Counter-rotating Taylor-Couette Flow Research Articles

Mean structure of the supercritical turbulent spiral in Taylor-Couette flow.

The large-scale laminar/turbulent spiral patterns that appear in the linearly unstable regime of counter-rotating Taylor-Couette flow are investigated from a statistical perspective by means of direct numerical simulation. Unlike the vast majority of previous numerical studies, we analyse the flow in periodic parallelogram-annular domains, following a coordinate change that aligns one of the parallelogram sides with the spiral pattern. The domain size, shape and spatial resolution have been varied and the results compared with those in a sufficiently large computational orthogonal domain with natural axial and azimuthal periodicity. We find that a minimal parallelogram of the right tilt significantly reduces the computational cost without notably compromising the statistical properties of the supercritical turbulent spiral. Its mean structure, obtained from extremely long time integrations in a co-rotating reference frame using the method of slices, bears remarkable similarity with the turbulent stripes observed in plane Couette flow, the centrifugal instability playing only a secondary role. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical transactions paper (Part 2)'.

Self-sustainment of coherent structures in counter-rotating Taylor–Couette flow

We investigate the local self-sustained process underlying spiral turbulence in counter-rotating Taylor–Couette flow using a periodic annular domain, shaped as a parallelogram, two of whose sides are aligned with the cylindrical helix described by the spiral pattern. The primary focus of the study is placed on the emergence of drifting–rotating waves (DRW) that capture, in a relatively small domain, the main features of coherent structures typically observed in developed turbulence. The transitional dynamics of the subcritical region, far below the first instability of the laminar circular Couette flow, is determined by the upper and lower branches of DRW solutions originated at saddle-node bifurcations. The mechanism whereby these solutions self-sustain, and the chaotic dynamics they induce, are conspicuously reminiscent of other subcritical shear flows. Remarkably, the flow properties of DRW persist even as the Reynolds number is increased beyond the linear stability threshold of the base flow. Simulations in a narrow parallelogram domain stretched in the azimuthal direction to revolve around the apparatus a full turn confirm that self-sustained vortices eventually concentrate into a localised pattern. The resulting statistical steady state satisfactorily reproduces qualitatively, and to a certain degree also quantitatively, the topology and properties of spiral turbulence as calculated in a large periodic domain of sufficient aspect ratio that is representative of the real system.

Second-Order Phase Transition in Counter-Rotating Taylor-Couette Flow Experiment.

In many basic shear flows, such as pipe, Couette, and channel flow, turbulence does not arise from an instability of the laminar state, and both dynamical states co-exist. With decreasing flow speed (i.e., decreasing Reynolds number) the fraction of fluid in laminar motion increases while turbulence recedes and eventually the entire flow relaminarizes. The first step towards understanding the nature of this transition is to determine if the phase change is of either first or second order. In the former case, the turbulent fraction would drop discontinuously to zero as the Reynolds number decreases while in the latter the process would be continuous. For Couette flow, the flow between two parallel plates, earlier studies suggest a discontinuous scenario. In the present study we realize a Couette flow between two concentric cylinders which allows studies to be carried out in large aspect ratios and for extensive observation times. The presented measurements show that the transition in this circular Couette geometry is continuous suggesting that former studies were limited by finite size effects. A further characterization of this transition, in particular its relation to the directed percolation universality class, requires even larger system sizes than presently available.

Patterns in Wall-Bounded Shear Flows

Experiments and numerical simulations have shown that turbulence in transitional wall-bounded shear flows frequently takes the form of long oblique bands if the domains are sufficiently large to accommodate them. These turbulent bands have been observed in plane Couette flow, plane Poiseuille flow, counter-rotating Taylor–Couette flow, torsional Couette flow, and annular pipe flow. At their upper Reynolds number threshold, laminar regions carve out gaps in otherwise uniform turbulence, ultimately forming regular turbulent–laminar patterns with a large spatial wavelength. At the lower threshold, isolated turbulent bands sparsely populate otherwise laminar domains, and complete laminarization takes place via their disappearance. We review results for plane Couette flow, plane Poiseuille flow, and free-slip Waleffe flow, focusing on thresholds, wavelengths, and mean flows, with many of the results coming from numerical simulations in tilted rectangular domains that form the minimal flow unit for the turbulent–laminar bands.

