13 12/04 M

1. Using suitable neat graphics, show that the boundary layer variables for the boundary layer around a circular cylinder of radius $a$ in a cross flow with velocity at infinity equal to $U$ and pressure at infinity $p_\infty$ are given by:
   
   $$x=a\theta \qquad y=r-a \qquad u=v_\theta \qquad v=v_r$$
   
   
   Write the three appropriate partial differential equations for the unsteady boundary layer flow around a circular cylinder in terms of the boundary layer variables above. Also write the boundary conditions at the wall and above the boundary layer, at $y/\sqrt{\nu}\approx\infty$. (Remember that $y/\sqrt{\nu}\approx\infty$ in the boundary layer solution should be taken to be equivalent to $y\approx0$ in the potential flow above it.) Assume an unsteady flow impulsively started from rest, where you can assume that outside the thin boundary layer, the flow is still given by the ideal flow solution. Solve the pressure field inside the boundary layer fully.

2. Rewrite the exact Navier-Stokes equations in polar coordinates, (the continuity equation and the $r$ and $\theta$ momentum equations) in terms of the boundary layer variables $x$, $y$, $u$, and $v$ and the radius of the cylinder $a$. (So $r$, $\theta$, $v_r$ and $v_\theta$ may no longer appear in the equations.) Carefully distinguish between $r$ (which may not appear in the results) and $a$. Compare these exact equations with the boundary layer equations. Explain for each discrepancy why the difference is small at high Reynolds numbers, where the boundary layer is thin.

3. 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 drag?

   Next, 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.

4. 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?

5. 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 $y$ versus $u$ graph (not picture). Remember the previous question while doing this!