Binary fluid flow simulations with free energy lattice Boltzmann methods

Stephan Simonis; Johannes Nguyen; Samuel J. Avis; Willy Dörfler; Mathias J. Krause

doi:10.3934/dcdss.2023069

Article Contents

2024, Volume 17, Issue 11: 3278-3294. Doi: 10.3934/dcdss.2023069

This issue Previous Article A lattice Boltzmann study of the thermodynamics of an interaction between two cavitation bubbles Next Article Preface

Binary fluid flow simulations with free energy lattice Boltzmann methods

1.
Institute for Applied and Numerical Mathematics (IANM), Karlsruhe Institute of Technology (KIT), Germany
2.
Lattice Boltzmann Research Group (LBRG), Karlsruhe Institute of Technology (KIT), Germany
3.
Department of Physics, Durham University, United Kingdom
4.
Institute of Mechanical Process Engineering and Mechanics (MVM), Karlsruhe Institute of Technology (KIT), Germany

^*Corresponding author: Stephan Simonis
^*Corresponding author: Stephan Simonis

Received: December 2022

Revised: February 2023

Early access: March 2023

Published: November 2024

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Abstract

We use free energy lattice Boltzmann methods to simulate shear and extensional flows of a binary fluid in two and three dimensions. To this end, two classical configurations are digitally twinned, namely a parallel-band device for binary shear flow and a four-roller apparatus for binary extensional flow. The free energy lattice Boltzmann method and the test cases are implemented in the open-source parallel C++ framework OpenLB and evaluated for several non-dimensional numbers. Characteristic deformations are captured, where breakup mechanisms occur for critical capillary regimes. Though the known mass leakage for small droplet-domain ratios and large Cahn numbers is observed, suitable mesh sizes show good agreement to analytical predictions and reference results.

Keywords:

Mathematics Subject Classification: Primary 65M22, 35Q30; Secondary 76D05, 76T99.

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Figure 1. Discrete velocity sets

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Figure 2. Geometric setup of numerical test cases for binary flow in two dimensions. Scales differ for the purpose of representation

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Figure 3. Deformation and inclination of a droplet in binary shear flow simulated with FRE LBM for varying capillary numbers

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Figure 4. Droplet breakup in 2D binary shear flow at $ C\!a = 3.5 $. Components $ C_{1}$ (red) and $ C_{2}$ (blue) are plotted at normalized time steps

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Figure 5. Droplet breakup in 3D binary shear flow at normalized time steps for $ R\!e = 0.0625$, $ C\!a = 0.42$, $ C\!h = 0.0379$, $ P\!e = 0.43$

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Figure 6. Deformation of a droplet in binary extensional flow simulated with FRE LBM for varying capillary numbers

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Figure 7. Droplet breakup in 2D binary extensional flow at normalized time steps for $ R\!e = 0.0625$, $ C\!a = 0.42$, $ C\!h = 0.057$, $ P\!e = 0.43$

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Figure 8. Static droplet test case in 2D computed with FRE LBM. The order parameter $ \phi $ is plotted over the cross section $ y = 0$ for the free energy functional $ \Psi^{(m)}$ with $ m = 2 $ and $ m = 6 $ at several time steps

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