Abstract
The modelling of liquid--vapour flow with phase transition
poses many challenges, both on the theoretical level, as well as on the
level of discretisation methods. Therefore, accurate mathematical models
and efficient numerical methods are required. In that, we focus on two
modelling approaches: the sharp-interface (SI) approach and the
diffuse-interface (DI) approach. For the SI-approach, representing the
phase boundary as a co-dimension-1 manifold, we develop and validate
analytical Riemann solvers for basic isothermal two-phase flow
scenarios. This ansatz becomes cumbersome for increasingly complex
thermodynamical settings. A more versatile multiscale interface solver,
that is based on molecular dynamics simulations, is able to accurately
describe the evolution of phase boundaries in the temperature-dependent
case. It is shown to be even applicable to two-phase flow of multiple
components. Despite the successful developments for the SI approach,
these models fail if the interface undergoes topological changes. To
understand merging and splitting phenomena for droplet ensembles, we
consider DI models of second gradient type. For these
Navier--Stokes--Korteweg systems, that can be seen as a third order
extension of the Navier--Stokes equations, we propose variants that are
more accessible to standard numerical schemes. More precisely, we
reformulate the capillarity operator to restore the hyperbolicity of the
Euler operator in the full system.
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