不可压Navier-Stokes方程的一阶有限元解法  

One-order Algorithm of Incompressible Navier-Stokes Equations in Finite Element Method

作　　者：卞正宁[1] 罗建辉[1] 

机构地区：[1]湖南大学土木工程学院,湖南长沙410082

出　　处：《湖南大学学报（自然科学版）》2013年第7期41-45,共5页Journal of Hunan University:Natural Sciences

基　　金：湖南省自然科学基金资助项目(08JJ3011)

摘　　要：不可压Navier-Stokes方程求解的困难之一在于如何确定压力场并且同时要满足不可压条件.压力项在连续性方程中并不出现,但是却对速度起约束作用.为了解决这一问题,对于粘性不可压流动,提出了以速度和应力为基本变量,不含压力项的一阶流体动力学方程系统及对应的积分形式.采用有限元方法,对于速度和应力进行同阶插值,对于非线性对流项,采用牛顿迭代法进行处理,对于时间项采用后向欧拉方法.基于FreeFem++平台,对两平行平板间的稳态粘性流动及二维非定常圆柱绕流进行了数值计算.分别通过和精确解及标准算例的对比,验证了方法的可行性和有效性.采用不含压力项的一阶系统,避免了连续性方程中不含压力项给不可压缩Navier-Stokes方程求解带来的困难.One of the difficulties of the numerical solution of incompressible Navier-Stokes equations is the determination of the pressure field and the fulfillment of the incompressibility condition. In fact, the pressure variable is not present in continuity equation, but a constraint for the velocity field is present. In this paper, the basic variables of velocity and stress were proposed for incompressible viscous fluid, a one- order fluid dynamics equation system without pressure term was proposed and its integral form was given to handle this problem. The stress and the velocity were interpolated by equal order finite element. The Newton iterative method was used to handle the nonlinear convective term. The backward Euler method was used to discretize the time term. A steady flow of incompressible viscous fluid between two infinite parallel plates and a Benchmark problem of incompressible viscous fluid flow around a cylinder were com- puted on the basis of FreeFem＋＋. The feasibility and the effectivity of the method were verified by com- paring with the analytic solution and the Benchmark results respectively. The difficulty of pressure term which is not present in continuity equation is circumvented by using one-order system without pressure term.

关 键 词：一阶系统 不可压缩流体 压力 应力精度 

分 类 号：O351.2[理学—流体力学]