How to approach calculating a stream function given a velocity profile on a grid
 

I have the velocity profiles for a 12x12 grid. So a 12x12 grid for the velocity in the x direction (u) and a 12x12 gride for velocity in the y direction(y) the stream function is defined as u=d(phi)/dy and v=d(phi)/dx. Where phi is the stream function.

This is a cavity flow: At the left, right, and bottom of the grid, the walls are stationary, therefore phi=0. To the right the change in phi (d(phi)) is the velocity in the y direction (v) times the change in distance going to the right (dx).

Denoting each grid line in the x direction as i, In summary:
d(phi) = dy * u ----> (phi)i+1 = (u)i*(dy)i+(phi)i

Thoughts on my approach?

[QUOTE=ComFlu;233904]I have the velocity profiles for a 12x12 grid. So a 12x12 grid for the velocity in the x direction (u) and a 12x12 gride for velocity in the y direction(y) the stream function is defined as u=d(phi)/dy and v=d(phi)/dx. Where phi is the stream function.


If u = d(phi)/dy then v = - d(phi)/dx. The minus sign is mandatory. [Or you can go with u = - d(phi)/dy and v = d(phi)/dx. If you carry the signs consistently throughout the derivations, either will work.]