Turbulent flow induced by elastorotational instability in viscoelastic Taylor–Couette flow (TCF) with Keplerian rotation is analogous to a turbulent accretion disk destabilized by magnetorotational instability. We examine this novel viscoelastic Keplerian turbulence via direct numerical simulations (DNS) for the shear Reynolds number ($Re$) ranging from $10^2$ to $10^4$. The observed characteristic flow structure consists of penetrating streamwise vortices with axial length scales much smaller than the gap width, distinct from the classic centrifugally induced Taylor vortices, which have axial lengths of the gap width. These intriguing vortices persist for the wide $Re$ range considered and give rise to intriguing scaling behaviour in key flow quantities. Specifically, the characteristic axial length of the penetrating vortices is shown to scale as $Re^{-0.22}$; the angular momentum transport scales as $Re^{0.42}$; the kinetic and elastic boundary-layer thicknesses based on angular velocity and hoop stress near the inner cylinder wall scale as $Re^{-0.48}$ and $Re^{-0.49}$, respectively. This implies that the viscoelastic Keplerian turbulence belongs to the classical turbulent regime of TCF with the Prandtl–Blasius-type boundary layer. Furthermore, we present an analytical relation between the viscous and elastic dissipation rates of kinetic energy and the angular momentum transport and in turn demonstrate its validity using our DNS data. This study has paved the way for future research to explore astrophysics-related Keplerian turbulence and angular momentum transport via the scaling relations of the analogous TCF of dilute polymeric solutions.

Reducing drag under high turbulence is a critical but challenging issue that has engendered great concern. This study utilizes hydrophilic tips in superhydrophobic (SHP) grooves to enhance the stability of plastron, which results in a considerable drag reduction ($DR$) up to 62 %, at Reynolds number ($Re$) reaching $2.79 \times 10^{4}$. The effect of the spacing width $w$ of the microgrooves on both $DR$ and flow structures is investigated. Experimental results demonstrate that $DR$ increases as either microgroove spacing $w$ or $Re$ increases. The velocity fields obtained using particle image velocimetry indicate that the air-filled SHP grooves induce a considerable wall slip. This slip significantly weakens the intensity of Taylor rolls, reduces local momentum transport, and consequently lowers drag. This phenomenon becomes more pronounced with increasing $w$. Furthermore, to quantify the multiscale relationship between global response and geometrical as well as driving parameters, $DR\sim (w, \phi _s, Re)$, a theoretical model is established based on angular momentum defect theory and magnitude estimate. It is demonstrated that a decrease in the surface solid fraction can reduce wall shear, and an increase in the groove width can weaken turbulence kinetic energy production, rendering enhanced slip and drag reduction. This research has implications for designing and optimizing turbulent-drag-reducing surfaces in various engineering applications, such as transportation and marine engineering.

Quasi-Keplerian flow, a special regime of Taylor–Couette co-rotating flow, is of great astrophysical interest for studying angular momentum transport in accretion disks. The well-known magnetorotational instability (MRI) successfully explains the flow instability and generation of turbulence in certain accretion disks, but fails to account for these phenomena in protoplanetary disks where magnetic effects are negligible. Given the intrinsic decrease of the temperature in these disks, we examine the effect of radial thermal stratification on three-dimensional global disturbances in linearised quasi-Keplerian flows under radial gravitational acceleration mimicking stellar gravity. Our results show a thermo-hydrodynamic linear instability for both axisymmetric and non-axisymmetric modes across a broad parameter space of the thermally stratified quasi-Keplerian flow. Generally, a decreasing Richardson or Prandtl number stabilises the flow, while a reduced radius ratio destabilises it. This work also provides a quantitative characterisation of the instability. At low Prandtl numbers $Pr$, we observe a scaling relation of the linear critical Taylor number $Ta_c\propto Pr^{-6/5}$. Extrapolating the observed scaling to high $Ta$ and low $Pr$ may suggest the relevance of the instability to accretion disks. Moreover, even slight thermal stratification, characterised by a low Richardson number, can trigger the flow instability with a small axial wavelength. These findings are qualitatively consistent with the results from a traditional local stability analysis based on short wave approximations. Our study refines the thermally induced linearly unstable transition route in protoplanetary disks to explain angular momentum transport in dead zones where MRI is ineffective.

