A class of higher order accurate schemes for fluid interface problems (PhD)

Abstract

The purpose of the work presented herein is to introduce novel numerical approaches, which can be used for various complex multi-phase fluid interface problems. We first aim to study the vortex shedding phenomena for the case of flow past a rotationally oscillating circular cylinder, at the initial stages and for the fully developed flows for different Reynolds numbers. Navier-Stokes (N-S) equations are solved using recently developed Higher Order Compact (HOC) scheme on a body fitted, non-uniform polar grid in such a way that surface of the cylinder forms the inner boundary of the domain. This study provides insight into the wake characteristics associated with vortex shedding phenomena from a rotationally oscillating circular cylinder due to fluid-structure interactions and consolidates previous related observations, often unaccounted for, found in the published literature. Then, a new modified HOC based finite difference scheme is developed in polar coordinates to solve elliptic partial differential equations (PDEs) in domains with discontinuities due to the presence of circular interfaces. After validating this scheme by solving elliptic PDEs, it is extended to N-S equations for simulating the flow past an impulsively started circular cylinder. Body fitted, non-uniform polar grid in constructed in such a way that surface of the cylinder is considered as an interface immersed in the fluid. All the flow characteristics including the von-Ka´rma´n vortex street are accurately captured. The most significant contribution of this work, however, is the development of a new generalized finite difference scheme for solving problems with arbitrary shaped fixed interfaces on a non body fitted, uniform Cartesian grid. Extensive validation is carried out by solving elliptic equations in two and three dimensions followed by its extension to solve incompressible, 2D Stokes flows. It produces excellent results for pressure term even if in a coarser grid like 32 × 32, and the results are significantly better than those of existing methods. Linear regression analysis is used to find order of accuracy of the scheme using grid refinement which shows that the proposed scheme has average second order accuracy. This idea is then extended to solve moving interface problems with discontinuities across time dependent moving interfaces. An unsteady, non linear general partial differential equation and Stefan’s problem are solved with a moving inter face in 1D. In 2D, heat equation is considered on a square domain with a circular interface whose radius is continuously changing with time. Crank-Nicolson (CN) type discretization is compared with a recently developed HOC scheme. HOC scheme is found to produce better results and the order of accuracy is also more than that of CN scheme. The scheme shows nearly second order accuracy and good agreement with the analytical solution.