Position of line vortex from Stream function

[600burger]

Homework Statement

So, the problem is 4 line vortices in a 2-d square. All have the same strength. I am asked to find the motion of the vortices in the flow of the others. Meaning, each vortex moves in the flow of the other three. I know the correct solution should be just a circular pattern but i can't seem to make it work out.

Homework Equations

I'm 'given' the stream function of a line vortex and allowed to set 'a' (the distance of each side on the square) to anything.

The Attempt at a Solution

So I can plot the correct stream lines for all four vorticies at once.

Then, excluding the vortex whose motion I am looking for, i find dpsi/dy = u and dpsi/dx = v.

From there, I should be able to setup and solve dx/dy = u and dy/dt = v as ODE's, correct?

I'm using a working runga kutta solver in MATLAB but the results are incorrect (a line y=x).

 

[mighty2000]

I would first verify that the stream function of a line vortex is indeed the correct formula to use in this situation. It is important to thoroughly understand the physical principles and assumptions behind the equations being used before attempting to solve a problem.

Next, I would check the boundary conditions and ensure that they are correctly incorporated into the equations. In this case, it seems that the boundary conditions include the distance of each side on the square, so it is important to make sure that this parameter is correctly included in the equations.

Additionally, I would double check the implementation of the equations in the MATLAB solver and make sure that all variables and parameters are correctly defined and used.

If the results still do not match the expected circular pattern, I would consider using a different solver or approach to solve the problem. It is also important to consider any potential simplifications or assumptions that can be made to make the problem more manageable and accurate.

Overall, as a scientist, it is important to approach problems with a critical and analytical mindset, constantly checking and verifying the equations and solutions being used.