Blasius Exact Solution
Rate of growth of L.B.L. is\(\frac{\delta}{x}=\frac{5.0}{\sqrt{\text{Re}_{x}}}\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ }\left(7.23\right)\)
Local skin friction coefficient:\(C_{f}=\frac{0.664}{\sqrt{\text{Re}_{x}}}\ \text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ }\left(7.24\right)\)
Average skin friction coefficient:\(\overset{\overline{}}{C_{f}}=C_{D}=\frac{1.328}{\sqrt{\text{Re}_{L}}}\text{\ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ \ \ \ }\text{\ \ }\text{\ \ \ \ }\text{\ \ }\text{\ \ }\left(7.25\right)\)
As an example to explain the laminar boundary layer, let us assume we
have the simplest laminar velocity profile which is\(\overset{\overline{}}{U}=\overset{\overline{}}{y}\)
We shall use the rate of growth formula inserted below;
\(\ \frac{\delta}{x}=\sqrt{\frac{2D_{o}}{\text{I\ }\text{Re}_{x}}}\)
\(\left.\ D_{o}=\frac{d\overset{\overline{}}{U}}{d\overset{\overline{}}{y}}\right\rceil_{y=0}=1\)
\(I=\int_{0}^{1}{\overset{\overline{}}{U}\left(1-\overset{\overline{}}{U}\right)d\overset{\overline{}}{y}=}\int_{0}^{1}{\overset{\overline{}}{y}\left(1-\overset{\overline{}}{y}\right)d\overset{\overline{}}{y}=}\frac{1}{6}\)
Then, \(\frac{\delta}{x}=\frac{3.464101615}{\sqrt{\text{Re}_{x}}}\)
Let us obtained the local skin friction coefficient using the formula
inserted below;
\({C_{f}}_{x}=2*I*\frac{\text{dδ}}{\text{dx}}=2*\frac{1}{6}\frac{d}{\text{dx}}\left\{\frac{3.464101615}{\sqrt{\text{Re}_{x}}}x\right\}=\frac{1}{3}\frac{d}{\text{dx}}\left\{3.464101615\frac{\nu^{0.5}}{{U_{\infty}}^{0.5}x^{0.5}}x\right\}\)
\({C_{f}}_{x}=\frac{1}{3}*3.646\left(\frac{\nu}{U_{\infty}}\right)^{0.5}\frac{d}{\text{dx}}x^{0.5}=\frac{1}{3}*3.646*0.5\left(\frac{\nu}{U_{\infty}}\right)^{0.5}1/x^{0.5}=\frac{\sqrt{3}}{3}\left(\frac{\nu}{U_{\infty}x}\right)^{0.5}\)
\({C_{f}}_{x}=\frac{0.5773502692}{\sqrt{\text{Re}_{x}}}\) , Commonly
written as\(\ {C_{f}}_{x}=\frac{0.577}{\sqrt{\text{Re}_{x}}}\)
The drag coefficient is more convenient in the aerodynamics researches,
in this way, we shall find out its expression for this profile;
\(\overset{\overline{}}{C_{f}}=\frac{1}{L}\ \int_{0}^{L}{{C_{f}}_{x}\text{dx}}=\frac{1}{L}\ \int_{0}^{L}{\frac{0.577}{\sqrt{\text{Re}_{x}}}\text{dx}}=\frac{1}{L}\ \int_{0}^{L}{0.577\frac{\nu^{0.5}}{{U_{\infty}}^{0.5}x^{0.5}}\text{dx}}\)
\(\overset{\overline{}}{C_{f}}=\frac{0.577}{L}\left(\frac{\nu}{U_{\infty}}\right)^{0.5}\int_{0}^{L}{x^{-0.5}\text{dx}}=\frac{0.577}{L}\left(\frac{\nu}{U_{\infty}}\right)^{0.5}\left.\ \frac{x^{0.5}}{0.5}\right|_{0}^{L}=\frac{0.577}{0.5}\left(\frac{\nu}{U_{\infty}}\right)^{0.5}\frac{L^{0.5}}{L}\)
\begin{equation}
\overset{\overline{}}{C_{f}}=1.154700538*\left(\frac{\nu}{U_{\infty}L}\right)^{0.5}=\frac{1.1547}{\sqrt{\text{Re}_{L}}}\nonumber \\
\end{equation}The same procedure can be used for other laminar velocity profile. We
tableted the famous velocity profiles and their characteristics below in
Table 2;
Table 2 Velocity Profile of Laminar Boundary Layer
Characteristics