In fluid dynamics, the vorticity transport equation can be derived by taking the curl of the Navier-Stokes equations. In 2D [$\boldsymbol \omega = (0,0,\omega)$], the vorticity transport equation can be written as
\begin{align}
\frac{\partial \boldsymbol\omega}{\partial t}+(\mathbf{u}\cdot\nabla)\boldsymbol\omega=\frac{1}{Re}\nabla^2\boldsymbol\omega
\end{align}
where the vorticity $\boldsymbol\omega$ is the curl of the velocity $\mathbf{u}$
\begin{align}
\boldsymbol\omega=\nabla\times\mathbf{u}
\end{align}
The curl of the curl vector identity
\begin{align}
\nabla \times \left( \nabla \times \mathbf{A} \right) = \nabla(\nabla \cdot \mathbf{A}) – \nabla^{2}\mathbf{A}\\\\
\nabla \times \left( \nabla \times \mathbf{u} \right) = \nabla(\nabla \cdot \mathbf{u}) – \nabla^{2}\mathbf{u}
\end{align}
is used together with the incompressible flow condition $\nabla\cdot\mathbf{u}=0$ to get a Poisson equation for the velocity field
\begin{align}
\nabla \times \boldsymbol \omega = – \nabla^{2}\mathbf{u}
\end{align}
This closes the system of equations for 2D incompressible flow using the vorticity-velocity formulation.
My question is how to proof, or get, the above Poisson equation from the vorticity transport equation. What I have tried is to apply the divergence operator to the whole vorticity transport equation which leads to
\begin{align}
\frac{\partial \nabla\cdot\boldsymbol\omega}{\partial t}+\nabla\cdot\left[(\mathbf{u}\cdot\nabla)\boldsymbol\omega\right]=\frac{1}{Re}\nabla\cdot\left(\nabla^2\boldsymbol\omega\right)
\end{align}
since $\nabla\cdot\boldsymbol\omega=0$
\begin{align}
\nabla\cdot\left[(\mathbf{u}\cdot\nabla)\boldsymbol\omega\right]&=\frac{1}{Re}\nabla^2\left(\nabla\cdot\boldsymbol\omega\right)\\\\
\nabla\cdot\left[(\mathbf{u}\cdot\nabla)\boldsymbol\omega\right]&=0
\end{align}
I guess that from the equation above somehow one can get the velocity Poisson equation, but I have not been able to solve this yet.
There is a vector identity which can be written as
\begin{align}
\nabla \times (\mathbf{A} \times \mathbf{B}) &\ =\ \mathbf{A}\ (\nabla \cdot \mathbf{B}) – \mathbf{B}\ (\nabla \cdot \mathbf{A}) + (\mathbf{B} \cdot \nabla) \mathbf{A} – (\mathbf{A} \cdot \nabla) \mathbf{B}\\\\
\nabla \times \left(\mathbf u \times \boldsymbol \omega \right) &= \mathbf u \left(\nabla \cdot \boldsymbol \omega\right) -\boldsymbol \omega \left(\nabla \cdot \mathbf u\right) + \left(\boldsymbol \omega \cdot \nabla \right) \mathbf u – \left(\mathbf u \cdot \nabla\right) \boldsymbol \omega
\end{align}
Which for 2D incompressible flow is
\begin{align}
\nabla \times \left(\mathbf u \times \boldsymbol \omega \right) &=- \left(\mathbf u \cdot \nabla\right) \boldsymbol \omega
\end{align}
So probably with the relation above the Poisson equation for the velocity field can be recovered, but I got stuck here. Any help would be very much appreciated.
Best Answer
The vorticity transport equation is obtained from the (dimensionless) Navier-Stokes momentum equation
$$\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla\mathbf{u}= -\nabla p \, + \, \frac{1}{Re} \nabla^2 \mathbf{u}.$$
Taking the curl of both sides and applying some vector identities we get
$$\frac{\partial \boldsymbol\omega}{\partial t} + \mathbf{u} \cdot \nabla\boldsymbol\omega= \boldsymbol\omega \cdot \nabla \mathbf{u} - \boldsymbol\omega \, (\nabla \cdot \mathbf{u}) \, + \, \frac{1}{Re} \nabla^2 \boldsymbol\omega.$$
The first term on the RHS (vortex-line stretching) vanishes in 2D flow and the second on the RHS vanishes for incompressible flow since $\nabla \cdot \mathbf{u} = 0,$ resulting in
$$\frac{\partial \boldsymbol\omega}{\partial t} + \mathbf{u} \cdot \nabla\boldsymbol\omega= \frac{1}{Re} \nabla^2 \boldsymbol\omega.$$
You have already derived the equation
$$\tag{*}-\nabla^2 \mathbf{u} = \nabla \times \boldsymbol\omega,$$
using the definition of vorticity, $ \boldsymbol\omega = \nabla \times \mathbf{u},$ and the incompressibility condition, $ \nabla \cdot \mathbf{u} = 0.$ These are independent of the momentum equation and vorticity transport equations.
You are not going to derive (*) from the vorticity transport equation. It is just a kinematic relationship that incorporates the incompressibility condition. The incompressibility condition expresses conservation of mass. The vorticity transport equation expresses conservation of angular momentum.