Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper: part 1.

In 1923, the Philosophical Transactions published G. I. Taylor's seminal paper on the stability of what we now call Taylor-Couette flow. In the century since the paper was published, Taylor's ground-breaking linear stability analysis of fluid flow between two rotating cylinders has had an enormous impact on the field of fluid mechanics. The paper's influence has extended to general rotating flows, geophysical flows and astrophysical flows, not to mention its significance in firmly establishing several foundational concepts in fluid mechanics that are now broadly accepted. This two-part issue includes review articles and research articles spanning a broad range of contemporary research areas, all rooted in Taylor's landmark paper. 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)'.

Experimental insights into elasto-inertial transitions in Taylor-Couette flows.

Since the seminal work of Taylor in 1923, Taylor-Couette (TC) flow has served as a paradigm to study hydrodynamic instabilities and bifurcation phenomena. Transitions of Newtonian TC flows to inertial turbulence have been extensively studied and are well understood, while in the past few years, there has been an increasing interest in TC flows of complex, viscoelastic fluids. The transitions to elastic turbulence (ET) or elasto-inertial turbulence (EIT) have revealed fascinating dynamics and flow states; depending on the rheological properties of the fluids, a broad spectrum of transitions has been reported, including rotating standing waves, flame patterns (FP), and diwhirls (DW). The nature of these transitions and the relationship between ET and EIT are not fully understood. In this review, we discuss experimental efforts on TC flows of viscoelastic fluids. We outline the experimental methods employed and the non-dimensional parameters of interest, followed by an overview of inertia, elasticity and elasto-inertia-driven transitions to turbulence and their modulation through shear thinning or particle suspensions. The published experimental data are collated, and a map of flow transitions to EIT as a function of the key fluid parameters is provided, alongside perspectives for the future work. 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)'.

Turbulent Taylor-Couette flow of dilute polymeric solutions: a 10-year retrospective.

This retrospective aims to present a coherent history of important findings in direct numerical simulations and experiments in turbulent Taylor-Couette (TC) flow of dilute polymeric solutions in the last decade. Specifically, the sequence of flow transitions due to a continuous increase of fluid elasticity from classical Newtonian, to inertially and in turn to elastically dominated, and finally to the inertialess purely elastic turbulence, is presented. In each elastically modified flow state, the drag modification, coherent flow structures, velocity and elastic stress statistics, mechanism of turbulent kinetic energy production, spectral features as well as the self-sustaining cycles of turbulence, are discussed. Finally, to provide a broader perspective, an overview of important similarities and differences between elastically induced turbulence in prototypical curvilinear and rectilinear shear flows including the curvature-free limit of TC flow, namely, the spanwise-rotating plane Couette flow, is presented. 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)'.

Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper: part 2.

In 1923, the Philosophical Transactions published G. I. Taylor's seminal paper on the stability of what we now call Taylor-Couette flow. In the century since the paper was published, Taylor's ground-breaking linear stability analysis of fluid flow between two rotating cylinders has had an enormous impact on the field of fluid mechanics. The paper's influence has extended to general rotating flows, geophysical flows and astrophysical flows, not to mention its significance in firmly establishing several foundational concepts in fluid mechanics that are now broadly accepted. This two-part issue includes review articles and research articles spanning a broad range of contemporary research areas, all rooted in Taylor's landmark paper. 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)'.

Flow regimes in a very wide-gap Taylor-Couette flow with counter-rotating cylinders.

In this study, we discuss the observed flow regimes in Taylor-Couette flow of radius ratio [Formula: see text] for various Reynolds numbers up to [Formula: see text]. We investigate the flow using a visualization method. The flow states in the centrifugally unstable flow are investigated in the case of counter-rotating cylinders and pure inner cylinder rotation. Beside classical known flow states as Taylor-vortex flow and wavy vortex flow, we observe a variety of new flow structures in the cylindrical annulus, especially for the transition to turbulence. Coexisting turbulent and laminar regions inside the system are observed. Turbulent spots and turbulent bursts are observed, as well as an irregular Taylor-vortex flow and non-stationary turbulent vortices. Especially, a single axially aligned columnar vortex between the inner and outer cylinder is found. The principal regimes observed in flow between independently rotating cylinders are summarized in a flow-regime diagram. 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)'.

Recent developments of turbulent emulsions in Taylor-Couette flow.

