Numerical Analysis of Compressible and Incompressible Turbulence Using High Order Finite Difference Method

本课题开发了基于高阶线法的直接数值模拟程序，并进行了验证，以分析可压缩湍流和不可压缩湍流。为了分析不可压缩湍流，采用变阶MAC方法求解由原始变量表示的Navier-Stokes方程。采用对流项的插值方法更精确地满足了能量守恒。其次，提出了求解不可压缩Navier-Stokes方程的新的非交错有限差分法，该方法在每个时间步满足连续性方程并得到光滑的压力场。将这种非交错有限差分方法推广到体拟合坐标系中。最后，采用高阶线法对二维均匀各向同性可压缩湍流进行了数值模拟。在弱激波湍流中，由内能提供的可压缩动能占主导地位，产生激波。另一方面，涡流运动加速流体，在强激波湍流中产生激波。

In this project, the direct numerical simulation codes based on the higher order method of lines are developed and the validations are carried out, in order to analyze the compressible and incompressible turbulences. To analyze the incompressible turbulence, the Navier-Stokes equations expressed by the primitive variables are solved by the variable order MAC method. The energy conservation is safisfied more exactly by using the interpolation method in the convective terms. Next, the new non-staggered finite difference method which satisfies the continuity equation at each time step and obtains the smooth pressure field, is proposed for solving the incompressible Navier-Stokes equations. This non-staggered finite difference method is extended to the body fitted coordinate system. Finaly, the 2-D homogeneous isotropic compressible turbulences are simulated by the higher order method of lines. In the weak shock turbulence, the compressible kinetic energy supplied by the internal energy is dominant and the shock waves are generated. On the other hand, the fluid is accelerated by the vortex motion and the shock waves are generated in the strong shock turbulence.