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:
D ω D t = ∂ ω ∂ t + ( u ⋅ ∇ ) ω = ( ω ⋅ ∇ ) u − ω ( ∇ ⋅ u ) + 1 ρ 2 ∇ ρ × ∇ p + ∇ × ( ∇ ⋅ τ ρ ) + ∇ × ( B ρ ) {\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
d ω i d t = ∂ ω i ∂ t + v j ∂ ω i ∂ x j = ω j ∂ v i ∂ x j − ω i ∂ v j ∂ x j + e i j k 1 ρ 2 ∂ ρ ∂ x j ∂ p ∂ x k + e i j k ∂ ∂ x j ( 1 ρ ∂ τ k m ∂ x m ) + e i j k ∂ B k ∂ x j {\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
D ω D t = ( ω ⋅ ∇ ) u + ν ∇ 2 ω {\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 。
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