Emulsions are common in many natural and industrial settings. Recently, much attention has been paid to understanding the dynamics of turbulent emulsions. This paper reviews some recent studies of emulsions in turbulent Taylor-Couette flow, mainly focusing on the statistics of the dispersed phase and the global momentum transport of the system. We first study the size distribution and the breakup mechanism of the dispersed droplets for turbulent emulsions with a low volume-fraction (dilute) of the dispersed phase. For systems with a high volume-fraction (dense) of the dispersed phase, we address the detailed response of the global transport (effective viscosity) of the turbulent emulsion and its connection to the droplet statistics. Finally, we will discuss catastrophic phase inversions, which can happen when the volume-fraction of the dispersed phase exceeds a critical value during dynamic emulsification. We end the manuscript with a summary and an outlook including some open questions for future research. 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)'.

End effects in low aspect ratio Taylor-Couette flow.

Taylor-Couette flow with a low aspect ratio cylinder suffers from end effects due to the finite-span of the gap between the cylinder sides and the secondary flow in the region below the inner cylinder. We experimentally explore these end effects by varying the cylinder aspect ratio between 6.67 and 40 for a range of wall gap widths and bottom gap heights. For these geometries, end effects (i.e. non-ideal Taylor-Couette flow) can be substantial due to both features of the finite-span and the bottom secondary flow. In some cases, the finite-span effects extended between 20% and 30% of the way into the Taylor-Couette flow region, and the secondary flow at the bottom accounted for nearly half of the total measured torque. By taking these effects into consideration, our high aspect ratio results agreed well with those obtained by Taylor (Taylor 1936 Proc. R. Soc. Lond. A 157, 546-564. (doi:10.1098/rspa.1936.0215)) at considerably higher aspect ratios. 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)'.

Generalized quasi-linear approximation and non-normality in Taylor-Couette spiral turbulence.

Taylor-Couette flow is well known to admit a spiral turbulence state in which laminar and turbulent patches coexist around the cylinder. This flow state is quite complex, with delicate internal structure, and it can be traced into certain regimes of linear stability. This behaviour is believed to be connected to the non-normality of the linear operator, which is itself a function of the control parameters. Using spiral turbulence in both linearly stable and unstable regimes, we investigate the effectiveness of the generalized quasi-linear approximation (GQL), an extension of quasi-linear theory designed to capture the essential aspects of turbulent flows. We find that GQL performs much better in the supercritical regime than the subcritical. By including only a small number of modes in the nonlinear interactions, GQL simulations maintain a turbulent-like state when in the supercritical regime. However, a much larger number is required to avoid returning to the laminar state when in the subcritical regime. 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)'.

Exploring the origin of turbulent Taylor rolls.

Since Taylor's seminal paper, the existence of large-scale quasi-axisymmetric structures has been a matter of interest when studying Taylor-Couette flow. In this article, we probe their formation in the highly turbulent regime by conducting a series of numerical simulations at a fixed Reynolds number [Formula: see text] while varying the Coriolis parameter to analyse the flow characteristics as the structures arise and dissipate. We show how the Coriolis force induces a one-way coupling between the radial and azimuthal velocity fields inside the boundary layer, but in the bulk, there is a two-way coupling that causes competing effects. We discuss how this complicates the analogy of narrow-gap Taylor-Couette to other convective flows. We then compare these statistics with a similar shear flow without no-slip boundary layers, showing how this double coupling causes very different effects. We finish by reflecting on the possible origins of turbulent Taylor rolls. 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)'.

Taylor-Couette flow for astrophysical purposes.

A concise review is given of astrophysically motivated experimental and theoretical research on Taylor-Couette flow. The flows of interest rotate differentially with the inner cylinder faster than the outer, but are linearly stable against Rayleigh's inviscid centrifugal instability. At shear Reynolds numbers as large as [Formula: see text], hydrodynamic flows of this type (quasi-Keplerian) appear to be nonlinearly stable: no turbulence is seen that cannot be attributed to interaction with the axial boundaries, rather than the radial shear itself. Direct numerical simulations agree, although they cannot yet reach such high Reynolds numbers. This result indicates that accretion-disc turbulence is not purely hydrodynamic in origin, at least insofar as it is driven by radial shear. Theory, however, predicts linear magnetohydrodynamic (MHD) instabilities in astrophysical discs: in particular, the standard magnetorotational instability (SMRI). MHD Taylor-Couette experiments aimed at SMRI are challenged by the low magnetic Prandtl numbers of liquid metals. High fluid Reynolds numbers and careful control of the axial boundaries are required. The quest for laboratory SMRI has been rewarded with the discovery of some interesting inductionless cousins of SMRI, and with the recently reported success in demonstrating SMRI itself using conducting axial boundaries. Some outstanding questions and near-future prospects are discussed, especially in connection with astrophysics. 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)'.

Unified torque scaling in counter-rotating suspension Taylor-Couette flow.

