2025/01/17 by Shangkun Weng, Hongwei Yuan, Weng, Shangkun +1
Engineering · Mathematics · #35L67 #35M12 #76H05 #76N15 #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Gas Dynamics and Kinetic Theory
paper · pdf · doi:10.48550/arxiv.2501.10125
openalex publication_date 2025/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.