Using the compressible Navier-Stokes equations as a model for heat transfer in solids

Conjugate heat transfer problems are of importance in many engineering applications, since flows are usually confined by some material with heat transfer properties. Whenever there is a temperature difference between the fluid and solid, heat will be transfered and change the flow properties in a non-trivial way (Giles 1997; Henshaw & Chand 2009; Lindström & Nordström 2010; Roe et al. 2008). The approach to solve the conjugate heat transfer problem in e.g., Giles (1997); Henshaw & Chand (2009); Lindström & Nordström (2010); Roe et al. (2008), is by coupling the Navier-Stokes equations for the fluid with the heat equation for the solid. This intuitive approach is appealing due to the apparent simplicity of the heat equation. The coupling is done in Giles (1997) by passing temperature gradients from the fluid to the solid, and temperature values from the solid to the fluid. Henshaw & Chand (2009) generalized the procedure and Lindström & Nordström (2010) showed how a fully coupled system leads to both well-posedness, improved stability properties and reduced time stepping restrictions. The approach with coupling to the heat equation requires different solvers for the fluid and solid domains. This complicates the programming, and conjugate heat transfer problems cannot easily be implemented in already existing multi-block Navier-Stokes codes. The solution methods can also differ, and even a third code which handles the communication between the different solvers might be needed. This complicates the programming additionally and brings in communication overhead which further slows down the computations. This work aims at a different approach to compute heat transfer problems. Rather than computing heat transfer in the solid by solving the heat equation, we will use the compressible Navier-Stokes equations. The benefit of using the Navier-Stokes equations for governing heat transfer is that conjugate heat transfer problems can easily be implemented into already existing multi-block Navier-Stokes codes because the same solver can be used everywhere. When prescribing zero velocities as initialand boundary data, the energy equation in the Navier-Stokes equations can be written in a form which strongly resembles the heat equation. We will derive analytical estimates for the difference in temperature distribution obtained from the heat equation and the Navier-Stokes equations. The estimates are accompanied by numerical simulations which corroborate the theory.