12 12/01

1. Derive the streamfunction of irrotational incompressible flow around a cylinder from solution of the PDE. The steps are similar to the ones used in class to derive the potential.

2. 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?

3. If the complex potential flow of a source and a line vortex equal
   
   $$F=\frac{Q}{2\pi}\ln(z) \qquad F={\rm i}\frac{\bar\Gamma}{2\pi}\ln(z)$$
   
   
   then what would be the real velocity potentials $\phi$? (use polar coordinates.) Differentiate to find the velocities and compare to questions 4.2 and 4.9.

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 100 m/s at sea level at an angle of attack of 10 degrees? What would be the drag?

5. What would be the circulation around the airfoil of the previous question?

6. Identify the boundary layer variables $x$, $y$, $u$, and $v$ for the case of a circular cylinder of radius $r_0$ in terms of the cylindrical variables $r$, $\theta$, $v_r$, and $v_\theta$.

7. Using the result of the previous question, write the continuity equation in cylindrical coordinates from table C.3 in terms of the boundary layer coordinates and comment on the differences from the boundary layer continuity equation. Is the difference small?

8. Similarly, write the $r$ and $\theta$ momentum equations of table C.5 in 2D and cross out the terms the boundary layer approximation ignores. Ignore gravity.