Nonlinear defects separating spiral waves in Taylor-Couette flow

Physical Review E, 1994

We present experimental results on novel pattern states which were observed in the Couette-Taylor flow subjected to an axial flow, in a wide range of the control parameters. Propagating Taylor vortices (PTV's), stationary spirals (SSP's), and moving spirals (MS's) were found as a result of a different symmetry breaking. These modes exhibit different wave-number selection. Novel states originating from an interaction between these patterns were also found. A "mixed phase" of PTV's and SSP's was identified. A "mode-competition" state, in which the PTV's and MS's are alternated in the column, is also described. Finally, a "disordered-Taylor-vortices" state was observed and characterized. PACS number(s}: 47.20.k, 47.60.+i, 47.54. +r I. INrRODUCnON The superposition of the circular Couette flow and the Poiseuille shear flow in the Taylor system with an axial flow, as has previously been shown by Snyder [1], results in a rich phase diagram which exhibits several types of periodic structures. The through flow at a relatively small Reynolds number value (Re &4) causes the Taylor vortices to drift downstream, and the flow takes the form of propagating Taylor vortices (PTV). Besides the PTV mode, which was studied in detail recently [2-5], the system also exhibits spiral modes which can be classified by the helicity and direction of propagation as was done by Biihler and Polifke [6]. A phase diagram which manifests instability modes of difFerent symmetries, such as the PTV and spiral modes, may exhibit interesting nonlinear dynamics in the vicinity of points or lines which separate difFerent instability regions. In such regions interactions between two or more instability modes are expected to be observed. The study of such interactions between modes is important because it can possibly shed light on pattern selection processes and on the scenario by which a macroscopic system becomes spatially and temporally disordered. We found that the Couette-Taylor system with a superimposed axial flow exhibits novel dynamical states which result from interactions between the PTV and two spiral modes. A wave number selection in these modes was also identified and studied.

The coiling of a thin filament of viscous fluid falling onto a surface is a common and easily reproducible hydrodynamic instability. Here we report for the first time that this instability can generate regular spiral patterns, in which air bubbles are trapped in the coil and then advected horizontally by the fluid spreading on the surface. We present a simple model that explains how these beautiful patterns are formed, and how the number of spiral branches and their curvature depends on the coiling frequency, the frequency of rotation of the coiling center, the total flow rate, and the thickness of the spreading fluid film.

New Journal of Physics, 2009

We present numerical simulations of closed wavy Taylor vortices and of helicoidal wavy spirals in the Taylor-Couette system. These wavy structures appearing via a secondary bifurcation out of Taylor vortex flow and out of spiral vortex flow, respectively, mediate transitions between Taylor and spiral vortices and vice versa. Structure, dynamics, stability and bifurcation behaviour are investigated in quantitative detail as a function of Reynolds numbers and wave numbers for counter-rotating as well as corotating cylinders. These results are obtained by solving the Navier-Stokes equations subject to axial periodicity for a radius ratio η = 0.5 with a combination of a finite differences method and a Galerkin method.

Physical Review E, 2013

Interactions between nearly bicritical modes in Taylor-Couette flow, which have been concerned with the framework of weakly nonlinear theory, are extended to fully nonlinear Navier-Stokes computation. For this purpose, a standard Newton solver for axially periodic flows is generalized to compute any mixed solutions having up to two phases, which typically arise from interactions of two spiral or Taylor vortex modes. Also, a simple theory is developed in order to classify the mixed solutions. With these methods, we elucidate pattern formation phenomena, which have been observed in a Taylor-Couette flow experiment. Focusing on the counter-rotating parameter range, all possible classes of interaction of various solutions with different azimuthal and axial wave numbers are considered within our computational restriction, and we observe numerous connection branches, e.g., footbridge solutions. Some of the mixed solutions result in a three-dimensional wavy spiral solution with axial relative periodicity or an axially doubly periodic toroidally closed vortex solution. The possible connection of the former solution family to spiral turbulence, which has been observed in highly counter-rotating Taylor-Couette flow, is discussed.

