  We present a novel technique for the imposition of non-linear entropy
conservative and entropy stable solid wall boundary conditions for the
compressible Navier-Stokes equations in the presence of an adiabatic wall, or a
wall with a prescribed heat entropy flow. The procedure relies on the formalism
and mimetic properties of diagonal-norm, summation-by-parts, and
simultaneous-approximation-term operators, and is a generalization of previous
works on discontinuous interface coupling [1] and solid wall boundary
conditions [2]. Using the method of lines, a semi-discrete entropy estimate for
the entire domain is obtained when the proposed numerical imposition of
boundary conditions are coupled with an entropy-conservative or entropy-stable
discrete interior operator. The resulting estimate mimics the global entropy
estimate obtained at the continuous level. The boundary data at the wall are
weakly imposed using a penalty flux approach and a
simultaneous-approximation-term technique for both the conservative variables
and the gradient of the entropy variables. Discontinuous spectral collocation
operators (mass lumped nodal discontinuous Galerkin operators), on high-order
unstructured grids, are used for the purpose of demonstrating the robustness
and efficacy of the new procedure for weakly enforcing boundary conditions.
Numerical simulations confirm the non-linear stability of the proposed
technique, with applications to three-dimensional subsonic and supersonic
flows. The procedure described is compatible with any diagonal-norm
summation-by-parts spatial operator, including finite element, finite
difference, finite volume, discontinuous Galerkin, and flux reconstruction
schemes.
