Suppose the Couette flow through a straight channel(no pressure drop between the inlet and outlet), the lower curve wall is fixed, and the upper straight wall is moving at the fixed velocity. How can we simulate the problem numerically? with the methods, such as SIMPLE or stream function-vorticity method? Thanks.

The way I simulated it was instead of having zero velocity at the top wall I would have it set to the velocity of the moving plate. Everything else is the same. If I remember correctly

If you are adventureous, look at http://webpages.charter.net/jtholdeman/IJNMFart1.pdf

Thanks for your suggestion. In this Couette flow, is the overall pressure drop should be zero? is the inlet or outlet pressure constant or variable along the channel height?

I didn't look at Jonas' link, so if I am repeating what it say I'm sorry. If you convert your equations over to the stream-function/vorticity equations your pressure term will drop out, so you won't have to worry about that. And, after solving for the velocities you can back solve for the pressure - if you need to.