Symmetry-breaking Hopf bifurcations to 1-, 2-, and 3-tori in small-aspect-ratio counterrotating Taylor-Couette flow

Ferrofluidic wavy Taylor vortices under alternating magnetic field

Many natural and industrial flows are subject to time-dependent boundary conditions and temporal modulations (e.g. driving frequency), which significantly modify the dynamics compared with their static counterparts. The present problem addresses ferrofluidic wavy vortex flow in Taylor-Couette geometry, with the outer cylinder at rest in a spatially homogeneous magnetic field subject to an alternating modulation. Using a modified Niklas approximation, the effect of frequency modulation on nonlinear flow dynamics and appearing resonance phenomena are investigated in the context of either period doubling or inverse period doubling. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper (part 1)'.

Bifurcation phenomena in Taylor-Couette flow in a very short annulus with radial through-flow

In this study, the non-linear dynamics of Taylor-Couette flow in a very small-aspect-ratio wide-gap annulus in a counter-rotating regime under the influence of radial through-flow are investigated by solving its full three-dimensional Navier-Stokes equations. Depending on the intensity of the radial flow, either an axisymmetric (pure [Formula: see text] mode) pulsating flow structure or an axisymmetric axially propagating vortex will appear subcritical, i.e. below the centrifugal instability threshold of the circular Couette flow. We show that the propagating vortices can be stably existed in two separate parameter regions, which feature different underlying dynamics. Although in one regime, the flow appears only as a limit cycle solution upon which saddle-node-invariant-circle bifurcation occurs, but in the other regime, it shows more complex dynamics with richer Hopf bifurcation sequences. That is, by presence of incommensurate frequencies, it can be appeared as 1-, 2- and 3-torus solutions, which is known as the Ruelle-Takens-Newhouse route to chaos. Therefore, the observed bifurcation scenario is the Ruelle-Takens-Newhouse route to chaos and the period doubling bifurcation, which exhibit rich and complex dynamics.

Four-Frequency Solution in a Magnetohydrodynamic Couette Flow as a Consequence of Azimuthal Symmetry Breaking

The occurrence of magnetohydrodynamic quasiperiodic flows with four fundamental frequencies in differentially rotating spherical geometry is understood in terms of a sequence of bifurcations breaking the azimuthal symmetry of the flow as the applied magnetic field strength is varied. These flows originate from unstable periodic and quasiperiodic states with broken equatorial symmetry, but having fourfold azimuthal symmetry. A posterior bifurcation gives rise to twofold symmetric quasiperiodic states, with three fundamental frequencies, and a further bifurcation to a four-frequency quasiperiodic state which has lost all the spatial symmetries. This bifurcation scenario may be favored when differential rotation is increased and periodic flows with m-fold azimuthal symmetry, m being a product of several prime numbers, emerge at sufficiently large magnetic field.

Wave propagation reversal for wavy vortices in wide-gap counter-rotating cylindrical Couette flow

We present a numerical study of wavy supercritical cylindrical Couette flow between counter-rotating cylinders in which the wavy pattern propagates either prograde with the inner cylinder or retrograde opposite the rotation of the inner cylinder. The wave propagation reversals from prograde to retrograde and vice versa occur at distinct values of the inner cylinder Reynolds number when the associated frequency of the wavy instability vanishes. The reversal occurs for both twofold and threefold symmetric wavy vortices. Moreover, the wave propagation reversal only occurs for sufficiently strong counter-rotation. The flow pattern reversal appears to be intrinsic in the system as either periodic boundary conditions or fixed end wall boundary conditions for different system sizes always result in the wave propagation reversal. We present a detailed bifurcation sequence and parameter space diagram with respect to retrograde behavior of wavy flows. The retrograde propagation of the instability occurs when the inner Reynolds number is about two times the outer Reynolds number. The mechanism for the retrograde propagation is associated with the inviscidly unstable region near the inner cylinder and the direction of the global average azimuthal velocity. Flow dynamics, spatio-temporal behavior, global mean angular velocity, and torque of the flow with the wavy pattern are explored.

