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:
- The boundary layer over a flat plate are three regions :laminar,
transition and turbulent region. It is worthy to mention that the
transition region analysis in the text book is limited and we hope the
explanation of the transition region in the present chapter could help
to the engineering students in better understanding of this region.
- The governing equations of fluid mechanics :mass, energy and momentum
of fluid had been derived in – full details.
- The turbulence is also illustrated in this chapter with mention of
important applications of the turbulent flow.
- The governing equations of hydrodynamics boundary layer had been
investigated also and then Von – Karman solution for this equation is
derived.
- We starting from the M.I.E. to derive an expression of the boundary
layer thickness, drag, skin friction coefficient, drag coefficient,
shear wall stress in terms of Reynolds number for laminar, transition
and turbulent regions.