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Coherent heat transport in two-dimensional penetrative Rayleigh–Bénard convection

Published online by Cambridge University Press:  15 June 2021

Zijing Ding Open the ORCID record for Zijing Ding [Opens in a new window]  and
Jian Wu Open the ORCID record for Jian Wu [Opens in a new window]
Zijing Ding*
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin150001, PR China
Jian Wu
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin150001, PR China
*
†Email address for correspondence: z.ding@hit.edu.cn
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Abstract

This paper investigates the steady coherent solutions, bifurcated from the linear stability of a stationary flow, in two-dimensional (2-D) penetrative convection. The results show that the thickness of the upper stably stratified layer, which is measured by $\theta _M$ (the dimensionless temperature at which the density is maximal, with $0\le \theta _M\le 1$), plays an important role in the linear and nonlinear dynamics. First, we investigate steady solutions of fixed aspect ratio $L=2{\rm \pi} /\alpha _c$ (where $\alpha _c$ is the critical wavenumber). The results show that the instability is supercritical when $\theta _M<0.4$ and is subcritical when $\theta _M>0.4$. When $\theta _M>0.4$, the results show that the type of solution depends on the Prandtl number ($Pr$). For instance, when $Pr\lesssim 2.4$ at $\theta _M=0.5$, the solution in one type of pair of convection cells does not exist, as the Rayleigh number $Ra$ exceeds a critical value due to a saddle-node bifurcation. When $Pr>2.4$, steady solutions can be found up to $Ra=10^{8}$ for all $\theta _M$, which exhibit the scaling of heat transfer (characterized by the Nusselt number $Nu$): $Nu\sim Ra^{1/4}$. Then, the optimal 2-D steady solutions are tracked up to $Ra=10^{9}$ by varying the aspect ratio $L$, which shows that heat transfer roughly follows the $Nu\sim Ra^{\gamma }$ ($\gamma \approx 1/3$) scaling in the regime of $10^{7}< Ra<10^{9}$. It is interesting that the optimal temperature field has an arm-like horizontal structure when $Pr<10$, while it has no significant horizontal structures when $Pr>10$. Thus, the mean temperature in the mixing region is higher at large $Pr$. The steady solutions show that $Nu\sim Pr^{-1/12}$ for $\theta _M=0$ in a certain range of $Pr$ by fixing the Rayleigh numbers, e.g. $1< Pr<10$ for 2-D optimal steady solutions at $Ra=10^{8}$ and $2< Pr<30$ for 2-D steady solutions of fixed aspect ratio at $Ra=10^{7}$. But when the Prandtl number is large or the upper stably stratified layer is thick, both the steady solutions of fixed aspect ratio and the 2-D optimal steady solutions are very weakly dependent on $Pr$.

JFM classification

Convection: Buoyancy-driven instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 920 , 10 August 2021 , A48
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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