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:
![](images/image066.gif) |
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![](images/image068.gif) |
(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
![](images/image070.gif) |
(11.14) |
Which is a boundary layer type of flow. For this flow, the only non-zero component of is
![](images/image074.gif) |
(11.15) |
Due to the inequalities (11.14), equation (11.15) becomes
![](images/image076.gif) |
(11.16) |
On the otherhand, in the equation for , the velocity-vorticity cross products term becomes
![](images/image082.gif) |
(11.17) |
Further
![](images/image084.gif) |
(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
![](images/image086.gif) |
(11.19) |
The Reynolds' equation for this situation reads as
![](images/image088.gif) |
(11.20) |
Comparing (11.19) and (11.20) we can write
![](images/image090.gif) |
(11.21) |
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