12 11/28 W

1. Go to the class airfoil programs page. Download Matlab program airfoil.m. Mark where a cylinder potential is set in a complex $\zeta$-plane. Also mark where this cylinder is mapped to a Joukowski airfoil in a complex $z$-plane, and list the formula used to get $z$ from $\zeta$ that achieves this mapping. You may observe that program airfoil.m is astonishingly simple for the complexity of the flow and graphics that it produces.

   Then in Matlab, select parameters that produce the flow around a slightly cambered Joukowski airfoil of roughly 10% thickness ratio at 15 degrees angle of attack. List your parameters. Plot out the picture of the streamlines and isobars. Next set the circulation to zero by setting variables Gamma and auto both to zero and replot to show the effect of not satisfying the Kutta condition. Do the streamlines still come off smoothly from the trailing edge? What happens to the pressure?

2. Streamlines around a cylinder or sphere for very low Reynolds number, very viscous Stokes flow look superficially the same as those for high Reynolds number ideal inviscid flows: both are symmetric front/rear. But do they really look the same? Find out. Unfortunately, Stokes flow around a cylinder is tricky. For truly low Reynolds numbers, the flow in the vicinity of the cylinder becomes negligible, so the streamlines become infinitely widely spaced. But Stokes flow around a sphere is more reasonable.

   So, use the Matlab program streamlines.m to first plot the streamlines for potential flow around a cylinder. Then create similar programs to plot the streamlines for both potential and Stokes flow around a sphere. You will need to put in the appropriate streamfunctions at the mesh points; potential flow around a sphere is in section 19.8, and Stokes flow in 21.8. You also need to change the values of the streamfunction that are plotted as streamlines; your streamlines should be equally spaced far upstream. The simplest way to do so is write the axysymmetric streamfunction far upstream, and then figure out what streamfunction values over there correspond to $y$ values equal to $\pm0.5\Delta{y},\pm1.5\Delta{y},\ldots$. Comment on the differences in streamline spacing between potential and Stokes flow near the surface. Also comment on the differences in streamline curvature one or two radii away from the sphere.

3. 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?

4. 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.

5. 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.