The first experimental results on pattern transitions in the co-rotation regime (i.e. the rotation ratio $\varOmega = \omega _o/\omega _i > 0$, where $\omega _i$ and $\omega _o$ are the angular speeds of the inner and outer cylinders, respectively) of the Taylor–Couette flow (TCF) are reported for a neutrally buoyant suspension of non-colloidal particles, up to a particle volume fraction of $\phi = 0.3$. While the stationary Taylor vortex flow (TVF) is the primary bifurcating state in dilute suspensions ($\phi \leq ~0.05$), the non-axisymmetric oscillatory states, such as the spiral vortex flow (SVF) and the ribbon (RIB), appear as primary bifurcations with increasing particle loading, with an overall de-stabilization of the primary bifurcating states (TVF/SVF/RIB) being found with increasing $\phi$ for all $\varOmega \geq ~0$. At small co-rotations ($\varOmega \sim 0$), the particles play the dual role of stabilization ($\phi < 0.1$) and destabilization ($\phi \geq ~0.1$) on the secondary/tertiary oscillatory states. The distinctive features of the ‘particle-induced’ spiral vortices are identified and contrasted with those of the ‘fluid-induced’ spirals that operate in the counter-rotation regime.

The effects of thermal convection on turbulence in accretion discs, and particularly its interplay with the magnetorotational instability (MRI), are of significant astrophysical interest. Despite extensive theoretical and numerical studies, such an interplay has not been explored experimentally. We conduct linear analysis of the azimuthal version of MRI (AMRI) in the presence of thermal convection and compare the results with our experimental data published before. We show that the critical Hartmann number ($Ha$) for the onset of AMRI is reduced by convection. Importantly, convection breaks symmetry between $m = \pm 1$ instability modes ($m$ is the azimuthal wavenumber). This preference for one mode over the other makes the AMRI wave appear as a ‘one-winged butterfly’.

We study the stationary Navier–Stokes equations in the region between two rotating concentric cylinders. We first prove that, for a small Reynolds number, if the fluid flow is axisymmetric and if its velocity is sufficiently small in the $L^\infty$-norm, then it is necessarily the Taylor–Couette–Poiseuille flow. If, in addition, the associated pressure is bounded or periodic in the $z$ axis, then it coincides with the well-known canonical Taylor–Couette flow. We discuss the relation between uniqueness and stability of such a flow in terms of the Taylor number in the case of narrow gap of two cylinders. The investigation in comparison with two Reynolds numbers based on inner and outer cylinder rotational velocities is also conducted. Next, we give a certain bound of the Reynolds number and the $L^\infty$-norm of the velocity such that the fluid is, indeed, necessarily axisymmetric. As a result, it is clarified that smallness of Reynolds number of the fluid in the two rotating concentric cylinders governs both axisymmetry and the Taylor–Couette–Poiseuille flow with the exact form of the pressure.

Direct numerical simulations of spiral Poiseuille flows in a narrow gap geometry are performed with the aim of identifying the mechanisms governing the dynamics of the axial friction coefficient. The investigation has explored a small portion of the Reynolds number–Taylor number phase space ($600 \leq Re \leq 5766$ and $1500 \leq Ta \leq 5000$), for which reference experimental results are available. The study is focused on the mechanism leading to the enhancement of the axial friction coefficient with the Taylor number when the Reynolds number is kept constant. The analysis of the spatial distribution of the Reynolds stress tensor and of the turbulent energy budget has evidenced the key role of the pressure–strain correlation in the energy transfer from the azimuthal to the axial component. The latter eventually determines the increase of the axial friction coefficient through the enhanced radial mixing of axial momentum. Data have also shown that the flow dynamics is heavily dependent on the $Ta/Re$ ratio, and different regimes develop (ranging from laminar to turbulent), each with peculiar behaviours.

Flow between axially rotating concentric cylinders is well known to exhibit rich dynamics. Hence, Taylor instabilities have been studied, both experimentally and theoretically, for many years. Although usually studied in the abstract, such geometries arise in a range of practical situations including drilling, when a drilling fluid flow enters a well via a pipe that is the centre body and returns via the annulus between the pipe and the borehole wall. In drilling, the centre body rotates and the annular flow contains rock cuttings. Here, we report the development of an Eulerian-Eulerian solver, based on OpenFOAM, that solves for this cuttings transport problem in the presence of both gravity and Taylor vortices. To check the reliability of the solver, we conduct a set of experiments spanning a wide range of complex flow regimes. We show that the model successfully predicts, in all regimes, the observed complex redistribution of particulates. However, for suspension flows under viscously dominated conditions, high particle concentrations and in rectilinear flow, particle pressure and normal stress differences are sufficient to capture particle migration. Results show that in more complex flows exemplified by the Taylor–Couette flow studied here, more realistic predictions of non-Brownian particle migration require inclusion of forces arising through the relative velocity of the two phases including lift forces originating both from inertia and particle rotation.

