14 12/3 F

1. Compute approximate values of the Reynolds number of the following flows:
   1. your car, assuming it drives;
   2. a passenger plane flying somewhat below the speed of sound (assume an aerodynamic chord of 30 ft);
   3. flow in a 1 cm water pipe if it comes out of the faucet at .5 m/s,
   In the last example, how fast would it come out if the Reynolds number is 1? How fast at the transition from laminar to turbulent flow?

2. 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:
   
   $$x=r_0\theta \qquad y=r-r_0 \qquad u=v_\theta \qquad v=v_r$$
   
   
   Write the appropriate equations for the unsteady boundary layer flow around a circular cylinder in terms of the boundary layer variables above.

3. 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 $r_0$.

4. Compare the equations you got above with the exact equations in polar coordinates, (the continuity equation and the $r$ and $\theta$ momentum equations). Explain for each discrepancy why the difference is small if the boundary layer is thin.

5. 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.

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.

7. 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?

8. 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!