涡量方程

涡量方程（英語：vorticity equation）是流体力学中描述流体质点涡量变化的方程。可压缩牛顿流体的涡量方程表达式为：

${\displaystyle {\begin{aligned}{\frac {D{\boldsymbol {\omega }}}{Dt}}&={\frac {\partial {\boldsymbol {\omega }}}{\partial t}}+(\mathbf {u} \cdot \nabla ){\boldsymbol {\omega }}\\&=({\boldsymbol {\omega }}\cdot \nabla )\mathbf {u} -{\boldsymbol {\omega }}(\nabla \cdot \mathbf {u} )+{\frac {1}{\rho ^{2}}}\nabla \rho \times \nabla p+\nabla \times \left({\frac {\nabla \cdot \tau }{\rho }}\right)+\nabla \times \left({\frac {B}{\rho }}\right)\end{aligned}}}$

其中D/Dt表示物质导数，u为流速，ρ为流体密度，p为压强，τ为粘性应力张量，B为流体所受外力。方程右边第一项表示涡旋伸展。使用爱因斯坦求和约定指标记号，上式又可写作

${\displaystyle {\begin{aligned}{\frac {d\omega _{i}}{dt}}&={\frac {\partial \omega _{i}}{\partial t}}+v_{j}{\frac {\partial \omega _{i}}{\partial x_{j}}}\\&=\omega _{j}{\frac {\partial v_{i}}{\partial x_{j}}}-\omega _{i}{\frac {\partial v_{j}}{\partial x_{j}}}+e_{ijk}{\frac {1}{\rho ^{2}}}{\frac {\partial \rho }{\partial x_{j}}}{\frac {\partial p}{\partial x_{k}}}+e_{ijk}{\frac {\partial }{\partial x_{j}}}\left({\frac {1}{\rho }}{\frac {\partial \tau _{km}}{\partial x_{m}}}\right)+e_{ijk}{\frac {\partial B_{k}}{\partial x_{j}}}\end{aligned}}}$

对于保守外力作用下的不可压缩流体，涡量方程可以简化为

${\displaystyle {\frac {D{\boldsymbol {\omega }}}{Dt}}=\left({\boldsymbol {\omega }}\cdot \nabla \right)\mathbf {u} +\nu \nabla ^{2}{\boldsymbol {\omega }}}$

其中ν为运动黏度，∇²为拉普拉斯算符。

参考文献

• Manna, Utpal; Sritharan, S. S. Lyapunov Functionals and Local Dissipativity for the Vorticity Equation in LTemplate:Isup and Besov spaces. Differential and Integral Equations. 2007, 20 (5): 581–598. 
• Barbu, V.; Sritharan, S. S. M-Accretive Quantization of the Vorticity Equation (PDF). Balakrishnan, A. V. (编). Semi-Groups of Operators: Theory and Applications. Boston: Birkhauser. 2000: 296–303 [2016-12-10]. （原始内容 (PDF)于2016-03-03）. 
• Krigel, A. M. Vortex evolution. Geophysical, Astrophysical Fluid Dynamics. 1983, 24: 213–223.