Finite-size Lagrangian coherent structures in a two-sided lid-driven cavity

The motion of small, rigid, nearly neutrally buoyant finite-size particles in a two-sided lid-driven cavity is calculated numerically. The rapid accumulation of particles into coherent structures which is observed experimentally is explained on the basis of single-particle one-way-coupled dynamics. Key is the transfer of the particle from regions of the incompressible flow occupied by chaotic streamlines to regions occupied by regular ones. The particle attractors are caused by lubrication forces which repel the particles from the moving boundaries. This mechanism is independent of particle inertia. Therefore, the particulate structures found represent a class of coherent structures which may be called finite-size coherent structures.

Figure 1

(a) Two-sided lid-driven cavity (top wall removed) showing half of the periodic domain (grey box) and the sliding lids (light blue). The origin of the coordinate system is located at the center of the periodic cell (black dot). (b) A finite-size spherical particle (red) at a distance $h$ from the wall (light blue) moves towards it. Within a distance $\overline{h}$ from the wall (dashed white line and grey layer) the Stokes lubrication force model is effective. Nondimensional lengths in units of $H$ (used in the simulations) are given in parentheses.

Figure 2

Schematic of the particle-boundary interaction in a three-dimensional view (a) and in two-dimensional projection (b). The lubrication force acting on the particle when its surface enters the lubrication region (gray volume) of thickness $a\overline{\epsilon }$ is shown as orange arrows. Red indicates that the particle is inside the lubrication region and the lubrication force is acting on the particle. In the lubrication-force model the particle is repelled from the wall upon approach and attracted upon departure. Green represents a particle outside of the lubrication layer ($F_L\equiv 0$).

Figure 3

Distribution of 4096 particles in the computational domain (convection cell) for $\text{Re}=400, \mathrm{\Gamma }=1.7, \lambda =2.73, \varrho =1$, and $\text{St}=5.\overline{5}\times {10}^{-4}$. (a) Random distribution at $t=0$. (b) Distribution at $t=2$. Particle size is not to scale.

Figure 4

(a), (b) Trajectories of inertial particles with $\varrho =0.9$ (red) and $\varrho =1.1$ (green) for $a=0.05$ integrating (3) and using (5). The closed streamline is shown in black. (a) $t\in [0,0.5], F_L\equiv 0$, and (b) $t\in [1.5,2], F_L\ne 0$. (c) Growth rate $\sigma \left(\varrho \right)$ of deviations from the limit cycle and linear fits (dashed) for $F_L\equiv 0$ and $a=0.01$ ($\circ$), 0.02 ($\square$), 0.03 ($\diamond$), 0.04 ($▵$), 0.05 ($▹$). Inset: slope $\gamma$ of the linear fits for $\sigma$ (dashed lines) as function of $\text{St}/\varrho$. (d) Attraction to the limit cycle for $a=0.058$ expressed by $d_P\left(t\right)$ for $F_L\equiv 0$ ($\square$) and $F_L\ne 0$ ($\circ$). Dashed lines are exponential fits $d_P=A{e}^{\sigma t}$.

Figure 5

Particle trajectories for $\varrho =1$ and different particle sizes, shown for $t\in [1.5,2]$, after an initial transient. (a) Chaotic trajectory (black) for $a=0.01$. (b) Periodic trajectories for $a=0.02$ and different initial conditions with period 4 (yellow), period 5 (cyan), and period 10 (red). (c), (d) Quasiperiodic orbits for $a=0.03$ (purple) and $a=0.04$ (green). (e) Periodic orbit for $a=0.05$ (orange). (f) Poincaré section at $x=-0.5$ of the attractors shown in panels (b)–(e). The crosses indicate the outermost KAM surface found in Ref. [9].

Figure 6

Correction factor $\alpha$ for lubrication drag forces exerted on a particle moving towards a rigid plane wall. Dashed line: exact solution [20] for a particle in creeping flow, circles: first-order lubrication approximation in creeping flow [28], squares: first-order lubrication approximation including the inertial contribution for ${\text{Re}}_p=3$ [48].

Figure 7

Trajectory of a particle with $a=0.05$ and $\varrho =0.9$ initialized at $\mathbf{y}=(0.5,0.2,0.59)$ with velocity $\dot{\mathbf{y}}=\mathit{0}$ (velocity mismatch). The instantaneous particle Reynolds number ${\text{Re}}_p$ is shown in the bottom panel. The particle-boundary interaction takes place between the two vertical lines.

Figure 8

Trajectory of a particle with $a=0.05$ and $\varrho =0.9$ initialized at $\mathbf{y}=(0.5,0.2,0.59)$ with velocity $\dot{\mathbf{y}}=\mathbf{0}$. The red line indicates the particle trajectory for $\overline{\epsilon }=0.5$; the black dashed line is for $\overline{\epsilon }=0.6$.