{"title":"埃蒙斯问题再探讨","authors":"Howard R. Baum, J. G. Quintiere","doi":"10.1007/s10694-024-01613-w","DOIUrl":null,"url":null,"abstract":"<p>The “Emmons Problem” is a foundation of fire science and gives a mathematical boundary layer solution to the burning of a vaporizing fuel from the surface of a flat plate immersed in a uniform flow of oxidizing gas. It approximates the Navier–Stokes equations assuming infinitely fast chemistry and ignores differential diffusion and thermal radiation. This allows “similarity” solutions to be developed and expressed in terms of the classic Blasius function. The current paper extends the solution, in mathematical form, to the entire domain far from the boundary layer and upstream of the leading edge. The introduction of conformal parabolic coordinates and use of the Howarth transformation allows the solution for the stream function to be expressed in exactly the same form as that found by Emmons and furnishes an explicit recipe for the pressure perturbation. The same coordinate transformations allow the exact solution of the full elliptic mixture fraction equation to be obtained, and the representation of the velocity components in terms of the stream function guarantees that the continuity equation is also satisfied exactly. Further, an exact solution to the transverse momentum equation is also displayed permitting the introduction of a crossflow into the spectrum of results obtained. In short, an analytic solution is found for the Emmons problem in the entire elliptic domain - upstream and in the far field.</p>","PeriodicalId":558,"journal":{"name":"Fire Technology","volume":null,"pages":null},"PeriodicalIF":2.3000,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The Emmons Problem Revisited\",\"authors\":\"Howard R. Baum, J. G. Quintiere\",\"doi\":\"10.1007/s10694-024-01613-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The “Emmons Problem” is a foundation of fire science and gives a mathematical boundary layer solution to the burning of a vaporizing fuel from the surface of a flat plate immersed in a uniform flow of oxidizing gas. It approximates the Navier–Stokes equations assuming infinitely fast chemistry and ignores differential diffusion and thermal radiation. This allows “similarity” solutions to be developed and expressed in terms of the classic Blasius function. The current paper extends the solution, in mathematical form, to the entire domain far from the boundary layer and upstream of the leading edge. The introduction of conformal parabolic coordinates and use of the Howarth transformation allows the solution for the stream function to be expressed in exactly the same form as that found by Emmons and furnishes an explicit recipe for the pressure perturbation. The same coordinate transformations allow the exact solution of the full elliptic mixture fraction equation to be obtained, and the representation of the velocity components in terms of the stream function guarantees that the continuity equation is also satisfied exactly. Further, an exact solution to the transverse momentum equation is also displayed permitting the introduction of a crossflow into the spectrum of results obtained. In short, an analytic solution is found for the Emmons problem in the entire elliptic domain - upstream and in the far field.</p>\",\"PeriodicalId\":558,\"journal\":{\"name\":\"Fire Technology\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.3000,\"publicationDate\":\"2024-07-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Fire Technology\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://doi.org/10.1007/s10694-024-01613-w\",\"RegionNum\":3,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"ENGINEERING, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fire Technology","FirstCategoryId":"5","ListUrlMain":"https://doi.org/10.1007/s10694-024-01613-w","RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"ENGINEERING, MULTIDISCIPLINARY","Score":null,"Total":0}
The “Emmons Problem” is a foundation of fire science and gives a mathematical boundary layer solution to the burning of a vaporizing fuel from the surface of a flat plate immersed in a uniform flow of oxidizing gas. It approximates the Navier–Stokes equations assuming infinitely fast chemistry and ignores differential diffusion and thermal radiation. This allows “similarity” solutions to be developed and expressed in terms of the classic Blasius function. The current paper extends the solution, in mathematical form, to the entire domain far from the boundary layer and upstream of the leading edge. The introduction of conformal parabolic coordinates and use of the Howarth transformation allows the solution for the stream function to be expressed in exactly the same form as that found by Emmons and furnishes an explicit recipe for the pressure perturbation. The same coordinate transformations allow the exact solution of the full elliptic mixture fraction equation to be obtained, and the representation of the velocity components in terms of the stream function guarantees that the continuity equation is also satisfied exactly. Further, an exact solution to the transverse momentum equation is also displayed permitting the introduction of a crossflow into the spectrum of results obtained. In short, an analytic solution is found for the Emmons problem in the entire elliptic domain - upstream and in the far field.
期刊介绍:
Fire Technology publishes original contributions, both theoretical and empirical, that contribute to the solution of problems in fire safety science and engineering. It is the leading journal in the field, publishing applied research dealing with the full range of actual and potential fire hazards facing humans and the environment. It covers the entire domain of fire safety science and engineering problems relevant in industrial, operational, cultural, and environmental applications, including modeling, testing, detection, suppression, human behavior, wildfires, structures, and risk analysis.
The aim of Fire Technology is to push forward the frontiers of knowledge and technology by encouraging interdisciplinary communication of significant technical developments in fire protection and subjects of scientific interest to the fire protection community at large.
It is published in conjunction with the National Fire Protection Association (NFPA) and the Society of Fire Protection Engineers (SFPE). The mission of NFPA is to help save lives and reduce loss with information, knowledge, and passion. The mission of SFPE is advancing the science and practice of fire protection engineering internationally.