The Study Of A Coupled Finite Element Method And Discontinuous Galerkin Method For The Simulation Of Flow Problems

The flow problems widely occur in many fields,such as the aerospace,polymer processing and production,ocean engineering,weather prediction,etc.The numerical simulation studies of the flow problems have great scientific significance and practical values.The real flow problems are complex,which always contain some physical phenomena,i.e.convection and diffusion,and will also encounter different fluids with evolving free surface.Therefore,developing the stable,efficient and accurate numerical method is a research hotspot all the time.Finite element method(FEM)has been one of the effective methods for solving flow problems,because of its ability to acquire high order numerical solutions,simply deal with the boundary conditions and irregular domains.However,when using the traditional FEM to directly solve the flow problems,it will encounter the LadyzhenskayaBabu?ka-Brezzi(LBB)constraint,the numerical instabilities caused by the convection dominated or the relatively high Weissenberg(We)numbers and it is also hard to exactly capture the moving interface.In order to settle these difficulties,the stabilized FEM was proposed.However,the selection of stabilization parameter is not easy,and the parameter may affect the results conspicuously.In recent years,the Discontinuous Galerkin(DG)method has also received a lot of attention due to its good capability to stably and accurately handle the convective problem and get high order numerical solutions.However,when solving the elliptic type equations,it needs to add extra variables or some penalty terms,which lead to the increase of computational costs and the complexity of program design.Therefore,in this dissertation,we aim to establish a coupled FEM-DG method based on the splitting scheme,which takes full advantages of these two methods,and simulate some typical flow problems via this coupled algorithm.The main work and conclusions of this dissertation are listed as follows:(1)When using FEM to directly solve the incompressible flows,there exists the LBB constraint.With the increase of Re number,it may cause the numerical instabilities.When using DG method to deal with the elliptic equation,it needs to introduce additional unknown variable or the penalty term.Thus,we propose a coupled FEM-DG algorithm based on the splitting scheme.As for the sub-equations via the splitting scheme,we take full advantages of FEM and DG method to solve them,i.e.,utilizing DG method to solve hyperbolic equation and FEM to handle Poisson and Helmholtz equations.It is able to use the equal order interpolation functions for the velocity and pressure to decrease the memory size and the complexity of programming.There is no need to resort to any stabilization terms.Moreover,comparing with the unified DG scheme,the coupled algorithm improves the computational efficiency.(2)When using FEM to deal with the Oldroyd-B non-Newtonian flow,the relatively high We numbers will cause some instabilities problem,and it is not easy to handle the area with stress singular point or the drastic change of stress.In this thesis,we further develop the coupled FEM-DG scheme to simulate some challenging viscoelastic Oldroyd-B flows at relatively high We numbers.As for the solution of non-Newtonian Navier-Stokes equations,we resort to the FEM-DG algorithm,and the Runge-Kutta Discontinuous Galerkin(RKDG)method is employed for the Oldroyd-B constitutive equation.In this coupled scheme,it is able to choose the equal order interpolation functions for the velocity,pressure and stress.There is also no need to add any stabilization terms or parameters and its ability to handle the problem with stress singular point is much better than stabilized FEM.(3)There exist two fluids with large density and viscosity ratios and significant changing free surface in the simulation of viscous two phase flow.When utilizing stabilized FEM to solve the viscous two phase flow,it needs to reinitialize the Level Set function in every time step.The Level Set method has the main drawback of main loss or gain.According to these problems,we build a combined FEM-DG-Level Set computational framework to simulate the viscous two phase flow in various situations.We use the coupled FEM-DG algorithm to solve the two phase Navier-Stokes equations and employ RKDG method for the solution of Level Set and its re-initialization equations.Moreover,a simple mass correction technique is also involved.Therefore,this combined algorithm is able to deal with the viscous two phase flow efficiently and accurately and guarantee the good mass conservation property.There is no need to reinitialize the Level Set function during a large time steps.(4)The polymer filling process mainly contains the viscoelastic-viscous two phase flow problem,which involves two fluids with different flow characteristic and it always encounter the irregular cavity.The simple power law model is not suitable to describe the rheological behavior of the polymer melt.Thus,according to the XPP constitutive equation,we further develop the combined FEM-DG-Level Set algorithm.The solution of the viscoelastic two phase Navier-Stokes equation is accomplished via the coupled FEM-DG scheme and we utilize RKDG method to deal with the XPP constitutive equation.This combined algorithm is able to solve the viscoelastic-viscous two phase flow stably and accurately.Moreover,it is successfully to simulate the complex polymer filling process for the socket with five irregular inserts,which will provide some numerical predictions for the industrial production.