13 11/28

This homework considers the flow around a flat plate airfoil and a symmetric Joukowski airfoil using conformal mapping from the flow around a cylinder.

The flow around the cylinder is described in a $z$-plane by the complex velocity potential:

$$F = U\left(z+\frac{r_0^2}{z}\right) -i V \left(z-\frac{r_0^2}{z}\right) + \frac{{\rm i}\Gamma}{2\pi}\ln(z/r_0)$$

where

$$z=x+{\rm i}y$$

Take the radius of the cylinder to be

$$r_0 = 1 + \epsilon$$

with value for $\epsilon$ as specified below. Take $U=1$ and $V=\tan(20^\circ)$ for an incoming flow at an angle of attack of 20 degrees. The values of $\Gamma$ to use are specified below.

The airfoil plane is described in terms of a coordinate

$$\zeta=x'+{\rm i}y'$$

which can be computed from $z$ using the shifted Joukowski transform

$$\zeta=z-\epsilon + \frac{1}{z-\epsilon}$$

Use a software package like matlab that can handle complex numbers, and create a polar coordinates mesh around the cylinder in the $z$-plane. (Which will correspond to a nonpolar mesh in the $\zeta$-plane.) In matlab, real() will evaluate the real part of a complex number and imag() the imaginary part.

1. Taking $\epsilon=0$ and $\Gamma=0$, evaluate the imaginary part of $F$, giving the streamfunction $\psi$, and then use a contour plotting package to plot the streamlines in the $z$-plane. You may need to first evaluate $x$ and $y$ as the real and imaginary parts of $z$. Verify that the streamlines describe the circulationless ideal flow around a cylinder.

2. Evaluate the pressure from Bernoulli, using the fact the the derivative of $F$ gives the complex conjugate velocity, so $\vert\vec v\vert^2=\vert{\rm d}F/{\rm d}z\vert^2$, (make sure to use the absolute value function here, not just the square, since $F$ is complex). Take the pressure zero at infinity. Plot isobars.

3. Now evaluate the $\zeta$ values, and then plot the streamlines in the $x',y'$-plane. Verify that you get the flow around a flat plate at an angle of attack, but that the Kutta condition is not satisfied.

4. Evaluate the circulation from the Kutta condition that ${\rm d}F/{\rm d}z$ must be zero at the trailing edge $z=r_0$.

5. Replot the streamlines in the $z$-plane.

6. Replot the streamlines in the $\zeta$-plane.

7. Set $\epsilon=0.1$. Replot the streamlines in the $z$-plane.

8. Replot the streamlines in the $\zeta$-plane.