\partial u_1}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\partial u_1}{\partial y}\right)+\frac{\partial}{\partial z}\left(\frac{\partial u_1}{\partial z}\right)\\ \frac{\partial}{\partial x}\left(\frac{\partial u_2}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\partial u_2}{\partial y}\right)+\frac{\partial}{\partial z}\left(\frac{\partial u_2}{\partial z}\right)\\ \frac{\partial}{\partial x}\left(\frac{\partial u_3}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\partial u_3}{\partial y}\right)+\frac{\partial}{\partial z}\left(\frac{\partial u_3}{\partial z}\right)\\ \end{matrix} \right] \end{split}\]
\[\begin{split} \notag \nabla\times\mathbf{U}=\left[ \begin{matrix} \frac{\partial u_3}{\partial y}-\frac{\partial u_2}{\partial z}\\ \frac{\partial u_1}{\partial z}-\frac{\partial u_3}{\partial x}\\ \frac{\partial u_2}{\partial x}-\frac{\partial u_1}{\partial y}\\ \end{matrix} \right] \end{split}\]
\[ \notag \mathbf{U}+\mathbf{V}=\mathbf{V}+\mathbf{U} \]
\[ \notag \alpha\mathbf{U}=\mathbf{U}\alpha \]
\[ \notag \mathbf{U}\cdot\mathbf{V}=\mathbf{V}\cdot\mathbf{U} \]
\[ \notag \mathbf{U}\times\mathbf{V}=-\mathbf{V}\times\mathbf{U} \]
\[ \notag \mathbf{U}\times\left(\mathbf{V}\times\mathbf{W}\right)\neq\left(\mathbf{U}\times\mathbf{V}\right)\times\mathbf{W} \]
\[ \notag \nabla\cdot\left(\nabla\times\mathbf{U}\right)=0 \]
\[ \notag \nabla\times\nabla\alpha=0 \]
\[ \notag \nabla (\alpha p)=\alpha\nabla p+p\nabla\alpha \]
\[ \notag \nabla \cdot(\alpha \mathbf{U})=\alpha\nabla\cdot \mathbf{U}+\mathbf{U} \cdot \nabla\alpha=\alpha\nabla\cdot\mathbf{U}+\nabla\alpha\cdot\mathbf{U} \]
\[ \notag \nabla \times (\alpha \mathbf{U})=\alpha\nabla\times \mathbf{U}+\left(\nabla\alpha\right) \times\mathbf{U} \]
\[ \notag \nabla (\alpha\bfU\cdot\bfV)=\alpha\bfU\cdot\nabla\bfV+\bfV\cdot\nabla\alpha\bfU \]
\[ \notag \nabla\alpha\bfU=\alpha\nabla\bfU+\bfU\nabla\alpha \]
\[ \notag \nabla\cdot(\mathbf{U} \mathbf{U})=\mathbf{U} \cdot \nabla \mathbf{U}+\mathbf{U} \nabla \cdot \mathbf{U} \]
\[ \notag \nabla\cdot(\alpha \tau)=\tau \cdot\nabla \alpha + \alpha \nabla \cdot \tau \]
\[ \notag \mathrm{tr}\left(\nabla\mathbf{U}\right)\bfI=\mathrm{tr}\left(\nabla\mathbf{U}^{\mathrm{T}}\right)\bfI=\left(\nabla\cdot\mathbf{U}\right)\bfI \]
\[ \notag \mathrm{tr}\left(\nabla\mathbf{U}+\nabla\mathbf{U}^{\mathrm{T}}\right)\bfI=2\mathrm{tr}\left(\nabla\mathbf{U}\right)\bfI=2\left(\nabla\cdot\mathbf{U}\right)\bfI \]
\[ \notag \nabla\cdot(\nabla\bfU)^T=\nabla(\nabla\cdot\bfU) \]
\[ \notag \nabla\cdot((\nabla\cdot\bfU)\bfI)=\nabla(\nabla\cdot\bfU) \]
\[\begin{split} \notag \nabla \cdot {\boldsymbol\tau} = \left[\begin{matrix} \frac{\partial\tau_{xx}}{\partial x}+\frac{\partial\tau_{yx}}{\partial y}+\frac{\partial\tau_{zx}}{\partial z} \\ \frac{\partial\tau_{xy}}{\partial x}+\frac{\partial\tau_{yy}}{\partial y}+\frac{\partial\tau_{zy}}{\partial z} \\ \frac{\partial\tau_{xz}}{\partial x}+\frac{\partial\tau_{yz}}{\partial y}+\frac{\partial\tau_{zz}}{\partial z} \end{matrix}\right] \end{split}\]
\[ \notag {\boldsymbol\tau}:{\boldsymbol\tau}=\tau_{11}\tau_{11}+\tau_{12}\tau_{12}+\tau_{13}\tau_{13}+ \tau_{21}\tau_{21}+\tau_{22}\tau_{22}+\tau_{23}\tau_{23}+ \tau_{31}\tau_{31}+\tau_{32}\tau_{32}+\tau_{33}\tau_{33} \]
\[ \notag |{\boldsymbol\tau}|=\sqrt{{\boldsymbol\tau}:{\boldsymbol\tau}} \]
\[ \notag |{\boldsymbol\tau}|^2={\boldsymbol\tau}:{\boldsymbol\tau} \]
\[\begin{split} \begin{equation}\notag \begin{split} |\nabla\nabla\bfU|^2 &=\left(\frac{\p u_i}{\p x_j \p x_k}\right)\left(\frac{\p u_i}{\p x_j \p x_k}\right)=|\nabla\nabla u_1|^2+|\nabla\nabla u_2|^2+|\nabla\nabla u_3|^2 \\\\ &=\left|\begin{matrix} \frac{\p u_1}{\p x \p x},\frac{\p u_1}{\p x \p y},\frac{\p u_1}{\p x \p z} \\ \frac{\p u_1}{\p y \p x},\frac{\p u_1}{\p y \p y},\frac{\p u_1}{\p y \p z} \\ \frac{\p u_1}{\p z \p x},\frac{\p u_1}{\p z \p y},\frac{\p u_1}{\p z \p z} \end{matrix}\right|^2 + \left|\begin{matrix} \frac{\p u_2}{\p x \p x},\frac{\p u_2}{\p x \p y},\frac{\p u_2}{\p x \p z} \\ \frac{\p u_2}{\p y \p x},\frac{\p u_2}{\p y \p y},\frac{\p u_2}{\p y \p z} \\ \frac{\p u_2}{\p z \p x},\frac{\p u_2}{\p z \p y},\frac{\p u_2}{\p z \p z} \end{matrix}\right|^2 + \left|\begin{matrix} \frac{\p u_3}{\p x \p x},\frac{\p u_3}{\p x \p y},\frac{\p u_3}{\p x \p z} \\ \frac{\p u_3}{\p y \p x},\frac{\p u_3}{\p y \p y},\frac{\p u_3}{\p y \p z} \\ \frac{\p u_3}{\p z \p x},\frac{\p u_3}{\p z \p y},\frac{\p u_3}{\p z \p z} \end{matrix}\right|^2 \end{split} \end{equation} \end{split}\]

