A stable and accurate scheme for solving the Stefan problem coupled with natural convection using the Immersed Boundary Smooth Extension method

Abstract

The dissolution of solids has created spectacular geomorphologies ranging from centimeter-scale cave scallops to the kilometer-scale "stone forests" of China and Madagascar. Mathematically, dissolution processes are modeled by a Stefan problem, which describes how the motion of a phase-separating interface depends on local concentration gradients, coupled to a fluid flow. Simulating these problems is challenging, requiring the evolution of a free interface whose motion depends on the normal derivatives of an external field in an ever-changing domain. Moreover, density differences created in the fluid domain induce self-generated convecting flows that further complicate the numerical study of dissolution processes. In this contribution, we present a numerical method for the simulation of the Stefan problem coupled to a fluid flow. The scheme uses the Immersed Boundary Smooth Extension method to solve the bulk advection-diffusion and fluid equations in the complex, evolving geometry, coupled to a θ-L scheme that provides stable evolution of the boundary. We demonstrate 3 rd -order temporal and pointwise spatial convergence of the scheme for the classical Stefan problem, and 2 nd -order temporal and pointwise spatial convergence when coupled to flow. Examples of dissolution of solids that result in high-Rayleigh number convection are numerically studied, and qualitatively reproduce the complex morphologies observed in recent experiments.

A hybrid level-set / embedded boundary method applied to solidification-melt problems

“…However, level-set methods are quite straightforward to implement, versatile enough to be combined with another method and their advantages and drawbacks, linked to the mathematical properties of the equations at play, have been studied quite thoroughly. Solutions have been found for applying an immersed boundary condition using a finite-difference treatment for the variables, for instance the LS-STAG method [36], the Immersed Boundary Smooth Extension [37] or the Ghost Fluid method [38]. Note that all these methods can be shown to still rely on smooth approximations of Dirac/Heaviside functions [14], and are thus only first-order accurate spatially.…”

A hybrid level-set / embedded boundary method applied to solidification-melt problems

“…However, level-set methods are quite straighforward to implement, versatile enough to be combined with another method and their advantages and drawbacks, linked to the mathematical properties of the equations at play, have been studied quite thoroughly. Solutions have been found for applying an immersed boundary condition using a finite-difference treatment for the variables, for instance the LS-STAG method [35], the Immersed Boundary Smooth Extension [36] or the Ghost Fluid method [37]. Note that all these methods can be shown to still rely on smooth approximations of Dirac/Heaviside functions [15], and are thus only first-order accurate spatially.…”

Improved phase-field models of melting and dissolution in multi-component flows

“…The evolution of the phases is then determined by a single set of equations that apply over the entire domain. Many other methods also model phase changes, such as enthalpy methods [59,57], level set methods [47,15], diffuse-domain approaches [39,1], or some immersed-boundary methods [42]. Yet phase-field models stand out for combining several key benefits:…”