Consider a steady, turbulent boundary layer on

Chapter 7, Problem 7.6

(choose chapter or problem)

Consider a steady, turbulent boundary layer on an isothermal flat plate of temperature \(T_{s}\). The boundary layer is “tripped” at the leading edge \(x=0\) by a fine wire. Assume constant physical properties and velocity and temperature profiles of the form

\(\frac{u}{u_{\infty}}=\left(\frac{y}{\delta}\right)^{1 / 7} \quad \text { and } \quad \frac{T-T_{\infty}}{T_{s}-T_{\infty}}=1-\left(\frac{y}{\delta_{t}}\right)^{1 / 7}\)

a) From experiment it is known that the surface shear stress is related to the boundary layer thickness by an expression of the form

\(\tau_{s}=0.0228 \rho u_{\infty}^{2}\left(\frac{u_{\infty} \delta}{\nu}\right)^{-1 / 4}\)

Beginning with the momentum integral equation (Appendix G), show that

\(\delta / x=0.376 R e_{x}^{-1 / 5}\)

Determine the average friction coefficient \(\bar{C}_{f, x}\).

(b) Beginning with the energy integral equation, obtain an expression for the local Nusselt number \(N u_{x}\) and use this result to evaluate the average Nusselt number \(\overline{N u}_{x} .\)