The self-similar solutions to the steady compressible Euler equations

We investigate the steady inviscid compressible self-similar flows which depend only on the polar angle in spherical coordinates. It is shown that besides the purely supersonic and subsonic self-similar flows, there exists purely sonic flows, Beltrami flows with a nonconstant proportionality factor and smooth transonic self-similar flows with large vorticity. For a constant supersonic incoming flow past an infinitely long circular cone, a conic shock attached to the tip of the cone will form, provided the opening angle of the cone is less than a critical value. We introduce the shock polar for the radial and polar components of the velocity and show that there exists a monotonicity relation between the shock angle and the radial velocity, which seems to be new and has not been observed before. If a supersonic incoming flow is self-similar with nonzero azimuthal velocity, a conic shock also forms attached to the tip of the cone. The downstream state may change smoothly from supersonic to subsonic, thus the shock can be supersonic-supersonic, supersonic-subsonic and supersonic-sonic where the shock front coincides with the sonic front.