Numerical simulation and stability analysis of free surface fluid flow problem

A numerical method to treat two-dimensional incompressible fluid flows with free surfaces is developed. By solving the primitive variables (velocities and pressure) of Navier-Stokes equations and the Poisson equation for pressure on a non-staggered grid in boundary-fitted coordinates, all the discretization and computation is done on the fixed, square grid regardless of the shape and movement of the physical boundary. A time-marching scheme is applied while the momentum equation is discretized using the Eulerian-Lagrangian method allowing a large time step with relatively small numerical diffusion. Since the pressure equation might have full Neumann boundary conditions (in problems without free surfaces), special care has been taken in discretization to satisfy the divergence theorem. Multigrid methods are used to improve the efficiency of indirect methods in solving the difference equations. The computer code has been applied to several problems to test its capability in solving the transient responses as well as the steady state solutions. The results show that this finite difference method is capable of treating full Navier-Stokes equation in a wide range of Reynolds number with or without moving boundaries.; Extended work has been done on the numerical linear stability analysis. By applying the similar numerical scheme to the linearized perturbed Navier-Stokes equations, the three dimensional instabilities of various flows with or without free surfaces are successfully predicted. Several simulations have been performed on problem like the well known Taylor instability of flows between two cylinders, and on the flows with free surface instability which can often be found in the coating processes. In order to verify the numerical results of the free surface problems, which the analytical solutions can hardly be derived, flow visualization by the method of streak photography has been carried out to obtain reliable data to compare with.