Figure 13 External flow illustrating the transition zone [11]
Firstly, let us develop an expression of turbulent boundary layer thickness at the transition region in terms of Reynolds number at the laminar boundary layer region as explained below;
However, for laminar zone, sine profile will be selected as a special case study;\(\overset{\overline{}}{U}=\sin{\frac{\pi}{2}\overset{\overline{}}{y}}\)
\(\delta_{L,t}=\frac{4.795\ x_{L,t}}{\sqrt{\text{Re}_{L,t}}}=4.79\left(\frac{\nu}{U_{\infty}\ x_{L,t}}\right)^{0.5}x_{L,t}=4.79\left(\frac{\nu}{U_{\infty}\ }\right)^{0.5}{x_{L,t}}^{0.5}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.37\right)\)
Multiplied and divided the above equation by\(\left(\frac{U_{\infty}}{\text{\ ν}}\right)^{1/2}\)
\(\delta_{L,t}=\frac{4.795\ x_{L,t}}{\sqrt{\text{Re}_{L,t}}}=4.79\left(\frac{\nu}{U_{\infty}\ }\right)^{0.5}{x_{L,t}}^{0.5}{*\left(\frac{U_{\infty}}{\text{\ ν}}\right)}^{\frac{1}{2}}\left(\frac{\nu}{\ U_{\infty}}\right)^{\frac{1}{2}}\)
\(\delta_{L,t}=4.79\ \left(\frac{\nu}{\ U_{\infty}}\right)^{1/2}{\text{Re}_{L,t}}^{0.5}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.38\right)\)
Thus,\(\delta_{T,t}=1.4\delta_{L,t}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.39\right)\)
\(\delta_{T,t}=1.4*4.79\ \left(\frac{\nu}{\ U_{\infty}}\right)^{\frac{1}{2}}{\text{Re}_{L,t}}^{0.5}=6.706\left(\frac{\nu}{\ U_{\infty}}\right)^{\frac{1}{2}}{\text{Re}_{L,t}}^{0.5}\)
\(\delta_{T,t}=6.706\left(\frac{\nu}{\ U_{\infty}}\right)^{\frac{1}{2}}{\text{Re}_{L,t}}^{0.5}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.40\right)\)
Secondly, let us find out a general expression of turbulent layer in the transition region;
\(\delta_{T,t}=\frac{0.38123\ x_{T,t}}{{\text{Re}_{T,t}}^{0.2}}=\frac{0.38123\ x_{T,t}}{\left(\frac{U_{\infty}\text{\ x}_{T,t}}{\text{\ ν}}\right)^{0.2}}=0.38123\left(\frac{\nu}{\ U_{\infty}}\right)^{0.2}{x_{T,t}}^{0.8}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.41\right)\)
By equalizing Eq. (7.40) with Eq. (7.41);
\(6.706\left(\frac{\nu}{\ U_{\infty}}\right)^{\frac{1}{2}}{\text{Re}_{L,t}}^{0.5}=0.38123\left(\frac{\nu}{\ U_{\infty}}\right)^{0.2}{x_{T,t}}^{0.8}\)
\({x_{T,t}}^{0.8}=\frac{6.706}{0.38123}\left(\frac{\nu}{\ U_{\infty}}\right)^{0.8}{\text{Re}_{L,t}}^{0.5}\)
\(x_{T,t}=36.023469\ \frac{\nu}{\ U_{\infty}}\ {\text{Re}_{L,t}}^{5/8\ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.42\right)\ \)
Finally, as aerodynamics researchers we are more interest in obtaining a formula for drag coefficient at the transition zone;
The effective length of turbulent layer is given by;\(L_{T}=L-X_{t}+X_{T,t}\)
\(D=\int_{0}^{L-X_{t}+X_{T,t}}{\tau_{w}\text{\ dx}}=\frac{1}{2}\rho{U_{\infty}}^{2}\int_{0}^{L-X_{t}+X_{T,t}}C_{f}\text{dx}\)
The local skin friction coefficient in the turbulent region can be obtained from the equation inserted below;
\(C_{f}=\frac{0.0593}{{\text{Re}_{x}}^{0.2}}\)
\(D=\frac{1}{2}\rho{U_{\infty}}^{2}\int_{0}^{L-X_{t}+X_{T,t}}\frac{0.0593}{{\text{Re}_{x}}^{0.2}}\text{dx}\)
\(D=\frac{0.0593}{2}\rho{U_{\infty}}^{2}\int_{0}^{L-X_{t}+X_{T,t}}\frac{\ \nu^{0.2}}{{{(U}_{\infty}\ x)}^{0.2}}\text{dx}\)
\(D=\frac{0.0593}{2}\rho{U_{\infty}}^{2}\int_{0}^{L-X_{t}+X_{T,t}}{\left(\frac{\nu}{\ U_{\infty}}\right)^{0.2}\ x^{-0.2}}\text{dx}\)
