The velocity profile for turbulent flow in circular pipes is often approximated by the \(1 / 7\) thpower law:

\[\bar{v}_{z}=v_{\max }(1-r / R)^{1 / 7}\]

Find an expression for the cross-sectionally averaged velocity.

A more general expression for the velocity profile is

\[\bar{v}_{z}=v_{\max }(1-r / R)^{1 / n}\]

where \(n\) is an index that is fitted as a function of the Reynolds number. The value of \(n\) increases with \(R e\), indicating that the profiles becomes steeper at large values of \(R e\).

Find an expression for the average velocity for this general profile.