Particle capture by drops in turbulent flow

We examine the process of particle capture by large deformable drops in turbulent channel flow. We simulate the solid-liquid-liquid three-phase flow with an Eulerian-Lagrangian method based on direct numerical simulation of turbulence coupled with a phase-field model, to capture the interface dynamics, and Lagrangian tracking of small (sub-Kolmogorov) particles. Drops have the same density and viscosity of the carrier liquid, and neutrally buoyant, quasi-inertialess, solid particles are one-way coupled with the other phases. Our results show that particles are transported towards the interface by jetlike turbulent motions and, once close enough, are captured by interfacial forces in regions of positive surface velocity divergence. These regions appear to be well correlated with high-enstrophy flow topologies that contribute to enstrophy production via vortex compression or stretching. Examining the turbulent mechanisms that bring particles to the interface, we have been able to derive a simple transport model for particle capture. The model is based on a single turbulent transport equation in which the only parameter scales with the turbulent kinetic energy of the fluid measured in the vicinity of the drop interface, and its predictions of the overall capture efficiency agree remarkably well with numerical results.

Figure 1

Qualitative rendering of the flow configuration. Drops are colored by the local curvature of the interface, the flow field is rendered by the fluid streaklines, and particles are visualized as blue spheres. The insets provide close-up views of one particle capture event (inset $I$) and of one interface-approaching particle in isolation (inset $II$), respectively. Gray arrows represent the particle velocity magnitude and render the motion of particles that move towards the interface. Red arrows represent the interfacial stress sampled by the particles at the time of adhesion. The color map in the top inset shows the spatial distribution of the order parameter $\varphi$: The interface is located at $\varphi =0$ (thick black line). The thin black lines represent the fluid layer within which $\mathrm{\Phi }$ transitions from $\varphi =-1$ (fluid) to $\varphi =+1$ (drop), as shown in the schematic on the right end of the inset, where the distribution of the capillary force $F_c$ over the interaction distance $\mathcal{D}$ is also rendered. Note that $\mathcal{D}$ is shorter than the distance over which $-1<\varphi <+1$.

Figure 2

PDF of the two-dimensional (2D) surface divergence, ${\nabla }_{2D}$, seen by the particles when they get captured by the interface. Regions of local flow expansion (velocity sources) are characterized by ${\nabla }_{2D}>0$; regions of local compression (velocity sinks) are characterized by ${\nabla }_{2D}<0$. Symbols refer to simulation results (circles: $\mathrm{St}=0.1$, triangles: $\mathrm{St}=0.8$), whereas the solid line refers to the PDF computed for tracer particles uniformly distributed over the entire interface of each drop. The PDF was computed starting at time $t^{+}\simeq 1000$ after particle injection and over a subsequent time interval $\mathrm{\Delta }{t}^{+}=2000$.

Figure 3

(a) Flow topology parameter $\mathcal{Q}$ on the channel midplane ($x^{+}-y^{+}$ plane); black solid lines identify the position of the drop interface (isolevel $\varphi =0$). (b), (c) Close-up view of the distribution of captured particles on the interface of the drop pair boxed in (a). Insets: (b) $\mathrm{St}=0.1$, (c) $\mathrm{St}=0.8$.

Figure 4

PDF of the topology parameter, $\mathcal{Q}$, seen by the particles when they get captured by the interface. Lines and symbols are as in Fig. 2. The PDF was computed starting at time $t^{+}\simeq 1000$ after particle injection and over a subsequent time interval $\mathrm{\Delta }{t}^{+}=2000$.

Figure 5

(a) Incompressible flow critical point topologies according to the classification scheme in [23]: Topologies falling in region $I$ are called unstable focus/compressing (UFC), while those falling in region $II$ are called stable focus/stretching (SFS). Topologies falling in region $III$ are called stable node/saddle/saddle (SN/S/S), while those falling in region $IV$ are called unstable node/saddle/saddle (UN/S/S). Further critical points can be identified along the $Q$ axis and the $D=0$ line, but their characterization is beyond the scope of this study. (b)–(d) Joint PDF of $Q$, $R$ conditionally sampled for fluid (b) at grid points over the entire interface of the drops, (c) at the position of the $\mathrm{St}=0.1$ particle when they get captured, and (d) at the position of the $\mathrm{St}=0.8$ particles when they get captured.

Figure 6

Time evolution of the number of particles captured by the interface. Symbols refer to simulation results (circles: $\mathrm{St}=0.1$; triangles: $\mathrm{St}=0.8$), whereas the solid line refers to the model provided by Eq. (27). The inset shows the instantaneous distribution of the captured particles over the interface of a drop, shown in isolation from the flow domain and colored using the local interface curvature (red: high positive curvature; blue: low negative curvature). Particles appear to sample the same interfacial regions, confirming the secondary role played by inertia.