Question

1. For flow in a circular tube, please evaluate the ratio of <Vz^2>/<Vz>^2 for (1) laminar flow (using parabolic velocity profile) and (2) turbulent flow (using 1/7- power expression).

2. For turbulent flow in smooth circular tubes, the above 1/7-power is not general. Empirical fitting of the curves of the time-smoothed velocity profiles suggested the following function, (Vz)/(Vz,max) = ((1-(r/R))^(1/n) , in which n depends on Reynolds number: near Re = 4000, n=6; near Re = 1.1*10^5, n=7; and near Re = 3.2*10^6, n=10. Show that the ratio of average to maximum velocity is, (Vz)/(Vz,max) = (((2n^2)/((n+1)(2n+1))) .

Answer 1