12 11/21

1. Using suitable neat graphics, show that the boundary layer variables for the boundary layer around a circular cylinder of radius $r_0$ in a cross flow with velocity at infinity equal to $U$ and pressure at infinity $p_\infty$ are given by:
   
   $$x=r_0\theta \qquad y=r-r_0 \qquad u=v_\theta \qquad v=v_r$$
   
   

2. Write the appropriate equations for the unsteady boundary layer flow around a circular cylinder in terms of the boundary layer variables above. Assuming that the potential flow outside the boundary layer is steady and unseparated, give all boundary conditions to be satisfied. Make sure to write them in terms of boundary layer variables only. Solve the pressure field inside the boundary layer.

3. For the same flow, rewrite the boundary continuity equation in terms of the polar coordinates $r$, $\theta$, $v_r$, $v_\theta$, and $p$. Compare this with the exact continuity equation in polar coordinates and explain why the difference is small if the boundary layer is thin. Also write the boundary layer $x$ and $y$ momentum equations in terms of the polar coordinates. Are they different from the exact momentum equations? Which of the two is most simplified?

4. For the same flow, consider the proposed solution (from Stokes’ second problem)
   
   $$u=u_e(x) {\rm erf}\left(y/\sqrt{4\nu t}\right)$$
   
   
   where $u_e$ is the potential flow slip velocity immediately above the boundary layer. The solution above satisfies the equation
   
   $$u_t = \nu u_{yy}$$
   
   
   Find out what the velocity $u_e$ must be. Also find the velocity $v$ inside the boundary layer. Note:
   
   $$\int_{\bar z=0}^z {\rm erf}(\bar z){ \rm d}\bar z = z {... ... + \frac{1}{\sqrt{\pi}} e^{-\bar z^2} - \frac{1}{\sqrt{\pi}}$$
   
   

5. Define a suitable boundary layer thickness for the proposed solution of the previous question. How does it vary with $x$ and $t$? Explain why the proposed solution is reasonable for very small times. Hint: Ask yourself, what happens to the magnitude of $u_{yy}$ when $t\to 0$? Does the same happen to the magnitudes of $u$, $v$, $u_x$, $u_y$. and $p_x$? Argue that for larger times, the proposed solution is no longer good. Base yourself here on results like those found on the program web page and links on that page like Shankar’s thesis and the Van Dommelen & Shen separation process as well as what you know about the proposed solution, such as, say, its boundary layer thickness.

6. According to potential flow theory, what would be the lift per unit span of a flat-plate airfoil of chord 2 m moving at 30 m/s at sea level at an angle of attack of 10 degrees? What would be the viscous drag if you compute it as if the airfoil is a flat plate aligned with the flow with that chord and the flow is laminar? Only include the shear stress over the last 98% of the chord, since near the leading edge the shear stress will be much different from an aligned flat plate. What is the lift to drag ratio? Comment on the value. Use $\rho=1.225$ kg/m$^3$ and $\nu=14.5\;10^{-6}$ m$^2$/s.