Vorticity in Turbulent Flows

Reynolds decomposition of vorticity leads to

with

Substituting this in vorticity equation (13.1) and performing the averaging technique yields the following expression for steady mean flow

(13.5)

In the above equation,

It is easy to prove that and , i.e., both mean and fluctuating vorticity are divergence free. From these and the continuity equation, we can derive the following

(13.6)

and

(since )
	(13.7)

RHS of (13.6) is analogous to Reynolds stress term in the momentum equation for mean flow. It is mean transport of through its interaction with . Equation (13.7) accounts for the gain or loss of mean vorticity caused by stretching and rotation of vorticity components due to fluctuating strain rates.

The term acts as a source or sink term in the equation for mean vorticity. This term is absent in a two-dimensional flow. If the flow is described by only plane, then and are trivial and only non-zero vorticity is . The vortex stretching term in a two-dimensional flow field thus becomes . In a two-dimensional flow field only and can be non-zero. Which tantamounts to for a two-dimensional flow field. In conclusion it can be said that a two-dimensional flow is incapable of stretching a vortex.

An important use of the contraction cone in the upstream of a wind tunnel is that it reduces irregularities of the flow. The mechanism is as follows. In contraction cone, the angular velocity in the direction of stretching becomes greater. Vortex stretching brings about a change in eddy scales (decreases). At the exit of the contraction cone high energy eddies are obtained. These eddies keep changing the scale. A development section follows the contraction cone in which turbulence decays rapidly following the cascade mechanism of energy transfer. Finally, turbulence level goes down to a very low value at the entry point of the test section which follows the development section.