# Vorticity Transport Equation

Since turbulent flows contain eddies that are rotational, a transport equation of local rotation (vorticity) of the fluid is briefly discussed.

The vorticity vector, $\stackrel{\to }{\omega }$ , is the curl of the velocity vector and is defined as follows: (1)
$\stackrel{\to }{\omega }=\nabla ×\stackrel{\to }{u}$
or (2)
${\omega }_{i}={ϵ}_{ijk}\frac{\partial {u}_{\text{k}}}{\partial {x}_{j}}sa$
The vorticity transport equation can be obtained by taking the curl of the momentum conservation equation: (3)

The U term becomes $\nabla ×\frac{\partial \stackrel{\to }{u}}{\partial t}=\frac{\partial }{\partial t}\left(\nabla ×\stackrel{\to }{u}\right)=\frac{\partial \stackrel{\to }{\omega }}{\partial t}$

The P term vanishes as $\nabla ×\nabla \cdot \text{p}=0$

The V term becomes $\nabla ×\left(\frac{\mu {\nabla }^{2}\stackrel{\to }{u}}{\rho }\right)=\frac{\mu }{\rho }{\nabla }^{2}\stackrel{\to }{\omega }$

The convective term in C can be written as: (4)
$\stackrel{\to }{u}\cdot \nabla \stackrel{\to }{u}=\frac{1}{2}\nabla \left(\stackrel{\to }{u}\cdot \stackrel{\to }{u}\right)-\stackrel{\to }{u}×\left(\nabla ×\stackrel{\to }{u}\right)=\frac{1}{2}\nabla \left({u}^{2}\right)-\stackrel{\to }{u}×\stackrel{\to }{\omega }$

The C term becomes $\nabla ×\left(\stackrel{\to }{u}\cdot \nabla \stackrel{\to }{u}\right)=\nabla ×\frac{1}{2}\nabla \left({u}^{2}\right)-\nabla ×\left(\stackrel{\to }{u}×\stackrel{\to }{\omega }\right)=\nabla ×\left(\stackrel{\to }{\omega }×\stackrel{\to }{u}\right)$

The C term can be rearranged as $\nabla ×\left(\stackrel{\to }{\omega }×\stackrel{\to }{u}\right)=\left(\stackrel{\to }{u}\cdot \nabla \right)\stackrel{\to }{\omega }-\left(\stackrel{\to }{\omega }\cdot \nabla \right)\stackrel{\to }{u}$

After substituting above terms, the vorticity transport equation can be obtained as: (5)
Using tensor notation, it is given by: (6)

The vortex stretching term, ${\omega }_{j}\frac{\partial {u}_{i}}{\partial {x}_{j}}$ , on the right-hand side of the equation corresponds to the enhancement of vorticity ${\omega }_{j}$ when a fluid element is stretched $\frac{\partial {u}_{i}}{\partial {x}_{j}}$ <0. In other words, when the cross section of the fluid element is decreased, the vorticity is increased. The image below shows the concept of vortex stretching. In the image, there are two cylindrical fluid elements within the streamwise fluid flow. The concept shows that compared to a thick element a thin element has stronger vorticity due to the angular momentum conservation.

In two-dimensional flow, the vortex stretching term ${\omega }_{j}\frac{\partial {u}_{i}}{\partial {x}_{j}}$ vanishes since ${\omega }_{x}=0$ , ${\omega }_{y}=0$ and $\frac{\partial }{\partial z}=0$ . Therefore, the vortex stretching term is essential to the energy cascade for three-dimensional turbulent flow.