Kinetic Energy Preserving and Entropy Stable Finite Volume Schemes for Compressible Euler and Navier-Stokes Equations (Communications in Computational Physics, 2013)
Centered numerical fluxes can be constructed for compressible Euler equationswhich preserve kinetic energy in the semi-discrete finite volume scheme. The essentialfeature is that the momentum flux should be of the form  where  are any consistent approximations to thepressure and the mass flux. This scheme thus leaves most terms in the numericalflux unspecified and various authors have used simple averaging. Here we enforceapproximate or exact entropy consistency which leads to a unique choice of all theterms in the numerical fluxes. As a consequence, a novel entropy conservative flux thatalso preserves kinetic energy for the semi-discrete finite volume scheme has been proposed.These fluxes are centered and some dissipation has to be added if shocks arepresent or if the mesh is coarse. We construct scalar artificial dissipation terms whichare kinetic energy stable and satisfy approximate/exact entropy condition. Secondly,we use entropy-variable based matrix dissipation flux which leads to kinetic energyand entropy stable schemes. These schemes are shown to be free of entropy violatingsolutions unlike the original Roe scheme. For hypersonic flows a blended scheme isproposed which gives carbuncle free solutions for blunt body flows. Numerical resultsfor Euler and Navier-Stokes equations are presented to demonstrate the performanceof the different schemes.