Anisotropic mesh adaptation for unsteady two-phase flow simulation with the Cahn-Hilliard Navier-Stokes model

We present an anisotropic mesh adaptation procedure based on Riemannian metrics for the simulation of two-phase incompressible flows with non-matching densities. The system dynamics are governed by the Cahn-Hilliard Navier-Stokes (CHNS) equations, discretized with mixed finite elements and implicit time-stepping. Spatial accuracy is controlled throughout the simulation by the \emph{global transient fixed-point method} from Alauzet \emph{et al.}, in which the simulation time is divided into sub-intervals, each associated with an adapted anisotropic mesh. The simulation is run in a fixed-point loop until convergence of each mesh–solution pair. Each iteration takes advantage of the previously computed solution and accurately predicts the flow variations. This ensures that the mesh always captures the fluid-fluid interface, and allows for a dynamic control of the interface thickness at a fraction of the computational cost compared to uniform or isotropic grids. Moreover, using a modest number of time sub-intervals reduces the transfer error from one mesh to another, which would otherwise eventually spoil the numerical solution. The overall adaptive procedure is verified with manufactured solutions and the well-known rising bubble benchmark.