Solving Aerodynamics Homework: Free Vortex near Infinite Plane

[jstrz13phys]

Homework Statement

A two dimensional free vortex is located near an infinite plane at a distance h above the plane. The pressure at infinity is p_\infty and the velocity at infinity is U_\infty parallel to the plane. Find the total force (per unit depth normal to the paper) on the plane if the pressure on the underside of the plane is p_\infty. The strength of the vortex is \Gamma. The fluid is incompressible and perfect. To what expression does the force simplify if h becomes very large?

Homework Equations

I know that I will eventually integrate pressure over an area to get the force. I just don't know where to start. I also thought about imposing a mirror image of the cortex under the plane to cancel out the y components of velocity. (Not sure about this)

The Attempt at a Solution

It is intuitive that as h increase the force becomes negligable. I just can't find the expression for the force.

 

[Astronuc]

The net force on the plate would result from the pressure differential across the plate, i.e on the bottom of the plate the pressure is constant P_infty, while above the plate, the vortex reduces the pressure locally, but the pressure would be P_infty as one moves further (laterally) from the vortex.

Can one determine the pressure field in and around the vortex?

 

[jstrz13phys]

I can use Bernoulls's equation to find the pressure above the plain right? Then integrate over the area of the vortex?

 

[Astronuc]

Is the 2-D vortex in a plane parallel to the plate, such that the axis is normal to the plate?

I'm trying to visualize a 2-D vortex, which I assume is circular in 2-D?

Ostensibly one would have a formula for the pressure within the vortex as a funtion of its rotational velocity.

 

[jstrz13phys]

I am not sure what you are asking, see the word document i have atached.

 

[Astronuc]

That helps.

There is a relationship between the velocity field around the vortex which is proportional to the vortex strength, \Gamma. Below the vortex, the flow field of the vortex is in the opposite direction of U-infty.

Is there a discussion in one's textbook that includes something like U_\theta = \frac{\Gamma}{2\pi{r}}, where r is the distance measure from the center of the vortex. I think then it is a matter of expressing the flow field around the vortex by r ~ h - y, where y is the elevation from the plane.

When h gets very large, the pressure equation should be that of Bernoulli's equation for a flow of U_infty and static pressure P_infty.

 

[jstrz13phys]

Thank you for your help, I have figured out the answer. I took the stream function for the superposition of flows and added a mirror image of the vortex. Then converted the stream function to rectangular coordinates. The took derivative wrt y then integrated pressurewrt to x this game me the lift = density X strength X Velocity (when h goes to infininty)