The steady inviscid compressible self-similar flows and the stability analysis

TL;DR

This paper studies steady inviscid compressible self-similar flows in spherical coordinates, revealing new flow types, shock behaviors, and analyzing their stability and existence, including flows with vorticity and complex transonic structures.

Contribution

It introduces new classes of self-similar flows, analyzes shock polar monotonicity, and proves existence and stability of smooth transonic flows with vorticity, advancing understanding of complex flow behaviors.

Findings

Existence of purely sonic, Beltrami, and smooth transonic flows.

Discovery of a monotonic relation between shock angle and radial velocity.

Proof of existence and uniqueness of smooth transonic flows with vorticity.

Abstract

We investigate the steady inviscid compressible self-similar flows which depends 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 proportionnality 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 not been observed before. If a supersonic incoming flow is self-similar with nonzero azimuthal velocity, a conic shock also form attached to the tip of the cone. The state at the downstream may change smoothly from supersonic to subsonic, thus the shock can be supersonic-supersonic, supersonic-subsonic and even supersonic-sonic where the shock front and the sonic front coincide. We further investigate the structural stability of smooth self-similar irrotational transonic flows and analyze the corresponding linear mixed type second order equation of Tricomi type. By exploring some key properties of the self-similar solutions, we find a multiplier and identify a class of admissible boundary conditions for the linearized mixed type second-order equation. We also prove the existence and uniqueness of a class of smooth transonic flows with nonzero vorticity which depends only on the polar and azimuthal angles in spherical coordinates.