Spectral Solution of High Speed Flows Over Blunt Bodies with Improved Boundary Treatment

Abstract

SummeryIn some fluid dynamics problems, like flow stability and transition, accurate solution of mean flow is required. While such an accurate solution cannot be obtained by means of ordinary finite difference approximations, spectral approximations are able to surmount. Furthermore, the low inherent dissipation of spectral methods is not enough to stabilize the method in the presence of shocks [1]. The spectral methods with shock-fitting have been shown to solve the inviscid blunt body problem accurately and efficiently. In addition, errors decay exponentially fast with the increase in the grid points [2].In this study, the chebyshev spectral collocation method is used to solve Euler equations over a blunt body in a supersonic flow fields. The axisymmetric and non-conservative form of Euler equations is considered as the governing equations. The initial shock shape is taken from an empirical formula developed for flow over spherically blunted cones experimentally [3]. The Rankine-Hugoniot relations are used for shock boundary condition in accordance with a proper compatibility relation, which carry information from the flow field to the shock. The shock acceleration is derived similar to the method found in [2] but with different formulation to treat the shock boundary condition. Zero normal velocity along with compatibility relations in a matrix form is used as a boundary condition for solid body, which simplifies the implementation. Since the outflow is supersonic (all four characteristics are leaving the flow field), no explicit boundary conditions are necessary; therefore the governing equations are used to obtain the boundary variables. At the symmetry line, the symmetry conditions are applied using spectral differentiation by means of the procedure found in [4].The sensitivity of spectral methods to boundary conditions, which can not be overstated, makes the boundary treatment the most difficult part of the solution. In contrast to [2] which uses different treatments for all types of boundaries, a uniform approach is used in the present study by extending the matrix method proposed by [5] for finite difference method to the spectral one. This procedure allows a more robust systematic solver. The solution is advanced in time using 4th order Runge-Kutta method. As a famous check on the validity of the spectral code, supersonic flow over a sphere is considered and the results for shock shape and body surface quantities are compared with the results of [5]; and the spectral accuracy is observed.