Flows coupled to phase change

2-D Rayleigh-Bénard with a melting boundary

This example is taken from appendix A.3 of Favier et al. (2019). We consider the convection flow between a heated lower plate and a moving phase boundary. There is no analytic form for the solution of this flow, so we can only perform a relative convergence study and compare to the results of Favier et al. (2019).

The initial condition consists of a linear temperature profile subject to a perturbation in the liquid phase. The melting temperature is set to \(T_m=0.5\) and the intial height of the interface is at \(x=0.5\). Precisely, the initial fields are

\[ T(x,y) = \begin{cases} 1 - x, & x\geq 0.5, \\ 1 - x + a \sin^2(2\pi x) \sin(4\pi y), & x < 0.5, \end{cases} \]

\[ \phi(x,y) = \frac{1 + \tanh((x - 0.5)/2\varepsilon)}{2}, \]

and zero velocity. Recall that \(x\) is the wall-normal direction in our formulation. The perturbation amplitude \(a\) is not specified in Favier et al. (2019), but we set it to 0.1.