Reynolds stress and vorticity

The instantaneous vorticity can be decomposed in the same manner as we did for velocity and pressure leading to

The equation for mean steady flow is:

	(11.13)

The dynamic significance to Reynolds stress is mainly associated with the interaction between the velocity and vorticity. This interaction is better understood by restricting ourselves to a 2D mean flow with

(11.14)

Which is a boundary layer type of flow. For this flow, the only non-zero component of is

(11.15)

Due to the inequalities (11.14), equation (11.15) becomes

(11.16)

On the otherhand, in the equation for , the velocity-vorticity cross products term becomes

(11.17)

Further

(11.18)

Substituting (11.17) and (11.18) into the momentum equation and neglecting the contribution of turbulence to the normal stress term and viscous effects, we obtain

(11.19)

The Reynolds' equation for this situation reads as

(11.20)

Comparing (11.19) and (11.20) we can write

(11.21)