Since the introduction of the logarithmic law of the wall more than 80 years ago, the equation for the mean velocity profile in turbulent boundary layers has been widely applied to model near-surface processes and parameterize surface drag. Yet the hypothetical turbulent eddies proposed in the original logarithmic law derivation and mixing length theory of Prandtl have never been conclusively linked to physical features in the flow. Here, we present evidence that suggests these eddies correspond to regions of coherent streamwise momentum known as uniform momentum zones (UMZs). The arrangement of UMZs results in a step-like shape for the instantaneous velocity profile, and the smooth mean profile results from the average UMZ properties, which are shown to scale with the friction velocity and wall-normal distance in the logarithmic region. These findings are confirmed across a wide range of Reynolds number and surface roughness conditions from the laboratory scale to the atmospheric surface layer.

The renormalization method proposed by Batchelor is used to derive gradient diffusion coefficients in Brownian suspensions of hard spheroidal particles with aspect ratio $\unicode[STIX]{x1D706}$ in the range $1\leqslant \unicode[STIX]{x1D706}\leqslant 3.5$. The theory is based on pairwise steric and hydrodynamic interactions, and the results are therefore valid for dilute suspensions such that $\unicode[STIX]{x1D706}^{2}\unicode[STIX]{x1D719}\ll 1$, where $\unicode[STIX]{x1D719}$ is the particle volume fraction. The driving force for gradient diffusion, i.e. the gradient in chemical potential, is larger for suspensions of spheroidal particles than for spheres at the same volume fraction. The hydrodynamic resistance also increases with aspect ratio, but the increase is weaker than that in the driving force. Consequently, at the same particle volume fraction, the increases in rates of gradient diffusion are greater for spheroidal particles than for spheres. The concentration-dependent gradient diffusion coefficient $D(\unicode[STIX]{x1D719},\unicode[STIX]{x1D706})$ is shown to be closely approximated by $D(\unicode[STIX]{x1D719},\unicode[STIX]{x1D706})=\unicode[STIX]{x1D709}_{m}D_{0}\{1+1.45\unicode[STIX]{x1D719}[1+0.259(\unicode[STIX]{x1D706}-1)+0.126(\unicode[STIX]{x1D706}-1)^{2}]\}$, which reduces to the result for spheres when $\unicode[STIX]{x1D706}=1$. Here, $D_{0}$ is the Stokes–Einstein diffusivity of a spherical particle with its radius equal to the longer dimension of the spheroidal particle, and $\unicode[STIX]{x1D709}_{m}D_{0}$ is the orientation-averaged diffusivity of an isolated spheroidal particle.

We investigate the interplay between large-scale patterns, so-called superstructures, in the fluctuation fields of temperature $\unicode[STIX]{x1D703}$ and vertical velocity $w$ in turbulent Rayleigh–Bénard convection at large aspect ratios. Earlier studies suggested that velocity superstructures were smaller than their thermal counterparts in the centre of the domain. However, a scale-by-scale analysis of the correlation between the two fields employing the linear coherence spectrum reveals that superstructures of the same size exist in both fields, which are almost perfectly correlated. The issue is further clarified by the observation that, in contrast to the temperature, and unlike assumed previously, superstructures in the vertical-velocity field do not result in a peak in the power spectrum of $w$. The origin of this difference is traced back to the production terms of the $\unicode[STIX]{x1D703}$ and $w$ variance. These results are confirmed for a range of Rayleigh numbers $Ra=10^{5}{-}10^{9}$; the superstructure size is seen to increase monotonically with $Ra$. Furthermore, the scale distribution of the temperature fluctuations in particular is pronouncedly bimodal. In addition to the large-scale peak caused by the superstructures, there exists a strong small-scale peak. This ‘inner peak’ is most intense at a distance of $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}$ from the wall and is associated with structures of size ${\approx}10\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}$, where $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}$ is the thermal boundary layer thickness. Finally, based on the vertical coherence relative to a reference height of $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}$, a self-similar structure is identified in the velocity field (vertical and horizontal components) but not in the temperature.

