Comparative numerical-experimental analysis of the universal impact of arbitrary perturbations on transport in three-dimensional unsteady flows

Numerical studies of three-dimensional (3D) time-periodic flow inside a lid-driven cylinder revealed that a weak perturbation of the noninertial state (Reynolds number $\text{Re}=0$) has a strong impact on the Lagrangian flow structure by inducing transition of a global family of nested spheroidal invariant surfaces into intricate coherent structures consisting of adiabatic invariant surfaces connected by tubes. These tubes provide paths for passive tracers to escape from one invariant surface to another. Perturbation is introduced in two ways: (i) weak fluid inertia by nonzero $\text{Re}\sim O\left({10}^{-3}\right)$; (ii) small disturbance of the external flow forcing. Both induce essentially the same dynamics, implying a universal response in the limit of a weak perturbation. Moreover, we show that the motion inside tubes possesses an adiabatic invariant. Long-term experiments were conducted using 3D particle-tracking velocimetry and relied on experimental imperfections as natural weak perturbations. This provided first experimental evidence of the tube formation and revealed close agreement with numerical simulations. We experimentally validated the universality of the perturbation response and, given the inevitability of imperfections, exposed the weakly perturbed state as the true “unperturbed state” in realistic systems.

Figure 1

Schematic of the flow forcing by piecewise steady translation of the bottom wall. (Inset) Trajectory of the bottom wall in the three-step forcing protocol adopted in this study. (Adapted from [25].)

Figure 2

Schematic of the laboratory setup for particle tracking by 3D-PTV.

Figure 3

Fundamental properties of the unperturbed flow. (a) Closed streamlines (symmetric about plans $x=0$ and $y=0$) of base flow ${\mathbf{u}}_s$; (b) invariant spheroids of the time-periodic flow in the noninertial limit $\text{Re}=0$. Simulation using rigid-wall boundary conditions. (Reproduced from [25].)

Figure 4

Continuous tracer paths (blue solid line) corresponding with a period-1 point (blue circle) on the period-1 line (red triangles). (a) $D=4$; (b) $D=14$. Simulations of the unperturbed flow using smooth boundary conditions.

Figure 5

Continuous tracer paths (blue solid line) corresponding with a period-2 point (blue circle) on the period-2 line (black/dark triangles). (a) $D=4$; (b) $D=14$. Period-1 lines (red/gray triangles) are included for reference. Simulations of the unperturbed flow using smooth boundary conditions.

Figure 6

Formation of tubes on the period-1 line (red/large markers) connecting adiabatic surfaces under weak perturbation $\left(D=14\right)$. (a) Fluid inertia $\left(\text{Re}=0.005\right)$; (b) top-wall motion $\left(\varepsilon =0.0005,{\theta }_0=\pi /6\right)$. Visualization by the 3D perspective and $r-z$ projection of the stroboscopic map of a single tracer (blue/small markers). Simulations using smooth boundary conditions.

Figure 7

Continuous tracer path (gray) of the single tracer outlining the period-1 tube (blue; indicated by arrow) on the period-1 line (red triangles) $\left(D=14\right)$. Perturbation by fluid inertia $\left(\text{Re}=0.005\right)$. Simulations using smooth boundary conditions.

Figure 8

Adiabatic invariance in period-1 tubes demonstrated for three concentric period-1 tubes (purple/gray/blue; inner/middle/outer) induced by fluid inertia $\left(\text{Re}=0.005\right)$. (a) $r-z$ projection; (b) cross section perpendicular to symmetry plane $\theta =\pi /6$; (c) adiabatic invariant $\Psi$ following (12) along the tubes (purple/gray/blue: lower/middle/upper). Simulations for $D=14$ using smooth boundary conditions.

Figure 9

Streamlines of the base flow obtained from laboratory experiments by particle tracking using 3D-PTV.

Figure 10

Period-1 lines obtained by 3D-PTV (blue/scattered markers) versus numerical simulations rigid-wall boundary condition (curves described by red markers) for $D=4$. (a) 3D perspective view; (b) top view. The black frame in panel (a) outlines symmetry plane $\theta =\pi /6$.

Figure 11

Formation of a period-1 tube on the period-1 line (curve described by red markers) demonstrated by the stroboscopic map of a single tracer (tube described by blue markers) for $D=4$. (a) Experiments using 3D-PTV; (b) numerical simulations using top-wall perturbation ($\varepsilon =0.0005$ and ${\theta }_0=\pi /6$) and rigid-wall boundary conditions.

Figure 12

Tube bifurcation at intersections of period-1 (red/dark gray) and period-2 (black) lines demonstrated by the stroboscopic map (blue/light gray) of a single tracer for $D=4$. Labels indicate period-1 $\left({\mathcal{P}}_1\right)$ and period-2 $\left(\mathcal{P}{2}_{1,2}\right)$ tubes; arrows indicate direction of motion. Numerical simulations using top-wall perturbation $\left(\varepsilon =0.0005,{\theta }_0=\pi /6\right)$ and rigid-wall boundary conditions.

Figure 13

Experimental evidence for period-2 behavior consistent with the formation of period-2 tubes for $D=4$. (a) Maps of individual tracers (distinguished by color/grayscale and marker size) centered on simulated period-2 lines (curves described by black markers); (b) periodwise alternation of a single tracer between positions ${\mathbf{X}}_{+}=\{{\mathbf{X}}_0,{\mathbf{X}}_2,\cdots \}$ and ${\mathbf{X}}_{-}=\{{\mathbf{X}}_1,{\mathbf{X}}_3,\cdots \}$ (large markers) on either side of symmetry plane $\theta =\pi /6$. The curve bisecting period-2 lines indicates period-1 line; solid lines in panel (b) indicate continuous tracer paths.

Figure 14

Tracer dynamics near the tube bifurcation on the period-1 line (curve described by red markers) demonstrated by the stroboscopic map (tubelike structures described by blue markers) of a single tracer for $D=4$. (a) Numerical simulations using top-wall perturbation and rigid-wall boundary conditions; (b) experiments using 3D-PTV. Shown are 3D perspective and $r-z$ projection.