Turbulent boundary layer
Unlike the L.B.L. there is only one well-known turbulent velocity profile which is known as the seventh root law profile that suggested by the Prandtl:
\(\overset{\overline{}}{U}={\overset{\overline{}}{y}}^{\frac{1}{7}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.26\right)\)
Local stream-function coefficient
Let us formulate an expression of the local stream-function coefficient in turbulent boundary layer.
\(\tau_{w}=0.0233\rho U^{\frac{7}{4}}\left(\frac{\nu}{y}\right)^{0.25}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.27\right)\)
\begin{equation} C_{f}=\frac{\tau_{w}}{\frac{1}{2}\rho{U_{\infty}}^{2}}=\frac{0.0233\rho U^{\frac{7}{4}}\left(\frac{\nu}{y}\right)^{0.25}\ }{\frac{1}{2}\rho{U_{\infty}}^{2}}=\frac{0.0466\ \rho U^{\frac{7}{4}}\left(\frac{\nu}{y}\right)^{0.25}\ }{\rho{U_{\infty}}^{2}}\nonumber \\ \end{equation}
\(at\ y=\delta\ \rightarrow U=U_{\infty}\)
\(C_{f}=\frac{0.0466\ \rho{U_{\infty}}^{\frac{7}{4}}\left(\frac{\nu}{y}\right)^{0.25}\ }{\rho{U_{\infty}}^{2}}=\frac{0.0466\nu^{0.25}}{{U_{\infty}}^{0.25}\delta^{0.25}}=\frac{0.0466}{{\text{Re}^{0.25}}_{\delta}}\)
\(C_{f}=\frac{0.0466}{{\text{Re}_{\delta}}^{0.25}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.28\right)\)
Rate of growth of turbulent boundary layer
The derivation is starting by writing the M.I.E. ;
\(\tau_{w}=\rho{U_{\infty}}^{2}\frac{d\theta}{\text{dx}}\)
\(\tau_{w}=\rho{U_{\infty}}^{2}I\frac{\text{dδ}}{\text{dx}}\)
\(I=\int_{0}^{1}{\overset{\overline{}}{U}\left(1-\overset{\overline{}}{U}\right)d\overset{\overline{}}{y}=\int_{0}^{1}{{\overset{\overline{}}{y}}^{\frac{1}{7}}\ \left(1-{\overset{\overline{}}{y}}^{\frac{1}{7}}\right)d\overset{\overline{}}{y}}}=\frac{7}{72}\)
\(\tau_{w}=\frac{7}{72}\rho{U_{\infty}}^{2}\frac{\text{dδ}}{\text{dx}}\rightarrow\frac{\tau_{w}}{\rho{U_{\infty}}^{2}}=\frac{7}{72}\frac{\text{dδ}}{\text{dx}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.29\right)\)
\(C_{f}=\frac{\tau_{w}}{\frac{1}{2}\rho{U_{\infty}}^{2}}\rightarrow\frac{1}{2}C_{f}=\frac{\tau_{w}}{\rho{U_{\infty}}^{2}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.17\right)\ \)
Let us equalize Eq. (7.17) with Eq. (7.29), we get\(\frac{7}{72}\frac{\text{dδ}}{\text{dx}}=\frac{1}{2}C_{f}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.30\right)\)
But we have already obtained\(C_{f}=\frac{0.0466}{{\text{Re}_{\delta}}^{0.25}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.28\right)\)
Substitute Eq. (7.28) in Eq. (7.30)
\(\frac{1}{2}\left\{\frac{0.0466}{{\text{Re}_{\delta}}^{0.25}}\right\}=\frac{7}{72}\frac{\text{dδ}}{\text{dx}}\)\(\rightarrow\frac{1}{2}\frac{0.0466\nu^{0.25}}{{U_{\infty}}^{0.25}\delta^{0.25}}=\frac{7}{72}\frac{\text{dδ}}{\text{dx}}\)This is 1st ODE will be solved simply using separation of variable method
\(\int_{0}^{x}{0.23965\left(\frac{\nu}{U_{\infty}}\right)^{0.25}\text{dx}}=\int_{0}^{0.25}\delta\text{dδ}\)\(\rightarrow\frac{4}{5}\delta^{5/4}=0.23965\left(\frac{\nu}{U_{\infty}}\right)^{1/4}\text{x\ }\)
\(\delta=0.3812325351\left(\frac{\nu}{U_{\infty}}\right)^{1/5}x^{4/5}=0.3812325351\left(\frac{\nu}{U_{\infty}}\right)^{0.2}\frac{x}{x^{0.2}}\ \)
