A lattice Boltzmann method for axisymmetric thermocapillary flows

Abstract

In this work, we develop a two-phase lattice Boltzmann method (LBM) to simulate axisymmetric thermocapillary flows. This method simulates the immiscible axisymmetric two-phase flow by an improved color-gradient model, in which the single-phase collision, perturbation and recoloring operators are all presented with the axisymmetric effect taken into account in a simple and computational consistent manner. An additional lattice Boltzmann equation is introduced to describe the evolution of the axisymmetric temperature field, which is coupled to the hydrodynamic equations through an equation of state. This method is first validated by simulations of Rayleigh-Bénard convection in a vertical cylinder and thermocapillary migration of a deformable droplet at various Marangoni numbers. It is then used to simulate the thermocapillary migration of two spherical droplets in a constant applied temperature gradient along their line of centers, and the influence of the Marangoni number (Ca), initial distance between droplets (S 0), and the radius ratio of the leading to trailing droplets (Λ) on the migration process is systematically studied. As M a increases, the thermal wake behind the leading droplet strengthens, resulting in the transition of the droplet migration from coalescence to non-coalescence; and also, the final distance between droplets increases with M a for the non-coalescence cases. The variation of S 0 does not change the final state of the droplets although it has a direct impact on the migration process. In contrast, Λ can significantly influence the migration process of both droplets and their final state: at low M a, decreasing Λ favors the coalescence of both droplets; at high M a, the two droplets do not coalesce eventually but migrate with the same velocity for the small values of Λ, and decreasing Λ leads to a shorter equilibrium time and a faster migration velocity.

Color-gradient lattice Boltzmann model with nonorthogonal central moments: Hydrodynamic melt-jet breakup simulations

“…Here, we set the parameter β to 0.7 to reproduce the correct interfacial behavior with as narrow an interface as possible [49,82,83]. For the current model, we can derive the following continuity and Navier-Stokes equations via Chapman-Enskog analysis [49,72,84] ∂ρ ∂t + ∇ · (ρu) = 0,…”

A mesh‐free radial basis function–based semi‐Lagrangian lattice Boltzmann method for incompressible flows

“…The lattice Boltzmann method (LBM) has been successfully applied in various flow problems, 1,2 including turbulent flow, 3 multiphase flow, [4][5][6] and fluid particle suspensions. 7 As compared with conventional numerical methods that directly solve the Navier-Stokes equations, the LBM has a simple formulation and a high paralleled ability.…”

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.…”