Chromo-dynamic multi-component lattice Boltzmann equation scheme for axial symmetry

J Spendlove,M Seaton,X Xu,T Schenkel,I Halliday

doi:10.1088/1751-8121/ab777f

Abstract

We validate the chromo-dynamic multi-component lattice Boltzmann equation (MCLBE) simulation for immiscible fluids with a density contrast against analytical results for complex flow geometries, with particular emphasis on the fundamentals of the method, i.e. compliance with inter-facial boundary conditions of continuum hydrodynamics. To achieve the necessary regimes for the chosen validations, we develop, from a three-dimensional, axially-symmetric flow formulation, a novel, two-dimensional, pseudo Cartesian, MCLBE scheme. This requires the inclusion in lattice Boltzmann methodology of a continuously distributed source and a velocity-dependent force density (here, the metric force terms of the cylindrical Navier–Stokes equations). Specifically, we apply our model to the problem of flow past a spherical liquid drop in Re = 0, Ca regime and, also, flow past a lightly deformed drop. The resulting simulation data, once corrected for the simulation’s inter-facial micro-current (using a method we also advance herein, based on freezing the phase field) show good agreement with theory over a small range of density contrasts. In particular, our data extend verified compliance with the kinematic condition from flat (Burgin et al 2019 Phys. Rev. E 100 043310) to the case of curved fluid–fluid interfaces. More generally, our results indicate a route to eliminate the influence of the inter-facial micro-current.

Highlights

Since 1991, when Gunstensen and Rothman [1, 2] invented the technique, a range of increasingly sophisticated multi-component lattice Boltzmann equation variants have evolved
All data presented and discussed correspond to maximum density contrasts Λ < 10
All the data we present with our novel scheme correspond to maximum density contrasts of Λ ≈ 10

Summary

Introduction

Since 1991, when Gunstensen and Rothman [1, 2] invented the technique, a range of increasingly sophisticated multi-component lattice Boltzmann (lB) equation variants have evolved. Current multi-component lattice Boltzmann equation (MCLBE) variants depart substantially from Gunstensen’s. They may be classified by the physical content of their fluid–fluid interface algorithm. We express the chromo-dynamic variant in colour-blind form, designating immiscible fluid components red and blue, to be described by distribution functions Ri(r, t) and Bi(r, t) where: fi(r, t) = Ri(r, t) + Bi(r, t). Here, with velocity-dependent forces which we expose by writing: Dα = Gα(r) + Fα(r, v), in which G might represent e.g. gravity, or the Lishchuk force (see equation (8) below) This separation is motivated by the definition of force-adjusted macroscopic observables: ρR(r, t) = Ri(r, t), ρB(r, t) = Bi(r, t)

Objectives

Results

Conclusion