Recently, relaxation methods have been developed to guarantee the preservation of a single global functional of the solution of an ordinary differential equation We generalize this approach to guarantee local entropy inequalities for finitely many convex functionals (entropies) and apply the resulting methods to the compressible Euler and Navier-Stokes equations. Based on the unstructured hp-adaptive SSDC framework of entropy conservative or dissipative semidiscretizations using summation-by-parts and simultaneous-approximation-term operators, we develop the first discretizations for compressible computational fluid dynamics that are primary conservative, locally entropy stable in the fully discrete sense under a usual CFL condition, explicit except for the parallelizable solution of a single scalar equation per element, and arbitrarily high-order accurate in space and time. We demonstrate the accuracy and the robustness of the fully-discrete explicit locally entropy-stable solver for a set of test cases of increasing complexity.

The continuum approach employing porous media models is a robust and efficient solution method in the area of the simulation of fixed-bed reactors. This paper applies the double-averaging methodology to refine the continuum approach, opening a way to alleviate its main limitations: space-invariant averaging volume and inaccurate treatment of the porous/fluid interface. The averaging operator is recast as a general space–time filter allowing for the analysis of commutation errors in a classic large eddy simulation (LES) formalism. An explicit filtering framework has been implemented to carry out an a posteriori evaluation of the unclosed terms appearing in the double-averaged Navier–Stokes (DANS) equations, also considering a space-varying filter width. Two resolved simulations have been performed. First, the flow around a single, stationary particle has been used to validate derived equations and the filtering procedure. Second, an LES of the turbulent flow in a channel partly occupied with a porous medium has been realized and filtered. The commutation error at the porous–fluid interface has been evaluated and compared to the prediction of two models. The significance of the commutation error terms is also discussed and assessed. Finally, the solver for DANS equations has been developed and used to simulate both of the studied geometries. The magnitude of the error associated with neglecting the commutation errors has been investigated, and an LES simulation combined with a porous drag model was performed. Very encouraging results have been obtained indicating that the inaccuracy of the drag closure overshadows the error related to the commutation of operators.

Nonlinear entropy stability analysis is used to derive entropy stable no-slip wall boundary conditions for the Eulerian model proposed by Svärd (Phys A Stat Mech Appl 506:350–375, 2018). The spatial discretization is based on entropy stable collocated discontinuous Galerkin operators with the summation-by-parts property for unstructured grids. A set of viscous test cases of increasing complexity are simulated using both the Eulerian and the classic compressible Navier–Stokes models. The numerical results obtained with the two models are compared, and similarities and differences are then highlighted. However, the differences are very small and probably smaller than what the current experimental technology allows to measure.

Nonlinear entropy stability analysis is used to derive entropy stable no-slip wall boundary conditions for the Eulerian model proposed by Svärd (Physica A: Statistical Mechanics and its Applications, 2018 ). and its spatial discretization based on entropy stable collocated discontinuous Galerkin operators with the summation-by-parts property for unstructured grids. A set of viscous test cases of increasing complexity are simulated using both the Eulerian and the classic compressible Navier-Stokes models. The numerical results obtained with the two models are compared, and differences and similarities are then highlighted. IntroductionThe classical compressible Navier-Stokes (CNS) equations can be derived based on the material (Lagrangian) derivative formulation [30]. In the Lagrangian sense, diffusion between gas pockets is non-existent, and thus, the continuity equation is hyperbolic. On the other hand, in the Eulerian model of Svärd [33], air molecules diffuse into other parts of the domain, and thus, the continuity equation is modeled as a parabolic equation.Generally speaking, entropy conservation and entropy stability are used to preserve the second law of thermodynamics in the mathematical sense, i.e., the