Calculate the mean velocity (um) for turbulent velocity distribution in a pipe.
um = Q/A
u = umax(y/R)1/7 (Gives velocity at a specified point in cross section of the pipe.
R = pipe diameter. y = distance from boundary layer. umax = distance at centre.
The Attempt at a Solution
I'll post a diagram of what I've done so far. As you can see, I've split the cross section of the pipe into rings of flow of thickness dr and radius r. So, dA = 2(pi)(r)(dr). dQ = u(dA)
Integrate in the limit 0 to R. Q = ∫ umax(y/R)1/7(2(pi)(r)(dr)).
How do I integrate the above? That's what's really confusing me. Is u constant and so can it factor out of the integral? The y term is really throwing me off. Thanks.