Three-dimensional finite amplitude electroconvection in dielectric liquids

Abstract

Charge injection induced electroconvection in a dielectric liquid lying between two parallel plates is numerically simulated in three dimensions (3D) using a unified lattice Boltzmann method (LBM). Cellular flow patterns and their subcritical bifurcation phenomena of 3D electroconvection are numerically investigated for the first time. A unit conversion is also derived to connect the LBM system to the real physical system. The 3D LBM codes are validated by three carefully chosen cases and all results are found to be highly consistent with the analytical solutions or other numerical studies. For strong injection, the steady state roll, polygon, and square flow patterns are observed under different initial disturbances. Numerical results show that the hexagonal cell with the central region being empty of charge and centrally downward flow is preferred in symmetric systems under random initial disturbance. For weak injection, the numerical results show that the flow directly passes from the motionless state to turbulence once the system loses its linear stability. In addition, the numerically predicted linear and finite amplitude stability criteria of different flow patterns are discussed.

Three-dimensional electroconvective vortices in cross flow

“…Both the rolling pattern and square pattern have the same non-dimensional linear growth rate (~0.896 as  in Eq. (18)), which agrees with the previously reported linear stability analysis[42] and the unified SRT LBM numerical model[49] (supplementary materials for validation). After about * 5 t = from the initial perturbation, the growth rate curves for square and rolling vortices patterns diverge.…”

“…6(d) verifies that the dynamic mode 0.908 r  =dominates the flow system. The growth rates of rolling and square patterns are similar to each other, which was also observed in the previous report[49]. Unstable dynamic modes visualized by z perturbation has the greatest of projection (b=1.826) on this eigenmode; therefore, it contains most of the energy of the system.…”

“…2(f-j)). To study the mechanism for the transition of 3D vortices to streamwise vortices, we consider the simplest scenario, i.e., the case where the equilibrium state is a single period square pattern, see I is the cathode current for the base state solution without EC vortices [38,49]; thus, 1 Ne  when EC vortices exist. Note that the use of current as the metric for EC convection has been used in the studies of related overlimiting current in electro-kinetic systems [48].…”

Two relaxation time lattice Boltzmann method coupled to fast Fourier transform Poisson solver: Application to electroconvective flow

“…Governing equations yield four non-dimensional parameters that describe the system's state [12,[17][18][19][20][21].…”

“…Other numerical models used to study EC phenomena include the particle-in-cell method [30], finite volume method with flux-corrected transport [31], total variation diminishing scheme [5,8,[14][15][16], and the method of characteristic [4]. Recently, Luo et al showed that a unified Lattice Boltzmann model (LBM) matches the linear and finite amplitude stability criteria of the subcritical bifurcation in EC flow [18][19][20][21] for both 2D and 3D flow scenarios. This unified LBM transforms the elliptic Poisson equation to a parabolic reaction-diffusion equation and introduces artificial coefficients to control the evolution of the electric potential, which may result in ambiguity in the numerical results.…”