11 HW 11

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. A Boeng 747 has a maximum take-off weight of about 400,000 kg and take-off speed of about 75 m/s. The wing span is 65 m. Estimate the circulation around the wing from the Kutta-Joukowski relation. This same circulation is around the trailing wingtip vortices. From that, ballpark the typical circulatory velocities around the trailing vortices, assuming that they have maybe a diameter of a quarter of the span. Compare to the typical take-off speed of a Cessna 52, 50 mph.

2. Model the two trailing vortices of a plane as two-dimensional point vortices (three-dimensional line vortices). Take them to be a distance $2\ell$ apart, and to be a height $h$ above the ground. Take the ground as the $x$-axis, and take the $y$-axis to be the symmetry axis midway between the vortices. Now:
   1. Identify the mirror vortices that represent the effect of the ground on the flow field. Make a picture of the $x,y$-plane with all vortices and their directions of circulation.
   2. Find the velocity at an arbitrary point $x$ on the ground due to all the vortices.
   3. From that, apply the Bernoulli law to find the pressure changes that the vortices cause at the ground. Sketch this pressure against $x$ for both $h$ significantly greater than $d$ and vice-versa.
   4. Also find the velocity that the right-hand non-mirror vortex R experiences due to the other vortices. In particular find the Cartesian velocity components $u_{\rm R}$ and $v_{\rm R}$ in terms of $\Gamma$, $h$ and $\ell$.

3. Continuing the previous question, the right non-mirror vortex R moves with the velocity that the other vortices induce:
   
   $$\frac{{\rm d}\ell}{{\rm d}t} = u_{\rm R} \qquad \frac{{\rm d}h}{{\rm d}t} = v_{\rm R}$$
   
   
   If you substitute in the found velocities and take a ratio to get rid of time, you get an expression for ${\rm d}h/{\rm d}\ell$. Integrate that expression using separation of variables to find the trajectory of the vortices with time. Accurately draw these trajectories in the $x,y$-plane, indicating any asymptotes. Do the vortices end up at the ground for infinite time, or do they stay a finite distance above it?

4. Describe transverse ideal (inviscid) flow around a circular cylinder of radius $r_0$ using a streamfunction approach. Write the partial differential equation to be satisfied by the streamfunction in an appropriate coordinate system. Also write the boundary conditions at the cylinder surface, $r=r_0$, and the boundary condition at infinity, $r\to\infty$.

5. Find the streamfunction of the previous question. To do so, guess it to be a single separation of variables term of the form $f(r)g(\theta)$. Assume an appropriate form for $g(\theta)$ and then find $f(r)$. Note: the equation that you get for $f(r)$ should have two different solutions of the form $r^n$ for some $n$.