\(\delta=\frac{0.3812325351x}{{\text{Re}_{x}}^{0.2}}\)
\(\frac{\delta_{T}}{x}=\frac{0.38123}{{\text{Re}_{x}}^{0.2}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.31\right)\)
  1. Characteristics of turbulent boundary layer
  2. Displacement thickness
\(\frac{\delta^{*}}{\delta}=\int_{0}^{1}{\left(1-\overset{\overline{}}{U}\right)d\overset{\overline{}}{y}}\)
\(\overset{\overline{}}{U}={\overset{\overline{}}{y}}^{\frac{1}{7}}\)
\(\frac{\delta^{*}}{\delta}=\int_{0}^{1}{\left(1-{\overset{\overline{}}{y}}^{\frac{1}{7}}\ \right)d\overset{\overline{}}{y}=\frac{1}{8}}\)
\(\delta^{*}=\frac{1}{8}*\delta=\frac{1}{8}*\frac{0.38123x}{{\text{Re}_{x}}^{0.2}}=\frac{0.04765\ }{{\text{Re}_{x}}^{0.2}}\text{\ x}\)
\(\delta^{*}=\frac{0.04765\ }{{\text{Re}_{x}}^{0.2}}\text{\ x}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.32\right)\ \)
Momentum thickness
\(\frac{\theta}{\delta}=\int_{0}^{1}{\overset{\overline{}}{U}\left(1-\overset{\overline{}}{U}\right)d\overset{\overline{}}{y}}=\int_{0}^{1}{{\overset{\overline{}}{y}}^{\frac{1}{7}}\left(1-{\overset{\overline{}}{y}}^{\frac{1}{7}}\right)d\overset{\overline{}}{y}}=\frac{7}{72}\)
Similarly, we get\(\ \theta=\frac{0.03706}{{\text{Re}_{x}}^{0.2}}\text{\ x}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.33\right)\)
Energy thickness
By the same approach, we shall get\(\delta^{**}=\frac{0.0667}{{\text{Re}_{x}}^{0.2}}\text{\ x}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.34\right)\)
Skin friction coefficient for turbulent boundary layer
\(C_{f}=\frac{0.0466}{{\text{Re}_{\delta}}^{0.25}}\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.28\right)\)
\(C_{f}=\frac{0.0466\nu^{0.25}}{{U_{\infty}}^{0.25}\delta^{0.25}}=0.0466*\left(\frac{\nu}{U_{\infty}}\right)^{0.25}*\left(\frac{1}{\delta}\right)^{0.25}\ \)
\(\frac{\delta}{x}=\frac{0.38123}{{\text{Re}_{x}}^{0.2}}\rightarrow\ \ \delta=\frac{0.38123}{{\text{Re}_{x}}^{0.2}}\text{x\ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.31\right)\)
Substitute Eq. (7.31) into Eq. (7.28), results;
\(C_{f}=0.0466*\left(\frac{\nu}{U_{\infty}}\right)^{0.25}*\left(\frac{1}{\frac{0.38123}{{\text{Re}_{x}}^{0.2}}x}\right)^{0.25}=0.0466*\left(\frac{\nu}{U_{\infty}}\right)^{0.25}*\frac{{{\text{Re}_{x}}^{0.2}}^{0.25}}{{{0.38123}^{0.25}\text{\ x}}^{0.25}}\)
\(C_{f}=0.05930470493*\left(\frac{\nu}{U_{\infty}}\right)^{0.25}\frac{{\text{Re}_{x}}^{0.05}}{x^{0.25}}=0.05930470493*\left(\frac{\nu}{U_{\infty}x}\right)^{0.25}{\text{Re}_{x}}^{0.05}\)
\({C_{f}}_{x}=\frac{0.05930470493}{{\text{Re}_{x}}^{0.2}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.35\right)\)
Finally, the wall shear stress will be;
\(\tau_{w}=\frac{1}{2}\rho{U_{\infty}}^{2}{C_{f}}_{x}=\frac{1}{2}\rho{U_{\infty}}^{2}\left\{\frac{0.05930}{{\text{Re}_{x}}^{0.2}}\right\}=0.02965*\frac{\rho{U_{\infty}}^{2}}{{\text{Re}_{x}}^{0.2}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.36\right)\)
Mixed (Transition) Boundary Layer Region
The schematic diagram of the external flow over a flat plate is inserted below;