Average velocity in turbulent pipe flow

[willm78]

When I first looked at this problem, I thought I'd be able to breeze through it. I sat down at my desk at 5pm and didn't get up until after 3am. I have the solution but am missing a step. I was able to find this exact question within a couple publications online but none show the complete solution.

Q: Assume the velocity profile in turbulent flow follows a power law, as in Eq. 3-7, with exponent 'n'. Derive anexpression for the ratio of the average velocity to the maximum velocity (v/vmax) as afunction of n. Plot v/vmax for n ranging from 0.1 to 0.5.

I hope someone might be able to help. This problem is driving me crazy! I have tried everything I know of and nothing has worked yet. Even when I integrate the part I'm stuck at on a calculator, the solution comes back wrong.

 

[willm78]

Solved it with substitution:

$\displaystyle \dfrac{\bar{v}(r)}{v_{max}}\, =\, \dfrac{2}{r_i^2}\,$$\displaystyle \displaystyle \int_0^{r_i}\, \left(1\, -\, \dfrac{r}{r_i}\right)^n\, r\, dr$

$\displaystyle u\, =\, 1\, -\, \dfrac{r}{r_i}\, \Rightarrow\, r\, =\, (1\, -\, u)\, r_i$

. . . . . . . . . . . .$\displaystyle dr\, =\, -r_i\, du$

\(\displaystyle \begin{align} \dfrac{\bar{v}(r)}{v_{max}}\, &=\, \dfrac{2}{r_i^2}\, \displaystyle{\int} \, u^n\, (1\, -\, u)\, r_i\, (-r_i)\, du

\\ \\&=\, \dfrac{2}{r_i^2} \, \left(\, \dfrac{r_i^2\, u^{n+2}}{n\, +\, 2}\, -\, \dfrac{r_i^2\, u^{n+1}}{n\, +\, 1}\, \right)

\\ \\&=\, \dfrac{2}{1} \, \left(\, \dfrac{u^{n+2}}{n\, +\, 2}\, -\, \dfrac{u^{n+1}}{n\, +\, 1}\, \right)

\\ \\&=\, \dfrac{2}{1} \, \left.\left(\, \dfrac{\left(1\, -\, \dfrac{r}{r_i}\right)^{n+2}}{n\, +\, 2}\, -\, \dfrac{\left(1\, -\, \dfrac{r}{r_i}\right)^{n+1}}{n\, +\, 1}\, \right)\, \right|_0^{r_i}

\\ \\&=\, 2\, \left(0\, -\, 0\, -\, \dfrac{1}{n\, +\,2}\, +\, \dfrac{1}{n\, +\, 1}\right)

\\ \\&=\, -\dfrac{2}{n\, +\, 2}\, +\, \dfrac{2}{n\, +\, 1}

\\ \\&=\, -\dfrac{2\, (n\, +\, 1)}{(n\, +\, 2)\, (n\, +\, 1)}\, +\, \dfrac{2\, (n\, +\, 2)}{(n\, +\, 1)\, (n\, +\, 2)}

\\ \\&=\, \dfrac{-(2n\, +\, 2)\, +\, (2n\, +\, 4)}{(n\, +\, 1)\, (n\, +\, 2)}

\\ \\&=\, \dfrac{-2n\, -\, 2\, +\, 2n\, +\, 4}{(n\, +\, 1)\, (n\, +\, 2)} \end{align}\)

$\displaystyle \boxed{ \dfrac{\bar{v}(r)}{v_{max}}\, =\,\dfrac{-2n\, -\, 2\, +\, 2n\, +\, 4}{(n\, +\, 1)\, (n\, +\, 2)} }$