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)