A fully-coupled framework for solving Cahn-Hilliard Navier-Stokes equations: Second-order, energy-stable numerical methods on adaptive octree based meshes

Abstract

We present a fully-coupled, implicit-in-time framework for solving a thermodynamically-consistent Cahn-Hilliard Navier-Stokes system that models two-phase flows. In this work, we extend the block iterative method presented in Khanwale et al. [Simulating two-phase flows with thermodynamically consistent energy stable Cahn-Hilliard Navier-Stokes equations on parallel adaptive octree based meshes, J. Comput. Phys. ( 2020)], to a fully-coupled, provably second-order accurate scheme in time, while maintaining energy-stability. The new method requires fewer matrix assemblies in each Newton iteration resulting in faster solution time. The method is based on a fully-implicit Crank-Nicolson scheme in time and a pressure stabilization for an equal order Galerkin formulation. That is, we use a conforming continuous Galerkin (cG) finite element method in space equipped with a residualbased variational multiscale (RBVMS) procedure to stabilize the pressure. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise, Rayleigh-Taylor instability, and lid-driven cavity flow problems. We analyze in detail the scaling of our numerical implementation.

Energy stable and conservative numerical schemes for simulating two-phase flows using Cahn-Hilliard Navier Stokes equations

“…We previously developed an energy-stable, fully-coupled, second order scheme in Khanwale et al (2020b). The coupled pressure-velocity (using equal order interpolation) is achieved using pressure stabilization based on the variational multi-scale method.…”

“…The coupled pressure-velocity (using equal order interpolation) is achieved using pressure stabilization based on the variational multi-scale method. While this approach (Khanwale et al, 2020b) is well-suited for systems affording naturally large timestep (or where steady state solutions are desired), the coupled non-linear solves become expensive for turbulent systems, especially when small timesteps are naturally required. Moreover, our analysis Khanwale et al (2020b) suggests that a fully-coupled pressure stabilized approach requires very careful design of preconditioners for efficient execution of large scale production simulations.…”

“…While this approach (Khanwale et al, 2020b) is well-suited for systems affording naturally large timestep (or where steady state solutions are desired), the coupled non-linear solves become expensive for turbulent systems, especially when small timesteps are naturally required. Moreover, our analysis Khanwale et al (2020b) suggests that a fully-coupled pressure stabilized approach requires very careful design of preconditioners for efficient execution of large scale production simulations. To address these issues we seek to improve the results of (Khanwale et al, 2020b) in three key aspects.…”

“…Moreover, our analysis Khanwale et al (2020b) suggests that a fully-coupled pressure stabilized approach requires very careful design of preconditioners for efficient execution of large scale production simulations. To address these issues we seek to improve the results of (Khanwale et al, 2020b) in three key aspects.…”

“…1. Second-order energy-stable projection scheme: We modify the time integration scheme presented in Khanwale et al (2020b) to decouple the pressure equation using a projection method while preserving second-order accuracy, energy-stability, and mass conservation.…”