Mathew Lewis

Topic

Large-scale structures in two-dimensional Rayleigh–Bénard convection driven by fixed heat fluxes

Department of Mathematics and Statistics

Date & location

• Friday, May 3, 2024
• 10:00 A.M.
• David Strong Building, Room C114

Examining Committee

Supervisory Committee

• Dr. David Goluskin, Department of Mathematics and Statistics, University of Victoria (Supervisor)
• Dr. Boualem Khouider, Department of Mathematics and Statistics, UVic (Member)

External Examiner

• Dr. Philipp Vieweg, Department of Applied Mathematics & Theoretical Physics, Cambridge University

Chair of Oral Examination

• Dr. Timothy Iles, Department of Pacific and Asian Studies, UVic

Abstract

Existence of large-scale structures in two-dimensional Rayleigh–Bénard convection is investigated in the case of boundaries that are no-slip and have fixed heat fluxes. Direct numerical simulations are carried out using the code Dedalus, which implements spectral methods. Simulations are carried out in a horizontally periodic domain, primarily with a horizontal period 20 times the layer height. The large-scale structure of interest is a pair of wide convection rolls. After finding one such two-roll state at fixed values of the Rayleigh number 𝑅 and Prandtl number 𝑃𝑟, the parameters are varied slowly in time to find two-roll states elsewhere in the 𝑅–𝑃𝑟 plane. Loss of a two-roll state occurs by transition to a four-roll state, which is detected using several criteria. The 𝑅–𝑃𝑟 plane is divided into one region where we have found two-roll states that persist, and one region where we have not. Along part of the boundary between these regions the two-roll states are steady, suggesting that their break-up is a linear instability. Elsewhere in the 𝑅–𝑃𝑟 plane the boundary is hard to locate precisely because the two-roll states are unsteady and can display metastable behaviour. The two-roll regime is found only when 𝑅 is sufficiently small and 𝑃𝑟 is sufficiently large, and these two-roll states are further classified as steady or unsteady. Contrasting our findings with simulations in the literature that have different boundary conditions and/or are three dimensional, we find that existence or nonexistence of large-scale structures is substantially affected by both thermal and velocity boundary conditions and by dimension. A simple model with one fitting parameter is found to capture the middle region of a wide roll at various parameter values, and partial results are presented towards using this model to understand the region in the 𝑅–𝑃 𝑟 plane at which two-roll states are found.