Trapping of inertial particles in a two-dimensional unequal-strength counterrotating vortex pair flow

The preferential accumulation of particles in turbulent flows occurs in many engineering and environmental applications. Recent research reveals that particle preferential accumulation is attributed to multiscale vortex structures in turbulent flows, which affect particle motion and cause particles to remain trapped in particular regions. In this study, the primary goal is to further investigate the mechanisms of particle trapping with emphasis on the effect of the vortex on particle motion. We specifically investigate particle trapping in a 2D unequal-strength counterrotating vortex pair (CVP) using a one-way coupled Euler–Lagrangian method. The small, rigid, spherical, dilute, and heavy particles are considered with the assumption that the particle Reynolds number is low. Using a geometric singular perturbation method to solve the particle motion, we first identify a particle-attracting ring in the potential CVP flow with a circulation ratio $\gamma \in (-0.65,0)$. The particle-attracting ring provides a simple mechanism to explain the occurrence of particle trapping, which is represented by a particle-clustering ring (PCR) in the CVP. We then conduct numerical simulations of particle motion for viscous CVP flows. In the simulations, the CVP is created through vortex–wall interaction, where a primary vortex induces the separation of the new discrete counterrotating vortex from the wall boundary, ultimately leading to the formation of the CVP. Particle trapping in the CVP flow is shown to be robust in the presence of viscosity. The trapping of particles in these viscous simulations has the same dynamical origin as the trapping phenomenon studied for potential CVP flows. The results of this research may help to further comprehend the mechanisms driving the preferential accumulation of particles in turbulent flows.

Figure 1

The potential CVP flow pattern consisting of a counterrotating vortex pair. (a) Streamline patterns. Vortex V1 has a circulation of ${\mathrm{\Gamma }}_1(>0)$, while vortex V2 has a circulation of ${\mathrm{\Gamma }}_2(<0)$. The dash-dotted axes remain fixed within the inertial reference frame, while the solid axes corotate with the CVP. The black lines with arrows denote the streamlines. In the CVP flow, distinct simply connected regions can be identified, enclosed by separatrix streamlines (indicated by black thick lines). The dashed black line represents $S_{max}$, corresponding to the separatrix streamline of vortex V2. (b)–(d) The velocity distribution with $\gamma$= $-0.2, -0.4$, and $-0.6$ in the reference rotating frame.

Figure 2

Stream function values of $S_{max}$ and $S_0$ for various $\gamma \in (-0.65,0)$. The inset image shows a schematic of the particle motion in the V2 core region at any $\gamma \in (-0.65,0)$. In region R1, $I\left(S\right)<0$, and the particles move outward towards $S_0$. In region R2, $I\left(S\right)>0$, and the particles move inward towards $S_0$.

Figure 3

PCRs and predicted streamlines $S_0$ and $S_{max}$ in the potential CVP flow for P1 $(\gamma =-0.2,\text{St}=0.01)$, P2 $(\gamma =-0.2,\text{St}=1.5)$, P3 $(\gamma =-0.2,\text{St}=2.5)$, P4 $(\gamma =-0.4,\text{St}=0.01)$, P5 $(\gamma =-0.4,\text{St}=0.5)$ and P6 $(\gamma =-0.4,\text{St}=1.0)$. The gray points represent particles. The coordinate origin is translated parallel to the $x$ axis. The centers of vortex V1 and V2 are located at (0.5,0) and ($-0.5$,0).

Figure 4

Particle trapping diagram for various $\gamma$ and St values. A filled circle indicates the occurrence of particle trapping. An open circle indicates no particle trapping. The solid line represents the critical ${\text{St}}_{\text{cr}}$ for the occurrence of particle trapping at various $\gamma$, which takes the form of the $\beta$ distribution function.

Figure 5

Schematic of the computational domain (not to scale) and boundary condition setup. The Lamb–Oseen vortex of circulation ${\mathrm{\Gamma }}_0$ is the initial primary vortex with a vortex core radius $a$ and a stand-off distance $b$ between the vortex center and the no-slip wall.

Figure 6

Detachments of the discrete opposite vortices from the wall boundary layer induced by the PV (the initial Lamb–Oseen vortex) for various Reynolds numbers at $\tau =3.0$. Note that the vortex is represented by the region of vorticity concentration, and the new discrete vortices are named in sequential order.

