Physical topology of three-dimensional unsteady flows with spheroidal invariant surfaces

Scope is the response of Lagrangian flow topologies of three-dimensional time-periodic flows consisting of spheroidal invariant surfaces to perturbation. Such invariant surfaces generically accommodate nonintegrable Hamiltonian dynamics and, in consequence, intrasurface topologies composed of islands and chaotic seas. Computational studies predict a response to arbitrary perturbation that is dramatically different from the classical case of toroidal invariant surfaces: said islands and chaotic seas evolve into tubes and shells, respectively, that merge into “tube-and-shell” structures consisting of two shells connected via (a) tube(s) by a mechanism termed “resonance-induced merger” (RIM). This paper provides conclusive experimental proof of RIM and advances the corresponding structures as the physical topology of realistic flows with spheroidal invariant surfaces; the underlying unperturbed state is a singular limit that exists only for ideal conditions and cannot be achieved in a physical experiment. This paper furthermore expands existing theory on certain instances of RIM to a comprehensive theory (supported by experiments) that explains all observed instances of this phenomenon. This theory reveals that RIM ensues from perturbed periodic lines via three possible scenarios: truncation of tubes by (i) manifolds of isolated periodic points emerging near elliptic lines or by either (ii) local or (iii) global segmentation of periodic lines into elliptic and hyperbolic parts. The RIM scenario for local segmentation includes a perturbation-induced change from elliptic to hyperbolic dynamics near degenerate points on entirely elliptic lines (denoted “virtual local segmentation”). This theory furthermore demonstrates that RIM indeed accomplishes tube-shell merger by exposing the existence of invariant surfaces that smoothly extend from the tubes into the chaotic shells. These phenomena set the response to perturbation—and physical topology—of flows with spheroidal invariant surfaces fundamentally apart from flows with toroidal invariant surfaces. Its entirely kinematic nature and reliance solely on continuity and solenoidality of the velocity field render the comprehensive theory and its findings universal and generically applicable for (arbitrary perturbation of) basically any incompressible flow—in fact any smooth solenoidal vector field—accommodating spheroidal invariant surfaces.

Figure 1

Representative flow configuration: time-periodic flow inside a cylinder of radius $R$ and height $H=2R$ driven by piecewise steady translations at velocity $U$ of the bottom wall (left) under angle ${\theta }_k=2k\pi /3$ ($k\in \{0,1,2\}$) with the $x$ axis (right).

Figure 2

Physical vs theoretical topology in flows with spheroidal invariant surfaces demonstrated by the stroboscopic map of a single tracer (left, 3D from identical viewpoints; right, $rz$ projection) in time-periodic cylinder flow driven by stepwise translations of the bottom wall at angles ${\theta }_k=2k\pi /3$ ($k\in \{0,1,2\}$) with the $x$ axis. (a) Simulated theoretical topology (unperturbed limit $\text{Re}=0$) of tracers (black) restricted to the chaotic sea in invariant surfaces (cyan/gray) and outlining intrasurface islands (black arrows). (b) Simulated physical topology of tube (magenta/gray) and shell (black) structures due to explicit perturbation by the top wall ($\text{Re}=0$). (c) Experimental topology originating from natural disturbances. Blue, curve periodic line; green star/yellow circle, degenerate points.

Figure 3

Laboratory setup for measurement of stroboscopic maps using 3DPTV. (a) Schematic of the setup (adapted from [20]). (b) Actual setup.

Figure 4

Buoyancy-driven background circulation of magnitude $v\sim O({10}^{-4}\mathrm{mm}/\mathrm{s})$ due to a $y$-wise temperature drop $\mathrm{\Delta }T\sim O\left(0.15\mathrm{K}\right)$ as measured by 3DPTV. (a) Top view. (b) Side view.

Figure 5

Simulated periodic lines in time-periodic cylinder flow for $\text{Re}=0$ and $D=4$. (a) Periodic lines inside the symmetry plane. (b) Periodic lines outside the symmetry plane. Blue/red (dark/light gray), elliptic/hyperbolic (segments of) periodic lines; green star and green star/yellow circle, degenerate points for ${\alpha }_p=2$ and $-2$, respectively; shaded plane (perspective view)/tilted line (top view), symmetry plane $\mathrm{\Theta }=\pi /6$.

