The velocity profile in turbulent flow in a pipe is often approximated by a one-seventh power law:

\[v_{z}=v_{\mathrm{c}}[1-(r / R)]^{1 / 7}\]

(a) Calculate the following: \(\langle vangle,\left\langle v^{2}\rightangle\), and \(\left\langle v^{3}\rightangle\).

(b) What error would be introduced if the rate of momentum flow were based on \(\langle vangle^{2}\) instead of \(\left\langle v^{2}\rightangle\) ?

(c) What error would be introduced if the rate of kinetic energy flow were based on \(\langle vangle^{3}\) instead of \(\left\langle v^{3}\rightangle\) ?