A weakly Compressible, Diffuse-Interface Model for Two-Phase Flows

Adam Kajzer,Jacek Pozorski

doi:10.1007/s10494-020-00164-8

Adam Kajzer, Jacek Pozorski

Open Access

https://doi.org/10.1007/s10494-020-00164-8

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Abstract

We present a novel mathematical model of two-phase interfacial flows. It is based on the Entropically Damped Artificial Compressibility (EDAC) model, coupled with a diffuse-interface (DI) variant of the so-called one-fluid formulation for interface capturing. The proposed EDAC-DI model conserves mass and momentum. We find appropriate values of the model parameters, in particular the numerical interface width, the interface mobility and the speed of sound. The EDAC-DI governing equations are of the mixed parabolic–hyperbolic type. For such models, the local spatial schemes along with an explicit time integration provide a convenient numerical handling together with straightforward and efficient parallelisation of the solution algorithm. The weakly-compressible approach to flow modelling, although computationally advantageous, introduces some difficulties that are not present in the truly incompressible approaches to interfacial flows. These issues are covered in detail. We propose a robust numerical solution methodology which significantly limits spurious deformations of the interface and provides oscillation-free behaviour of the flow fields. The EDAC-DI solver is verified quantitatively in the case of a single, steady water droplet immersed in gas. The pressure jump across the interface is in good agreement with the theoretical prediction. Then, a study of binary droplets coalescence and break-up in two chosen collision regimes is performed. The topological changes are solved correctly without numerical side effects. The computational cost incurred by the stiffness of the governing equations (due to the finite speed of sound and the interface diffusion term) can be overcome by a massively parallel execution of the solver. We achieved an attractively short computation time when our EDAC-DI code is executed on a single, desktop-type Graphics Processing Unit.

Highlights

The usual way of modelling the low speed and incompressible flows, called later on as the truly incompressible approach, is based on the law of momentum conservation and the assumption that the speed of sound cs, in comparison to the convective velocity scale, is high enough to be considered as infinite
The choice of the Mach number is mainly dictated by the computation of the surface tension: to avoid significant pressure oscillations in the vicinity of the interface one has to take into account the locations where |∇φ| > 10−3∕Δx
Our choice is dictated by the rich variety of physical phenomena governed by the Entropically Damped Artificial Compressibility (EDAC)-DI equations and we look for overall robustness of the numerical methodology

Summary

Introduction

The usual way of modelling the low speed and incompressible flows, called later on as the truly incompressible approach, is based on the law of momentum conservation and the assumption that the speed of sound cs , in comparison to the convective velocity scale, is high enough to be considered as infinite. The advantages of this traditional approach are: (1) only one velocity scale is present and (2) the density is constant. The second feature is especially advantageous in the case of two-phase flows where the densities of particular fluids differ—the density varies only at the fluidfluid interface, while it remains constant in the bulk of each phase. Fewfold speed-ups (in comparison to a single desktop CPU) are reported in the literature and the main reason is the elliptic type of the equations

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