The dynamic interplay between surface and subsurface flow in the presence of a permeable boundary was investigated using low and high frame-rate particle-image velocimetry measurements in a refractive-index-matching flow environment. Two idealized permeable wall models were considered. Both models contained five layers of cubically packed spheres, but one exhibited a smooth interface with the flow, while the other embodied a hemispherical surface topography. The relationship between the large-scale turbulent motions overlying the permeable walls and the small-scale turbulence just above, and within, the walls was explored using instantaneous and statistical analyses. Although previous studies have indirectly identified the potential existence of amplitude modulation in permeable-wall turbulence (a phenomenon identified in impermeable-wall turbulence whereby the outer large scales modulate the intensity of the near-wall, small-scale turbulence), the present effort provides direct evidence of its existence in flow over both permeable walls considered. The spatio-temporal signatures of amplitude modulation were also characterized using conditional averaging based on zero-crossing events. This analysis highlights the connection between large-scale regions of high/low streamwise momentum in the surface flow, downwelling/upwelling across the permeable interface and enhancement/suppression of small-scale turbulence, respectively, just above and within the permeable walls. The presence of bed roughness is found to intensify the strength and penetration of flow into the permeable bed modulated by large-scale structures in the surface flow, and linked to possible roughness-formed channelling effects and shedding of smaller-scale flow structures from the roughness elements.

Two-dimensional (2-D) numerical simulations based on the Eulerian–Lagrangian method that take droplet break-up into account are conducted to clarify the mean structure of gaseous detonation laden with a dilute water spray. The premixed mixture is a slightly diluted stoichiometric hydrogen–oxygen mixture at low pressure. The simulated results are analysed via 2-D flow fields and statistical Favre spatiotemporal averaging techniques. Gaseous detonation with water droplets (WD) propagates stably with a velocity decrease compared with the dry Chapman–Jouguet speed. The mean structure of gaseous detonation with dilute water spray shares a similar structure as the one without water spray. However, the hydrodynamic thickness is changed due to the interaction with water spray. Overall interphase exchanges (mass, momentum and energy) that take place within the hydrodynamic thickness induce a decrease of the detonation velocity and lower the level of fluctuations downstream of the mean leading shock wave. Droplet break-up occurs downstream of the induction zone and in our case, the water vapour from the evaporation of water spray does not affect the reactivity of gaseous detonation. The laminar master equation for gaseous detonation laden with inert WD shows that the hydrodynamic thickness should rely on the gaseous sound speed, and works well as the working mixture is weakly unstable and its cellular structure is regular. The droplet flow regimes and break-up modes have also been determined. The characteristic lengths of detonation and interphase exchanges have been ordered under the present simulation conditions and have been shown to be intimately intertwined.

Transition to chaos through a cascade of period doublings of the primary $1/2$ synchronization mode is discovered in steady approaching flow around a forced inline oscillating cylinder near a plane boundary at a Reynolds number $(Re)$ of 175. The transition occurs well within the otherwise synchronized region (known as the Arnold tongue) in the frequency and amplitude space of the oscillating cylinder, creating two parameter strips of desynchronized flows within the Arnold tongue. Five orders of period doublings from mode $1/2$ to mode $16/32$ are revealed by progressively increasing the frequency resolution in the simulation. The ratio of frequency intervals of two successive period-doubling modes asymptotes towards the first Feigenbaum constant, reaching a value of 4.52 at mode of $16/32$. Additional three-dimensional simulations demonstrate the existence of period doubling with a regular spanwise flow structure similar to regular mode B of steady flow around an isolated cylinder. Although transition to chaos through cascades of period doublings is primarily reported for the primary $1/2$ synchronization mode, it is also observed for other synchronization modes $(p/q)$ (Tang et al., J. Fluid Mech., vol. 832, 2017, pp. 146–169), where $p$ and $q$ are integers with a non-reducible $p/q$, such as $2/3$. The physical mechanisms responsible for the present period-doubling bifurcations and transition to chaos through cascades of period doublings are ascribed to the interaction of asymmetric vortex shedding from the cylinder (due to a geometric asymmetry) and the boundary layer developed on the plane boundary, through specifically designed numerical tests.

Numerical simulations of the response flow in a fluid-filled rotating cube that is subjected to precessional forcing are examined over a wide range of rotation, precession and forcing frequencies. The responses are shown to correspond to resonantly excited inertial modes of the rotating cube that have the same spatio-temporal symmetry as the precessional forcing and, under certain conditions, the response flow loses stability via symmetry breaking that is intricately associated with a triadic resonance between the forced flow and two free inertial modes whose spatio-temporal symmetries do not coincide with that of the precessional forcing.