Physical Review Letters, 1999

The effect of weak inhomogeneity on spiral wave dynamics is studied within the framework of the two-dimensional complex Ginzburg-Landau equation description. The inhomogeneity gives spatial dependence to the frequency of spiral waves. This provides a mechanism for the formation of a dominant spiral domain that suppresses other spiral domains. The spiral vortices also slowly drift in the inhomogeneity, and results for the velocity are given. [S0031-9007(98)08283-0]

Physical Review E, 2011

We present numerical calculations of the linearized Navier-Stokes equations for axially extended and axially localized spiral structures in the Taylor-Couette system. The eigenvalue surface for spiral vortices with azimuthal wave number M = 2 shows significantly more structure than that for vortices with M = 0 and 1. Islands are found in parameter space where axially periodic vortex perturbations can grow. Bicriticality of different axial wave numbers is observed. Furthermore, parameter islands of absolute instability are found where wave packets consisting of near-critical extended perturbations can grow and expand via oppositely moving fronts. Some results are compared with those of the Ginzburg-Landau approximation.

Spiral waves are investigated in oscillatory media exhibiting period-doubling bifurcations. In the period-doubled and chaotic regimes, the rotational symmetry of the spiral wave is broken. The loss of symmetry takes the form of synchronization defect lines where the phase of the local oscillation changes by multiples of 2p. The internal structure and geometry of these synchronization defects is studied and a discussion of the possible types of defect lines is presented. [S0031-9007(97)

Fluid Dynamics Research, 2014

This work investigates the transition between different traveling helical waves (spirals, SPIs) in the setup of differentially independent rotating cylinders. We use direct numerical simulations to consider an infinite long and periodic Taylor-Couette apparatus with fixed axial periodicity length. We find so-called mixed-cross-spirals (MCSs), that can be seen as nonlinear superpositions of SPIs, to establish stable footbridges connecting SPI states. While bridging the bifurcation branches of SPIs, the corresponding contributions within the MCS vary continuously with the control parameters. Here discussed MCSs presenting footbridge solutions start and end in different SPI branches. Therefore they differ significantly from the already known MCSs that present bypass solutions (Altmeyer and Hoffmann 2010 New J. Phys. 12 113035). The latter start and end in the same SPI branch, while they always bifurcate out of those SPI branches with the larger mode amplitude. Meanwhile, these only appear within the coexisting region of both SPIs. In contrast, the footbridge solutions can also bifurcate out of the minor SPI contribution. We also find they exist in regions where only one of the SPIs contributions exists. In addition, MCS as footbridge solution can appear either stable or unstable. The latter detected transient solutions offer similar spatio-temporal characteristics to the flow establishing stable footbridges. Such transition processes are interesting for pattern-forming systems in general because they accomplish transitions between traveling waves of different azimuthal wave numbers and have not been described in the literature yet.

Physical Review E, 2009

A comprehensive numerical exploration of secondary finite-amplitude solutions in small-gap Taylor-Couette flow for high counter-rotating Reynolds numbers is provided, using Newton-Krylov methods embedded within arclength continuation schemes. Two different families of rotating waves have been identified: short axial wavelength subcritical spirals ascribed to centrifugal mechanisms and large axial scale supercritical spirals and ribbons associated with shear dynamics in the outer linearly stable radial region. This study is a first step taken in order to provide the inner structure of the skeleton of equilibria that may be responsible for the intermittent regime usually termed as spiral turbulence that has been reported by many experimentalists in the past.

Physics of Fluids, 2005

To investigate the nature of nonlinear waves appearing in an axially rotating pipe, we have performed a series of time-depending, three-dimensional numerical simulations of the incompressible Navier-Stokes equations in a rotating long pipe. As a difference with some previous works on the subject, which look for several given types of traveling wave solutions in pipes of infinite length, we leave the flow to evolve freely after a pressure difference is set between two points, one on each end of the finite rotating pipe. We use a recently developed numerical method that allows us to simulate numerically the three-dimensional flow produced in a pipe when Dirichlet boundary condition for the pressure is given on part of the inlet and outlet sections of the pipe. This technique is further improved here so that the pressure is only fixed at just one point on each one of the open boundaries of the pipe. Thus, no restrictions on the flow properties are given in these sections, allowing the free entrance and exit of possible waves through the pipe. We find that packets of traveling spiral waves are formed for values of the Reynolds numbers based on both the axial and the azimuthal velocities just above the critical ones given by the linear stability theory. These traveling waves have the same characteristics predicted by the linear stability theory and produce no significant mean flux defect. As the values of these parameters are increased above their critical values, the spiral waves become more involved and their amplitude increase, giving rise to a significant axial mean flow defect. For sufficiently high Reynolds numbers, we detect the apparition of spiral waves traveling also upstream, in agreement with the stability analysis for absolute instabilities. At the end, these traveling waves appearing above the onset for absolute instabilities transform into a standing spiral wave superimposed to the rotating Hagen-Poiseuille flow.