边界层转换阶段直接数值模拟的影响矩阵法

Influent Matrix Method of Direct Numerical Simulation for Transition Boundary Layer

提出一种求解时间相关、三维不可压完全Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程的影响矩阵法。考虑流动区域为上半空间的平极边界层流动，在水平方向，利用周期性条件，采用Ｆｏｕｒｉｅｒ离散，在半无穷方向，定义了一种特殊映像并利用Ｃｈｅｂｙｓｈｅｖ多项式展开。由此将Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程转化为以水平波数（ｋ１，ｋ２）为参数的一组法向Ｈｅｌｍｈｏｌｔｚ方程组，同时可更有效地求解压力。本文讨论了连续问题的影响矩阵法与原问题的等价性。影响矩阵法的离散形式，以及影响矩阵法的算法实现等。最后给出了一维Ｈｅｌｍｈｏｌｔｙ问题的数值实验结果。

This paper presents an influent matrix method (IMM) of solving three dimensional, unsteady and incompressible full Navier Stokes equations. Boundary layer transition problem in upper half space is studied. By means of periodic conditions in the horizontal directions, Fourier spectral discretization is used. One special transform is defined and the unknown function is expanded on Chebyshev polynomials in the normal direction. The most obvious one of advantages for IMM is that the Navier Stokes equations can be changed into one set of the Helmholtz equations with the pair of parameters of (k 1, k 2), the horizontal wavenumbers. The equivalence of the continuous problem and the IMM equations, the discrete form of the IMM equations, and the code performance of the IMM are discussed. Numerical experiments are fulfilled on one dimensional Helmholtz problem.