An appropriate turbulent pipe flow velocity profile is [ mathbf{V}=u_{c}left(frac{R-r}{R} ight)^{1 / n} hat{mathbf{i}} ] where (u_{c}=)

An appropriate turbulent pipe flow velocity profile is

\[ \mathbf{V}=u_{c}\left(\frac{R-r}{R}\right)^{1 / n} \hat{\mathbf{i}} \]

where \(u_{c}=\) centerline velocity, \(r=\) local radius, \(R=\) pipe radius, and \(\hat{\mathbf{i}}=\) unit vector along pipe centerline. Determine the ratio of average velocity, \(\bar{u}\), to centerline velocity, \(u_{c}\), for

(a) \(n=4\),

(b) \(n=6\),

(c) \(n=8\),

(d) \(n=10\). Compare the different velocity profiles.