A linear stability analysis of circular Couette flow of a Bingham fluid subjected to axisymmetric perturbations was presented by Peng & Zhu (J. Fluid Mech., vol. 512, 2004, pp. 21–45) and Landry et al. (J. Fluid Mech., vol. 560, 2006, pp. 321–353). Here, we consider the stability of this flow with respect to a finite amplitude perturbation. We focus particularly on the case where the basic flow has an unyielded fluid layer on outer cylinder. A weakly nonlinear stability analysis is developed for a wide gap and a narrow gap. A third-order Ginzburg–Landau equation is derived, and the influence of the different nonlinearities on bifurcation features is investigated in detail. The results indicate that: (i) the nonlinear inertial terms act in favour of pitchfork supercritical bifurcation and the nonlinear yield stress terms promote a subcritical bifurcation; (ii) for a range of Bingham numbers $B$, the extent of which depends on the radius ratio and outer Reynolds number, the nonlinear yield stress terms are dominant and the primary bifurcation is subcritical. The amplitude analysis indicates that in the supercritical bifurcation regime, near the threshold, when the nonlinear inertial terms are dominant, the amplitude decreases slightly with increasing $B$. Once the nonlinear yield stress terms start to become significant, the equilibrium amplitude increases substantially with increasing $B$. Similar trends are observed for Taylor vortex strength. Finally, the erosion of the static layer is analysed. It is shown that the nonlinear yield stress terms play a significant role.

Direct numerical simulations are performed to explore the effects of the rotating direction of the vertically asymmetric rough wall on the transport properties of Taylor–Couette (TC) flow, up to a Taylor number of ${Ta} = 2.39\times 10^{7}$. It is shown that, compared with the smooth wall, the rough wall with vertical asymmetric strips can enhance the dimensionless torque ${Nu}_{\omega }$. More importantly, at high Ta, clockwise rotation of the inner rough wall (where the fluid is sheared by the steeper slope side of the strips) results in a significantly greater torque enhancement compared to counter-clockwise rotation (where the fluid is sheared by the smaller slope side of the strips), due to the larger convective contribution to the angular velocity flux. However, the rotating direction has a negligible effect on the torque at low Ta. The larger torque enhancement caused by the clockwise rotation of the vertically asymmetric rough wall at high Ta is then explained by the stronger coupling between the rough wall and the bulk, attributed to the larger biased azimuthal velocity towards the rough wall at the mid-gap of the TC system, the increased turbulence intensity manifested by larger Reynolds stress and a thinner boundary layer, and the more significant contribution of the pressure force on the surface of the rough wall to the torque.

Turbulent shear flows are abundant in geophysical and astrophysical systems and in engineering-technology applications. They are often riddled with large-scale secondary flows that drastically modify the characteristics of the primary stream, preventing or enhancing mixing, mass and heat transfer. Using experiments and numerical simulations, we study the possibility of modifying these secondary flows by using superhydrophobic surface treatments that reduce the local shear. We focus on the canonical problem of Taylor–Couette flow, the flow between two coaxial and independently rotating cylinders, which has robust secondary structures called Taylor rolls that persist even at significant levels of turbulence. We generate these structures by rotating only the inner cylinder of the system, and show that an axially spaced superhydrophobic treatment can weaken the rolls through a mismatching surface heterogeneity, as long as the roll size can be fixed. The minimum hydrophobicity of the treatment required for this flow control is rationalized, and its effectiveness beyond the Reynolds numbers studied here is also discussed.

Axisymmetric steady solutions of Taylor–Couette flow at high Taylor numbers are studied numerically and theoretically. As the axial period of the solution shortens from approximately one gap length, the Nusselt number goes through two peaks before returning to laminar flow. In this process, the asymptotic nature of the solution changes in four stages, as revealed by the asymptotic analysis. When the aspect ratio of the roll cell is approximately unity, the solution has the Nusselt number proportional to the quarter power of the Taylor number, and captures quantitatively the characteristics of the classical turbulence regime. By shortening the axial period the Nusselt number can even reach the experimental value around the onset of the ultimate turbulence regime. However, at higher Taylor numbers, the theoretical predictions eventually underestimate the experimental values. An important consequence of the asymptotic analyses is that the mean angular momentum should become uniform in the core region unless the axial wavelength is too short. The theoretical scaling laws deduced for the steady solutions can be carried over to Rayleigh–Bénard convection.

