Critical Stokes number for the capture of inertial particles by recirculation cells in two-dimensional quasisteady flows

Inertial particles are often observed to be trapped, temporarily or permanently, by recirculation cells which are ubiquitous in natural or industrial flows. In the limit of small particle inertia, determining the conditions of trapping is a challenging task, as it requires a large number of numerical simulations or experiments to test various particle sizes or densities. Here, we investigate this phenomenon analytically and numerically in the case of heavy particles (e.g., aerosols) at low Reynolds number, to derive a trapping criterion that can be used both in analytical and numerical velocity fields. The resulting criterion allows one to predict the characteristics of trapped particles as soon as single-phase simulations of the flow are performed. Our analysis is valid for two-dimensional particle-laden flows in the vertical plane, in the limit where the particle inertia, the free-fall terminal velocity, and the flow unsteadiness can be treated as perturbations. The weak unsteadiness of the flow generally induces a chaotic tangle near heteroclinic or homoclinic cycles if any, leading to the apparent diffusion of fluid elements through the boundary of the cell. The critical particle Stokes number ${\text{St}}_c$ below which aerosols also enter and exit the cell in a complex manner has been derived analytically, in terms of the flow characteristics. It involves the nondimensional curvature-weighted integral of the squared velocity of the steady fluid flow along the dividing streamline of the recirculation cell. When the flow is unsteady and $\text{St}>{\text{St}}_c$, a regular motion takes place due to gravity and centrifugal effects, like in the steady case. Particles driven towards the interior of the cell are trapped permanently. In contrast, when the flow is unsteady and $\text{St}<{\text{St}}_c$, particles wander in a chaotic manner in the vicinity of the border of the cell, and can escape the cell.

Figure 1

Sketch of the stagnation points $A$ and $B$ of the steady flow ${\mathbf{u}}_0$, joined by a separatrix streamline [panel (i)]. Vector $\mathbf{n}$ is the unit vector perpendicular to ${\mathbf{u}}_0$, such that $\left({\mathbf{u}}_0,\mathbf{n}\right)$ is counterclockwise about the $z$ axis. Some examples which will be treated in this paper: circular cell (ii), cavity flow (iii), and backward facing step (iv).

Figure 2

Sketch of the invariant manifolds of the dynamics of particles in the cavity flow (a),(b) and in the backward facing step flow (c),(d), when particles are assumed to slip on the walls. In the steady case, or in the unsteady case with $\text{St}>{\text{St}}_c$ (that is $D>D_c$), manifolds do not intersect and take the form (a) and (c) if gravity dominates inertial curvature effects: the motion is regular and particles entering the cell cannot exit. In the unsteady case with $\text{St}<{\text{St}}_c$ (that is $D<D_c$), manifolds take the form (b) and (d): the motion is chaotic; some particles can enter and exit the cell in a complex manner. Case (e) corresponds to a regular motion when centrifugal effect dominates. Panel (f) is a sketch of a trajectory wandering in the stochastic zone near the separatrix.

Figure 3

Continuous lines: critical Stokes numbers ${\text{St}}_c^{+}$ (solid line) and ${\text{St}}_c^{-}$ (dashed line), for the circular separatrices of the analytical cell flow (14), obtained from the theoretical expressions (19). Symbols: the critical Stokes numbers obtained from numerical simulations of a large number of particle trajectories in the same flow discretized on a mesh. Numerical values of the parameters in these simulations: ${\rho }_p/{\rho }_f=1000,\text{Fr}=0.1$, and $\epsilon =0.05$.

Figure 4

Test case: typical clouds of particles transported in the vicinity of the unsteady cell flow (14), and initially injected near point $A$, with $y>0$. Six particle diameters are used, corresponding to six different values of the Stokes number. All Stokes numbers are supercritical for the lower separatrix (that is, ${\text{St}}_c^{-}<\text{St}$). Hence particles that reach the lower circular separatrix $AB$ are too inertial to reenter the cell. Numerical values of the parameters in these simulations: ${\rho }_p/{\rho }_f=1000,\text{Fr}=0.1$, and $\epsilon =0.05$.

Figure 5

Critical diameter obtained from the asymptotic theory (solid line: cavity; dashed line: backward-facing step). Symbols indicate the diameters obtained by tracking a large number of inertial particles (circles: cavity; triangles: backward-facing step). Numerical values of the parameters: $\omega =26.2\mathrm{rad}/\mathrm{s}$; $\epsilon =0.05$.

Figure 6

Typical clouds of particles moving in the vicinity of the cavity flow. The thick line is the dividing streamline $AB$ of the steady flow. Colors indicate particles' diameter $D$, in microns. All particles are injected near $A$, and their density is ${\rho }_p=100{\mathrm{kg}/\mathrm{m}}^3$. Those for which $D>D_c=26.5$ microns have a regular dynamics and cannot exit. Other parameters: $U_{\infty }^0=0.13\mathrm{m}/\mathrm{s}$, cavity size $1\text{cm}\times 2\text{cm},\omega =26.2\mathrm{rad}/\mathrm{s}$, and $\epsilon =0.05$.

Figure 7

Typical clouds of particles moving in the vicinity of the backward facing step. The thick line is the dividing streamline $AB$ of the steady flow. Colors indicate particles' diameter $D$, in microns. All particles are injected near $A$, and their density is ${\rho }_p=100{\mathrm{kg}/\mathrm{m}}^3$. Those for which $D>D_c=34.5$ microns (red particles) drop in the cell and are trapped permanently. They have a regular dynamics. Those for which $D<D_c$ (blue) have a chaotic dynamics near the heteroclinic cycle and can exit in a complex manner. Other parameters: $U_{\infty }^0=0.17\mathrm{m}/\mathrm{s}$, vertical step size $L=0.01\text{m},\omega =25.2\mathrm{rad}/\mathrm{s}$, and $\epsilon =0.05$.