Numerical calculation of hypersonic flow by the spectral method

Abstract

1.INTRODUCTION Recently much attention has been paid to the hypersonic flow problem such as Space Shuttle and Space Plane. We have conducted the experiments on the hypersonic flow by using the shock tunnel with Mach number of 8 at Nagoya University. In particular, the data on the aerodynamic coefficients of the Shuttle-like body and the shape of bow shock have been collected. In parallel with the above experiments, the hypersonic flow has been numerically simulated. We have so far used the finite difference method and the finite volume method to calculate the hypersonic flow. The Euler equations, the thin layer Navier-Stokes equations, and the PNS equations have been successfully applied to this flow. Those results showed reasonable agreement with the experiment. In the present paper we employed another approach, the spectral method(1). The spectral method is a promising method to solve the fluid dynamics problem(2,3). In particular, the turbulent flow problems have so far been successfully solved. In addition to this, the Spectral method was applied to other velocity regime flow such as transonic and hypersonic flows. In the fluid dynamics problems, the finite difference method has been extensively used due to its simplicity and easy extension to higher order accuracy. Furthermore, recently the TVD method became popular to obtain the sharp shock wave in the hyperbolic type flow. However, most researchers in other fields get used to the finite element method due to its generality. Recently the finite element method has been applied to the fluid flow problems. Its strong point is that the grid can be easily generated in the complicated place. In the finite element method the interpolation function is valid within each cell. On the other hand, in the spectral method the overall region is approximated by one function which is expanded by truncated orthogonal functions. This method has several variations. One of them is referred to as the spectral element method which is similar to the finite element method. Generally, the spectral method is said to be more accurate than any other method when compared on the basis of the same number of grid points. Here we applied the collocation spectral method to the three-dimensional flow at the hypersonic flow regime. The similar flows were already simulated by several researchers where promising results were obtained. We will also show several