To model the wave propagation more accurately in moderate water area,a numerical model was established under the non-staggered grids based on the hyperbolic mild slope equations derived by Copeland.

Considering the kinematic characteristics of streamflow and cooling water, and based on the control equations of cooling water in the non-orthogonal curvilinear coordinate, a 2-D mathematic model for cooling water is presented with non-staggered grids in the non-orthogonal curvilinear coordinate.

However, its time accuracy on non-staggered grids and moving grids has not been carefully investigated.

In order to cure the pressure fluctuation encountered in solving N S equations on a nonstaggered grid, a complete pressure correction method was extended for the solution of incompressible flow in 3 D non orthogonal curvilinear coordinate.

With the non staggered grids,a three dimensional quasi state laser remelting problem,considering heat transfer and fluid flow,is numerically simulated.

A 2 D depth averaged numerical model in boundary fitted coordinates for non staggered grids, taking contravariant velocity as an independent variable, is developed.

The governing equations are discretizated with non-staggered grid by means of finite volume method.

The paper uses a high order method to solve incompressible viscous complex flows based on a domain decomposition and matched method,, The primitive variable and pressure Poisson equation formulation are employed for the numerical solutions of the incompressible Navier-Stokes equations, in tensor forms, on non-staggered grids.

This paper presents numerical solution for governing equation of flows in curvilinear coordinate system with nonstaggered grids.