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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.16909 |
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| _version_ | 1866911126811312128 |
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| author | Kang, Shifan Long, Bingsong Yuan, Hairong |
| author_facet | Kang, Shifan Long, Bingsong Yuan, Hairong |
| contents | In this paper, we establish a mathematical theory on statement and validation of the hypersonic similarity law within the framework of Radon measure solutions of steady compressible Euler equations. We consider two scenarios: (1) two-dimensional steady non-isentropic compressible Euler flows past an infinitely long slender curved wedge; (2) three-dimensional steady non-isentropic compressible Euler flows past an infinitely long axisymmetric cone.
It turns out that, for the hypersonic flow passing through a slender body with tiny slenderness $τ$, if the parameter $K\doteq M_{\infty}τ$ is fixed, by taking $τ\to 0$ (i.e., the Mach number of the upcoming flow $M_{\infty} \to \infty$), the flow field structures (after scaling) no longer depend on the body's shape and the Mach number $M_{\infty}$ independently, but only on $K$ and adiabatic index $γ$ of the polytropic gas. Mathematically, for non-isentropic Euler flows, we find a new system of hypersonic small-disturbance equations to describe steady compressible hypersonic flows passing slender bodies. We demonstrate that as $ τ\to0$, under suitable non-dimensional scalings, the Radon measure solutions of the original problems of hypersonic flow passing bodies converge to those of corresponding hypersonic small-disturbance problems. The explicit forms of the Radon measure solutions derived for the two scenarios effectively simplify the convergence analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16909 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hypersonic similarity law for steady compressible Euler flows past slender bodies within the framework of Radon measure solutions Kang, Shifan Long, Bingsong Yuan, Hairong Analysis of PDEs In this paper, we establish a mathematical theory on statement and validation of the hypersonic similarity law within the framework of Radon measure solutions of steady compressible Euler equations. We consider two scenarios: (1) two-dimensional steady non-isentropic compressible Euler flows past an infinitely long slender curved wedge; (2) three-dimensional steady non-isentropic compressible Euler flows past an infinitely long axisymmetric cone. It turns out that, for the hypersonic flow passing through a slender body with tiny slenderness $τ$, if the parameter $K\doteq M_{\infty}τ$ is fixed, by taking $τ\to 0$ (i.e., the Mach number of the upcoming flow $M_{\infty} \to \infty$), the flow field structures (after scaling) no longer depend on the body's shape and the Mach number $M_{\infty}$ independently, but only on $K$ and adiabatic index $γ$ of the polytropic gas. Mathematically, for non-isentropic Euler flows, we find a new system of hypersonic small-disturbance equations to describe steady compressible hypersonic flows passing slender bodies. We demonstrate that as $ τ\to0$, under suitable non-dimensional scalings, the Radon measure solutions of the original problems of hypersonic flow passing bodies converge to those of corresponding hypersonic small-disturbance problems. The explicit forms of the Radon measure solutions derived for the two scenarios effectively simplify the convergence analysis. |
| title | Hypersonic similarity law for steady compressible Euler flows past slender bodies within the framework of Radon measure solutions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2508.16909 |