14 12/02

1. Reconsider the unsteady boundary layer flow around a circular cylinder in terms of the boundary layer equations. 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.

2. 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 component $v$ inside the boundary layer. Note:
   
   $$\int_{\bar z=0}^z {\rm erf}(\bar z){ \rm d}\bar z = z {... ... + \frac{1}{\sqrt{\pi}} e^{-\bar z^2} - \frac{1}{\sqrt{\pi}}$$
   
   

3. 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.

4. 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.

5. Assume that a flow enters a two dimensional duct of constant area. If no boundary layers developed along the wall, the centerline velocity of the flow would stay constant. Assuming that a Blasius boundary layer develops along each wall, what is the correct expression for the centerline velocity?

6. Continuing the previous question. Approximate the Blasius velocity profile to be parabolic up to $\eta=3$, and constant from there on. At what point along the duct would you estimate that developed flow starts based on that approximation? Sketch the velocity profile at this point, as well as at the start of the duct and at the point of the duct where the range $0\le\eta\le3$ corresponds to $\frac18$ of the duct height accurately in a single graph. Remember the previous question while doing this!