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TWiki> Main Web>TWikiUsers>RichardHoward>RichardHowardLESwithQ>RichardHowardydependence (2010-07-22, NeilAshton)

TWiki> Main Web>TWikiUsers>RichardHoward>RichardHowardLESwithQ>RichardHowardydependence (2010-07-22, NeilAshton)

The instantaneous velocity perturbations in a 2D (in the mean) turbulent boundary layer only vary in the wall normal direction because the flow is unconstrained in the other directions and therefore any velocity perturbation is unnaffected by a change of position in the streamwise or spanwise direction.

At the wall, and the "no-slip" condition applies so and .

Continuity also requires that

From this it can be seen that and hence

Looking at leading order terms in the vicinity of the wall we can therefore say

Thus we can now show dependence of the Reynolds stress perturbations:

we can also look at the dominant terms in some other useful quantities

where the following quantities become small due to

i) lack of streamwise and spanwise dependence , , ,

ii) continuity

so the term ends up being independent of the direction.

There is also a suggestion for the following quantity:

strictly speaking has no dependence due to the assumption that there is no dependence of fluctuations on the streamwise and spanwise directions . However the streamwise and spanwise dependences are one order lower than all the equivalent terms in . In the case of LES where we are always working locally it is reasonable to expect that the dependence of is at least one order greater than the dependence of .

If we make the approximation that first order terms in and retain their dependence and second order terms lose their dependence we can propose approximate conditions such as:

and

and hence

Strictly speaking and however the fact that has less dependence on the streamwise and spanwise directions than suggests that it has more of a residual dependence than

Topic revision: r3 - 2010-07-22 - 16:45:53 - NeilAshton

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