Entropy–stable discontinuous Galerkin approximation with summation–by–parts property for the incompressible Navier–Stokes/Cahn–Hilliard system

Abstract

We develop an entropy stable two-phase incompressible Navier-Stokes/Cahn-Hilliard discontinuous Galerkin (DG) flow solver method. The model poses the Cahn-Hilliard equation as the phase field method, a skew-symmetric form of the momentum equation, and an artificial compressibility method to compute the pressure. We design the model so that it satisfies an entropy law, including free-and no-slip wall boundary conditions with non-zero wall contact angle. We then construct a high-order DG approximation of the model that satisfies the SBP-SAT property. With the help of a discrete stability analysis, the scheme has two modes: an entropy conserving approximation with central advective fluxes and the Bassi-Rebay 1 (BR1) method for diffusion, and an entropy stable approximation with an exact Riemann solver for advection and interface stabilization added to the BR1 method. The scheme is applicable to, and the stability proofs hold for, three-dimensional unstructured meshes with curvilinear hexahedral elements. We test the convergence of the schemes on a manufactured solution, and their robustness by solving a flow initialized from random numbers. In the latter, we find that a similar scheme that does not satisfy an entropy inequality had 30% probability to fail, while the entropy stable scheme never does. We also solve the static and rising bubble test problems, and to challenge the solver capabilities we compute a three-dimensional pipe flow in the annular regime.

A Novel Robust Strategy for Discontinuous Galerkin Methods in Computational Fluid Mechanics: Why? When? What? Where?

“…But using the SBP property of the DGSEM-LGL operators, it is possible to apply ideas similar to Fisher et al and construct a novel DGSEM with LGL quadrature, that is discretely L 2 -stable for the nonlinear Burgers' equation, without the assumption on exact evaluation of the integrals [111]. These first results have been extended and compounded upon for the compressible Euler equations [114][115][116][117][118], the shallow water equations [63,[119][120][121], the compressible Navier-Stokes equations [32,122,124], non-conservative multi-phase problems [124], magnetohydrodynamics [125,126], relativistic Euler [127], relativistic magnetohydrodynamics [128], the Cahn-Hilliard equations [129], incompressible Navier-Stokes (INS) [130], and coupled Cahn-Hilliard and INS [131] among many other complex PDE models and DG discretization types e.g., [132].…”

A High-Order Discontinuous Galerkin Solver for Multiphase Flows

“…Since we use the SIP method, we use solution averages to couple inter-element fluxes, U = {{U }}. All the integrals involved in (18) are computed discretely similar to those in (16), i.e.,…”

“…Specifically, we use a Discontinuous Galerkin Spectral Element Method (DGSEM) [2] that allows the generation of provably stable schemes [8]. These schemes provide enhanced robustness when compared to classical high-order methods [17][18][19][20]. As far as the temporal discretisation is concerned, we use an efficient implicit-explicit approach that permits maintaining the time step restriction of a typical one phase Navier-Stokes solver.…”

A free–energy stable p–adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equation

“…Furthermore, another advantage of the DGSEM exploited in this work is that the approximation order can vary across elements. There is a limited amount of publications concerning the use of discontinuous Galerkin methods in order to solve the Cahn-Hilliard equation [8][9][10][11][12][13][14] and there is still a variety of aspects regarding the efficiency, robustness and accuracy that should be addressed.…”