Solving Turbulent Flow Velocity Distribution in a Pipe

[mathfied]

Hi Guys.
I have an absurd problem and no idea to solve it. Please could you help me?

$$\frac{u}{\hat u} = \left [ \frac{y}{R} \right ]^{1/n} = \left [1- \frac{r}{R} \right ]^{1/n}$$

Suppose the turbulent flow velocity distribution in a pipe of Radius R can be described by the equation above (Power Law relation).

y= distance from wall
r= radial distance from the axis
$$\hat u$$ = velocity on the axis

If $$\bar u$$ is the space mean average velocity in the pipe, show that:

$$\frac{\bar u}{\hat u} = \frac{2n^2}{(n+1)(2n+1)}$$

Any solutions please? I know I haven't posted my solutions but that's simply because I have no idea on this question.

 

[ideasrule]

Typically, "average velocity" is defined so that average velocity * area = volume flow, because that's the most useful definition of the "average". So the first step is to find the volume flow across this pipe, using integration.