Question 10.P.7: The shear at a location 2 m from the leading edge of a flat ......
The shear at a location 2 m from the leading edge of a flat plate was measured as 2.1 N/m². Assuming the flow to be turbulent from the start determine if air at 20°C was flowing over the plate (i) the velocity of air (ii) the boundary layer thickness and (iii) the velocity at 15 mm above the plate. ρ = 1.205 kg/m³, v = 15.06 × 10^{–6} m²/s
Using equation (10.2.3) C_{f x}=0.0594/\mathrm{Re}_{x}^{\mathrm{~0.2}},\,\tau_{w}=C_{f x}(1/2)\ \rho \,u^{2},
Equating 2.1=\frac{0.0594\times(15.06\times10^{-6})^{0.2}}{u_{\infty}^{0.2}2^{0.2}}\times\frac{1}{2}\times1.205\;u_{\infty}^{2}, Solving,
u_{\infty}{}^{1.8}=621.04\ \mathrm{or}\ u_{\infty}=35.623\ \mathrm{m}/\mathrm{s} \\ {\mathrm{Re}}=35.623\times2/15.06\times10^{-6}=4.808\times10^{6},
Turbulent hence the use of equation (10.2.3) is justified. Using equation (10.2.1),
{\frac{u}{u_{\infty}}}=\left({\frac{y}{\delta}}\right)^{1/7} \quad\quad (10.2.1)
\delta=0.382x/\mathrm{Re}_{x}^{\mathrm{~0.2}}=0.382\times\ 2/(4.808\times10^{6})^{0.2} \\ =0.0352\ \mathrm{m~~or~}\,\delta=35.2\,\mathrm{mm}
If the velocity profile is assumed as
{\frac{u}{u_{\infty}}}=\left({\frac{y}{\delta}}\right)^{1/7}
∴\qquad\quad \mathrm{u}=35.623\ (15/35.2)^{1/7}=32.05\ \mathrm{m/s}