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

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.