FE² multi-scale framework for the two-equation model of transient heat conduction in two-phase media

In the study of transient heat conduction in heterogeneous two-phase media, the local thermal non-equilibrium condition calls for the use of a two-equation model to appropriately describe different temperatures in the two phases. We propose for the two-equation model an FE² multi-scale framework that is capable of addressing nonlinear conduction problems. The FE² framework consists of volume-averaged macroscale equations, well-defined microscale problems, and the information exchange between the two scales. Compared to a traditional FE² method for the one-equation model, the proposed FE² framework introduces an additional source term at the macroscale that is upscaled from the microscale interfacial heat transfer. At variance with the tangent matrices (i.e., effective conductivity) of the heat flux, the tangent matrices of the interfacial heat transfer depend on the microscopic length scale. The proposed FE² framework is validated against single-scale direct numerical simulations, and some numerical examples are employed to demonstrate its potential.