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Penetrative convection: heat transport with marginal stability assumption

Published online by Cambridge University Press:  05 April 2023

Zijing Ding Open the ORCID record for Zijing Ding [Opens in a new window]  and
Zhen Ouyang
Zijing Ding*
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, PR China
Zhen Ouyang
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, PR China
*
†Email address for correspondence: z.ding@hit.edu.cn
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Abstract

This paper investigates heat transport in penetrative convection with a marginally stable temporal-horizontal-averaged field or background field. Assuming that the background field is steady and is stabilised by the nonlinear perturbation terms, we obtain an eigenvalue problem with an unknown background temperature $\tau$ by truncating the nonlinear terms. Using a piecewise profile for $\tau$, we derived an analytical scaling law for heat transport in penetrative convection as $Ra\rightarrow \infty$: $Nu=(1/8)(1-T_M)^{5/3}Ra^{1/3}$ ($Nu$ is the Nusselt number; $Ra$ is the Rayleigh number and $T_M$ corresponds to the temperature at which the density is maximal). A conditional lower bound on $Nu$, under the marginal stability assumption, is then derived from a variational problem. All the solutions to the full system should deliver a higher heat flux than the lower bound if they satisfy the marginal stability assumption. However, data from the present direct numerical simulations and previous optimal steady solutions by Ding & Wu (J. Fluid Mech., vol. 920, 2021, A48) exhibit smaller $Nu$ than the lower bound at large $Ra$, indicating that these averaged fields are over-stabilised by the nonlinear terms. To incorporate a more physically plausible constraint to bound heat transport, an alternative approach, i.e. the quasilinear approach is invoked which delivers the highest heat transport and agrees well with Veronis's assumption, i.e. $Nu\sim Ra^{1/3}$ (Astrophys. J., vol. 137, 1963, p. 641). Interestingly, the background temperature $\tau$ yielded by the quasilinear approach can be non-unique when instability is subcritical.

JFM classification

Convection: Buoyancy-driven instability Mathematical Foundations: Variational methods Nonlinear Dynamical Systems: Bifurcation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 960 , 10 April 2023 , A26
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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