Numerical Investigation Of The Incompressible Electrofluids

The Navier-Stokes(NS)equations are equations used to describe fluid flow and have important applications in many fields of science and engineering.Numerical research on the NS equations has a long history,and a large amount of literature has been accumulated,but there are still difficult problems to overcome.The incompressible electrodynamic fluid model is a couple of the NS equations and the electrodynamic equation,and its numerical study integrates the difficulties from both equations,so is more challenging.This dissertation aims to design and analyze efficient numerical methods with specific advantages for these two sets of equations.The first part of the dissertation considers numerical methods for the incompressible NS equations.The main goal is to construct and analyze highly efficient time stepping schemes having the fast convergence to steady-state solutions.First-and second-order numerical schemes are constructed based on the exponential scalar auxiliary variable(SAV)method.The unconditional stability of the scheme is proved strictly.We also provide an error analysis for the first-order scheme.The convergence order of the scheme is verified numerically which is consistent with the theoretical prediction.Furthermore,the spatial discretization using finite element method and spectral Fourier method is described in detail.The proposed schemes are validated by simulating some benchmark problems,which demonstrate the robustness and efficiency of the new method.The second part is dedicated to the realization of a comparative study of the ability to capture steady-state solutions for a number of schemes.The purpose is to investigate their performance in computing steady solutions,and comparing the convergence speed to the correct solution.Our analysis and simulations show that for some existing methods,the time step size strongly affects the convergence speed and the accuracy of numerical solutions.In particular,our numerical experiments show that some scalar auxiliary variables and projection methods result in very slow convergence,or even worse converging to an incorrect steady-state solution.The cause of this phenomena is analyzed,and the way to overcome this difficulty is discussed.In the third part we study the numerical method for an electrohydrodynamics model which describes the phenomenon of electric convection arising from charge injection on the boundary of the insulating liquid.The model is a coupling of the NS equations,charge transfer equation,and potential energy equation.Based on the methods that we proposed in the previous chapters for the NS equations,a class of stable numerical schemes is constructed for the electrohydrodynamics model.The schemes decouple the charge density and potential energy from the NS equations,which are unconditionally stable.Numerical experiments show that the proposed schemes achieve the expected convergence rate,and can be used to efficiently simulate the changes of flow field and electric field induced by the electrical convection.Finally,we further extend the numerical approach to variable density electrohydrodynamics.An unconditionally stable scheme is constructed,analyzed,and validated.