Question 10.4: During flow over a flat plate the laminar boundary layer und...
During flow over a flat plate the laminar boundary layer undergoes a transition to turbulent boundary layer as the flow proceeds in the downstream. It is observed that a parabolic laminar profile is finally changed into a 1/7th power law velocity profile in the turbulent regime. Find out the ratio of turbulent and laminar boundary layers if the momentum flux within the boundary layer remains constant.
Assume width of the boundary layers be a. Then momentum flux is
A=\int u \rho u a d y=\rho U_{\infty}^{2} a \delta \int\left(\frac{u}{U_{\infty}}\right)^{2} d \eta
where \eta=y / \delta
For laminar flow, \frac{u}{U_{\infty}}=2 \eta-\eta^{2}
A_{ lam }=\rho U_{\infty}^{2} a \delta_{ lam } \int_{0}^{1}\left(4 \eta^{2}-4 \eta^{3}+\eta^{4}\right) d \eta
=\rho U_{\infty}^{2} a \delta_{\operatorname{lam}}\left[\frac{4}{3} \eta^{3}-\eta^{4}+\frac{\eta^{5}}{5}\right]_{0}^{1}
=\frac{8}{15} \rho U_{\infty}^{2} a \delta_{\text {lam }}
For 1/7th power law turbulent profile,
\frac{\bar{u}}{U_{\infty}}=\eta^{1 / 7}
A_{\text {turb }}=\rho U_{\infty}^{2} a \delta_{\text {turb }} \int_{0}^{1}\left(\eta^{1 / 7}\right)^{2} d \eta
=\rho U_{\infty}^{2} a \delta_{\text {turb }} \int_{0}^{1} \eta^{2 / 7} d \eta
=\rho U_{\infty}^{2} a \delta_{\text {turb }}\left[\frac{7}{9} \eta^{9 / 7}\right]_{0}^{1}=\frac{7}{9} \rho U_{\infty}^{2} a \delta_{\text {turb }}
Comparing the momentum fluxes,
\frac{\delta_{\text {turb }}}{\delta_{\text {lam }}}=\frac{72}{105}
It is to be noted that generally turbulent boundary layer grows faster than the laminar boundary layer when a completely turbulent flow is considered from the leading edge. However the present result is valid at transition for a constant momentum flow.