Three-dimensional electroconvective vortices in cross flow

This study focuses on the three-dimensional (3D) electrohydrodynamic flow instability between two parallel electrodes driven by unipolar charge injection with and without cross flow. Lattice Boltzmann method with a two-relaxation time model is used to compute flow patterns. In the absence of cross flow, the base-state solution is hydrostatic, and the electric field is one-dimensional. With strong charge injection and high electrical Rayleigh number, the system exhibits electroconvective vortices. Disturbed by perturbation patterns, such as rolling pattern, square pattern, and hexagon pattern, the flow develops corresponding to the most unstable mode. The growth rate and pattern transitions are studied using dynamic mode decomposition of the transient numerical solutions. The interactions between cross flow and electroconvective vortices lead to suppression and disappearance of structures with velocity components in the direction of cross flow, while the other components are not affected. Surprisingly, the transition from a 3D to a 2D flow pattern enhances the convective charge transport, marked by an increase in the electric Nusselt number. Hysteresis in the 3D to 2D transition is characterized by the nondimensional parameter Y, a ratio of the electrical force term to the viscous term in the momentum equation.

Figure 1

Hydrostatic solution comparison of the TRT LBM and fast Poisson solver [87, 97], unified SRT LBM [52], and the analytical solution [92, 107] for $C=0.1$ and $C=10$, $F_e=4000$. (a) Charge density and (b) electric field.

Figure 2

Contours of $u_z$ at $z=H/2$ for the equilibrium states (a–e) without cross flow, and (f–j) with cross flow sufficient for pattern transition. For different electrical Rayleigh numbers, domain sizes, and initial perturbations (initial conditions), square patterns, oval patterns, hexagon patterns, and mixed patterns are established. Strong cross flow in the $x$ direction is applied to the equilibrium states (a–e), resulting in the 3D transition to 2D streamwise vortices.

Figure 3

Contours of $u_z^{*}$ at $z=H/2$ for initial perturbation of (a) a square pattern and (b) a rolling pattern with $T=170$, $C=10$, $M=10$, and $F_e=3500$.

Figure 4

Time evolution of maximum $u_z^{*}$ for the rolling pattern and square pattern in (a) a linear scale and (b) a logarithmic scale. Both patterns have similar growth rates (∼0.896) in the linear growth region $\left(t^{*}=0\sim 5\right)$. The DMD algorithm-based solutions in the interval $t^{*}=0–2.5$ project the state as shown at $t^{*}=$ 3.75 using Eq. (35).

Figure 5

(a) Eigenvalues of the discrete-time mapping matrix $\mathbf{A}$ and (b) logarithmic mapping of eigenvalues of $\mathbf{L}$. The eigenvalues outside the unit circle, whose logarithmic value has a real component ${\omega }_r$ greater than 0, represent the unstable dynamic modes. The logarithmic mapping of the eigenvalue $\mathbf{L}$ indicates the growth rate of each dynamic mode.

Figure 6

Unstable dynamic modes visualized by $u_z$ at $z=H/2$. (a–c) Three unstable DMD modes; (c) perturbation has the greatest of projection ($b=1.826$) on this eigenmode; therefore, it contains most of the energy of the system; (d) the simulated $u_z$ at $z=H/2$. The growth rate of mode (c) ${\omega }_r=0.908$ is close to the general growth rate of the entire system (∼0.896) observed in Fig. 4.

Figure 7

Time evolution of maximum $u_z^{*}$ and Ne for (a, c) Couette cross flow and (b, d) Poiseuille cross flow. Maximum $u_z^{*}$ have similar growth rates (∼0.896) in the linear growth region $\left(t^{*}=0\sim 5\right)$. The square pattern initial perturbation scheme [Eqs. (24, 25, 26)] is used. For strong cross flow, the systems develop into longitudinal rolling patterns. For the weak cross flow, the systems develop into oblique 3D structures with both transverse and longitudinal structures.

Figure 8

Contours of $u_z^{*}$ for $z=H/2$, $t=7.5$: (a) Couette flow $u_{\mathrm{wall}}^{*}=2.24$; (b) Poiseuille flow $u_{\mathrm{center}}^{*}=2.84$. The color map corresponds to the values of $u_z$, which is also given by the vertical axis.

