INFINITE-THIN SHOCK LAYER SOLUTIONS FOR STATIONARY COMPRESSIBLE CONICAL FLOWS AND NUMERICAL RESULTS VIA FOURIER SPECTRAL METHOD

We consider the problem of steady uniform supersonic inviscid flows passing straight solid cones with attack angles, by studying Radon measure solutions with infinite-thin shock layers for the nonisentropic compressible Euler equations, particularly for the Chaplygin gas and limiting hypersonic flows. This approach also provides a generalized Newton-Busemann law on the distribution of pressures on the cone's boundary. The construction of Radon measure solutions is reduced to solving a system of nonlinear ordinary differential equations (ODE) for weights of Dirac measures supported on the edge of the cone on the unit 2-sphere. It is further reduced to solving nonnegative periodic solutions of a singular and nonlinear nonautonomous ODE. A numerical algorithm based on the combination of the Fourier spectral method and Newton's method is developed to solve the ODE. The numerical simulations for different attack angles exhibit proper theoretical properties and excellent accuracy. This research demonstrates the benefits of Radon measure solutions and could be useful for engineering applications in hypersonic aerodynamics.