Phase-field-based lattice Boltzmann model for axisymmetric multiphase flows

Abstract

In this paper, a phase-field-based lattice Boltzmann (LB) model is proposed for axisymmetric multiphase flows. Modified equilibrium distribution functions and some source terms are properly added into the evolution equations such that multiphase flows in the axisymmetric coordinate system can be described. Different from previous axisymmetric LB multiphase models, the added source terms that arise from the axisymmetric effect contain no additional gradients, and therefore the present model is much simpler. Furthermore, through the Chapmann-Enskog analysis, the axisymmetric Chan-Hilliard equation and Navier-Stokes equations can be exactly derived from the present model. The model is also able to deal with flows with density contrast. A variety of numerical experiments, including planar and curve interfaces, an elongation field, a static droplet, a droplet oscillation, breakup of a liquid thread, and dripping of a liquid droplet under gravity, have been conducted to test the proposed model. It is found that the present model can capture accurate interface and the numerical results of multiphase flows also agree well with the analytical solutions and/or available experimental data.

A lattice Boltzmann method for axisymmetric multicomponent flows with high viscosity ratio

“…During the contraction of the liquid ligament, a pair of droplets can be found at the end of the ligament, and they exhibit an increasing trend to pinch-off from the middle portion as the wave number increases. Therefore, it is expected that a liquid ligament can break up into multiple droplets as long as it is sufficiently long, consistent with the previous findings [85,86,36]. In addition, we also quantify the sizes of the main and satellite droplets at various wave numbers, which are plotted in Fig.12.…”

“…In each of the simulations below, r = 0 represents the axis of symmetry, and the singularity will occur at r = 0 because of the terms containing r −1 [54,36]. To avoid the singularity, we set the first lattice line at r = 0.5δ x and apply the symmetry boundary condition to a ghost lattice line positioned at r = −0.5δ x (see Fig.1): where Q is an arbitrary node at the first fluid line, and P is the symmetric ghost node of Q.…”

“…The densities of red and blue fluids are both fixed at ρ c = 1. Following Liang et al [36], the interface profile is initialized as Next, we examine the influence of the droplet size on the oscillation period. µ R fixed at 0.05.…”

An improved axisymmetric lattice Boltzmann flux solver for axisymmetric isothermal/thermal flows

“…Since the first axisymmetric lattice Boltzmann (LB) model proposed by Halliday et al, successive models were then developed to truly recover the desired macroscopic equations Furthermore, different attempts were devoted to simplifying the treatments of source terms by either reducing the number of the source terms or avoiding the computations of spatial gradients therein . Moreover, the applicability of the axisymmetric LBM has also been well verified by practical axisymmetric flow tests …”

“…5,6,[12][13][14][15][16] Moreover, the applicability of the axisymmetric LBM has also been well verified by practical axisymmetric flow tests. 15,[17][18][19][20][21][22][23] Despite its widespread applications, LBM suffers from some drawbacks, such as its limitation to simple geometry and uniform mesh, constraint to viscous flows, and the intrinsic tie-up between the time interval and the mesh spacing. 24,25 Morover, the implementation of boundary constraints of the second or the third types (ie, the Neumann condition and Robin conditions) for the LB method is still challenging, especially for curved boundaries.…”