\(D=\frac{0.0593}{2}\rho{U_{\infty}}^{2}\left(\frac{\nu}{\ U_{\infty}}\right)^{0.2}\int_{0}^{L-X_{t}+X_{T,t}}{\ x^{-0.2}}\text{dx}\)
\(D=\frac{0.0593}{2}\rho{U_{\infty}}^{2}\left(\frac{\nu}{\ U_{\infty}}\right)^{0.2}\frac{\left(L-X_{t}+X_{T,t}\right)^{0.8}}{0.8}\)
\(D=\frac{0.0593}{2*0.8}\rho{U_{\infty}}^{2}\left(\frac{\nu}{\ U_{\infty}}\right)^{0.2}\left(L-X_{t}+X_{T,t}\right)^{0.8}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.43\right)\)
Let us first simplify \(\left(L-X_{t}+X_{T,t}\right)^{0.8}\) by multiplying and divided it by\(\left(\frac{U_{\infty}}{\text{\ ν}}\right)\)
\(\left(L-X_{t}+X_{T,t}\right)^{0.8}=\left(\frac{\nu}{\ U_{\infty}}*\frac{U_{\infty}}{\text{\ ν}}\left(L-X_{t}+X_{T,t}\right)\right)^{0.8}=\left(\frac{\nu}{\ U_{\infty}}*\left(\text{Re}_{L}-\text{Re}_{t}+\text{Re}_{T,t}\right)\right)^{0.8}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.44\right)\)
\(\therefore D=\frac{0.0593}{2*0.8}\rho{U_{\infty}}^{2}\left(\frac{\nu}{\ U_{\infty}}\right)^{0.2}\left(\frac{\nu}{\ U_{\infty}}*\left(\text{Re}_{L}-\text{Re}_{t}+\text{Re}_{T,t}\right)\right)^{0.8}\)
\(\therefore D=\frac{0.0593}{2*0.8}\rho{U_{\infty}}^{2}\left(\frac{\nu}{\ U_{\infty}}\right)^{0.2}\left(\frac{\nu}{\ U_{\infty}}\right)^{0.8}\left(\text{Re}_{L}-\text{Re}_{t}+\text{Re}_{T,t}\right)^{0.8}\)
\(\therefore D=\frac{0.0593}{2*0.8}\rho{U_{\infty}}^{2}\frac{\nu}{\ U_{\infty}}\left(\text{Re}_{L}-\text{Re}_{t}+\text{Re}_{T,t}\right)^{0.8}\)
\(\therefore D=0.0370625\ \rho U_{\infty}\nu\left(\text{Re}_{L}-\text{Re}_{t}+\text{Re}_{T,t}\right)^{0.8}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.45\right)\)
The drag coefficient is nothing but the drag divided by\(\frac{1}{2}\rho{U_{\infty}}^{2}S\)
\(C_{d}=\frac{D}{\frac{1}{2}\rho{U_{\infty}}^{2}S}=\frac{0.0370625\ \rho U_{\infty}\nu\left(\text{Re}_{L}-\text{Re}_{t}+\text{Re}_{T,t}\right)^{0.8}}{\frac{1}{2}\rho{U_{\infty}}^{2}L*1}\)
\(C_{d}=\frac{2*0.0370625\ \nu\left(\text{Re}_{L}-\text{Re}_{t}+\text{Re}_{T,t}\right)^{0.8}}{U_{\infty}L}\)
\(C_{d}=0.074125\left(\frac{\text{\ ν}}{U_{\infty}L}\right)\left(\text{Re}_{L}-\text{Re}_{t}+\text{Re}_{T,t}\right)^{0.8}\)
\(C_{d}=\frac{0.074125}{\text{Re}_{L}}\left(\text{Re}_{L}-\text{Re}_{t}+\text{Re}_{T,t}\right)^{0.8}\)
Since \(\text{Re}_{T,t}=\frac{U_{\infty}\ X_{T,t}}{\text{\ ν}}\)
The equivalent length of turbulent layer in the transition zone\({(X}_{T,t})\) is given for sine laminar profile as derived previously,
\(x_{T,t}=36.023469\ \frac{\nu}{\ U_{\infty}}\ {\text{Re}_{L,t}}^{5/8\ }\)
\(\text{Re}_{T,t}=\frac{U_{\infty}\ }{\text{\ ν}}X_{T,t}=\frac{U_{\infty}\ }{\text{\ ν}}36.023469\ \frac{\nu}{\ U_{\infty}}\ {\text{Re}_{L,t}}^{5/8\ }\)
\(\text{Re}_{T,t}=36.023469\ \ {\text{Re}_{L,t}}^{5/8\ }\)
In this way, we finally get
\(C_{d}=\frac{0.074125}{\text{Re}_{L}}\left(\text{Re}_{L}-\text{Re}_{t}+\text{Re}_{T,t}\right)^{0.8}\)
\(C_{d}=\frac{0.074125}{\text{Re}_{L}}\left(\text{Re}_{L}-\text{Re}_{t}+36.023469\ \ {\text{Re}_{L,t}}^{5/8\ }\right)^{0.8}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(7.46\right)\)
Chapter Summary and study guide
The present chapter demonstrates the mathematical analysis of the hydrodynamics boundary layer over a flat plate. The main important points can be summarized in the following points: