Study On The Generalized Finite Difference Method For Fluid Interface Problem

The Stokes interface problem is a classical class of fluid interface problems,which is based on the study of multiphase fluid flow and its interface behavior.In this thesis,a collocated meshless method based on the Taylor series approximation,namely the generalized finite difference method(GFDM),is used to solve the steady-state Stokes interface problem.In order to overcome the problem of numerical oscillation of pressure that occurs in the Stokes interface problem,this thesis firstly verifies the stability and accuracy of the mixed boundary condition(MBC)of the Stokes problem.This scheme only adds mixed boundary conditions to the Stokes equations,that is,the projection of the momentum equations onto the normal vectors outside the boundary,without making any other changes to the control equations.In this thesis,the accuracy and stability of the GFDM for solving the Stokes problem is illustrated by six numerical arithmetic examples over square regions as well as more complex computational domains.In particularly,the numerical results show the high accuracy of the GFDM in solving the Stokes problem and the errors can reach second-order convergence of all the cases with the complex region.Based on the study of the MBC scheme of the Stokes problem,a new scheme of the Stokes interface problem is investigated in this thesis,that is,adding mixed boundary and mixed interface conditions to the boundary and interface to increase the pressure information,respectively.In addition,in order to solve the Stokes interface problem more conveniently,this thesis transforms it into two coupled Stokes non-interface subproblems which are coupled with interface conditions on the subregion,where the interface appears as the boundary of the two subproblems.Since the main idea of GFDM is to approximate the derivatives of each order of the unknown variables by a linear combination of the values of the functions of the nearby nodes,this makes it easy to discretize the proposed new scheme by GFDM.In order to test the accuracy of GFDM in solving this problem and the applicability of the proposed new scheme,numerical examples are provided in the thesis for two simple interfaces and four complex interfaces,such as polygon and amoeba.The numerical results are stable and accurate,also show the superiority of GFDM in dealing with the Stokes interface problem with complex interfaces.This confirms the feasibility of GFDM for solving fluid interface problems and it will be the beginning of solving more complex fluid interface problems in the future.