The effect of trailing-edge shape on the self-propulsive performance of three-dimensional flexible plates is studied numerically. In our study, the trailing edges of the plates are symmetric chevron shapes, and the trailing-edge angle $\unicode[STIX]{x1D719}$ varies from $30^{\circ }$ (concave plate) to $150^{\circ }$ (convex plate). Under different bending stiffnesses $K$, three regimes of the propulsive performance in terms of propulsive velocity $U$ and efficiency $\unicode[STIX]{x1D702}$ as a function of $\unicode[STIX]{x1D719}$ are identified. When $K$ is small, moderate and large, the square, convex and concave plate achieves the best performance, respectively. Analyses of vortical structures and velocity fields show that usually the jet behind the plate with the best performance is longest. Besides, the inclination angle of the jet may be small. The different propulsive performances at small and moderate $K$ are mainly attributed to the phase lag of the trailing edge. The force acting on the plate is analysed and it is found that the thrust force is mainly contributed by the normal force. If $U$, $\unicode[STIX]{x1D702}$ and $K$ are rescaled by the normal force and the area moment of the plate, the curves for different $\unicode[STIX]{x1D719}$ almost collapse into a single curve when the bending stiffness coefficient is small or moderate. The scaling confirms that the normal force should be the characteristic fluid force at small or moderate $K$ and the $\unicode[STIX]{x1D719}$ effect is governed by the area moment. The findings may shed some light on the propulsive performance of aquatic animals.

A series of laboratory experiments were conducted to investigate the characteristics of a dense gravity current flowing down an inclined slope into a quiescent two-layer stratification. The presence of the pycnocline causes the gravity current to split and intrude into the ambient at two distinct levels of neutral buoyancy, as opposed to the classical description of gravity currents in stratified media as being either a pure underflow or interflow. The splitting behaviour is observed to be dependent on the Richardson number ($Ri_{\unicode[STIX]{x1D70C}}$) of the gravity current, formulated as the ratio of the excess density and the ambient stratification. For low $Ri_{\unicode[STIX]{x1D70C}}$, underflow is more dominant, while at higher $Ri_{\unicode[STIX]{x1D70C}}$ interflow is more dominant. As $Ri_{\unicode[STIX]{x1D70C}}$ increases, however, we find that the splitting behaviour eventually becomes independent of $Ri_{\unicode[STIX]{x1D70C}}$. Additionally, we have also identified two different types of waves that form on the pycnocline in response to the intrusion of the gravity current. An underflow-dominated regime causes a pycnocline displacement where the speed of the wave crest is locked to the gravity current, whereas an interflow-dominated regime launches an internal wave that moves much faster than the gravity current head or interfacial intrusion.

The Burnett- and super-Burnett-order constitutive relations are derived for homogeneously sheared gas–solid suspensions by considering the co-existence of ignited and quenched states and the anisotropy of the second moment of velocity fluctuations ($\unicode[STIX]{x1D648}=\langle \boldsymbol{C}\boldsymbol{C}\rangle ,C$ is the fluctuation or peculiar velocity) – this analytical work extends our previous works on dilute (Saha & Alam, J. Fluid Mech., vol. 833, 2017, pp. 206–246) and dense (Alam et al., J. Fluid Mech., vol. 870, 2019, pp. 1175–1193) gas–solid suspensions. For the combined ignited–quenched theory at finite densities, the second-moment balance equation, truncated at the Burnett order, is solved analytically, yielding expressions for four invariants of $\unicode[STIX]{x1D648}$ as functions of the particle volume fraction ($\unicode[STIX]{x1D708}$), the restitution coefficient ($e$) and the Stokes number ($St$). The phase boundaries, demarcating the regions of (i) ignited, (ii) quenched and (iii) co-existing ignited–quenched states, are identified via an ordering analysis, and it is shown that the incorporation of excluded-volume effects significantly improves the predictions of critical parameters for the ‘quenched-to-ignited’ transition. The Burnett-order expressions for the particle-phase shear viscosity, pressure and two normal-stress differences are provided, with their Stokes-number dependence being implicit via the anisotropy parameters. The roles of ($St,\unicode[STIX]{x1D708},e$) on the granular temperature, the second-moment anisotropy and the nonlinear transport coefficients are analysed using the present theory, yielding quantitative agreements with particle-level simulations over a wide range of ($St,\unicode[STIX]{x1D708}$) including the bistable regime that occurs at $St\sim O(5)$. For highly dissipative particles ($e\ll 1$) that become increasingly important at large Stokes numbers, it is shown that the Burnett-order solution is not adequate and further higher-order solutions are required for a quantitative agreement of transport coefficients over the whole range of control parameters. The latter is accomplished by developing an approximate super-super-Burnett-order theory for the ignited state ($St\gg 1$) of sheared dense gas–solid suspensions in the second part of this paper. An extremum principle based on viscous dissipation and dynamic friction is discussed to identify ignited–quenched transition.

