Rayleigh-B\'enard convection with a melting boundary

We study the evolution of a melting front between the solid and liquid phases of a pure incompressible material where fluid motions are driven by unstable temperature gradients. In a plane layer geometry, this can be seen as classical Rayleigh-B\'enard convection where the upper solid boundary is allowed to melt due to the heat flux brought by the fluid underneath. This free-boundary problem is studied numerically in two dimensions using a phase-field approach, classically used to study the melting and solidification of alloys, which we dynamically couple with the Navier-Stokes equations in the Boussinesq approximation. The advantage of this approach is that it requires only moderate modifications of classical numerical methods. We focus on the case where the solid is initially nearly isothermal, so that the evolution of the topography is related to the inhomogeneous heat flux from thermal convection, and does not depend on the conduction problem in the solid. From a very thin stable layer of fluid, convection cells appears as the depth -- and therefore the effective Rayleigh number of the layer increases. The continuous melting of the solid leads to dynamical transitions between different convection cell sizes and topography amplitudes. The Nusselt number can be larger than its value for a planar upper boundary, due to the feedback of the topography on the flow, which can stabilize large-scale laminar convection cells.Comment: 36 pages, 16 figure

Bistability in Rayleigh-B\'enard convection with a melting boundary

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-B\'enard 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

Interaction entre un écoulement convectif et un front de changement de phase solide-liquide

Les processus de fusion et solidification trouvent leurs applications dans l’industrie de la fonderie et sont omniprésents dans la nature. Dans les deux domaines, ces processus sont souvent couplés à un écoulement qui affecte les dynamiques de fonte. Les travaux présentés dans cette thèse se focalisent sur le problème où une interface de fusion inter-agit avec un écoulement dans une configuration idéalisée et à l’aide de simulations numériques. Dans cette étude, une méthode de champ de phase couplée avec une méthode de pénalisation est utilisée. La méthode de champ de phase permet de suivre l’avancement du front alors que la méthode de pénalisation assure une vitesse nulle dans le solide. Deux codes sont utilisés pour la résolution des équations, le premier est un code mixte pseudo-spectral différences finies d’ordre quatre et le second est le code open-souce pseudo-spectral DEDALUS. Deux configurations sont étudiées. La première décrit une configuration où le solide et le liquide sont délimités par deux parois horizontales telle que le solide est refroidi par le haut et le liquide réchauffé par le bas. On s'intéresse ici à la dynamique du système pendant la phase de fusion ainsi que les stabilités de ces systèmes. Par ailleurs, la deuxième configuration consiste en un solide immergé dans un écoulement chaud et unidirectionnel. On s'intéresse ici au temps de disparition de l'objet ainsi que sa forme pendant la phase de fusion. Pour finir, une étude d’invariance galiléenne est effectuée entre un objet fixe (écoulement liquide) et un objet en mouvement (liquide au repos)Melting or solidification processes are important in casting industries and are ubiquitous in nature. In both cases, those processes can be coupled to a flow which can dramatically impact the melting or solidification dynamics. The work presented in this thesis focuses on the general problem where a melting boundary interacts with a flow in an idealized setting through computational simulations. A phase-field method coupled with a volume penalization method is used to study this problem. The phase-field method allows for the tracking of the solid-liquid interface while the volume penalization method ensures that the solid phase remains stationary. A fourth-order pseudo-spectral finite differences code and the open source pseudo-spectral code, DEDALUS, are used to solve the corresponding equations. Two distinct settings are studied. The first one consists of bounding the solid and the liquid in between two horizontal plates while being cooled from aboveand heated from below. The melting dynamics as well as the equilibrium of such systems were studied and several scaling laws were found. The second setting describes the melting of a solid immersed in a warm horizontally flowing liquid in between two plates. A scaling law for the vanishing time of the object is obtained which is similar to previous studies about eroding and dissolving object. Galilean invariance between configurations with a fixed solid (in a flowing liquid) and a moving solid (in a static liquid) is finally shown, which opens new avenues to study sedimentation and melting particles, a configurationof geophysical interes