Computational fluid dynamics: steady 2D flow

[Feodalherren]

1. Homework Statement

Homework Equations

CFD

The Attempt at a Solution

I'm a bit confused by this question.

So at first what I do for the problem on the left, I find the changes in the velocities in X and and Y on all four sides.

I notice that the values on the diagram to the left are higher on top than they are on the bottom, therefore I conclude that v must be "up".
I also notice that the values are higher on the right than they are on the left, therefore I conclude that u is to the right.

For the diagram on the right: I notice that bottom > top, therefore v is down.
Right > left, therefore u is to the right.Are these assumptions correct?

After that it's a fairly simple problem but this first step has me confused as to what I'm supposed to be doing.

 

[Chestermiller]

I think what they want you to do is to find the y velocity components at the centers of the horizontal faces, and the x velocity components at the centers of the vertical faces, and then average to get the velocity components at the center of the cell.

Chet

 

[Feodalherren]

I don't understand. The solution says "Ψtop>Ψbottom, therefore U is to the right" and "Ψright>Ψleft therefore V is down".
This is completely counter intuitive and seems to be taken out of the thin air. What are they basing this on?

 

[Chestermiller]

I don't understand. The solution says "Ψtop>Ψbottom, therefore U is to the right" and "Ψright>Ψleft therefore V is down".
This is completely counter intuitive and seems to be taken out of the thin air. What are they basing this on?

$$v_x=\frac{\partial \psi}{\partial y}$$
$$v_y=-\frac{\partial \psi}{\partial x}$$

 

[Feodalherren]

That makes a million times more sense. Thanks.