A numerical method for two-phase flow with its application to the Kelvin–Helmholtz instability problem

Abdullah Shah*, Muhammad Sohaib, Li Yuan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


This article uses the artificial compressibility-based high-order finite difference method to simulate the two-phase Kelvin–Helmholtz (KH) instability problem. The equations are based on the mass-conserving Allen–Cahn equation coupled with the incompressible Navier–Stokes equations. One of the advantages of the artificial compressibility approach is that many high-order numerical schemes based on hyperbolic conservation law can be applied. A fifth-order weighted essentially non-oscillatory (WENO) scheme is used for discretizing the convective terms while a standard central finite difference scheme is used for the viscous and surface tension terms. The system of equations is then solved using the Beam-Warming approximate factorization technique. For validation, the effects of both single- and double-mode sinusoidal perturbations on the Kelvin–Helmholtz instability dynamics are analyzed. When there is a single-mode sinusoidal perturbation, the interface roll-up at the center of the domain. Additionally, the role of the surface tension parameter in the instability's dynamics is investigated. The development of Kelvin–Helmholtz instability is shown to be sensitive to the surface tension value. The analysis of grid convergence is also performed to capture the interface dynamics at varying resolutions. The comparison of the computed results is in good agreement with those from the literature. It is observed that the proposed technique effectively resolves the dynamics of the chaotically distorted interfaces of the Kelvin–Helmholtz instability in two-phase flow.

Original languageEnglish
Article number107334
JournalCommunications in Nonlinear Science and Numerical Simulation
StatePublished - Oct 2023

Bibliographical note

Publisher Copyright:
© 2023 Elsevier B.V.


  • ADI scheme
  • Allen–Cahn Navier–Stokes model
  • Artificial compressibility
  • Dual time–stepping
  • Kelvin–Helmholtz (KH) instability
  • WENO scheme

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Applied Mathematics


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