  Centered numerical fluxes can be constructed for compressible Euler equations
which preserve kinetic energy in the semi-discrete finite volume scheme. The
essential feature is that the momentum flux should be of the form $f^m_\jph =
\tp_\jph + \avg{u}_\jph f^\rho_\jph$ where $\avg{u}_\jph = (u_j + u_{j+1})/2$
and $\tp_\jph, f^\rho_\jph$ are {\em any} consistent approximations to the
pressure and the mass flux. This scheme thus leaves most terms in the numerical
flux unspecified and various authors have used simple averaging. Here we
enforce approximate or exact entropy consistency which leads to a unique choice
of all the terms in the numerical fluxes. As a consequence novel entropy
conservative flux that also 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 are present or if the mesh is coarse. We
construct scalar artificial dissipation terms which are kinetic energy stable
and satisfy approximate/exact entropy condition. Secondly, we use entropy-
variable based matrix dissipation flux which leads to kinetic energy and
entropy stable schemes. These schemes are shown to be free of entropy violating
solutions unlike the original Roe scheme. For hypersonic flows a blended scheme
is proposed which gives carbuncle free solutions for blunt body flows.
Numerical results for Euler and Navier-Stokes equations are presented to
demonstrate the performance of the different schemes.
