Conjugate heat transfer using modified interface conditions for the Navier-Stokes equations

This paper evaluates the use of the compressible Navier-Stokes equations, with prescribed zero velocities, as a model for heat transfer in solids. In particular in connection with conjugate heat transfer problems. We derive estimates, and show how to choose and scale the coefficients of the energy part in the Navier-Stokes equations, such that the difference between the energy equation and the heat equation is minimal. A rigorous analysis of the physical interface conditions for the conjugate heat transfer problem is performed and energy estimates are derived in non-standard Lequivalent norms. The numerical schemes are proven energy stable with the physical interface conditions and the stability of the schemes are independent of the order of accuracy. We have performed computations of a conjugate heat transfer problem in two different ways. One where, as traditionally, the heat transfer in the solid is governed by the heat equation. The other where the heat transfer in the solid is governed by the Navier-Stokes equations. The simulations are compared and the numerical results corroborate the theoretical results.