if the navier-stokes equations are transformed into non-orthogonal boundary-fitted coordinate system, then, can we use the normal finite difference scheme to solve it? or we may need some sort of modification on the finite difference scheme before solving it? if possible please suggest any reference on this topic. thanks in advance for any advice. regards, yfyap

(1). Have you looked at the standard cfd book, such as "computational fluid mechanics and heat transfer", by Dale Anderson, John Tannehill and Richard Pletcher?

Please have a look to

Fengyan Shi's paper, 2000

He was solving Boussinesq Equations on Curvilinear Coordinate system, the main point is to use the Contraviant velocity components, otherwise the equation will be very complicated in that coordinate system.

The FDM schemes there are basically the same yet.