Application Of Projection Method On Moving Grid Problems

As an important algorithm for incompressible flows, the projection method has become more and more popular in recent years because of its high efficiency in solving unsteady flow problems. Thanks to the theoretical analysis based on semi-discretized schemes and the numerical experiments on staggered grids, the time accuracy of projection method has been improved to second and even higher order. However, its time accuracy on non-staggered grids and moving grids has not been carefully investigated.It is important to develop accurate and efficient numerical algorithm on non-staggered grids since it is more simple and convenient than staggered grids to be applied in complex geometry. However, direct implementation of projection method on non-staggered grids causes numerical oscillation on pressure field. The approximate projection method with Armfield correction could suppress the numerical oscillation, but it introduces an artificial source term related to fourth order derivative of pressure into the discrete continuity equation. We show that this artificial source term will decrease time accuracy of the method to first order. A Double Periodic Shear Layer (DPSL) test case is used to make a comparison among the projection method on staggered grids, the exact projection method on non-staggered grids and the method talked above. It verifies that the artificial source term is responsible for the accuracy decrease.Moving grid problems prevail in many aspects of science and engineering problems. Currently, the applications of projection method on moving grids mainly focus on the simulation and research of specific flow problems, rather than the analysis of time accuracy. In this thesis, two benchmark test cases (DPSL and force flow) are used to verify the second order accuracy of projection method, indicating that the method with second order accuracy on static grids will not lose its accuracy when the grids are moving. As for the application of our algorithm, we apply projection method to moving non-staggered grids by studying a channel flow problem with moving boundary conditions, and investigate several features of the flow field (Such as the periodic features of flow, distributions of velocity and pressure field, streamlines of disturbance velocity field, and the evolution of wall friction with time). Our work could help to understand the physics of such flow field and illustrate the capability of projection method in solving moving grid problems. It could also provide reference for further development of highly accurate projection method and future research on more complicated unsteady moving grid problems.