{"title":"Simple shape model for normal shock trains in straight channels","authors":"Fangyou Yu, Tinglong Huang, Hao Chen, Qifan Zhang, Lianjie Yue","doi":"10.1007/s00162-025-00736-2","DOIUrl":null,"url":null,"abstract":"<div><p>Normal shock trains are a flow phenomenon of significance to ramjet engines, but it remains unclear what its structure is decided by and how it evolves with the incoming Mach number. To seek a theoretical explanation, the minimum entropy production principle is generalized to the quasi-steady behavior of normal shock trains in two-dimensional straight channels with uniform incoming flow. Numerical simulations are also performed to validate the model together with the data collected from public literature. The analysis suggests that the flow parameters of a normal shock train depend on the inviscid shock-shock interactions rather than the local boundary-layer separations, though the angles of two incident shocks should still be equal as similar to the case that complies with the free-interaction theory. The shock feet’s positions, meanwhile, are allowed to be coincident or not, free from the entropy restriction. This freedom of position explains why both symmetric and partially asymmetric normal shock trains could be found previously. Further theoretical calculations reveal the inclinations of two incident shocks increase first and then decrease with the incoming Mach number, peaking at 48.570 degrees when the Mach number reaches 1.753. It is also indicated that the Mach number range allowing for a normal shock train is 1.652 to 2.254, giving evidence for past observations.</p></div>","PeriodicalId":795,"journal":{"name":"Theoretical and Computational Fluid Dynamics","volume":"39 1","pages":""},"PeriodicalIF":2.2000,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theoretical and Computational Fluid Dynamics","FirstCategoryId":"5","ListUrlMain":"https://link.springer.com/article/10.1007/s00162-025-00736-2","RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MECHANICS","Score":null,"Total":0}
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Abstract
Normal shock trains are a flow phenomenon of significance to ramjet engines, but it remains unclear what its structure is decided by and how it evolves with the incoming Mach number. To seek a theoretical explanation, the minimum entropy production principle is generalized to the quasi-steady behavior of normal shock trains in two-dimensional straight channels with uniform incoming flow. Numerical simulations are also performed to validate the model together with the data collected from public literature. The analysis suggests that the flow parameters of a normal shock train depend on the inviscid shock-shock interactions rather than the local boundary-layer separations, though the angles of two incident shocks should still be equal as similar to the case that complies with the free-interaction theory. The shock feet’s positions, meanwhile, are allowed to be coincident or not, free from the entropy restriction. This freedom of position explains why both symmetric and partially asymmetric normal shock trains could be found previously. Further theoretical calculations reveal the inclinations of two incident shocks increase first and then decrease with the incoming Mach number, peaking at 48.570 degrees when the Mach number reaches 1.753. It is also indicated that the Mach number range allowing for a normal shock train is 1.652 to 2.254, giving evidence for past observations.
期刊介绍：
Theoretical and Computational Fluid Dynamics provides a forum for the cross fertilization of ideas, tools and techniques across all disciplines in which fluid flow plays a role. The focus is on aspects of fluid dynamics where theory and computation are used to provide insights and data upon which solid physical understanding is revealed. We seek research papers, invited review articles, brief communications, letters and comments addressing flow phenomena of relevance to aeronautical, geophysical, environmental, material, mechanical and life sciences. Papers of a purely algorithmic, experimental or engineering application nature, and papers without significant new physical insights, are outside the scope of this journal. For computational work, authors are responsible for ensuring that any artifacts of discretization and/or implementation are sufficiently controlled such that the numerical results unambiguously support the conclusions drawn. Where appropriate, and to the extent possible, such papers should either include or reference supporting documentation in the form of verification and validation studies.
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