Figure 6

Tube-shell merger (left) due to formation of isolated periodic point ${\mathbf{x}}_{*}$ (magenta sphere) near the elliptic line (blue), characterized by $\left|\alpha \left(z\right)\right|<|{\alpha }_p|=2$ (right), demonstrated by the perturbed canonical map (left) of a single tracer (black, time progression upwards) released at ${\mathbf{x}}_0$ (red star). Gray horizontal plane, $W_{\text{2D}}$ of ${\mathbf{x}}_{*}$; cyan funnel, invariant surfaces describing tube-and-shell structures.

Figure 7

Tube-shell merger due to segmentation of periodic lines demonstrated by the perturbed canonical map (left) of a single tracer (black, time progression upwards) released at ${\mathbf{x}}_0$ (red star) near the periodic line (vertical axis; blue/red, (dark/light gray) elliptic/hyperbolic) characterized by $\alpha \left(z\right)$ (right). (a) Local segmentation for ${\alpha }_p=2$. (b) Local segmentation for ${\alpha }_p=-2$. (c) Global segmentation for ${\alpha }_p=2$. (d) Global segmentation for ${\alpha }_p=-2$. Dashed line in $\alpha \left(z\right)$, parabolic limit ${\alpha }_p=\pm 2$; light/dark gray vertical planes, stable/unstable manifolds of the hyperbolic line segment; cyan, diverging upright cylinders, invariant surfaces describing tube-and-shell structures.

Figure 8

Tube-shell merger due to virtual local segmentation of the elliptic periodic line (right; vertical axis) characterized by $\alpha \left(z\right)$ (left) with the parabolic point at $\tilde{z}=0$ demonstrated by the perturbed canonical map (right) of a single tracer (black, time progression upwards) released at ${\mathbf{x}}_0$ (red star). (a) ${\alpha }_p=2$. (b) ${\alpha }_p=-2$. Dynamics near blue (dark gray) and red (light gray) line segments remains elliptic and becomes hyperbolic, respectively, upon perturbation. Dashed line in $\alpha \left(z\right)$, parabolic limit ${\alpha }_p=\pm 2$.

Figure 9

Manifestations of RIM in cylinder flow due to segmented periodic lines (color coding and labeling as before) demonstrated by the stroboscopic map of a single tracer. (a) Simulated map. (b) Experimental map. Parabolic points I–III and IV concern actual (${\alpha }_p=-2$) and virtual (${\alpha }_p=2$) segmentation, respectively; time progression is from IV to I.

Figure 10

Further investigation of RIM due to segmented periodic lines in Fig. 9. (a) Continuation of the simulated stroboscopic map of a single tracer in Fig. 9 (black) at parabolic points I (cyan/light gray) and IV (magenta/medium gray) forward and backward in time, respectively. (b) Experimental chaotic shell at point I. (c) Coincidence of the experimental (blue/dark gray) and simulated (cyan/light gray) chaotic shell.

Figure 11

Highly localized manifestation of RIM at point IV in Fig. 9 due to virtual segmentation upon perturbation: switching between tube segments (black and magenta/medium gray) via the hyperbolic layer at point IV (green/light gray marker).

Figure 12

Tube bifurcation by period-2 resonance (${\alpha }_p=-2$) at the parabolic point (green star) between elliptic (blue/dark gray) and hyperbolic (red/light gray) segments of the period-1 line in the symmetry plane (shaded) demonstrated by the stroboscopic map of a single tracer. (a) Simulated map. (b) Experimental map.

Figure 13

Hyperbolic portions of invariant surfaces for RIM due to segmentation of the periodic line (dashed) at parabolic points (stars). (a) Invariant surface for global segmentation (cyan/light gray) for a tracer (black) expelled into a chaotic shell and neighboring surface (blue/dark gray). (b) Invariant surfaces for local segmentation (cyan/light gray and red/dark gray) for tracers (black/gray) dots expelled into/entrained from the shell. Magenta (dark gray)/green (light gray), circles connecting elliptic/hyperbolic surface portions.

Figure 14

Lower bound ${\alpha }_{\text{min}}$ of the range for discriminant $\alpha$ according to (D5) resulting in qualitative change near the elliptic line from elliptic to hyperbolic dynamics upon perturbation. (a) ${\alpha }_{\text{min}}$ for $a_1=-b_2$. (b) ${\alpha }_{\text{min}}$ for generic $(a_1,b_2)$ [the dashed line indicates the simplified case in panel (a)].