Abstract
The Navier–Stokes–Korteweg (NSK) equations are a classical diffuse-interface model for compressible two-phase flows. As direct numerical simulations based on the NSK system are quite expensive and in some cases even impossible, we consider a relaxation of the NSK system, for which robust numerical methods can be designed. However, time steps for explicit numerical schemes depend on the relaxation parameter and therefore numerical simulations in the relaxation limit are very inefficient. To overcome this restriction, we propose an implicit–explicit asymptotic-preserving finite volume method. We prove that the new scheme provides a consistent discretization of the NSK system in the relaxation limit and demonstrate that it is capable of accurately and efficiently computing numerical solutions of problems with realistic density ratios and small interfacial widths.
Highlights
There are in general two approaches to describe the behavior of multi-phase fluids, the sharp interface (SI) and the diffuse interface (DI) approach
One idea to loosen the strict coupling between interfacial width and capillary forces is to introduce an additional Cahn–Hilliard or Allen–Cahn type equation for a new phase field variable. This was done for example in [1,5,42]. Another ansatz to avoid some of the difficulties for the NSK systems suggests to introduce a relaxation of the NSK system, in which the third-order term is replaced by a first-order term and a Poisson equation, that defines a new phase field parameter, see, e.g., [37]
This restriction is a huge drawback for numerical simulations and we are interested in finding a way to circumvent this restriction
Summary
There are in general two approaches to describe the behavior of multi-phase fluids, the sharp interface (SI) and the diffuse interface (DI) approach. This was done for example in [1,5,42] Another ansatz to avoid some of the difficulties for the NSK systems suggests to introduce a relaxation of the NSK system, in which the third-order term is replaced by a first-order term and a Poisson equation, that defines a new phase field parameter, see, e.g., [37]. This model is parametrized by a so-called Korteweg parameter α.
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