dev(tau)\(=\tau-\frac{1}{3}\mathrm{tr}\left(\tau\right)\mathbf{I}\)

dev2(tau)\(=\tau-\frac{2}{3}\mathrm{tr}\left(\tau\right)\mathbf{I}\)

symm(gradU)\( =\frac{\nabla\bfU+\nabla\bfU^T}{2}\)

twoSymm(gradU)\( =\nabla\bfU+\nabla\bfU^T\)

dev(twoSymm(gradU))\(=\nabla\bfU+\nabla\bfU^T-\frac{1}{3}\mathrm{tr}\left(\nabla\bfU+\nabla\bfU^T\right)\mathbf{I}=\nabla\bfU+\nabla\bfU^T-\frac{2}{3}\left(\nabla\cdot\mathbf{U}\right)\bfI\)

dev(symm(gradU))\(=\frac{\nabla\bfU+\nabla\bfU^T}{2}-\frac{1}{3}\mathrm{tr}\left(\frac{\nabla\bfU+\nabla\bfU^T}{2}\right)\mathbf{I}=\frac{\nabla\bfU+\nabla\bfU^T}{2}-\frac{1}{3}\left(\nabla\cdot\mathbf{U}\right)\bfI\)

tr(tau)\( =\tau_{xx}+\tau_{yy}+\tau_{zz}\)

sph(tau)\(=\frac{1}{3}\left(\tau_{xx}+\tau_{yy}+\tau_{zz}\right)\)

skew(gradU)\(=\frac{\nabla\bfU-\nabla\bfU^T}{2}\)

magSqrGradGrad(U)\(=\left|\nabla\nabla\bfU\right|^2\)

det(tau)\(=|\tau|\)

