Entropy-stable discontinuous Galerkin approximation with summation-by-parts property for the incompressible Navier-Stokes equations with variable density and artificial compressibility

Abstract

We present a provably stable discontinuous Galerkin spectral element method for the incompressible Navier-Stokes equations with artificial compressibility and variable density. Stability proofs, which include boundary conditions, that follow a continuous entropy analysis are provided. We define a mathematical entropy function that combines the traditional kinetic energy and an additional energy term for the artificial compressiblity, and derive its associated entropy conservation law. The latter allows us to construct a provably stable split-form nodal Discontinuous Galerkin (DG) approximation that satisfies the summation-by-parts simultaneous-approximation-term (SBP-SAT) property. The scheme and the stability proof are presented for general curvilinear three-dimensional hexahedral meshes. We use the exact Riemann solver and the Bassi-Rebay 1 (BR1) scheme at the inter-element boundaries for inviscid and viscous fluxes respectively, and an explicit low storage Runge-Kutta RK3 scheme to integrate in time. We assess the accuracy and robustness of the method by solving the Kovasznay flow, the inviscid Taylor-Green vortex, and the Rayleigh-Taylor instability.

References in corpus (5)

• Split Form Nodal Discontinuous Galerkin Schemes with Summation-By-Parts Property for the Compressible Euler Equations
• On the use of kinetic energy preserving DG-schemes for large eddy simulation
• Entropy-stable discontinuous Galerkin approximation with summation-by-parts property for the incompressible Navier-Stokes/Cahn-Hilliard system
• A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation
• On the role of numerical dissipation in stabilising under-resolved turbulent simulations using discontinuous Galerkin methods

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• Entropy-Stable Gauss Collocation Methods for Ideal Magneto-Hydrodynamics
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• A modified equation analysis for immersed boundary methods based on volume penalization: applications to linear advection-diffusion and high-order discontinuous Galerkin schemes
• Riemann solvers and pressure gradients in Godunov-type schemes for variable density incompressible flows
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• High-order flux reconstruction method for the hyperbolic formulation of the incompressible Navier-Stokes equations on unstructured grids
• High--order discontinuous Galerkin approximation for a three--phase incompressible Navier--Stokes/Cahn--Hilliard model