Research On Variable Splitting Method For The Incompressible Flow Problems

The Navier-Stokes equations and its coupling equations are kinematic equa-tions that describe the momentum conservation of viscous incompressible fluid,they reflect the basic mechanical law of viscous fluid flow and are of great significance in fluid mechanics.They are widely used in scientific and engineering fields,such as atmospheric movement,ocean currents,bearing lubrication,blood flow,reservoir simulation,military war,aerospace and so on.Due to the limitations of incompressibility constrants,the existence of nonlinear phenomena and the shape irregularity of the fluid flow domain,it is extremely difficult to find the exact solutions of the Navier-Stokes equations and its coupling equations.However,numerical solutions of these equations can be obtained by the way of numerical simulation,and the existence state of the corresponding numerical solutions can be furtherly understood.As we all know,the variable velocity and pressure are coupled through incompressible constraints.It shows the contradiction between the huge scale of solving the problems and the limited storage space.Therefore,in order to lower the scale of solving the problems and save the storage space,we need to establish some efficient numerical algorithms to decouple velocity and pressure,and resort to parallel computing method to realise large-scale numerical simulation of incompressible flow problems for the purpose of attaining a profound understanding of fluid movement.This is also the significance of research on variable splitting method for incompressible flow problems in this paper.Based on the previous work,we make a further study on the variable splitting method of incompressible flow problems.The main research work is presented as follows:(1)The local and parallel Uzawa finite element method for solving the stationary generalized Navier-Stokes equations is given.The nonlinear term of the generalized Navier-Stokes equations can be linearized by using the Oseen scheme.In addition,so as to reduce the workload of solving the equations and save the working time,the Uzawa finite element method is adopted to decouple velocity and pressure,which can transform the large-scale incompressible flow problem into a small-scale problem.Firstly,it is theoretically proved that the Uzawa finite element method based on the Oseen scheme converges geometrically,and we find that the crispation number is a constant independent of the mesh size.Secondly,a local and parallel Uzawa finite element method for solving the generalized Navier-Stokes equations in this paper is developed through combining with the parallel computing method based on fully overlapping domain decomposition techniques and the Uzawa finite element method based on the Oseen scheme.According to the unique characterisitics of this method,the corresponding parallel computing function can be realised by modifying slightly the existing serial code instead of recoding.Finally,numerical experiments are conducted to compare the advantages and disadvantages of the local and parallel Uzawa finite element method,the Uzawa finite element method and the traditional finite element method in terms of CPU time and convergence order,so as to verify the effectiveness and high efficiency of the proposed methods in this paper.The results show that the Uzawa finite element method has better convergence than the traditional finite element method.The local and parallel Uzawa finite element method is more efficient than the Uzawa finite element method.(2)The parallel rotational pressure projection method is given to solve the nonstationary coupled Navier-Stokes/Navier-Stokes equations.The spatial non-iterative Oseen scheme and the first-order backward Euler scheme are used to deal with the trilinear terms and time derivative terms of the coupled Navier-Stokes/Navier-Stokes equations,respectively.Moreover,the rotational pressure projection method is used to directly decouple velocity and pressure.They can be solved in three equations of prediction,projection and correction,which omits the iterative process repeatedly,so that the workload of solving the equations can be reduced and the computational efficiency can be further improved.At first,it is proved that the rotational pressure projection method can make the convergence order of velocity reach the first order in theory.Then,based on the basic idea of fully overlapping domain decomposition techniques,a basic scheme of the parallel rotational pressure projection method for solving the coupled Navier-Stokes/Navier-Stokes equations is proposed by means of parallel computing method.At last,we compare accuracy and computing time between the parallel and serial rotational pressure projection method and the traditional decoupled fractional time-stepping method through numerical experiments with the exact solution problem and the submarine mountain problem.We find that the parallel rotational pressure projection method is the most effective method in three methods,the serial rotational pressure projection method is more convergent than the decoupled fractional time-stepping method.