Torque measurements are reported for the Taylor-Couette flow of a neutrally buoyant non-colloidal suspension in the counter-rotation regime up to a particle volume fraction of [Formula: see text]. A unified scaling relation for the dimensionless torque (the pseudo-Nusselt number) of the form [Formula: see text] is shown to hold over a range of Taylor numbers [Formula: see text] that covers primary, secondary and tertiary bifurcating states; here, [Formula: see text] is the reduced Taylor number, [Formula: see text] is the critical Taylor number at primary bifurcation and [Formula: see text] is the relative viscosity of the suspension. Possible effects of flow transitions and inhomogeneous distribution of particles on torque scaling are discussed. 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)'.

Taylor-Couette flow of hard-sphere suspensions: overview of current understanding.

Although inertial particle-laden flows occur in a wide range of industrial and natural processes, there is both a lack of fundamental understanding of these flows and continuum-level governing equations needed to predict transport and particle distribution. Towards this effort, the Taylor-Couette flow (TCF) system has been used recently to study the flow behaviour of particle-laden fluids under inertia. This article provides an overview of experimental, theoretical and computational work related to the TCF of neutrally buoyant non-Brownian suspensions, with an emphasis on the effect of finite-sized particles on the series of flow transitions and flow structures. Particles, depending on their size and concentration, cause several significant deviations from Newtonian fluid behaviour, including shifting the Reynolds number corresponding to transitions in flow structure and changing the possible structures present in the flow. Furthermore, particles may also migrate depending on the flow structure, leading to hysteretic effects that further complicate the flow behaviour. The current state of theoretical and computational modelling efforts to describe the experimental observations is discussed, and suggestions for potential future directions to improve the fundamental understanding of inertial particle-laden flows are provided. 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)'.

Transition to chaos and modal structure of magnetized Taylor-Couette flow.

Taylor-Couette flow (TCF) is often used as a simplified model for complex rotating flows in the interior of stars and accretion discs. The flow dynamics in these objects is influenced by magnetic fields. For example, quasi-Keplerian flows in Taylor-Couette geometry become unstable to a travelling or standing wave in an external magnetic field if the fluid is conducting; there is an instability even when the flow is hydrodynamically stable. This magnetorotational instability leads to the development of chaotic states and, eventually, turbulence, when the cylinder rotation is sufficiently fast. The transition to turbulence in this flow can be complex, with the coexistence of parameter regions with spatio-temporal chaos and regions with quasi-periodic behaviour, involving one or two additional modulating frequencies. Although the unstable modes of a periodic flow can be identified with Floquet analysis, here we adopt a more flexible equation-free data-driven approach. We analyse the data from the transition to chaos in the magnetized TCF and identify the flow structures related to the modulating frequencies with dynamic mode decomposition; this method is based on approximating nonlinear dynamics with a linear infinite-dimensional Koopman operator. With the use of these structures, one can construct a nonlinear reduced model for the transition. 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)'.

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)'.

Dispersion and mixing dynamics of complex oil-in-water emulsions in Taylor-Couette flows.

The seminal study by G. I. Taylor (1923) has inspired generations of work in exploring and characterizing Taylor-Couette (TC) flow instabilities and laid the foundation for research of complex fluid systems requiring a controlled hydrodynamic environment. Here, TC flow with radial fluid injection is used to study the mixing dynamics of complex oil-in-water emulsions. Concentrated emulsion simulating oily bilgewater is radially injected into the annulus between rotating inner and outer cylinders, and the emulsion is allowed to disperse through the flow field. The resultant mixing dynamics are investigated, and effective intermixing coefficients are calculated through measured changes in the intensity of light reflected by the emulsion droplets in fresh and salty water. The impacts of the flow field and mixing conditions on the emulsion stability are tracked via changes in droplet size distribution (DSD), and the use of emulsified droplets as tracer particles is discussed in terms of changes in the dispersive Péclet, Capillary and Weber numbers. For oily wastewater systems, the formation of larger droplets is known to yield better separation during a water treatment process, and the final DSD observed here is found to be tunable based on salt concentration, observation time and mixing flow state in the TC cell. 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)'.

Arrow-shaped elasto-inertial rotating waves.

We present direct numerical simulations of the Taylor-Couette flow of a dilute polymer solution when only the inner cylinder rotates and the curvature of the system is moderate ([Formula: see text]). The finitely extensible nonlinear elastic-Peterlin closure is used to model the polymer dynamics. The simulations have revealed the existence of a novel elasto-inertial rotating wave characterized by arrow-shaped structures of the polymer stretch field aligned with the streamwise direction. This rotating wave pattern is comprehensively characterized, including an analysis of its dependence on the dimensionless Reynolds and Weissenberg numbers. Other flow states having arrow-shaped structures coexisting with other types of structures have also been identified for the first time in this study and are briefly discussed. 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)'.

