12 11/18

1. Write the complex velocity potential $F$ for nose flow. From its derivative, identify the location of the stagnation point. Identify the streamfunction from $F$. From it, find the value of the streamfunction at the stagnation point. From that, find the equation for the wall streamline, which passes through the stagnation point. From that, find the thickness of the nose far downstream. From the derivative of $F$, find the square magnitude of the velocity in terms of polar coordinates. Write the pressure on the wall as a function of the polar angle $\theta$ by writting the equation for the wall streamline in polar coordinates and plugging it in the square velocity.

2. Draw the streamlines of the uniform flow $F=Uz$, with $U$ a positive real constant. Now define a new coordinate $\zeta=\sqrt{z}$. Plot the streamlines of the same flow $F$ in the $\zeta$-plane.

3. The flow around a circular cylinder is
   
   $$F = U\left(z + \frac1z\right)$$
   
   
   Apply the Joukowski transformationw
   
   $$\zeta = z + \frac{1}{z}$$
   
   
   to find a new coordinate $\zeta$. How does the cylinder flow in the $z$-plane look in the $\zeta$ plane.

   That was too boring. Take the flow around a circle of radius 2,
   

   $$F = U\left(z + \frac4z\right)$$
   
   
   and apply again the Joukowski transformationw
   
   $$\zeta = z + \frac{1}{z}$$
   
   
   Show that the circle transforms to an ellipse in the $\zeta$ plane by setting $z=2e^{{\rm i}\theta}$ on the surface of the circle and transforming that. What is the aspect ratio of the ellipse? Find the velocity at the top and bottom points of the ellipse and compare with the value $2U$ that applies for a circle. To find the velocity, use
   
   $$v_{\xi} - {\rm i}v_{\eta} = \frac{{\rm d}F}{{\rm d}\zeta} = \frac{{\rm d}F}{{\rm d}z} \Big/ \frac{{\rm d}\zeta}{{\rm d}z}$$
   
   
   where $v_{\xi}$ and $v_{\eta}$ are the velocity components of the flow in the $\zeta$ plane.

4. Read through the cylinder flow program cylinder.m. Save it as type ``all files'' and run it in Matlab or Octave 3.0 or higher. Make plots to show the effect of changing the angle-of-attack parameter and of the circulation. In particular, plot the flow for the values of the circulation for which the stagnation points are at opposite sides of the cylinder, for which they are apart by a 90$^\circ$ angle, for which they coincide, and for which they are completely off the cylinder. Use screen capture or the print command to make hardcopies of the plots. Try help print if needed.

5. Read through the cylinder flow program airfoil.m. Save it as type ``all files'' and run it in Matlab or Octave 3.0 or higher. Do a “clear all” first to get rid of the cylinder stuff. Make plots to show zero thickness airfoil. Change the radius of the cylinder to produce a symmetric Joukowski airfoil with and without lift. Destroy the top/bottom symmetry of the conformal mapping to create camber. Be sure to create a reasonable cambered airfoil shape, thickness ratio about 15%, angle of attack about 15 degrees.

6. Airfoils have a suction peak at the nose. Read through and download pressure.m. Adjust to get a good picture. Mark both the stagnation point and the suction peak in the picture. Comment on the pressure at the trailing edge. How big is it?