New techniques are developed for solving multi-phase flows in unbounded domains using the Diffuse Interface Model in 1-D. They extend two open boundary conditions originally designed for the Navier–Stokes equations. The non-dimensional formulation of the DIM generalizes the approach to any fluid. The equations support a steady state whose analytical approximation close to the critical point depends only on temperature. This feature enables the use of detectors at the boundaries switching between conventional boundary conditions in bulk phases and a multi-phase strategy in interfacial regions. Moreover, the latter takes advantage of the steady state approximation to minimize the interface–boundary interactions. The techniques are applied to fluids experiencing a phase transition and where the interface between the phases travels through one of the boundaries. When the interface crossing the boundary is fully developed, the technique greatly improves results relative to cases where conventional boundary conditions can be used. Limitations appear when the interface crossing the boundary is not a stable equilibrium between the two phases: the terms responsible for creating the true balance between the phases perturb the interior solution. Both boundary conditions present good numerical stability properties: the error remains bounded when the initial conditions or the far field values are perturbed. For the PML, the influence of its main parameters on the global error is investigated to make a compromise between computational costs and maximum error. The approach can be extended to multiple spatial dimensions.
Desmarais, J. L., & Kuerten, J. G. M. (2014). Open boundary conditions for the Diffuse Interface Model in 1-D. Journal of computational physics, 263, 393-418. https://doi.org/10.1016/j.jcp.2014.01.032