Competition between drag and Coulomb interactions in turbulent particle-laden flows using a coupled-fluid–Ewald-summation based approach

We present a numerical study on inertial electrically charged particles suspended in a turbulent carrier phase. Fluid-particle interactions are accounted for in an Eulerian-Lagrangian (EL) framework and coupled to a Fourier-based Ewald summation method, referred to as the particle-particle-particle–mesh $\left({\mathrm{P}}^3\mathrm{M}\right)$ method, to accurately capture short- and long-range electrostatic forces in a tractable manner. The EL ${\mathrm{P}}^3\mathrm{M}$ method is used to assess the competition between drag and Coulomb forces for a range of Stokes numbers and charge densities. Simulations of like- and oppositely charged particles suspended in a two-dimensional Taylor-Green vortex and three-dimensional homogeneous isotropic turbulence are reported. It is found that even in dilute suspensions, the short-range electric potential plays an important role in flows that admit preferential concentration. Suspensions of oppositely charged particles are observed to agglomerate in the form of chains and rings. Comparisons between the particle-mesh method typically employed in fluid-particle calculations and ${\mathrm{P}}^3\mathrm{M}$ are reported, in addition to one-point and two-point statistics to quantify the level of clustering as a function of Reynolds number, Stokes number, and nondimensional electric settling velocity.

Figure 1

Plot of the $L_2$-norm of the electrostatic force with the electric field computed as a function of the ${\mathrm{P}}^3\mathrm{M}$ constant $\beta$ that appears in Eq. (22) solved using (a) spectral difference and (b) finite difference for the PM method $\left(--\right)$, the hybrid method $(\cdots )$, and the ${\mathrm{P}}^3\mathrm{M}$ method (symbols) with interpolation order 0 to 4 from top to bottom, respectively.

Figure 2

Instantaneous particle position in the Taylor-Green vortex at $t/{\tau }_f=1000$ using the ${\mathrm{P}}^3\mathrm{M}$ method with increasing electric settling velocity $v_c^{*}=0$, 0.1, and 1 (increasing from left to right) and $\text{St}/{\mathrm{St}}_{\text{cr}}=0.1$, 1, and 100 (increasing from top to bottom). Grayscale represents the electric-field magnitude [ranging from 0 (white) to 10 $000\mathrm{V}/\mathrm{m}$ (black)]. Particles are colored by their charge: neutral (black), positive (red), and negative (blue).

Figure 3

Zoomed-in view of the Taylor-Green vortex computed using the ${\mathrm{P}}^3\mathrm{M}$ method with $v_c^{*}=0.1,\mathrm{St}={\mathrm{St}}_{\text{cr}}$, and (a) $t/{\tau }_f=1$ and (b) $t/{\tau }_f=10$. The color scheme is the same as in Fig. 2.

Figure 4

Instantaneous particle position in the Taylor-Green vortex at $t/{\tau }_f=1000$ using the PM method with increasing electric settling velocity $v_c^{*}=0$, 0.1, and 1 (increasing from left to right) and $\text{St}/{\mathrm{St}}_{\text{cr}}=0.1$, 1, and 100 (increasing from top to bottom). The color scheme is the same as in Fig. 2.

Figure 5

Variation of $D$ with nondimensional Coulomb velocity for ${\mathrm{St}}_{\eta }=0.5\left(◊\right)$, 0.$9(+)$, 1.$0(+)$, 6.$2(\square )$, 7.$0(\square )$, 25.$0(◯)$, and 28.$0(◯)$ with like-charged particles for (a) ${\text{Re}}_{\lambda }=25.8({\mathrm{P}}^3\mathrm{M}$ method), (b) ${\text{Re}}_{\lambda }=25.8$ (PM method), (c) ${\text{Re}}_{\lambda }=43.5({\mathrm{P}}^3\mathrm{M}$ method), and (d) ${\text{Re}}_{\lambda }=43.5$ (PM method).

Figure 6

Variation of $D$ with nondimensional Coulomb velocity for ${\mathrm{St}}_{\eta }=0.5\left(◊\right)$, 0.$9(+)$, 1.$0(+)$, 6.$2(\square )$, 7.$0(\square )$, 25.$0(◯)$, and 28.$0(◯)$ with oppositely charged particles for (a) ${\text{Re}}_{\lambda }=25.8({\mathrm{P}}^3\mathrm{M}$ method), (b) ${\text{Re}}_{\lambda }=25.8$ (PM method), (c) ${\text{Re}}_{\lambda }=43.5({\mathrm{P}}^3\mathrm{M}$ method), and (d) ${\text{Re}}_{\lambda }=43.5$ (PM method).

Figure 7

Radial distribution functions for ${\mathrm{Re}}_{\lambda }=43.5$ and (a) ${\text{St}}_{\eta }=0.5$, (b) ${\text{St}}_{\eta }=0.5$, (c) ${\text{St}}_{\eta }=1.0$, (d) ${\text{St}}_{\eta }=1.0$, (e) ${\mathrm{St}}_{\eta }=7.0,\mathrm{and}\left(\mathrm{f}\right){\mathrm{St}}_{\eta }=7.0$ with $v_c^{*}=0(-)$, 0.$1\left(—–\right)$, 0.$25\left(-·\right)$, 0.$5\left(--\right)$, and 1.$0(\cdots )$ using the ${\mathrm{P}}^3\mathrm{M}$ method, with (a), (c), and (e) oppositely charged particles and (b), (d), and (f) like-charged particles.

Figure 8

The RDF near contact with $v_c^{*}=0\left(◊\right)$, 0.$1(+)$, 0.$25\left(\mathrm{\Delta }\right)$, 0.$5(\square )$, and 1.$0(◯)$ using the ${\mathrm{P}}^3\mathrm{M}$ method, with (a) oppositely charged particles and (b) like-charged particles.