Hydrodynamic instabilities in the interstage of turbomachinery and their relation to fan broadband noise are investigated over wide operating conditions. By applying compressible linear stability analysis to the velocity profiles of NASA's high-speed fan rig, source diagnostic test (SDT), three distinct unstable regimes are found: the first mode amplifies disturbances over a wide frequency range including the rotor speed. The second mode rotates much slower and develops inside the outer-wall boundary layer. The third mode spins in the opposite direction. An approach equivalent to dynamic mode decomposition (DMD) successfully extracts these modes from a database of improved delayed detached-eddy simulation (IDDES) solving the SDT geometry at approach and cut-back conditions: the extracted DMD modes generally capture the characteristics of eigenfunctions predicted by the linear stability analysis. At the approach condition, the first two unstable modes seem to interact inside the outer-wall boundary layer, while at the cut-back condition, disturbances associated with the first regime migrate away from the wall. Moreover, outer-wall pressure fluctuations of the IDDES database are Fourier decomposed in the interstage: coherent structures extracted using proper orthogonal decomposition are found to be dominated by duct acoustic modes, and they can be mapped not only to interaction tones but also to broadband noise likely associated with the interaction between disturbances in the rotor wakes and quasi-stationary structures near the outer wall. This interaction noise appears to be radiated to both inlet and exhaust with similar azimuthal-mode distributions; however, the importance relative to other noise sources decreases with the engine speed.

Turbulent Taylor–Couette flow between two concentric independently rotating cylinders with a radius ratio of $\eta = 0.1$ is studied experimentally. While the scope is to study the counter-rotating cases between both cylinders, the radial and azimuthal velocity components are recorded at different horizontal planes with high-speed particle image velocimetry. The parametric study considered a set of different shear Reynolds numbers in the range of $20\,000 \leq Re_s \leq 1.31 \times 10^5$, with different rotation ratios of $-0.06 \leq \mu \leq +0.008$. The observed flow fields had a clear dependence on the rotation ratio, where flow patterns evolved with a more pronounced axial dependence. The angular momentum transport is computed as a result of the recorded flow fields and given by a quasi-Nusselt number. The dependence of the Nusselt number on the different rotation ratios shows a maximum for the low counter-rotating case and $\mu _{max}$ is found between $-0.011 < \mu _{max} < -0.0077$. The Nusselt number decreases for stronger counter-rotation until a minimum is reached, where it tends to increase again for higher counter-rotation rates. The space–time behaviour of the turbulent flow showed the existence of patterns propagating from the inner region towards the outer region for all studied counter-rotating cases. In addition, patterns have been found that tend to propagate from the outer region towards the inner region with a novel character at high counter-rotation cases. These patterns enhance the angular momentum transport where a second maximum in the transport mechanism has to be expected.

The electromagnetically driven anticyclonic flow in the wide gap of a rotating concentric spheres system is studied experimentally and numerically in the laminar regime. The working fluid is an electrolyte contained in the spherical gap. The outer sphere rotates at constant angular speed, and the inner sphere is at rest. The electromagnetic stirring is generated due to the interaction of a direct current, which is injected radially through ring-shaped electrodes located at the equatorial zone of each sphere, and a dipolar magnetic field produced by a permanent magnet located inside the inner sphere. Experimental velocity fields in the equatorial plane were obtained with particle image velocimetry. Additionally, full three-dimensional numerical simulations were performed. The high shearing at the equatorial region promotes an instability that can be perceived as a triggering mechanism of four tornado-like vortical structures tilted and entangled in the polar direction. The instability structure rotates in either the cyclonic or the anticyclonic direction, depending on the flow parameters, namely, the angular velocity of the outer sphere and the applied electric current. To the best knowledge of the authors, this is the first time the phenomenon under study is reported with electromagnetic forcing. The numerical results agree quantitatively with the experimental observations.

Immiscible and incompressible liquid–liquid flows are considered in a Taylor–Couette geometry and analysed by direct numerical simulations coupled with the volume-of-fluid method and a continuum surface force model. The system Reynolds number $Re \equiv r_i \omega _i d / \nu$ is fixed to $960$, where the single-phase flow is in the steady Taylor vortex regime, whereas the secondary-phase volume fraction $\varphi$ and the system Weber number $We \equiv \rho r_i^2 \omega _i^2 d / \sigma$ are varied to study the interactions between the interface and the Taylor vortices. We show that different Weber numbers lead to two distinctive flow regimes, namely an advection-dominated regime and an interface-dominated regime. When $We$ is high, the interface is easily deformed because of its low surface tension. The flow patterns are then similar to the single-phase flow, and the system is dominated mainly by advection (advection-dominated regime). However, when $We$ is low, the surface tension is so large that stable interfacial structures with sizes comparable to the cylinder gap can exist. The background velocity field is modulated largely by these persistent structures, thus the overall flow dynamics is governed by the interface (interface-dominated regime). The effect of the interface on the global system response is assessed by evaluating the Nusselt number $Nu_{\omega }$ based on the non-dimensional angular velocity transport. It shows non-monotonic trends as functions of the volume fraction $\varphi$ for both low and high $We$. We explain how these dependencies are closely linked to the velocity and interfacial structures.

