Efficient lattice Boltzmann method for electrohydrodynamic solid-liquid phase change

Melting in the presence of electrohydrodynamic (EHD) flow driven by the Coulomb force in dielectric phase change material is numerically studied. A model is developed for the EHD flow in the solid-liquid phase change process. The fully coupled equations including mechanical equations, electrical equations, energy equations, and the continuity equations in the solid-liquid interface are solved using a unified lattice Boltzmann model (LBM). Firstly, the numerical model is validated by several cases in the hydrostatic state, and all LBM results are found to be highly consistent with analytical solutions. Besides, our LBM code is able to reproduce the step changes in the distribution of charge density and electric field due to the discontinuous distribution of physical properties at the interface. Then, a systematical investigation is conducted on various nondimensional parameters, including electric Rayleigh number $T$, Prandtl number Pr, and Stefan number St. Results are presented for the transient evolutions of temperature, fluid flow, charge density fields, and liquid fraction. Four flow stages in the melting process together with three kinds of flow instabilities are observed. It is found that the electric field has significant influence on the melting, especially at high $T$ and Pr and low St. Over the tested cases, a maximum melting time saving of around 50% is found.

Figure 1

Schematic diagram of EHD melting problem and related boundary condition; sketch map in right side illustrates the conditions in the solid-liquid interface.

Figure 2

Comparison between LBM results and analytical solutions for melting by heat conduction problem: (a) temperature distributions at four transient moments; (b) the evolution of liquid fraction at different Stefan numbers St.

Figure 3

Validation of LBM model for Poisson equation across the solid-liquid interface under permittivity ratio ${\varepsilon }_r=2$: (a) electric potential $\varphi$ distributions, and (b) electric field at $y$-direction $E_y$ for different interface position $f_l$.

Figure 4

Validation of LBM model for electric field–space charge coupled problem across the solid-liquid interface under permittivity ratio ${\varepsilon }_r=2$ and mobility ratio $K_r=1$: (a) charge density distributions, and (b) electric field at $y$-direction $E_y$ for different interface positions $f_l$.

Figure 5

Validation of LBM model at hydrostatic state across the solid-liquid interface under permittivity ratio ${\varepsilon }_r=2$ and mobility ratio $K_r=10$: (a) charge density distributions, and (b) electric field at $y$-direction $E_y$ for different interface position $f_l$.

Figure 6

Melting in the presence of electroconvective flow at $T=1000$, $\mathrm{Pr}=1$, $\mathrm{St}=0.1$. The time history of (a) liquid fraction $f_l$ with four contours of phase field placed from left to right, (b) normalized peak velocity $V_{\max }$ with streamlines, (c) mean Nusselt number Nu at the hot wall with four contours of temperature fields, and (d) mean Coulomb force F with four contours of charge density fields.

Figure 7

The liquid fraction $f_l$ vs the nondimensional time $t$ for different electric Rayleigh number $T$ at $\mathrm{Pr}=1$, $\mathrm{St}=0.1$.

Figure 8

Time evolutions of the peak velocity $V_{\max }$ in the melting process for the cases $T=800$ and $T=1400$ at $\mathrm{Pr}=1$, $\mathrm{St}=0.1$. A spectral analysis is added at the total melting state for $T=1400$ to prove the quasiperiodic type flow.

Figure 9

Temperature field, charge density distribution, and streamlines (from left to right) of EHD melting at five representative moments for the case $T=800$: (a) $t=2.6$; (c) $t=3.5$; (e) $t=4.0$; (g) $t=4.9$; (i) $t=5.5$. For the case $T=1400$: (b) $t=2.0$; (d) $t=2.8$; (f) $t=3.3$; (h) $t=4.9$; (j) $t=5.0$. Other parameters are set as $\mathrm{Pr}=1$, $\mathrm{St}=0.1$.

Figure 10

Time evolutions of liquid fraction $f_l$ with different Prandtl numbers Pr at $T=1000$ and $\mathrm{St}=0.1$: (a) with the effect of EHD forces, and (b) without EHD forces.

Figure 11

Temperature field, charge density distribution, and phase field with streamlines of EHD melting at three transient moments for the case $\mathrm{Pr}=0.5$ (a) $t=1.2$; (c) $t=2.0$; (e) $t=2.6$. For the case $\mathrm{Pr}=5$: (b) $t=4.0$; (d) $t=7.0$; (f) $t=13.5$. Other parameters are set as $T=1000$, $\mathrm{St}=0.1$.

Figure 12

Time evolutions of total liquid fraction $f_l$ with different Stefan numbers St at $T=1000$ and $\mathrm{Pr}=1$: (a) with the effect of EHD forces, and (b) without EHD forces.

Figure 13

Temperature field, charge density distribution, and phase field with streamlines of EHD melting at three transient moments for the case $\mathrm{St}=0.01$: (a) $t=20$; (c) $t=22$; (e) $t=42$. For the case $\mathrm{St}=1$: (b) $t=0.50$; (d) $t=0.62$; (f) $t=0.71$. Other parameters are set as $T=1000$, $\mathrm{Pr}=1$.