The steady inviscid compressible self-similar flows and the stability analysis
Shangkun Weng, Hongwei Yuan
TL;DR
This work analyzes three-dimensional steady Euler flows past an infinitely long circular cone under a self-similar ansatz that depends only on the polar angle. It develops a comprehensive framework combining reduced ODEs for irrotational and rotational flows, conic shock analysis via Rankine–Hugoniot and shock polar concepts, and a stability theory for mixed-type (Tricomi) equations using multipliers and energy estimates. The authors prove the existence and uniqueness of smooth transonic self-similar flows, including cases with nonzero vorticity, and establish a monotone relation between shock angle and radial velocity, clarifying the downstream state transitions (supersonic–supersonic, supersonic–subsonic, and supersonic–sonic). They further construct a class of smooth transonic flows with nonzero vorticity depending on $(\theta,\varphi)$ and prove the associated stability via a deformation–curl decomposition, showing that the sonic surface forms a conic surface close to the background. Together, these results extend transonic flow theory in 3D Euler to self-similar, swirl-containing configurations and provide rigorous stability analyses and existence/uniqueness results for both irrotational and rotational self-similar flows. The work has potential implications for understanding three-dimensional conical shocks and the role of swirl in stabilizing transonic self-similar solutions.
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.