innerSqr(tau)\(=\tau\cdot\tau=\left[\begin{matrix} \tau_{xx}\tau_{xx}+\tau_{xy}\tau_{xy}+\tau_{xz}\tau_{xz}, \tau_{xx}\tau_{xy}+\tau_{xy}\tau_{yy}+\tau_{xz}\tau_{yz}, \tau_{xx}\tau_{xz}+\tau_{xy}\tau_{yz}+\tau_{xz}\tau_{zz} \\ \tau_{xx}\tau_{xy}+\tau_{xy}\tau_{yy}+\tau_{xz}\tau_{yz}, \tau_{xy}\tau_{xy}+\tau_{yy}\tau_{yy}+\tau_{yz}\tau_{yz}, \tau_{xy}\tau_{xz}+\tau_{yy}\tau_{yz}+\tau_{yz}\tau_{zz} \\ \tau_{xx}\tau_{xz}+\tau_{xy}\tau_{yz}+\tau_{xz}\tau_{zz}, \tau_{xy}\tau_{xz}+\tau_{yy}\tau_{yz}+\tau_{yz}\tau_{zz}, \tau_{xz}\tau_{xz}+\tau_{yz}\tau_{yz}+\tau_{zz}\tau_{zz} \end{matrix}\right]\)

cof(tau) \(=\left[ \begin{matrix} \tau_{yy}\tau_{zz} - \tau_{zy}\tau_{yz} & \tau_{zx}\tau_{yz} - \tau_{yx}\tau_{zz} & \tau_{yx}\tau_{zy} - \tau_{yy}\tau_{zx}\\ \tau_{xz}\tau_{zy} - \tau_{xy}\tau_{zz} & \tau_{xx}\tau_{zz} - \tau_{xz}\tau_{zx} & \tau_{xy}\tau_{zx} - \tau_{xx}\tau_{zy}\\ \tau_{xy}\tau_{yz} - \tau_{xz}\tau_{yy} & \tau_{yx}\tau_{xz} - \tau_{xx}\tau_{yz} & \tau_{xx}\tau_{yy} - \tau_{yx}\tau_{xy}\\ \end{matrix} \right] \)

inv(tau)\(=\tau^{-1}\)

invariantI(tau)\( =\mathrm{tr} \left(\tau\right)\)

invariantII(tau) \(=\tau_{xx}\tau_{yy}+\tau_{yy}\tau_{zz}+ \tau_{xx}\tau_{zz}-\tau_{xy}\tau_{yx}-\tau_{yz}\tau_{zy}-\tau_{xz}\tau_{zx} \)

invariantIII(tau)\( =\mathrm{det} \left(\tau\right)\)

tau.T()\(=\tau^{T}\)

U & V\(=\bfU\cdot\bfV\)

U ^ V\(=\bfU \times\bfV\)

U * V\(=\bfU\bfV\)

tau & tau\(=\tau\cdot\tau\)

tau && tau\(=\tau :\tau\)

sign(a)\(\mathrm{sgn}(a)\)

log(a)\(=\mathrm{ln}(a)\)

log10(a)\(=\mathrm{log}(a)\)

\(\mathbf{F}\frac{\mathrm{kg}\cdot\mathrm{m}}{\mathrm{s}^2}\)

压力 \(p \frac{\mathrm{kg}}{\mathrm{m} \cdot\mathrm{s}^2}\)

湍流动能 \(k \frac{\mathrm{m}^2}{\mathrm{s}^2}\)

湍流动能耗散率 \(\varepsilon \frac{\mathrm{m}^2}{\mathrm{s}^3}\)

湍流频率 \(\omega \frac{1}{\mathrm{s}}\)

运动粘度 \(\nu \frac{\mathrm{m}^2}{\mathrm{s}}\)

动力粘度 \(\mu \frac{\mathrm{kg}}{\mathrm{m}\cdot\mathrm{s}}\)

加速度 \(\mathbf{A}\frac{\mathrm{m}}{\mathrm{s}^2}\)

动量\(m\bfU\) \(\frac{\mathrm{kg}\cdot\mathrm{m}}{\mathrm{s}}\)

动量密度\(\rho\bfU\) \(\frac{\mathrm{kg}}{\mathrm{m}^2\cdot\mathrm{s}}\)

比能密度\(\rho E\) \(\frac{\mathrm{kg}}{\mathrm{m}\cdot\mathrm{s}^2}\)

比内能密度\(\rho e\) \(\frac{\mathrm{kg}}{\mathrm{m}\cdot\mathrm{s}^2}\)

比气体常数\(R\) \(\frac{\mathrm{m}^2}{\mathrm{s}^2 \cdot\mathrm{K}}\)

定容比热容\(C_v\) \(\frac{\mathrm{m}^2}{\mathrm{s}^2 \cdot\mathrm{K}}\)