Deterministic and stochastic bifurcations in two-dimensional electroconvective flows

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We study numerically a sequence of eddies in two-dimensional electrohydrodynamics (EHD) flows of a dielectric liquid, driven by an electric potential difference between a hyperbolic blade electrode and a flat plate electrode (or the blade–plate configuration). The electrically driven flow impinges on the plate to generate vortices, which resemble Moffatt eddies (Moffatt, J. Fluid Mech., vol. 18, 1964, pp. 1–18). Such a phenomenon in EHD was first reported in the experimental work of Perri et al. (J. Fluid Mech., vol. 900, 2020, A12). We conduct direct numerical simulations of the EHD flow with three Moffatt-type eddies in a large computational domain at moderate electric Rayleigh numbers ( $T$ , quantifying the strength of the electric field). The ratios of size and intensity of the adjacent eddies are examined, and they can be compared favourably to the theoretical prediction of Moffatt; interestingly, the quantitative comparison is remarkably accurate for the two eddies in the far field. Our investigation also shows that a larger $T$ strengthens the vortex intensity, and a stronger charge diffusion effect enlarges the vortex size. A sufficiently large $T$ can further result in an oscillating flow, consistent with the experimental observation. In addition, a global stability analysis of the steady blade–plate EHD flow is conducted. The global mode is characterised in detail at different values of $T$ . When $T$ is large, the confinement effect of the geometry in the centre region may lead to an increased oscillation frequency. This work contributes to the quantitative characterisation of the Moffatt-type eddies in EHD flows.

“…2015; Zhang 2016; Feng et al. 2021). Since a strong injection regime has been considered in this work, we take .…”

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Moffatt eddies in electrohydrodynamics flows: numerical simulations and analyses

Sun

Zhang

2022

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“…However, in these algorithms, some free parameters must be introduced, such as the embedding dimension and delay time. To characterize the chaotic motions in the EC in a more systematic way and avoid assigning free parameters, the method of Wolf et al (1985) has been widely used recently (Feng et al 2021;Huang et al 2021). Moreover, Li et al (2020) first introduced this method to investigate the transition process from laminar to chaotic flow in ETC.…”

Section: Bifurcations and Chaos In Etcmentioning

confidence: 99%

Stability and bifurcation of annular electro-thermo-convection

Luo

Jiang

et al. 2023

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We numerically investigated the global linear instability and bifurcations in electro-thermo-convection (ETC) of a dielectric liquid confined in a two-dimensional (2-D) concentric annulus subjected to a strong unipolar injection. Seven kinds of solutions exist in this ETC system due to the complex bifurcations, i.e. saddle-node, subcritical and supercritical Hopf bifurcations. These bifurcation routes constitute at most four solution branches. Global linear instability analysis and energy analysis were conducted to explain the instability mechanism and transition of different solutions and to predict the local instability regions. The linearized lattice Boltzmann method (LLBM) for global linear instability analysis, first proposed by Pérez et al. (Theor. Comput. Fluid Dyn., vol. 31, 2017, pp. 643–664) to analyse incompressible flows, was extended here to solve the whole set of coupled linear equations, including the linear Navier–Stokes equations, the linear energy equation, Poisson's equation and the linear charge conservation equation. A multiscale analysis was also performed to recover the macroscopic linearized Navier–Stokes equations from the four different discrete lattice Boltzmann equations (LBEs). The LLBM was validated by calculating the linear critical value of 2-D natural convection; it has an error of 1.39% compared with the spectral method. Instability with global travelling wave behaviour is a unique behaviour in the annulus configuration electrothermohydrodynamic system, which may be caused by the baroclinity. Finally, the chaotic behaviour was quantitatively analysed through calculation of the fractal dimension and Lyapunov exponent.

“…In particular, Wu et al (2013) studied the critical bifurcation of EHD flow in a 2-D finite container without considering the charge diffusion effect. Deterministic and stochastic bifurcations in EHD flow of a dielectric liquid between two parallel plates was investigated by Feng et al (2021). They tried to reduce the discrepancy of the linear instability criteria between experiment and theory by considering stochastic boundary conditions.…”

Section: Stability Analysis Of Ehd Flowmentioning

confidence: 99%

“…Deterministic and stochastic bifurcations in EHD flow of a dielectric liquid between two parallel plates was investigated by Feng et al. (2021). They tried to reduce the discrepancy of the linear instability criteria between experiment and theory by considering stochastic boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Numerical analyses of wire-plate electrohydrodynamic flows

Vázquez

Zhang

2023

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We present numerical analyses of two-dimensional electrohydrodynamic (EHD) flows of a dielectric liquid between a wire electrode and two plate electrodes with a Poiseuille flow, using direct numerical simulation and global stability analysis. Both conduction and injection mechanisms for charge generation are considered. In this work we focused on the intensity of the cross-flow and studied the EHD flows without a cross-flow, with a weak cross-flow and with a strong cross-flow. (1) In the case without a cross-flow, we investigated its nonlinear flow structures and linear dynamics. We found that the flow in the conduction regime is steady, whereas the flow in the injection regime is oscillatory, which can be explained by a global stability analysis. (2) The EHD flow with a weak cross-flow is closely related to the flow phenomena in an electrostatic precipitator (ESP). Our analyses indicate that increasing the cross-flow intensity or the electric Reynolds number leads to a less stable flow. Based on these results, we infer that one should adopt a relatively low voltage and weak cross-flow in the wire-plate EHD flow to avoid flow instability, which may hold practical implications for ESP. (3) The case of strong cross-flow is examined to study the EHD effect on the wake flow. By comparing the conventional cylindrical wake with the EHD wake in linear and nonlinear regimes, we found that the EHD effect brings forward the vortex shedding in wake flows. Besides, the EHD effect reduces the drag coefficient when the cross-flow is weak, but increases it when it is strong.