In 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. The spectral methods with shock-fitting have been shown to solve the in-viscid blunt body problem accurately and efficiently. In addition, errors decay exponentially fast with the increase in the grid points. In this study, the chebyshev spectral collocation method is used to solve Euler equations over cylinder and sphere in a supersonic flow fields. The axisymmetric and non-conservative form of Euler equations is considered as the governing equations. The sensitivity of spectral methods to boundary conditions, which can not be overstated, makes the boundary treatment the most difficult part of the solution. 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 used by Kopriva 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. The solution is advanced in time using 4th order Runge-Kutta method. Finally, supersonic flows over cylinder and sphere are solved to show the capability of the present approach. The body surface quantities and the shock shapes compare well with previous investigators. In addition, the accuracy of the spectral collocation method is demonstrated.
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