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The phenomenon of drag reduction induced by injection of bubbles into a turbulent carrier fluid has been known for a long time; the governing control parameters and underlying physics is, however, not well understood. In this paper, we use three-dimensional numerical simulations to uncover the effect of deformability of bubbles injected in a turbulent Taylor–Couette flow on the overall drag experienced by the system. We consider two different Reynolds numbers for the carrier flow, i.e.  $Re_{i}=5\\times 10^{3}$ and $Re_{i}=2\\times 10^{4}$ ; the deformability of the bubbles is controlled through the Weber number, which is varied in the range $We=0.01{-}2.0$ . Our numerical simulations show that increasing the deformability of bubbles (that is, $We$ ) leads to an increase in drag reduction. We look at the different physical effects contributing to drag reduction and analyse their individual contributions with increasing bubble deformability. Profiles of local angular velocity flux show that, in the presence of bubbles, turbulence is enhanced near the inner cylinder while attenuated in the bulk and near the outer cylinder. We connect the increase in drag reduction to the decrease in dissipation in the wake of highly deformed bubbles near the inner cylinder.

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.

The magnetorotational instability (MRI) is thought to be a powerful source of turbulence and momentum transport in astrophysical accretion discs, but obtaining observational evidence of its operation is challenging. Recently, laboratory experiments of Taylor-Couette flow with externally imposed axial and azimuthal magnetic fields have revealed the kinematic and dynamic properties of the MRI close to the instability onset. While good agreement was found with linear stability analyses, little is known about the transition to turbulence and transport properties of the MRI. We here report on a numerical investigation of the MRI with an imposed azimuthal magnetic field. We show that the laminar Taylor-Couette flow becomes unstable to a wave rotating in the azimuthal direction and standing in the axial direction via a supercritical Hopf bifurcation. Subsequently, the flow features a catastrophic transition to spatio-temporal defects which is mediated by a subcritical subharmonic Hopf bifurcation. Our results are in qualitative agreement with the PROMISE experiment and dramatically extend their realizable parameter range. We find that as the Reynolds number increases defects accumulate and grow into turbulence, yet the momentum transport scales weakly.

The flow physics of inertio-elastic turbulent Taylor–Couette flow for a radius ratio of $0.5$ in the Reynolds number ($Re$) range of $500$ to $8000$ is investigated via direct numerical simulation. It is shown that as $Re$ is increased the turbulence dynamics can be subdivided into two distinct regimes: (i) a low $Re \\leqslant 1000$ regime where the flow physics is essentially dominated by nonlinear elastic forces and the main contribution to transport and mixing of momentum, stress and energy comes from large-scale flow structures in the bulk region and (ii) a high $Re \\geqslant 5000$ regime where inertial forces govern the flow physics and the flow dynamics is mainly governed by small-scale flow structures in the near-wall region. Flow–microstructure coupling analysis reveals that the elastic Görtler instability in the near-wall region is triggered via significant polymer extension and commensurately high hoop stresses. This instability gives rise to small-scale elastic vortical structures identified as elastic Görtler vortices which are present at all $Re$ considered. In fact, these vortices develop herringbone streaks near the inner wall that have a longer average life span than their Newtonian counterparts due to their elastic origin. Examination of the budgets of mean streamwise enstrophy, mean kinetic energy, turbulent kinetic energy and Reynolds shear stress demonstrates that increasing fluid inertia hinders the generation of elastic stresses, leading to a monotonic reduction of the elastic-related effects on the flow physics.

We provide experimental measurements for the effective scaling of the Taylor–Reynolds number within the bulk $\\mathit{Re}_{\\unicode[STIX]{x1D706},\\mathit{bulk}}$ , based on local flow quantities as a function of the driving strength (expressed as the Taylor number $\\mathit{Ta}$ ), in the ultimate regime of Taylor–Couette flow. We define $Re_{\\unicode[STIX]{x1D706},bulk}=(\\unicode[STIX]{x1D70E}_{bulk}(u_{\\unicode[STIX]{x1D703}}))^{2}(15/(\\unicode[STIX]{x1D708}\\unicode[STIX]{x1D716}_{bulk}))^{1/2}$ , where $\\unicode[STIX]{x1D70E}_{bulk}(u_{\\unicode[STIX]{x1D703}})$ is the bulk-averaged standard deviation of the azimuthal velocity, $\\unicode[STIX]{x1D716}_{bulk}$ is the bulk-averaged local dissipation rate and $\\unicode[STIX]{x1D708}$ is the liquid kinematic viscosity. The data are obtained through flow velocity field measurements using particle image velocimetry. We estimate the value of the local dissipation rate $\\unicode[STIX]{x1D716}(r)$ using the scaling of the second-order velocity structure functions in the longitudinal and transverse directions within the inertial range – without invoking Taylor’s hypothesis. We find an effective scaling of $\\unicode[STIX]{x1D716}_{\\mathit{bulk}}/(\\unicode[STIX]{x1D708}^{3}d^{-4})\\sim \\mathit{Ta}^{1.40}$ , (corresponding to $\\mathit{Nu}_{\\unicode[STIX]{x1D714},\\mathit{bulk}}\\sim \\mathit{Ta}^{0.40}$ for the dimensionless local angular velocity transfer), which is nearly the same as for the global energy dissipation rate obtained from both torque measurements ( $\\mathit{Nu}_{\\unicode[STIX]{x1D714}}\\sim \\mathit{Ta}^{0.40}$ ) and direct numerical simulations ( $\\mathit{Nu}_{\\unicode[STIX]{x1D714}}\\sim \\mathit{Ta}^{0.38}$ ). The resulting Kolmogorov length scale is then found to scale as $\\unicode[STIX]{x1D702}_{\\mathit{bulk}}/d\\sim \\mathit{Ta}^{-0.35}$ and the turbulence intensity as $I_{\\unicode[STIX]{x1D703},\\mathit{bulk}}\\sim \\mathit{Ta}^{-0.061}$ . With both the local dissipation rate and the local fluctuations available we finally find that the Taylor–Reynolds number effectively scales as $\\mathit{Re}_{\\unicode[STIX]{x1D706},\\mathit{bulk}}\\sim \\mathit{Ta}^{0.18}$ in the present parameter regime of $4.0\\times 10^{8}<\\mathit{Ta}<9.0\\times 10^{10}$ .