Sophisticated measurements of fluid velocity near to an undulating air–water boundary have traditionally been confined to the laboratory setting. Developments in camera technology and the opening of novel modes of analysis have allowed for sensitive measurements of the current profile in the ocean’s uppermost layer. Taking advantage of the Research Platform R/P FLIP as a ‘laboratory at sea’, here we present first-of-their-kind thermal and polarimetric camera-based observations of wave orbital velocities and mean shear flows in the upper centimetres of the ocean surface layer. Measurements reveal a well-defined logarithmic layer as seen in laboratory measurements and described by classical surface layer theory; however, substantial spread of observations is found at low levels of wind forcing, where the Stokes drift of swell may have a substantial impact on the near-surface current profile. A novel application of short time window Fourier transforms allows for the estimation of near-surface wave orbital velocity magnitudes. These are found to be in general agreement with the prescriptions of linear wave theory, although observations diverge from theory at high levels of wind forcing where the interface is subject to surface wave breaking. Finally, the surface gravity wave phase-coherent short wave growth is presented and discussed in the context of hydrodynamic wave and airflow modulation.

We propose a predictive model of the turbulent burning velocity $S_{T}$ in homogeneous isotropic turbulence (HIT) based on Lagrangian statistics of propagating surfaces. The propagating surfaces with a constant displacement speed are initially arranged on a plane, and they evolve in non-reacting HIT, behaving like the propagation of a planar premixed flame front. The universal constants in the model of $S_{T}$ characterize the enhancement of area growth of premixed flames by turbulence, and they are determined by Lagrangian statistics of propagating surfaces. The flame area is then modelled by the area of the propagating surfaces at a truncation time. This truncation time signals the statistical stationary state of the evolutionary geometry of the propagating surfaces, and it is modelled by an explicit expression using limiting conditions of very weak and strong turbulence. Another parameter in the model of $S_{T}$ characterizes the effect of fuel chemistry on $S_{T}$, and it is pre-determined by the very few available data points of $S_{T}$ from experiments or direct numerical simulation (DNS) in weak turbulence. The proposed model is validated using three DNS series of turbulent premixed flames with various fuels. The model prediction of $S_{T}$ generally agrees well with DNS in a wide range of premixed combustion regimes, and it captures the basic trends of $S_{T}$ in terms of the turbulence intensity, including the linear growth in weak turbulence and the ‘bending effect’ in strong turbulence.

The morphology of nodes generated by the interaction of discontinuities in steady two-dimensional inviscid flows is examined. The fluids considered are Bethe–Zel’dovich–Thompson (BZT) fluids, which feature negative values of the fundamental derivative of gas dynamics in the vapour phase. The operating conditions correspond to the non-classical gas-dynamic regime where expansion shocks, compression fans and composite waves are admissible in addition to the classical compression shocks and expansion fans. Interactions caused by the crossing, overtaking and splitting of compression/expansion shocks, along with the refraction of these through a contact discontinuity, are analysed here. The well-established method of wave curves is applied to non-classical wave curves, revealing a variety of interaction patterns that are simply not admissible in classical gas dynamics. It is shown that shock waves can be reflected, transmitted and refracted as Prandtl–Meyer fans or composite waves. Based on numerical evidence, the splitting (and consequently the Mach reflection) of an expansion shock seems to be disallowed. Theoretical considerations on the admissibility of such configurations are also provided. The present analysis is relevant to applications potentially involving supersonic flows of BZT fluids, e.g. organic Rankine cycle power systems, and can also be used in front-tracking algorithms for general equations of state.