Figure 9

Eigenvalues ${\lambda }_i$ for $u_z^{*}$ in linear growth region $\left(t^{*}=0–7\right)$ for Couette cross flow [(a) ${u^{*}}_{\mathrm{wall}}=2.20$ and (c) ${u^{*}}_{\mathrm{wall}}=2.24$]. Three additional unstable dynamic modes in ${u^{*}}_{\mathrm{wall}}=2.20$ case change the equilibrium solution from a rolling pattern to oblique 3D structures. The corresponding eigenvectors sliced at $z=H/2$ are shown in (b) mode $m_1$ and (d) mode $m_2$ and ${\overline{m}}_2$.

Figure 10

Eigenvalues ${\lambda }_i$ for $u_z^{*}$ in linear growth region $\left(t^{*}=0–6.25s\right)$ for Poiseuille cross flow [(a) ${u^{*}}_{\mathrm{center}}=2.80$ and (c) ${u^{*}}_{\mathrm{center}}=2.84$]. An additional pair of conjugate unstable dynamic modes in the ${u^{*}}_{\mathrm{center}}=2.80$ case change the equilibrium solution from a rolling pattern to an oblique 3D structure pattern. The corresponding eigenvectors sliced at $z=H/2$ are shown in (b) mode $m_3$ and (d) mode ${\overline{m}}_3$.

Figure 11

Time evolution of maximum $u_z^{*}$ after a finite velocity is applied to the upper wall. For small ${u^{*}}_{\mathrm{wall}}$, the maximum $u_z^{*}$ decreases and reaches a new equilibrium state where oblique 3D structures are observed. For large ${u^{*}}_{\mathrm{wall}}$, the maximum $u_z^{*}$ decreases down to the rolling pattern where longitudinal rolls dominate, after a nonlinear transition. Bifurcation occurs at ${u^{*}}_{\mathrm{wall}}=3.88$.

Figure 12

Time evolution of maximum $u_z^{*}$ after a uniform body force is applied to the flow field. For small ${u^{*}}_{\mathrm{center}}$, the maximum $u_z^{*}$ decreases and reaches a new equilibrium state where oblique 3D structures are observed. For large ${u^{*}}_{\mathrm{center}}$, the maximum $u_z^{*}$ decreases to the rolling pattern values where longitudinal rolls dominate, after a nonlinear transition. The bifurcation occurs at ${u^{*}}_{\mathrm{center}}=3.96$.

Figure 13

Eigenvalues ${\lambda }_i$ for $u_z^{*}$ in the transition region for (a) Couette-type cross flow, ${u^{*}}_{\mathrm{wall}}=3.84$, and (c) Poiseuille-type cross flow, ${u^{*}}_{\mathrm{center}}=3.88$. Unstable dynamic modes change the equilibrium solution from a square pattern to oblique 3D structures. The corresponding eigenvectors sliced at $z=H/2$ are shown in (b), mode $m_4-{\overline{m}}_9$, and in (d), mode $m_{10}-{\overline{m}}_{12}$.

Figure 14

E igenvalues ${\lambda }_i$ for $u_z^{*}$ in the transition region for (a) Couette-type cross flow with ${u^{*}}_{\mathrm{wall}}=3.88$, and (c) Poiseuille type cross flow with ${u^{*}}_{\mathrm{center}}=4.16$. The unstable dynamic modes yield the rolling structures.

Figure 15

Contours of valley and peak velocity $u_z^{*}$ for (a, b) Couette cross flow (${u^{*}}_{\mathrm{wall}}=3.88$) and (c, d) Poiseuille cross flow (${u^{*}}_{\mathrm{center}}=4.16$).

Figure 16

Isosurfaces of charge density for (a) square pattern without cross flow, (b) ${u^{*}}_{\mathrm{wall}}=3.84$, (c) ${u^{*}}_{\mathrm{center}}=3.92$, and (d) strong cross flow or rolling pattern. Streamlines (e) without cross flow (circulation streamline) and (f) with strong cross flow (helical streamline). Cross flow is in the $x$ direction.

Figure 17

Hysteresis loop of $\mathrm{Ne}$ vs. $Y$ for Couette cross flow. The bifurcation thresholds are $Y_c=772.73$, $Y_f=438.14$.

Figure 18

Hysteresis loop of $\mathrm{Ne}$ vs. $Y$ for Poiseuille-type cross flow. The bifurcation thresholds are $Y_c=300.75$, $Y_f=213.90$.