Let us consider a fully developed channel flow, without loss of generality
Assumption: All derivatives with respect to x are zero except
Mean velocity field
The averaged NS equations give
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(18.1) |
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(18.2) |
Integrating 2 between 0 and y , with no-slip boundary condition, we get
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(18.3) |
Thus
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(18.4) |
also,
From (18.1) and (18.4)
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(18.5) |
Integrating (18.5) between 0 and y , we get
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(18.6) |
Set |
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Where, is the friction velocity. The equation (18.6) becomes
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(18.7) |
At y =h , at the center of the channel, due to symmetry,
Thus from (18.7), we get
Equation (18.7) becomes
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(18.8) |
put
Now equation (18.8) reads as
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(18.9) |
Where
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