We extend upon the known flow transitions in neutrally buoyant particle-laden Taylor–Couette flows by accessing higher suspension Reynolds numbers $(Re_{{susp}} \\sim O(10^3))$ in a geometry with radius ratio $\\eta = 0.917$ and aspect ratio $\\varGamma = 21.67$. Flow transitions for several particle volume fractions ($0 \\leq \\phi \\leq 0.40$) are investigated by means of flow visualization experiments, in a flow driven by a rotating inner cylinder. Despite higher effective ramp rates, we observe non-axisymmetric patterns, such as spirals, in the presence of particles. A novel observation in our experiments is the azimuthally localized wavy vortex flow, characterized by waviness present on a fraction of the otherwise axisymmetric Taylor vortices. The existence of this flow state suggests that in addition to the already established, destabilizing effect of particles, they may also inhibit the growth of instabilities. Flow topologies corresponding to higher-order transitions in particle-laden suspensions appear to be qualitatively similar to those observed in single-phase flows. A key difference, however, is the visible reduction in the appearance of a second, incommensurate frequency at higher particle loadings, which could have implications for the onset of chaos. Simultaneous torque measurements allow us to estimate an empirical scaling law between the Nusselt number ($Nu_{\\omega }$), the Taylor number ($Ta$) and the relative viscosity ($\\chi ^{e}): Nu_{\\omega } \\propto Ta^{0.24} \\chi ^{e \\, 0.41}$. The scaling exponent of $Ta$ is non-trivially independent of the particle loading. Apparently, particles do not trigger a qualitative change in the nature of angular momentum transfer between the cylinders.

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.

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.

This paper describes the hysteresis in the torque for Taylor–Couette flow in the turbulent flow regime for different shear Reynolds numbers, aspect ratios and boundary conditions. The hysteresis increases with decreasing shear Reynolds number and becomes more pronounced as the aspect ratio is increased from 22 to 88. Measurements conducted in two different Taylor–Couette set-ups depict the effect of the flow conditions at the ends of the cylinders on the flow hysteresis by showing reversed hysteresis behaviour. In addition, the flow structure in the different branches of the hysteresis loop was investigated by means of stereoscopic particle image velocimetry. The results show that the dominant flow structures differ in shape and magnitude depending on the branch of the hysteresis loop. Hence, it can be concluded that the geometry could have an effect on the hysteresis behaviour of turbulent Taylor–Couette flow, but its occurrence is related to a genuine change in the flow dynamics.

Recent experiments have reported a novel transition to elasto-inertial turbulence in the Taylor–Couette flow of a dilute polymer solution. Unlike previously reported transitions, this newly discovered scenario, dubbed vortex merging and splitting (VMS) transition, occurs in the centrifugally unstable regime and the mechanisms underlying it are two-dimensional: the flow becomes chaotic due to the proliferation of events where axisymmetric vortex pairs may be either created (vortex splitting) or annihilated (vortex merging). In this paper, we present direct numerical simulations, using the finitely extensible nonlinear elastic-Peterlin (FENE-P) constitutive equation to model the polymer dynamics, which reproduce the experimental observations with great accuracy and elucidate the reasons for the onset of this surprising dynamics. Starting from the Newtonian limit and increasing progressively the fluid's elasticity, we demonstrate that the VMS dynamics is not associated with the well-known Taylor vortices, but with a steady pattern of elastically induced axisymmetric vortex pairs known as diwhirls. The amount of angular momentum carried by these elastic vortices becomes increasingly small as the fluid's elasticity increases and it eventually reaches a marginal level. When this occurs, the diwhirls become dynamically disconnected from the rest of the system and move independently from each other in the axial direction. It is shown that vortex merging and splitting events, along with local transient chaotic dynamics, result from the interactions among diwhirls, and that this complex spatio-temporal dynamics persists even at elasticity levels twice as large as those investigated experimentally.