Large-eddy simulation of counter-rotating Taylor–Couette flow: The effects of angular velocity and eccentricity

In the present work, turbulent flow in the annulus of a counter-rotating Taylor-Couette (CRTC) system is studied using large-eddy simulation. The numerical methodology employed is validated, for both the mean and second-order statistics, with the direct numerical simulation (DNS) data available in the literature, for a range of Reynolds numbers from 500 to 4000. Thereafter, turbulent flow occurring in this system at Reynolds numbers of 8000 and 16000 are studied, and the results obtained are analyzed using mean and second-order statistics, vortical structures, velocity vector plots and power energy spectra. Further, the spatio-temporal variation of azimuthal velocity, extracted near the inner cylinder, shows the existence of herringbone like patterns similar to that observed in the previous studies. The effect of eccentricity of the inner cylinder with respect to the outer cylinder is studied, on the turbulent flow in the CRTC system, for two different eccentricity ratios of 0.2 and 0.5 and for two different Reynolds numbers of 1500 and 4000. The results of the eccentric CRTC are analyzed using contours of pressure, mean and second-order statistics, velocity vectors, vortical structures, and turbulence anisotropy maps. It is observed from the eccentric CRTC simulations that the smaller-gap region seems to contain higher amplitude fluctuations and more vortical structures when compared with the larger-gap region. The mean turbulent kinetic energy contours do not change qualitatively with the Reynolds number, however, quantitatively a higher turbulent kinetic energy is observed in the higher Reynolds number case of 4000.

DNS of Taylor–Couette flow between counter-rotating cylinders at small radius ratio

A counter-rotating Taylor-Couette flow with relatively small radius ratios of $\eta$ = 0.2--0.5 was investigated over a wide range of the Reynolds number, from laminar to turbulent regime, by means of three-dimensional direct numerical simulations. We investigated the $\eta$ dependence of the flow structure and determined a critical value between $\eta$ = 0.2 and 0.3, below which, the stable outer cylinder side exhibited a modal structure that was different from the Taylor-vortex flow on the inner side. At $\eta \geq$ 0.3, the Taylor-vortex on the unstable inner side dominated the entire flow field between the cylinders, whose footprints were observed in the vicinity of the outer cylinder wall. However, for $\eta$ = 0.2, the influence from the inner side was limited up to the centre of the cylinder gap. Moreover, on the stable outer cylinder side, there appeared a modal structure that was axially homogeneous, azimuthally periodic, and similar to the Tollmien-Schlichting (TS) instability wave. As the Reynolds number increased with a fixed $\eta$ = 0.2, the modal structure changed its azimuthal wavenumber and thickened radially in the wall unit. Although the Reynolds shear stress on the outer side remained approximately zero, the intensity of the velocity fluctuations was comparable to the Taylor-vortex flows in the central part.

Subcritical Equilibria in Taylor-Couette Flow

Nonlinear equilibrium states characterized by strongly localized vortex pairs are calculated in the linearly stable parameter region of counterrotating Taylor-Couette flow. These subcritical states are rotating waves whose region of existence is consistent with the critical threshold for relaminarization observed in experiments. For sufficiently rapid outer cylinder rotation the solutions extend beyond the static inner cylinder case to corotation, thus exceeding, for the first time, the boundary defined by the inviscid Rayleigh's stability criterion.

Axisymmetric propagating vortices in centrifugally stable Taylor–Couette flow

AbstractWe present numerical as well as experimental results of axisymmetric, axially propagating vortices appearing in counter-rotating Taylor–Couette flow below the centrifugal instability threshold of circular Couette flow without additional externally imposed forces. These propagating vortices are periodically generated by the shear flow near the Ekman cells that are induced by the non-rotating end walls. These axisymmetric vortices propagate into the bulk towards mid-height, where they get annihilated by rotating, non-propagating defects. These propagating structures appear via a supercritical Hopf bifurcation from axisymmetric, steady vortices, which have been discovered recently in centrifugally stable counter-rotating Taylor–Couette flow (Abshagen et al., Phys. Fluids, vol. 22, 2010, 021702). In the nonlinear regime of the Hopf bifurcation, contributions of non-axisymmetric modes also appear.

Fully nonlinear mode competitions of nearly bicritical spiral or Taylor vortices in Taylor-Couette flow

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.

