# 欧拉方程 (流体动力学)

## 守恆形式（分量）

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

• 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；
• ${\displaystyle \otimes }$代表張量積

${\displaystyle {\partial (\rho u_{j}) \over \partial t}+\sum _{i=1}^{3}{\partial (\rho u_{i}u_{j}) \over \partial x_{i}}+{\partial p \over \partial x_{j}}=0,}$

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

## 守恆形式（向量）

${\displaystyle {\frac {\partial \mathbf {m} }{\partial t}}+{\frac {\partial \mathbf {f} _{x}}{\partial x}}+{\frac {\partial \mathbf {f} _{y}}{\partial y}}+{\frac {\partial \mathbf {f} _{z}}{\partial z}}=0,}$

${\displaystyle {\mathbf {m} }={\begin{pmatrix}\rho \\\rho u\\\rho v\\\rho w\\E\end{pmatrix}};}$
${\displaystyle {\mathbf {f} _{x}}={\begin{pmatrix}\rho u\\p+\rho u^{2}\\\rho uv\\\rho uw\\u(E+p)\end{pmatrix}};\qquad {\mathbf {f} _{y}}={\begin{pmatrix}\rho v\\\rho uv\\p+\rho v^{2}\\\rho vw\\v(E+p)\end{pmatrix}};\qquad {\mathbf {f} _{z}}={\begin{pmatrix}\rho w\\\rho uw\\\rho vw\\p+\rho w^{2}\\w(E+p)\end{pmatrix}}.}$

## 非守恆形式（通量雅可比矩陣）

${\displaystyle {\frac {\partial \mathbf {m} }{\partial t}}+\mathbf {A} _{x}{\frac {\partial \mathbf {m} }{\partial x}}+\mathbf {A} _{y}{\frac {\partial \mathbf {m} }{\partial y}}+\mathbf {A} _{z}{\frac {\partial \mathbf {m} }{\partial z}}=0.}$

${\displaystyle \mathbf {A} _{x}={\frac {\partial \mathbf {f} _{x}(\mathbf {s} )}{\partial \mathbf {s} }},\qquad \mathbf {A} _{y}={\frac {\partial \mathbf {f} _{y}(\mathbf {s} )}{\partial \mathbf {s} }},\qquad \mathbf {A} _{z}={\frac {\partial \mathbf {f} _{z}(\mathbf {s} )}{\partial \mathbf {s} }}.}$

### 理想氣體的通量雅可比矩陣

H為：

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

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

### 線性化形式

${\displaystyle {\frac {\partial \mathbf {m} }{\partial t}}+\mathbf {A} _{x,0}{\frac {\partial \mathbf {m} }{\partial x}}+\mathbf {A} _{y,0}{\frac {\partial \mathbf {m} }{\partial y}}+\mathbf {A} _{z,0}{\frac {\partial \mathbf {m} }{\partial z}}=0,}$

### 線性化一維的非耦合波方程

${\displaystyle {\frac {\partial \mathbf {m} }{\partial t}}+\mathbf {A} _{x,0}{\frac {\partial \mathbf {m} }{\partial x}}=0.}$

${\displaystyle \mathbf {A} _{x,0}=\mathbf {P} \mathbf {\Lambda } \mathbf {P} ^{-1},}$
${\displaystyle \mathbf {P} =\left[\mathbf {r} _{1},\mathbf {r} _{2},\mathbf {r} _{3}\right]=\left[{\begin{array}{c c c}1&1&1\\u-a&u&u+a\\H-ua&{\frac {1}{2}}u^{2}&H+ua\\\end{array}}\right],}$
${\displaystyle \mathbf {\Lambda } ={\begin{bmatrix}\lambda _{1}&0&0\\0&\lambda _{2}&0\\0&0&\lambda _{3}\\\end{bmatrix}}={\begin{bmatrix}u-a&0&0\\0&u&0\\0&0&u+a\\\end{bmatrix}}.}$

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

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

## 注釋

1. ^ Anderson, John David. Computational fluid dynamics: the basics with applications. New York, NY: McGraw-Hill. 2010 [2022-07-21]. ISBN 978-0-07-001685-9. OCLC 711810200. （原始内容存档于2022-07-21） （英语）.
2. ^ Christodoulou, Demetrios. The Euler Equations of Compressible Fluid Flow. Bulletin of the American Mathematical Society. 2007-06-18, 44 (4). ISSN 0273-0979. doi:10.1090/S0273-0979-07-01181-0 （英语）.
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.