Taylor-Couette flow in the narrow-gap limit.

A Cartesian representation of the Taylor-Couette system in the vanishing limit of the gap between coaxial cylinders is presented, where the ratio, [Formula: see text], of the angular velocities, [Formula: see text] and [Formula: see text], of the inner and the outer cylinders, respectively, affects its axisymmetric flow structures. Our numerical stability study finds remarkable agreement with previous studies for the critical Taylor number, [Formula: see text], for the onset of axisymmetric instability. The Taylor number [Formula: see text] can be expressed as [Formula: see text], where [Formula: see text] (the rotation number) and [Formula: see text] (the Reynolds number) in the Cartesian system are related to the average and the difference of [Formula: see text] and [Formula: see text]. The instability sets in the region [Formula: see text], while the product of [Formula: see text] and [Formula: see text] is kept finite. Furthermore, we developed a numerical code to calculate nonlinear axisymmetric flows. It is found that the mean flow distortion of the axisymmetric flow is antisymmetric across the gap when [Formula: see text], while a symmetric part of the mean flow distortion appears additionally when [Formula: see text]. Our analysis also shows that for a finite [Formula: see text] all flows with [Formula: see text] approach the [Formula: see text] axis, so that the plane Couette flow system is recovered in the vanishing gap limit. 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)'.

Transitions in Taylor-Couette flow of concentrated non-colloidal suspensions.

Taylor-Couette flow of concentrated non-colloidal suspensions with a rotating inner cylinder and a stationary outer one is numerically investigated. We consider suspensions of the bulk particle volume fraction ϕb = 0.2, 0.3 with the ratio of annular gap to the particle radius ε = 60 confined in a cylindrical annulus of the radius ratio (i.e. ratio of inner and outer radii) η = 0.877. Numerical simulations are performed by applying suspension-balance model and rheological constitutive laws. To observe flow patterns caused by suspended particles, the Reynolds number of the suspension, based on the bulk particle volume fraction and the rotating velocity of the inner cylinder, is varied up to 180. At high Reynolds number, modulated patterns undiscovered in the flow of a semi-dilute suspension emerge beyond a wavy vortex flow. Thus, a transition occurs from the circular Couette flow via ribbons, spiral vortex flow, wavy spiral vortex flow, wavy vortex flow and modulated wavy vortex flow for the concentrated suspensions. Moreover, friction and torque coefficients for suspensions are estimated. It turns out that suspended particles significantly enhance the torque on the inner cylinder while reducing friction coefficient and the pseudo-Nusselt number. In particular, the coefficients are reduced in the flow of more dense suspensions. 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)'.

Routes to turbulence in Taylor-Couette flow.

Fluid flows between rotating concentric cylinders exhibit two distinct routes to turbulence. In flows dominated by inner-cylinder rotation, a sequence of linear instabilities leads to temporally chaotic dynamics as the rotation speed is increased. The resulting flow patterns occupy the whole system and sequentially lose spatial symmetry and coherence in the transition process. In flows dominated by outer-cylinder rotation, the transition is abrupt and leads directly to turbulent flow regions that compete with laminar ones. We here review the main features of these two routes to turbulence. Bifurcation theory rationalizes the origin of temporal chaos in both cases. However, the catastrophic transition of flows dominated by outer-cylinder rotation can only be understood by accounting for the spatial proliferation of turbulent regions with a statistical approach. We stress the role of the rotation number (the ratio of Coriolis to inertial forces) and show that it determines the lower border for the existence of intermittent laminar-turbulent patterns. 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)'.

A parameter study of strato-rotational low-frequency modulations: impacts on momentum transfer and energy distribution.

Previous comparisons of experimental data with nonlinear numerical simulations of density stratified Taylor-Couette (TC) flows revealed nonlinear interactions of strato-rotational instability (SRI) modes that lead to periodic changes in the SRI spirals and their axial propagation. These pattern changes are associated with low-frequency velocity modulations that are related to the dynamics of two competing spiral wave modes propagating in opposite directions. In the present paper, a parameter study of the SRI is performed using direct numerical simulations to evaluate the influence of the Reynolds numbers, the stratification, and of the container geometry on these SRI low-frequency modulations and spiral pattern changes. The results of this parameter study show that the modulations can be considered as a secondary instability that are not observed for all SRI unstable regimes. The findings are of interest when the TC model is related to star formation processes in accretion discs. 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)'.