# 欧拉方程 (流体动力学)

## 守恆形式（分量）

\begin{align} &{\partial\rho\over\partial t}+ \nabla\cdot(\rho\bold u)=0\\[1.2ex] &{\partial\rho{\bold u}\over\partial t}+ \nabla\cdot(\bold u\otimes(\rho \bold \bold u))+\nabla p=0\\[1.2ex] &{\partial E\over\partial t}+ \nabla\cdot(\bold u(E+p))=0, \end{align}

• ρ為流體的質量密度
• u 為流體速度向量，分量為uvw
• E = ρ e + ½ ρ ( u2 + v2 + w2 )為每一單位容量所含的總能量，其中e為流體每一單位容量所含的內能
• p為壓力；
• $\otimes$代表張量積

${\partial(\rho u_j)\over\partial t}+ \sum_{i=1}^3 {\partial(\rho u_i u_j)\over\partial x_i}+ {\partial p\over\partial x_j} =0,$

$\rho\left( \frac{\partial}{\partial t}+{\bold u}\cdot\nabla \right){\bold u}+\nabla p=0$

## 守恆形式（向量）

$\frac{\partial \bold m}{\partial t}+ \frac{\partial \bold f_x}{\partial x}+ \frac{\partial \bold f_y}{\partial y}+ \frac{\partial \bold f_z}{\partial z}=0,$

${\bold m}=\begin{pmatrix}\rho \\ \rho u \\ \rho v \\ \rho w \\E\end{pmatrix};$
${\bold f_x}=\begin{pmatrix}\rho u\\p+\rho u^2\\ \rho uv \\ \rho uw\\u(E+p)\end{pmatrix};\qquad {\bold f_y}=\begin{pmatrix}\rho v\\ \rho uv \\p+\rho v^2\\ \rho vw \\v(E+p)\end{pmatrix};\qquad {\bold f_z}=\begin{pmatrix}\rho w\\ \rho uw \\ \rho vw \\p+\rho w^2\\w(E+p)\end{pmatrix}.$

## 非守恆形式（通量雅可比矩陣）

$\frac{\partial \bold m}{\partial t} + \bold A_x \frac{\partial \bold m}{\partial x} + \bold A_y \frac{\partial \bold m}{\partial y} + \bold A_z \frac{\partial \bold m}{\partial z} = 0.$

$\bold A_x=\frac{\partial \bold f_x(\bold s)}{\partial \bold s}, \qquad \bold A_y=\frac{\partial \bold f_y(\bold s)}{\partial \bold s}, \qquad \bold A_z=\frac{\partial \bold f_z(\bold s)}{\partial \bold s}.$

### 理想氣體的通量雅可比矩陣

H為：

$H = \frac{E}{\rho} + \frac{p}{\rho},$

$a=\sqrt{\frac{\gamma p}{\rho}} = \sqrt{(\gamma-1)\left[H-\frac{1}{2}\left(u^2+v^2+w^2\right)\right]}.$

### 線性化形式

$\frac{\partial \bold m}{\partial t} + \bold A_{x,0} \frac{\partial \bold m}{\partial x} + \bold A_{y,0} \frac{\partial \bold m}{\partial y} + \bold A_{z,0} \frac{\partial \bold m}{\partial z} = 0,$

### 線性化一維的非耦合波方程

$\frac{\partial \bold m}{\partial t} + \bold A_{x,0} \frac{\partial \bold m}{\partial x} =0.$

$\mathbf{A}_{x,0} = \mathbf{P} \mathbf{\Lambda} \mathbf{P}^{-1},$
$\mathbf{P}= \left[\bold r_1, \bold r_2, \bold r_3\right] =\left[ \begin{array}{c c c} 1 & 1 & 1 \\ u-a & u & u+a \\ H-u a & \frac{1}{2} u^2 & H+u a \\ \end{array} \right],$
$\mathbf{\Lambda} = \begin{bmatrix} \lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & \lambda_3 \\ \end{bmatrix} = \begin{bmatrix} u-a & 0 & 0 \\ 0 & u & 0 \\ 0 & 0 & u+a \\ \end{bmatrix}.$

$\mathbf{w}= \mathbf{P}^{-1}\mathbf{m},$

$\frac{\partial \mathbf{w}}{\partial t} + \mathbf{\Lambda} \frac{\partial \mathbf{w}}{\partial x} = 0$

## 注釋

1. ^ Anderson, John D. (1995), Computational Fluid Dynamics, The Basics With Applications. ISBN 0-07-113210-4
2. ^ Christodoulou, Demetrios. The Euler Equations of Compressible Fluid Flow. Bulletin of the American Mathematical Society. 2007-10, 44 (4): 581–602 [June 13, 2009]. doi:10.1090/S0273-0979-07-01181-0.
3. ^ 見Toro (1999)

## 資料來源及延伸閱讀

• Batchelor, G. K. An Introduction to Fluid Dynamics. Cambridge University Press. 1967. ISBN 0521663962.
• Thompson, Philip A. Compressible Fluid Flow. New York: McGraw-Hill. 1972. ISBN 0070644055.
• Toro, E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag. 1999. ISBN 3-540-65966-8.