Question

5. One of the simplest ways to recapture the Reynolds stress is to time-average the conservation form of the axial momentum equation in Cartesian coordinates. (a) Start by expanding (xial momentum in conservation form) Then using the continuity equation, show that the result is identical to the traditional form: +1, (steady, incompressible, axial momentum form) (b) By substituting the Reynolds decomposed variables, u +,p-p+p', into the conservation form, time-averaging, cancelling terms that average out, and assuming that streamwise variations in 'u' are much slower than the cross-streamwise variations in u'v', show that Hint: The time-averaged products of mean and fluctuating quantities, such as '',vu', and v all vanish identically. Similarly, the time-averaged fluctuating quantities vanish unless they are squared or multiplied by other fluctuating quatntities. Finally, the time-averaged mean quantities are the mean quantities.

Answer 1