Hypersonic flow onto a large curved wedge and the dissipation of shock wave
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910921229598720 |
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| author | Hu, Dian Qu, Aifang |
| author_facet | Hu, Dian Qu, Aifang |
| contents | The flow field with a Mach number larger than 5 is named hypersonic flow. In this paper, we explore the existence of smooth flow field after shock for hypersonic potential flow past a curved smooth wedge with neither smallness assumption on the height of the wedge nor that it is a BV perturbation of a line. The asymptotic behaviour of the shock is also analysed. We prove that for given Bernoulli constant of the incoming flow, there exists a sufficient large constant such that if the Mach number of the incoming flow is larger than it, then there exists a global shock wave attached to the tip of the wedge together with a smooth flow field between it and the wedge. The state of the flow after shock is in a neighbourhood of a curve that is determined by the wedge and the density of the incoming flow. If the slope of the wedge has a positive limit as $x$ goes to infinity, then the slope of the shock tends to that of the self-similar case that the same incoming flow past a straight wedge with slope of the limit. Specifically, we demonstrate that if the slope of the wedge is parallel to the incoming flow at infinity, the strength of the shock will diminish to zero at infinity. The restrictions on the surface of a wedge have been greatly relaxed compared to the previous works on supersonic flow past wedges. The method employed in this paper is characteristic decomposition, and the existence of the solution is obtained by finding an invariant domain of the solution based on geometry structures of the governing equations. The ideas and methods presented here may be applicable to other problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10059 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hypersonic flow onto a large curved wedge and the dissipation of shock wave Hu, Dian Qu, Aifang Analysis of PDEs 35B20, 35D30, 35Q31, 35L65, 76J20, 76L05, 76N10 The flow field with a Mach number larger than 5 is named hypersonic flow. In this paper, we explore the existence of smooth flow field after shock for hypersonic potential flow past a curved smooth wedge with neither smallness assumption on the height of the wedge nor that it is a BV perturbation of a line. The asymptotic behaviour of the shock is also analysed. We prove that for given Bernoulli constant of the incoming flow, there exists a sufficient large constant such that if the Mach number of the incoming flow is larger than it, then there exists a global shock wave attached to the tip of the wedge together with a smooth flow field between it and the wedge. The state of the flow after shock is in a neighbourhood of a curve that is determined by the wedge and the density of the incoming flow. If the slope of the wedge has a positive limit as $x$ goes to infinity, then the slope of the shock tends to that of the self-similar case that the same incoming flow past a straight wedge with slope of the limit. Specifically, we demonstrate that if the slope of the wedge is parallel to the incoming flow at infinity, the strength of the shock will diminish to zero at infinity. The restrictions on the surface of a wedge have been greatly relaxed compared to the previous works on supersonic flow past wedges. The method employed in this paper is characteristic decomposition, and the existence of the solution is obtained by finding an invariant domain of the solution based on geometry structures of the governing equations. The ideas and methods presented here may be applicable to other problems. |
| title | Hypersonic flow onto a large curved wedge and the dissipation of shock wave |
| topic | Analysis of PDEs 35B20, 35D30, 35Q31, 35L65, 76J20, 76L05, 76N10 |
| url | https://arxiv.org/abs/2409.10059 |