Dynamics of ferrofluidic flow in the Taylor-Couette system with a small aspect ratio

We investigate fundamental nonlinear dynamics of ferrofluidic Taylor-Couette flow - flow confined be-tween two concentric independently rotating cylinders - consider small aspect ratio by solving the ferro-hydrodynamical equations, carrying out systematic bifurcation analysis. Without magnetic field, we find steady flow patterns, previously observed with a simple fluid, such as those containing normal one- or two vortex cells, as well as anomalous one-cell and twin-cell flow states. However, when a symmetry-breaking transverse magnetic field is present, all flow states exhibit stimulated, finite two-fold mode. Various bifurcations between steady and unsteady states can occur, corresponding to the transitions between the two-cell and one-cell states. While unsteady, axially oscillating flow states can arise, we also detect the emergence of new unsteady flow states. In particular, we uncover two new states: one contains only the azimuthally oscillating solution in the configuration of the twin-cell flow state, and an-other a rotating flow state. Topologically, these flow states are a limit cycle and a quasiperiodic solution on a two-torus, respectively. Emergence of new flow states in addition to observed ones with classical fluid, indicates that richer but potentially more controllable dynamics in ferrofluidic flows, as such flow states depend on the external magnetic field.

Ring-bursting behavior en route to turbulence in narrow-gap Taylor-Couette flows

We investigate the Taylor-Couette system where the radius ratio is close to unity. Systematically increasing the Reynolds number, we observe a number of previously known transitions, such as one from the classical Taylor vortex flow (TVF) to wavy vortex flow (WVF) and the transition to fully developed turbulence. Prior to the onset of turbulence, we observe intermittent bursting patterns of localized turbulent patches, confirming the experimentally observed pattern of very short wavelength bursts (VSWBs). A striking finding is that, for a Reynolds number larger than that for the onset of VSWBs, a new type of intermittently bursting behavior emerges: patterns of azimuthally closed rings of various orders. We call them ring-bursting patterns, which surround the cylinder completely but remain localized and separated in the axial direction through nonturbulent wavy structures. We employ a number of quantitative measures including the cross-flow energy to characterize the ring-bursting patterns and to distinguish them from the background flow. These patterns are interesting because they do not occur in the wide-gap Taylor-Couette flow systems. The narrow-gap regime is less studied but certainly deserves further attention to gain deeper insights into complex flow dynamics in fluids.

Quasiperiodic routes to chaos in confined two-dimensional differential convection

The complete cascade of bifurcations from steady to chaotic convection, as the Rayleigh number is varied, is considered numerically inside an air-filled differentially heated cavity. The system is assumed to be two-dimensional and is invariant under a generalized reflection about the center of the cavity. In the neighborhood of several codimension-two points, two main routes emerge, characterized by different symmetries of the first oscillatory eigenstate. Along these two competing routes, different sequences of bifurcations and symmetry breakings lead from the steady base flow to the hyperchaotic regime. Several families of two- and three-frequency tori have been identified via the computation of the leading Lyapunov exponents. Modal structures extracted from time series reveal the occurrence of slow internal oscillations in the center of the cavity and faster wall modes confined to vertical boundary layers. Further quasiperiodicity windows have been detected on each route. The different regimes eventually disappear in a boundary crisis in favor of a single, globally symmetric, hyperchaotic regime.

Three-dimensional tori and Arnold tongues

This study analyzes an Arnold resonance web, which includes complicated quasi-periodic bifurcations, by conducting a Lyapunov analysis for a coupled delayed logistic map. The map can exhibit a two-dimensional invariant torus (IT), which corresponds to a three-dimensional torus in vector fields. Numerous one-dimensional invariant closed curves (ICCs), which correspond to two-dimensional tori in vector fields, exist in a very complicated but reasonable manner inside an IT-generating region. Periodic solutions emerge at the intersections of two different thin ICC-generating regions, which we call ICC-Arnold tongues, because all three independent-frequency components of the IT become rational at the intersections. Additionally, we observe a significant bifurcation structure where conventional Arnold tongues transit to ICC-Arnold tongues through a Neimark-Sacker bifurcation in the neighborhood of a quasi-periodic Hopf bifurcation (or a quasi-periodic Neimark-Sacker bifurcation) boundary.