Converting vorticity/velocity fields from cartesian to cylindrical co-ordinates

Oct 2013
2
0
Australia
Hi there,

I'm trying to determine the velocity equations, given a known vorticity distribution, and the requirement that the continuity equation be satisfied. In Cartesian, the equations are

w_z = dv/dx - du/dy
0 = du/dx + dv/dy

where (u,v) are the velocities in the (x,y) directions, respectively. I obtained functions for the velocities fairly easily, using a Gaussian distribution for w_z.

I was trying to do the same thing in an axisymmetric cylindrical coordinate system, so that the vorticity in the r-z plane is the same as in the x-y plane as above. Here

w_theta = dv/dz - du/dr
0 = dv/dr + v/r + du/dz

where (u,v) are the velocities in the (z,r) directions, respectively. However, I'm having a lot of trouble trying to determine the (z,r) velocities, given the same distribution as above, due to the extra term in the continuity equation. Using the same velocity fields in the (z,r) plane as the (x,y) plane produces the desired vorticity, but then continuity is not satisfied.

If anyone can give me any pointers, it would be greatly appreciated.
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Hey chrisg.

It sounds like what you need is a change of co-ordinates from one co-ordinate system to another. Have you by any chance come across tensor mathematics and differential geometry?
 
Oct 2013
2
0
Australia
Hi chiro, thanks for your response!

I've worked a little with tensors, but I don't think they will help in this case. In the Cartesian set up, the geometry is that of a "tube" of vorticity. In the axisymmetric cylindrical system, the geometry becomes a "ring" of vorticity. So changing the co-ordinate system from Cartesian to cylindrical will give me the definition for a "tube" in cylindrical co-ordinates, rather than a "ring".
 

chiro

MHF Helper
Sep 2012
6,608
1,263
Australia
Can you make a transformation that projects the space onto that ring? (In other words, the map will not be bijective, only a projection)?