数值求解不可压缩流动的投影方法研究进展

在过去的20多年中, 投影方法通过速度和压力的解耦计算, 获得了比全耦合方法更高的计算效率, 这个显著优点使之得以广泛应用. 目前, 在计算非定常不可压缩流动的原始变量形式的数值方法中, 投影方法得到了越来越广泛的应用. 本文根据投影方法的构造思路,将众多的投影方法分成了3类, 即: Helmholtz-Hodge分解类投影方法、算子分裂类投影方法和局部连续投影方法, 并详细的介绍了3类投影方法的发展历程和求解步骤. 从投影方法的求解过程不难发现, 通过速度和压力的解耦计算, 提高了投影方法的计算效率, 但同时也给投影方法的时间精度分析带来了困难, 并长期成为大家争论的焦点. 普遍认为, 速度的时间精度比较容易达到高阶, 但是压力一般来说只有一阶精度. 但通过对3类投影的对比分析后, 我们认为, 局部连续投影方法将有助于澄清目前投影方法存在的相关争议, 并使得发展高阶精度的投影方法在理论上和技术上成为可能. 

REVIEW ON THE PROJECTION METHODS IN THE NUMERICAL SOLUTION OF THE INCOMPRESSIBLE FLOW

Because of the decoupling of the velocity and the pressure computation,the projection method is much more efficient than the fully coupled procedures.This notable advantage has attracted great attention, and many improved projection methods have been developed during the past 20 years.The projection methods are currently among the most popular methods for solving viscous incompressible flow based on the primitive variable formulations.According to their processes of construction,the projection methods are classified into three types in the present paper,namely,the Helmholtz-Hodge decomposition projection methods,the operator splitting projection methods and the local continuous projection methods.Their development and solution procedures axe introduced in details.From the solution procedures,it is found that the velocity-pressure decoupling makes it difficult to analyze the accuracy of the projection method,which has often been a subject for debating.Generally speaking,high order convergence in time for the velocity can be readily obtained,while the computed pressure is typically only first order accurate in time.However,by comparing and analyzing the three types of the projection methods,we show that the local continuous projection methods make it possible to develop high order accurate projection methods both theoretically and practically,which may clarify some misunderstandings about the accuracy of projection methods.