Linear vs nonlinear transport during chaotic advection in fluid flows.

The goal of this study is explicit demarcation of the region of validity of a linear canonical representation for chaotic advection of Lagrangian fluid parcels in "chaotic seas" in two-dimensional (2D) and three-dimensional (3D) time-periodic fluid flows governed by Hamiltonian mechanics. The concept of lobe dynamics admits exact and unique geometric demarcation of this region and, inherently, distinction of the portions of chaotic seas with essentially linear vs nonlinear Lagrangian transport. This, furthermore, admits explicit establishment of a topological equivalence between the (embedded) Hamiltonian structure of the Lagrangian dynamics in 2D (3D) flows and their canonical form. The linear transport region in physical space encompasses four adjacent subregions that each corresponds to one of the four quadrants in canonical space and may exchange material with their environment in two essentially nonlinear ways. First, exchange between quadrants within the linear transport region and, second, exchange with the exterior of this region. Both forms of exchange can be linked to specific subsets of material elements defined by interacting lobes and combined give rise to circulation through the quadrants of the linear transport region that systematically exchanges the material with the exterior.

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.

Direct experimental visualization of the global Hamiltonian progression of two-dimensional Lagrangian flow topologies from integrable to chaotic state.

Countless theoretical/numerical studies on transport and mixing in two-dimensional (2D) unsteady flows lean on the assumption that Hamiltonian mechanisms govern the Lagrangian dynamics of passive tracers. However, experimental studies specifically investigating said mechanisms are rare. Moreover, they typically concern local behavior in specific states (usually far away from the integrable state) and generally expose this indirectly by dye visualization. Laboratory experiments explicitly addressing the global Hamiltonian progression of the Lagrangian flow topology entirely from integrable to chaotic state, i.e., the fundamental route to efficient transport by chaotic advection, appear non-existent. This motivates our study on experimental visualization of this progression by direct measurement of Poincaré sections of passive tracer particles in a representative 2D time-periodic flow. This admits (i) accurate replication of the experimental initial conditions, facilitating true one-to-one comparison of simulated and measured behavior, and (ii) direct experimental investigation of the ensuing Lagrangian dynamics. The analysis reveals a close agreement between computations and observations and thus experimentally validates the full global Hamiltonian progression at a great level of detail.

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 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 Re∼O(10(-3)); (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.

Finite-time transport in volume-preserving flows.

Finite-time transport between distinct flow regions is of great relevance to many scientific applications, yet quantitative studies remain scarce to date. The primary obstacle is computing the evolution of material volumes, which is often infeasible due to extreme interfacial stretching. We present a framework for describing and computing finite-time transport in n-dimensional (chaotic) volume-preserving flows that relies on the reduced dynamics of an (n-2)-dimensional "minimal set" of fundamental trajectories. This approach has essential advantages over existing methods: the regions between which transport is investigated can be arbitrarily specified; no knowledge of the flow outside the finite transport interval is needed; and computational effort is substantially reduced. We demonstrate our framework in 2D for an industrial mixing device.

Observability of periodic lines in three-dimensional lid-driven cylindrical cavity flows.

This study employs three-dimensional particle-tracking velocimetry (3D-PTV) for experimental investigation of the existence and properties of periodic lines in 3D lid-driven time-periodic flows inside a cylindrical cavity. These periodic lines, consisting of material points that periodically return to their initial position, play a central role in the transport properties of laminar flows, yet their existence has so far been demonstrated only in numerical simulations. The formation and characteristics of periodic lines are inextricably linked with spatiotemporal symmetries of the flow. 3D-PTV measurements determined that relevant symmetries, identified with previous symmetry analyses, are satisfied within experimental error bounds. These measurements subsequently isolated periodic lines in the designated symmetry planes, thus offering first experimental evidence of their physical existence and their fundamental reliance on symmetries. Experimental periodic lines are topologically equivalent to those in simulated flows with identical symmetries and exhibit the same response to changes in forcing conditions. The laboratory experiments by these observations bridge the gap from theoretical and numerical predictions on periodic lines to real 3D flows.

Topology of advective-diffusive scalar transport in laminar flows.

The present study proposes a unified Lagrangian transport template for topological description of advective fluid transport and advective-diffusive scalar transport in laminar flows. The key to this unified description is the expression of scalar transport as purely advective transport by the total scalar flux. This admits generalization of the concept of transport topologies known from laminar mixing studies to scalar transport. The study is restricted to two-dimensional systems and the fluid and scalar transport topologies, as a consequence, prove to be Hamiltonian. The unified Lagrangian transport template is demonstrated and investigated for a heat-transfer problem with nonadiabatic boundaries, representing generic scalar transport with permeable boundaries. The fluid and thermal transport topologies under steady conditions both accommodate islands (constituting isolated flow and thermal regions) that undergo Hamiltonian disintegration into chaotic seas upon introducing time periodicity. The thermal transport topology invariably comprises transport conduits that connect the nonadiabatic boundaries and facilitate wall-wall and wall-fluid heat transfer. For steady conditions these transport conduits are regular; for time-periodic conditions these conduits may, depending on degree of diffusion, be regular or chaotic. Regular conduits connect nonadiabatic walls only with specific flow regions; chaotic heat conduits establish connection (and thus heat exchange) of the nonadiabatic walls with the entire flow domain.

Merger of coherent structures in time-periodic viscous flows.

Inertia-induced changes in transport properties of an incompressible viscous time-periodic flow are studied in terms of the topological properties of volume-preserving maps. In the noninertial limit, the flow admits one constant of motion and thus relates to a so-called one-action map. However, the invariant surfaces corresponding to the constant of motion are topologically equivalent to spheres rather than the common case of tori. This has fundamental ramifications for the effect of inertia and leads to a new kind of response scenario: resonance-induced merger of coherent structures.