"Wake Turbulence" Problem

When a heavy airliner is taking off $($or landing$),$ large wingtip vortices $($see figure below$)$ are formed as

the flow from the high pressure bottom side of the wing "spills" over to the lower pressure top surface of

the wing. Winglets can mitigate the formation of this vortex $($also shown below$),$ but such vortices can

still be hazardous to smaller general aviation aircraft. If a small airplane flies through the trailing vortex,

the pilot can lose control of the aircraft. The Federal Aviation Administration and National Transportation

Safety Board refer to this phenomenon as "wake turbulence", and there are procedures in place to ensure

sufficient separation between aircraft.

For this assignment, approximate the wingtip vortices as two free vortices separated by a distance $b$ and

the strength of each vortex is $+-\frac{}{2}.$ To model the interaction with the ground, a pair of mirror vortices

should also be included, as shown in the figure. You can use an approximate span of $b=30m$ and a

circulation of $=400\frac{m^2}{s}.$ For simplicity, you can neglect the three$-$dimensionality and assume that the

flow is planar.

A$)$ For an initial height above the ground of

$h=30m,$ determine what velocity $($speed and

direction$)$ at the center of each vortex. Also

use a computer to plot several streamlines to

help visually show the flow behavior.

B$)$ Using a computer, plot the trajectory $($position

over time$)$ of the wingtip vortices as they

interact with each other and the ground. The

vortices can be assumed to move with the

local fluid velocity. You can also assume that

the potential flow solution is a valid solution

even though the flow is technically unsteady.

C$)$ Use a computer to plot the pressure field and velocity fields at the the initial time, $t=0$ and a

later time $t=20s.$

D$)$ Comment on how long such vortices may take to clear a runway? Note that the Lafayette airport

runway is $150$ feet wide. How would these times be affected if there were a crosswind?