Topic: Fluid Mechanics
The velocity distribution in a turbulent boundary layer is given by $u/v=(y/\delta)^{1/7}$. What is the displacement thickness $\delta^{*}$
- A. $\delta$
- B. $\delta/7$
- C. $7\delta/8$
- D. $\delta/8$
Answer: D. $\delta/8$
Explanation: Displacement thickness is given by $\delta^{*} = \int_{0}^{\delta}(1 - \frac{u}{v})dy$. Substituting the given velocity profile $\frac{u}{v} = (\frac{y}{\delta})^{1/7}$ into the integral yields $\int_{0}^{\delta}(1 - (\frac{y}{\delta})^{1/7})dy$. Evaluating this integral gives $[y - \frac{7}{8}\frac{y^{8/7}}{\delta^{1/7}}]_{0}^{\delta} = \delta - \frac{7}{8}\delta = \frac{\delta}{8}$.