Figure 7

The Reynolds number of the SV as ${\text{Re}}_{\text{sv}}={\mathrm{\Gamma }}_{\text{sv}}/\left(4\pi \nu \right)$ versus the Reynolds number Re. The open circles denote the numerical results, and the dashed line is the corresponding linear fit by Eq. (10).

Figure 8

Trajectories of the PV centroid for different Reynolds numbers: (a) $\text{Re}=500$ and (b) $\text{Re}=2000$. For the inviscid case, the PV is convected parallel to the wall with the drift velocity by the field of its image below the wall [57]. The trajectory of the PV centroid is presented as a dashed line. The inset images show the vorticity field $\omega$ at various instants. The complete transient evolutions of the vorticity field for $\text{Re}=500$ and $\text{Re}=2000$ are visualized in Supplemental Material [58] and Material [59], respectively.

Figure 9

Time evolution of the primary vortex circulation ${\mathrm{\Gamma }}_{\text{PV}}$ and circulation ratio ${\gamma }^{*}={\mathrm{\Gamma }}_{\text{SV}}/{\mathrm{\Gamma }}_{\text{PV}}$ for $\text{Re}=500$ and $\text{Re}=2000$.

Figure 10

Instantaneous distributions of particles and the vorticity field for the case of $\text{Re}=500$ and $\text{St}=0.1$ at several instants: (a) $\tau =0.5$, (b) $\tau =2.5$, (c) $\tau =4.5$, and (d) $\tau =8$. The corresponding particle motion is visualized in Supplemental Material [61]. The gray points represent particles in the flow.

Figure 11

Instantaneous distributions of particles and the vorticity field for the case of $\text{Re}=2000$ and $\text{St}=0.1$ at several instants: (a) $\tau =0.5$, (b) $\tau =2.5$, (c) $\tau =4.5$, and (d) $\tau =12$. The corresponding particle motion is visualized in Supplemental Material [64]. The gray points represent particles in the flow.

Figure 12

Instantaneous distributions of particles and the vorticity field for different Reynolds numbers ($\text{Re}=500$ and 2000) and particle Stokes numbers ($\text{St}=1.0$ and 5.0) at the same time for $\tau =4.5$. The particle distributions for $\text{St}=0.1$ and the Reynolds numbers $\text{Re}=500$ and 2000 at $\tau =4.5$ are presented in Figs. 10 and 11.

Figure 13

Theoretically predicted $S_0$ and $S_{max}$ for a PCR in the viscous CVP flow represented by the vorticity field (a) with $\text{Re}=500$ and $\text{St}=0.1$ at $\tau =4.5$ and (b) with $\text{Re}=2000$ and $\text{St}=0.1$ at $\tau =4.5$. The gray points represent particles.

Figure 14

Definitions of the normal component $F_n$ and tangential component $F_t$ of the forces on a particle. The arrows point in the positive direction.

Figure 15

Evolution of the normal and tangential components of the forces acting on the particle ($\text{St}=0.1$) in a single cycle of periodic particle motion. The inset images show the forces acting on the particle and the particle position relative to the vortex (represented by the region of vorticity concentration) at several instants, ${\tau }_1, {\tau }_2, {\tau }_3$, and ${\tau }_4$. At instants ${\tau }_1$ and ${\tau }_3$, the normal composite force ${\text{CF}}_n$ on the particle is balanced. At instants ${\tau }_2$ and ${\tau }_4$, the tangential composite force ${\text{CF}}_t$ on the particle is balanced. The subscripts $n$ and $t$ denote the normal and tangential components of the force, respectively.

Figure 16

Trajectories of the vortex centroid in cases of different Reynolds numbers: (a) ${\text{Re}}_0=1000$ and (b) ${\text{Re}}_0=5000$. The solid lines are the results calculated by the present simulation method, and the open circles represent the DNS results [56].

Figure 17

Vortex centroid trajectories for a set of Reynolds numbers: (a) $\text{Re}=1000$, (b) $\text{Re}=2500$, and (c) $\text{Re}=4000$. The solid line shows the results for the coarser mesh, and the scatter points represent the results for the finer mesh.

Figure 18

Wall shear stress ${\tau }_w$ distributions at the no-slip wall for $\text{Re}=1000$ at different instants ($\tau =0.3$, and 0.6). The solid lines are the present simulation results, and the open circles are the results of Gargano et al. [39].