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

TL;DR

This work addresses the challenge of efficiently simulating unsteady two-phase flows governed by the Cahn-Hilliard-Navier-Stokes (CHNS) model by introducing an anisotropic mesh adaptation framework based on time-dependent Riemannian metrics. A global transient fixed-point strategy drives space-time mesh adaptation, producing a sequence of optimized, anisotropic meshes that concentrate resolution along the fluid-fluid interface while controlling interpolation error in the phase field. The CHNS solver is discretized monolithically with FE spaces $P_2$-$P_1$ for velocity-pressure and $P_1$-$P_1$ for $(\phi,\mu)$, with implicit $BDF2$ time stepping and Newton iterations; Hessian recovery via Polynomial Preserving Recovery guides metric construction. Verification with manufactured solutions and application to rising-bubble benchmarks demonstrate significant mesh reduction (compared to uniform grids) and accurate interface thickness control, validating the approach and its potential for efficient, high-fidelity multiphase simulations. The method provides a practical path toward scalable CHNS simulations in complex geometries and time-dependent scenarios, with future enhancements including conservative transfer and higher-order time-space adaptivity.

Abstract

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.

Figures (13)

• Figure 1: Space-time mesh adapted to a Rayleigh-Taylor instability. The time axis is sliced into ${n_I}$ sub-intervals, on which ${n_T}$ time steps each are computed.
• Figure 2: Verification of the CHNS solver: spatial (left) and temporal (right) convergence.
• Figure 3: Convergence study for the manufactured problem \ref{['eq:MMS_heat']}. The time step decreases from one convergence curve to the next, but is kept constant on a given convergence curve. Evolution of the error for ${n_I} = 4$ or $16$ intervals as the average mesh density $\mathcal{N}_{avg}$ increases.
• Figure 4: Top left: the manufactured solution $\phi_\text{ref}$ at $t = 0$ and $t = 1$. Middle and bottom left: adapted meshes for the time sub-intervals $1$ and $2$ (out of 2) for a prescribed mesh density $\mathcal{N}_{avg} = 6.4$M. These meshes contain 13,965 and 13,987 vertices respectively. Right: adapted meshes for the intervals $1$, $64$ and $128$ (out of 128) and $\mathcal{N}_{avg} = 100$k. These meshes contain 13,341, 13,254 and 13,321 vertices respectively.
• Figure 5: Convergence curves for the manufactured problem \ref{['eq:MMS_heat']} at fixed time step $\Delta t = 1/1024$. Left: error for a fixed prescribed average mesh density $\mathcal{N}_{avg}$ and increasing number of time sub-intervals. Right: comparison between error at fixed number of intervals and increasing average mesh density vs. fixed mesh density and increasing number of time intervals.