The vorticity equation

It is possible to derive an equation for the conservation of vorticity by taking the curl of the Navier-Stokes equation. This leads to

	(13.1)

So far we have worked with the momentum equation. Let us focus at the equation for the conservation of vorticity as well.

We note that is skew-symmetric and is symmetric. It can be shown that the product of a skew-symmetric tensor and a symmetric tensor is zero. Furthermore,

With the introduction of these in to Eqn. (13.1) and properly calculating the Kronecker-deltas, we obtain

The above equation, may be set to zero since vorticity is divergence free.

The final form of equation (13.1) may be expressed as

(13.2)

Now   is split up into two parts

(13.3)

The second term on the rhs of (13.3) becomes

Since j and k are dummy indices, they can be interchanged to obtain

Once again interchanging the indices j and k in , we obtain a change in sign because is skew symmetric. Hence we find

This is true if is zero. As a consequences, Equation (13.2) becomes

(13.4)

The term is the vortex stretching term. The vortex stretching takes place in the accelerating flow in a wind tunnel contraction

Figure 13.1

Vortex stretching brings about a change in length scale. Refer to equation (12.6) as well !