An Exactly Curl-Free Staggered Semi-Implicit Finite Volume Scheme for a First Order Hyperbolic Model of Viscous Two-Phase Flows with Surface Tension

In this paper, we present a pressure-based semi-implicit numerical scheme for a first order hyperbolic formulation of compressible two-phase flow with surface tension and viscosity. The numerical method addresses several complexities presented by the PDE system in consideration: (i) The presence of involution constraints of curl type in the governing equations requires explicit enforcement of the zero-curl property of certain vector fields (an interface field and a distortion field); the problem is solved by adopting a set of compatible discrete curl and gradient operators on a staggered grid, allowing to preserve the Schwarz identity of cross-derivatives exactly at the discrete level. (ii) Since the complexity of the studied PDE system does not allow the explicit computation of its exact eigenvalues, reliable and precise analytical estimates are provided. (iii) The evolution equations feature highly nonlinear stiff algebraic source terms which are used for the description of viscous interactions as emergent behaviour of an elasto-plastic solid in the stiff strain relaxation limit; such source terms are reliably integrated with a novel and very efficient semi-analytical technique proposed for the first time in this paper. (iv) In the low-Mach number regime, standard explicit density-based Godunov-type schemes lose efficiency and accuracy; the issue is addressed by means of a simple semi-implicit, pressure-based, split treatment of acoustic and non-acoustic waves, again using staggered grids that recover the implicit solution for a single scalar field (the pressure) through a sequence of symmetric-positive definite linear systems that can be efficiently solved via the conjugate gradient method.