A diffuse boundary method for phase boundaries in viscous compressible flow

Many physical systems of interest involve the close interaction of a flow in a domain with complex, time-varying boundaries. Treatment of boundaries of this nature is cumbersome due to the difficulty in explicitly tracking boundaries that may exhibit topological transitions and high curvature. Such conditions can also lead to numerical instability. Diffuse boundary methods such as the phase field method are an attractive way to describe systems with complex boundaries, but coupling such methods to hydrodynamic flow solvers is nontrivial. This work presents a systematic approach for coupling flow to arbitrary implicitly-defined diffuse domains. It is demonstrated that all boundary conditions of interest can be expressed as suitable fluxes, noting that angular momentum flux is necessary in order to account for cases such as the no-slip condition. Moreover, it is shown that the diffuse boundary formulation converges exactly to the sharp interface solution, resulting in a well-defined error bound. The method is applied in a viscous compressible flow solver with block-structured adaptive mesh refinement, and the convergence properties are shown. Finally, the efficacy of the method is demonstrated by coupling to other classical flow problems (vortex shedding), problems in solidification (coupling to dendritic growth), and flow through eroding media (coupling to the Allen Cahn equation).