Work

New bounds for heat transport in internally heated convection at infinite Prandtl number

Year: 2025

Type: article

Abstract: We prove new bounds on the heat flux out of the bottom boundary, FB, for a fluid at infinite Prandtl number, heated internally between isothermal parallel plates, under two kinematic boundary conditions. In uniform internally heated convection, the supply of heat equally leaves the domain entirely by conduction in the absence of a flow. When the heating, quantified by the Rayleigh number, R, becomes sufficiently large, turbulent convection ensues, reducing the heat leaving the domain through the bottom boundary. In the case of no-slip boundary conditions, with the background field method, we prove that FB≳R−2/3−R−1/2⁡log(1−R−2/3), up to a positive constant independent of the Rayleigh and Prandtl numbers. Whereas between stress-free boundaries we show that, FB≳R−40/29−R−35/29⁡log(1−R−40/29). We perform a numerical study of the system in two dimensions up to a Rayleigh number of 5 × 109 using the spectral solver Dedalus. The numerical investigations indicate that FB∼R−0.092 and FB∼R−0.12 for the two kinematic boundary conditions respectively. The gap between the bounds and simulations, along with the constructions used in the proofs highlight the potential for further optimisation of bounds for FB in internally heated convection.

Source: Journal of Mathematical Physics

DOI: 10.1063/5.0209505

Authors: Ali Arslan, Rubén E. Rojas

Institutions: ETH Zurich, Institute of Geophysics

Language: English

Topic: Fluid Dynamics and Turbulent Flows

Subfield: Computational Mechanics

Field: Engineering

Domain: Physical Sciences

Open Access status: hybrid

Funders: Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung, HORIZON EUROPE European Research Council

Awards: G1612724619, Dynamics of the Earth's core under the plesio-geostrophy paradigm