Measurements of mean flow and turbulence characteristics in high-Reynolds number counter-rotating Taylor-Couette flow

Particle image velocimetry was used for measuring the velocity and Reynolds stress distributions in the latitudinal plane of counter-rotating Taylor-Couette flow at high Reynolds numbers (Re). The ratio of outer to inner cylinder angular velocity, μ, varied between −10.79 and −0.68, and Rei based on the inner cylinder velocity ranged between 2635 and 40 446, substantially extending previously available data. The results were used for examining scaling trends, especially the effects of Re and μ on the mean flow and turbulence statistics. We showed that using a kind of “inner wall” scaling, μ was the primary parameter controlling the normalized profiles of mean velocity, Reynolds stresses, TKE production and dissipation rates. Re effects on the scaled profiles were much smaller. Increasing μ flattened the mean azimuthal velocity profiles in the center of the annulus, increased the radial velocity gradients near the walls, and moved the radial point at which the velocity changed sign towards the outer cylinder. The flow also became more turbulent and a log layer with increasing extent developed near the inner wall. All the Reynolds stress components, along with the TKE production and dissipation rates peaked near the inner wall. Raising μ extended the high turbulence levels deeper into the annulus. At low μ, the stabilizing effect of the outer cylinder kept the flow in the outer regions laminar. Only when the magnitude of the inner cylinder angular velocity equaled or exceeded that of the outer one, the Reynolds stresses remained significant across the entire measurement range, and started increasing also near the outer cylinder. The azimuthal energy spectra confirmed these trends and showed that the changes to turbulence levels occurred at a broad range of scales. Furthermore, for low μ, the instantaneous vorticity fields were dominated by nearly parallel, elongated, counter-rotating vorticity contours, reminiscent of inclined counter-rotating vortex pairs. At high μ, more randomly distributed structures were generated near both walls, and eventually filled the whole annulus.

Families of subcritical spirals in highly counter-rotating Taylor-Couette flow

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.

Localized spirals in Taylor-Couette flow

We present a type of spiral vortex state that appears from a supercritical Hopf bifurcation below the linear instability of circular Couette flow in a Taylor-Couette system with rigid end plates. These spirals have been found experimentally as well as numerically as "pure" states but also coexist with "classical" spirals (or axially standing waves for smaller systems) which typically appear from linear instability in counterrotating Taylor-Couette flow. These spiral states have an axial distribution of the strongly localized amplitude in the vicinity of the rigid end plates that confine the system in the axial direction. Furthermore, they show significantly different oscillation frequencies compared to the critical spiral frequencies. Despite the localization of the amplitude near the ends, the states appear as global states with spirals that propagate either toward the middle from each end of the system or vice versa. In contrast to classical spirals, these states exhibit a spatial or a spatiotemporal reflection symmetry.

The effect of physical boundaries on oscillatory bifurcation in counterrotating Taylor–Couette flow

The results of an experimental study on the bifurcation structure of oscillatory modes in counterrotating Taylor–Couette flow with stationary end plates are presented. It is shown that the cylinder length L acts as an important geometric control parameter of the system. As a result of a supercritical Hopf bifurcation it is found that for an aspect ratio Γ=L/d>16 (d gap width) only spiral vortices appear in basic laminar flow. For Γ<10.5 spiral vortices are almost entirely replaced by two types of standing waves called SW0 and SWπ as supercritical oscillatory flow. Experimental evidence is presented that the mode exchange between standing waves SW0 and SWπ is governed by underlying Ekman induced vortices which appear as a result of stationary end plates in the flow. In this regime spiral vortices appear only from a sub- or supercritical symmetry breaking bifurcation of the standing waves. Within an “intermediate regime” between 10.5⩽Γ⩽16 spiral vortices are found to be the predominant primary oscillatory flow but small stability intervals of standing waves are also observed. Surprisingly, the experimentally determined critical Reynolds number is found to deviate not more than 2% from the numerical values for all values of aspect ratio even though they are calculated under the assumption of infinite axial length. Moreover, the critical oscillation frequency is also in agreement with the numerical values and is independent from Γ.

Standing waves in flow between finite counterrotating cylinders.

Experimental evidence for standing waves resulting from a supercritical Hopf bifurcation that appears as the first pattern-forming instability in counterrotating Taylor-Couette flow is presented. Depending on the aspect ratio two different types of standing waves, denoted as SW0 and SW(pi), could be observed. Both modes have an azimuthal wave number m=1 but differ in symmetry. While for SW(pi), a spatiotemporal glide-reflection symmetry could be found, SW0 is purely spatial reflection symmetric. The transition between the two modes is found to be organized in a cusp bifurcation unfolded by variations of the aspect ratio. The "classical" spiral vortex flow appears in this control parameter regime only as a result of a secondary steady bifurcation from SW0. This transition is found to be either subcritical or supercritical. The experimentally observed bifurcation structure has been predicted by theory of Hopf bifurcation to spiral vortex flow in finite counterrotating Taylor-Couette systems.