{"title":"Transitions in a Poiseuille-Rayleigh-Bénard flow in a vertical slender long duct","authors":"Raúl Rechtman, Alejandra García-Morales, Guadalupe Huelsz","doi":"10.1016/j.euromechflu.2024.01.012","DOIUrl":null,"url":null,"abstract":"<div><p>The flow of air inside a vertical slender long duct with a temperature difference between the horizontal walls characterized by the Rayleigh number <span><math><mrow><mi>R</mi><mi>a</mi></mrow></math></span> and a pressure gradient in the horizontal direction characterized by the Reynolds number <span><math><mrow><mi>R</mi><mi>e</mi></mrow></math></span> is studied using the lattice Boltzmann method. In this case, the longitudinal rolls found in ducts with a width larger than the height cannot develop and the flow remains quasi-two-dimensional. Therefore, a two-dimensional approach is used, considering periodic boundary conditions on the vertical walls. For <span><math><mrow><mi>R</mi><msub><mrow><mi>a</mi></mrow><mrow><mi>c</mi></mrow></msub><mo><</mo><mi>R</mi><mi>a</mi><mo>≤</mo><mn>2</mn><mo>.</mo><mn>0</mn><mspace></mspace><mo>×</mo><mn>1</mn><msup><mrow><mn>0</mn></mrow><mrow><mn>5</mn></mrow></msup></mrow></math></span> with <span><math><mrow><mi>R</mi><msub><mrow><mi>a</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math></span> the critical Rayleigh number for thermal convection, there are two transitions at <span><math><mrow><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mn>0</mn></mrow></msub><mrow><mo>(</mo><mi>R</mi><mi>a</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub><mrow><mo>(</mo><mi>R</mi><mi>a</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mn>0</mn></mrow></msub><mo><</mo><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub></mrow></math></span>. The first transition at <span><math><mrow><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span> has a sharp decrease in the average Nusselt number and for <span><math><mrow><mn>0</mn><mo><</mo><mi>R</mi><mi>e</mi><mo><</mo><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span> the temperature difference between the horizontal walls is more important than the pressure gradient. The second transition at <span><math><mrow><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub></mrow></math></span> marks the appearance of a conductive Poiseuille flow with no thermal convection. For <span><math><mrow><mn>1</mn><mo>.</mo><mn>0</mn><mo>×</mo><mn>1</mn><msup><mrow><mn>0</mn></mrow><mrow><mn>5</mn></mrow></msup></mrow></math></span> ¡ <span><math><mrow><mi>R</mi><mi>a</mi></mrow></math></span> there is a third transition at <span><math><mrow><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>M</mi></mrow></msub></mrow></math></span>, <span><math><mrow><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mn>0</mn></mrow></msub><mo><</mo><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>M</mi></mrow></msub><mo><</mo><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub></mrow></math></span>. For <span><math><mrow><mi>R</mi><mi>a</mi><mo><</mo></mrow></math></span> <span><math><mrow><mn>1</mn><mo>.</mo><mn>0</mn><mo>×</mo><mn>1</mn><msup><mrow><mn>0</mn></mrow><mrow><mn>5</mn></mrow></msup></mrow></math></span> and <span><math><mrow><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mn>0</mn></mrow></msub><mo><</mo><mi>R</mi><mi>e</mi><mo><</mo><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub></mrow></math></span>, and for <span><math><mrow><mn>1</mn><mo>.</mo><mn>0</mn><mo>×</mo><mn>1</mn><msup><mrow><mn>0</mn></mrow><mrow><mn>5</mn></mrow></msup></mrow></math></span> <span><math><mrow><mo><</mo><mi>R</mi><mi>a</mi></mrow></math></span> and <span><math><mrow><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>M</mi></mrow></msub><mo><</mo><mi>R</mi><mi>e</mi><mo><</mo><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub></mrow></math></span>, the pressure gradient dominates over the pressure gradient. Both the temperature difference and the pressure gradient are important for <span><math><mrow><mn>1</mn><mo>.</mo><mn>0</mn><mo>×</mo><mn>1</mn><msup><mrow><mn>0</mn></mrow><mrow><mn>5</mn></mrow></msup><mo><</mo><mi>R</mi><mi>a</mi></mrow></math></span> and <span><math><mrow><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mn>0</mn></mrow></msub><mo><</mo><mi>R</mi><mi>e</mi><mo><</mo><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>M</mi></mrow></msub></mrow></math></span> with an appreciable decrease of the average Nusselt number at <span><math><mrow><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span> and <span><math><mrow><mi>R</mi><msub><mrow><mi>e</mi></mrow><mrow><mi>M</mi></mrow></msub></mrow></math></span>.</p></div>","PeriodicalId":11985,"journal":{"name":"European Journal of Mechanics B-fluids","volume":null,"pages":null},"PeriodicalIF":2.5000,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"European Journal of Mechanics B-fluids","FirstCategoryId":"5","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0997754624000219","RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MECHANICS","Score":null,"Total":0}
引用次数: 0
Abstract
The flow of air inside a vertical slender long duct with a temperature difference between the horizontal walls characterized by the Rayleigh number and a pressure gradient in the horizontal direction characterized by the Reynolds number is studied using the lattice Boltzmann method. In this case, the longitudinal rolls found in ducts with a width larger than the height cannot develop and the flow remains quasi-two-dimensional. Therefore, a two-dimensional approach is used, considering periodic boundary conditions on the vertical walls. For with the critical Rayleigh number for thermal convection, there are two transitions at and , . The first transition at has a sharp decrease in the average Nusselt number and for the temperature difference between the horizontal walls is more important than the pressure gradient. The second transition at marks the appearance of a conductive Poiseuille flow with no thermal convection. For ¡ there is a third transition at , . For and , and for and , the pressure gradient dominates over the pressure gradient. Both the temperature difference and the pressure gradient are important for and with an appreciable decrease of the average Nusselt number at and .
期刊介绍:
The European Journal of Mechanics - B/Fluids publishes papers in all fields of fluid mechanics. Although investigations in well-established areas are within the scope of the journal, recent developments and innovative ideas are particularly welcome. Theoretical, computational and experimental papers are equally welcome. Mathematical methods, be they deterministic or stochastic, analytical or numerical, will be accepted provided they serve to clarify some identifiable problems in fluid mechanics, and provided the significance of results is explained. Similarly, experimental papers must add physical insight in to the understanding of fluid mechanics.