Hypersonic Similarity Law For Steady Compressible Euler Flows Past Slender Bodies Within the Framework of Radon Measure Solutions

In this paper, we develop a mathematical theory for the statement and validation of the hypersonic similarity law within the framework of Radon measure solutions to the steady compressible Euler equations. We investigate two scenarios: (1) two-dimensional steady nonisentropic compressible Euler flows past infinitely long slender curved wedges, and (2) three-dimensional steady nonisentropic compressible Euler flows past infinitely long axisymmetric cones. We find that for hypersonic flow over a slender body with a small slenderness parameter (Formula presented.), if the parameter (Formula presented.) is fixed, then as (Formula presented.) (which corresponds to the Mach number of the incoming flow (Formula presented.)), the flow field structures, after scaling, become independent of the body's shape and the Mach number (Formula presented.). Instead, they depend solely on (Formula presented.) and the adiabatic exponent (Formula presented.) of the polytropic gas. Mathematically, we revisit the classical hypersonic small-disturbance equations within the framework of Radon measure solutions, enabling explicit construction of solutions with concentration boundary layers. We demonstrate that as (Formula presented.), under suitable nondimensional scalings, the Radon measure solutions of the original hypersonic flow problems converge to those of the corresponding hypersonic small-disturbance problems. The explicit forms of the Radon measure solutions obtained for the two scenarios facilitate the convergence analysis.