Bistability in Rayleigh-Bénard convection with a melting boundary

Abstract

A pure and incompressible material is confined between two plates such that it is heated from below and cooled from above. When its melting temperature is comprised between these two imposed temperatures, an interface separating liquid and solid phases appears. Depending on the initial conditions, freezing or melting occurs until the interface eventually converges towards a stationary state. This evolution is studied numerically in a two-dimensional configuration using a phase-field method coupled with the Navier-Stokes equations. Varying the control parameters of the model, we exhibit two types of equilibria: diffusive and convective. In the latter case, Rayleigh-Bénard convection in the liquid phase shapes the solid-liquid front, and a macroscopic topography is observed. A simple way of predicting these equilibrium positions is discussed and then compared with the numerical simulations. In some parameter regimes, we show that multiple equilibria can coexist depending on the initial conditions. We also demonstrate that, in this bi-stable regime, transitioning from the diffusive to the convective equilibrium is inherently a nonlinear mechanism involving finite amplitude perturbations.

Nonlinear interactions between an unstably stratified shear flow and a phase boundary

“…The jump behaviour is in contrast to what happens near Ra c in the classical RBC (Chandrasekhar 2013) and is in good agreement with the analysis of Davis et al (1984), who predicted a subcritical bifurcation at Ra = Ra c . Similar behaviour at Ra = Ra c has been reported in previous experiments (Dietsche & Müller 1985) and DNS studies (Esfahani et al 2018;Purseed et al 2020).…”

“…The effective Ra for this case is Ra e ≈ 6.5 × 10 6 and the steady state N = 16.27, which is larger than N = 12.07 for classical RBC (Johnston & Doering 2009). These results are in qualitative agreement with the findings of Favier et al (2019) and Purseed et al (2020). The increase in the heat flux compared with classical RBC is because the non-planar phase boundary 'locks in' the convection cells, thereby delaying the onset of unsteady convection (Favier et al 2019).…”

“…This results in higher heat transport than in the classical RBC in two dimensions, where the flow becomes unsteady at Ra ≈ 7.5 × 10 5 (Toppaladoddi, Succi & Wettlaufer 2015a). Purseed et al (2020) considered a more general situation where the melting point of the solid lies between the temperatures imposed at the upper and lower boundaries and studied the bistability close to the onset of convection which was first predicted by Davis et al (1984).…”

Topography generation by melting and freezing in a turbulent shear flow

“…2019; Purseed et al. 2020). Other methods that simultaneously solve for the evolution of a fluid phase and a solid phase include the enthalpy method (Ulvrová et al.…”

“…Our numerical model solves the evolution of the fluid and solid phases simultaneously using the phase-field method. The phase-field method is a one-domain two-phase fixed-grid method that was originally developed by the metallurgy community for relatively smooth flows (Wang et al 1993;Karma & Rappel 1998;Beckermann et al 1999), but which was recently applied to the case of vigorous convective flows (Favier et al 2019;Purseed et al 2020). Other methods that simultaneously solve for the evolution of a fluid phase and a solid phase include the enthalpy method (Ulvrová et al 2012), the level-set method (Gibou et al 2007), the lattice-Boltzmann method (Rabbanipour Esfahani et al 2018) and two-domain moving-boundary methods (Ulvrová et al 2012).…”