[Paper Review] Further investigation of the spurious interface fragmentation in multiphase Smoothed Particle Hydrodynamics

Kamil Szewc, Michał T. Lewandowski|arXiv (Cornell University)|Feb 25, 2016

Fluid Dynamics Simulations and Interactions3 references3 citations

TL;DR

This paper challenges the widespread use of artificial interface correction terms in multiphase Smoothed Particle Hydrodynamics (SPH), arguing that the so-called 'spurious interface fragmentation' is a physically correct manifestation of Kelvin-Helmholtz instability when surface tension is neglected. Using stability analysis and SPH/VOF simulations, it demonstrates that such corrections can introduce non-physical solutions and explains the empirically observed inverse relationship between the correction parameter ε and smoothing length h as a direct consequence of hydrodynamic stability theory.

ABSTRACT

This article presents results of further investigation of the problem of spurious interface fragmentation in the multiphase SPH. In order to remove arising instabilities, many authors introduced the artificial interface correction procedure. In the present paper we show that the interface instabilities are physical and the introduction of the interface correction procedure may leads to non-physical solutions. We also explain the puzzling relation between the parameters $\varepsilon$ and $h$. The analysis is performed on the basis of the stability analysis and numerical calculations using SPH and Volume Of Fluid (VOF) approach.

Motivation & Objective

• To investigate the physical origin of interface instabilities in multiphase SPH when surface tension is neglected.
• To challenge the validity of artificial interface correction terms commonly used to stabilize simulations.
• To clarify the empirical relationship ε_min ∝ 1/h observed in prior studies.
• To demonstrate that surface tension, though small, plays a critical stabilizing role in the Sussman bubble test case.
• To advocate for cautious use of interface correction procedures in SPH simulations.

Proposed method

• Conducts a linearized Kelvin-Helmholtz stability analysis on a two-fluid interface with uniform velocity and density contrast.
• Derives a dispersion relation incorporating an artificial normal stress term to model the interface correction (ε-term).
• Uses non-dimensionalized numerical units (R = 1 cm, g = 1 cm/s², ρ = 1 g/cm³) for consistency with prior benchmarks.
• Performs SPH simulations with and without the interface correction term, comparing results to VOF method outputs.
• Analyzes the dependence of the minimal stable ε (ε_min) on the smoothing length h using numerical experiments.
• Derives the theoretical relation ε_min ∝ 1/h from the stability condition, confirming numerical observations.

Experimental results

Research questions

• RQ1 Is the interface fragmentation observed in SPH simulations with negligible surface tension a numerical artifact or a physical instability?
• RQ2 What is the true physical role of the artificial interface correction term (ε-term) in stabilizing the SPH interface?
• RQ3 Why does the minimal stable ε value scale inversely with the smoothing length h, as empirically observed?
• RQ4 Does surface tension, despite its small magnitude in the Sussman test case, play a significant stabilizing role?
• RQ5 Can the observed ε_min ∝ 1/h relationship be derived analytically from hydrodynamic stability theory?

Key findings

• The interface instabilities observed in SPH simulations with negligible surface tension are physically correct and correspond to the onset of Kelvin-Helmholtz instability.
• The artificial interface correction term (ε-term) does not correct a physical flaw but masks a real instability, potentially leading to non-physical solutions.
• The empirically observed scaling ε_min ∝ 1/h is analytically explained by the stability condition derived from the Kelvin-Helmholtz analysis.
• Surface tension, although small in the Sussman test case, is sufficient to stabilize the interface and prevent fragmentation.
• The interface correction term acts primarily as a numerical 'beautifier' rather than a physical stabilizer, and should be used with caution.