By varying the oil volume fraction, the microscopic droplet size and the macroscopic rheology of emulsions are investigated in a Taylor–Couette turbulent shear flow. Although here oil and water in the emulsions have almost the same physical properties (density and viscosity), unexpectedly, we find that oil-in-water (O/W) and water-in-oil (W/O) emulsions have very distinct hydrodynamic behaviours, i.e. the system is clearly asymmetric. By looking at the micro-scales, the average droplet diameter hardly changes with the oil volume fraction for O/W or for W/O. However, for W/O it is about $50\,\%$ larger than that of O/W. At the macro-scales, the effective viscosity of O/W is higher when compared to that of W/O. These asymmetric behaviours are expected to be caused by the presence of surface-active contaminants from the walls of the system. By introducing an oil-soluble surfactant at high concentration, remarkably, we recover the symmetry (droplet size and effective viscosity) between O/W and W/O emulsions. Based on this, we suggest a possible mechanism responsible for the initial asymmetry and reach conclusions on emulsions where interfaces are fully covered by the surfactant. Next, we discuss what sets the droplet size in turbulent emulsions. We uncover a $-6/5$ scaling dependence of the droplet size on the Reynolds number of the flow. Combining the scaling dependence and the droplet Weber number, we conclude that the droplet fragmentation, which determines the droplet size, occurs within the boundary layer and is controlled by the dynamic pressure caused by the gradient of the mean flow, as proposed by Levich (Physicochemical Hydrodynamics, Prentice-Hall, 1962), instead of the dynamic pressure due to turbulent fluctuations, as proposed by Kolmogorov (Dokl. Akad. Nauk. SSSR, vol. 66, 1949, pp. 825–828). The present findings provide an understanding of both the microscopic droplet formation and the macroscopic rheological behaviours in dynamic emulsification, and connects them.

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.

Three-dimensional elastic turbulence in Taylor–Couette flows of dilute polymer solutions has been realized and thoroughly investigated via direct numerical simulations. A novel flow transition pathway from elastically dominated turbulence to solitary vortex pairs (or diwhirls) and eventually to elastic turbulence is observed by decreasing the fluid inertia ($Re$) over seven orders of magnitude, i.e. from $Re=1000$ to $0.0001$. The dominant spatio-temporal flow features in the elastic turbulence regime are those of large-scale unsteady diwhirls and small-scale axial and azimuthal travelling waves in the outer and inner halves of the gap, respectively. Moreover, it is conclusively shown that production of turbulent kinetic energy in purely elastic turbulence solely arises due to the stochastic nature of polymer stretch/relaxation. Overall, based on this comprehensive numerical investigation, the differences in the underlying fluid physics that give rise to turbulent fluctuations in elastically dominated and purely elastic turbulence have been delineated.

This paper is concerned with the optimal upper bound on mean quantities (torque, dissipation and the Nusselt number) obtained in the framework of the background method for the Taylor–Couette flow with a stationary outer cylinder. Along the way, we perform the energy stability analysis of the laminar flow, and demonstrate that below radius ratio $0.0556$, the marginally stable perturbations are not the axisymmetric Taylor vortices but rather a fully three-dimensional flow. The main result of the paper is an analytical expression of the optimal bound as a function of the radius ratio. To obtain this bound, we begin by deriving a suboptimal analytical bound using analysis techniques. We use a definition of the background flow with two boundary layers, whose relative thicknesses are optimized to obtain the bound. In the limit of high Reynolds number, the dependence of this suboptimal bound on the radius ratio (the geometrical scaling) turns out to be the same as that of numerically computed optimal bounds in three different cases: (1) the perturbed flow only satisfies the homogeneous boundary conditions but need not be incompressible; (2) the perturbed flow is three-dimensional and incompressible; (3) the perturbed flow is two-dimensional and incompressible. We compare the geometrical scaling with the observations from the turbulent Taylor–Couette flow, and find that the analytical result indeed agrees well with the available direct numerical simulations data. In this paper, we also dismiss the applicability of the background method to certain flow problems and therefore establish the limitation of this method.