The dispersion of a neutral solute band by electrokinetic flow in polyelectrolyte layer (PEL)-grafted rectangular/slit microchannels is theoretically studied. The flow is assumed to be both steady and fully developed and a first-order irreversible reaction is considered at the wall to account for probable surface adsorption of solutes. Considering low electric potentials, analytical solutions are obtained for electric potential, fluid velocity and solute concentration. Special solutions are also obtained for the case without wall adsorption. To track the dispersion properties of the solute band, the generalized dispersion model is adopted by considering the exchange, the convection and the dispersion coefficients. The solutions developed are validated by comparing the results with the predictions of finite-element-based numerical simulations. Even though the solutions can take any form of initial solute concentration into account, the results are presented by considering a solute band of rectangular shape. The results reveal that, while the short-term transport coefficients are strongly affected by the initial concentration profile, the long-term values are not dependent upon the initial conditions. In addition, it is shown that the mass transport coefficients are strong functions of the channel aspect ratio; hence, approximating a rectangular geometry by the space between two parallel plates may lead to considerable errors in the estimation of mass transport characteristics. This is particularly important for the dispersion coefficient for which the long-term values for a slit microchannel are quite different from those for a rectangular channel of very high aspect ratio. It is also illustrated that the exchange and convection coefficients increase on increasing the Damköhler number, whereas the opposite is true for the dispersion coefficient. The convection and dispersion coefficients are generally increasing functions of the PEL fixed charge density and the PEL thickness and decreasing functions of the PEL friction coefficient. Last but not least, a thicker electric double layer is found to provide a larger degree of solute dispersion, which is the opposite of that observed in a microchannel with bare walls.

We consider interactions between surface and interfacial waves in a two-layer system. Our approach is based on the Hamiltonian structure of the equations of motion, and includes the general procedure for diagonalization of the quadratic part of the Hamiltonian. Such diagonalization allows us to derive the interaction cross-section between surface and interfacial waves and to derive the coupled kinetic equations describing spectral energy transfers in this system. Our kinetic equation allows resonant and near-resonant interactions. We find that the energy transfers are dominated by the class III resonances of Alam (J. Fluid Mech., vol. 691, 2012, pp. 267–278). We apply our formalism to calculate the rate of growth for interfacial waves for different values of wind velocity. Using our kinetic equation, we also consider the energy transfer from wind-generated surface waves to interfacial waves for the case when the spectrum of the surface waves is given by the JONSWAP spectrum and interfacial waves are initially absent. We find that such energy transfer can occur along a time scale of hours; there is a range of wind speeds for the most effective energy transfer at approximately the wind speed corresponding to white capping of the sea. Furthermore, interfacial waves oblique to the direction of the wind are also generated.

This article shows the importance of flow compressibility on the heat transfer in confined impinging jets, and how it is driven by both the Mach number and the wall heat flux. Hence, we present a collection of cases at several Mach numbers with different heat-flux values applied at the impingement wall. The wall temperature scales linearly with the imposed heat flux and the adiabatic wall temperature is found to be purely governed by the flow compression. Especially for high heat-flux values, the non-constant wall temperature induces considerable differences in the thermal conductivity of the fluid. This phenomenon has to date not been discussed and it strongly modulates the Nusselt number. In contrast, the heat transfer coefficient is independent of the varying thermal properties of the fluid and the wall heat flux. Furthermore, we introduce the impingement efficiency, which highlights the areas of the wall where the temperature is influenced by compressibility effects. This parameter shows how the contribution of the flow compression to raising the wall temperature becomes more dominant as the heat flux decreases. Thus, knowing the adiabatic wall temperature is indispensable for obtaining the correct heat transfer coefficient when low heat-flux values are used, even at low Mach numbers. Lastly, a detailed analysis of the dilatation field also shows how the compressibility effects only affect the heat transfer in the vicinity of the stagnation point. These compressibility effects decay rapidly further away from the flow impingement, and the density changes along the developing boundary layer are caused instead by variable inertia effects.

We investigate the instability of finite-amplitude progressive ring waves in deep water, which are radiated by the time-periodic oscillation of a half-submerged sphere (with radius $r_{1}$), under the influence of gravity and surface tension. We use direct numerical simulations of fully nonlinear wave–body interactions to quantify the temporal–spatial evolution of the base and perturbed outgoing ring wave fields, from which the stability of ring waves is analysed. The numerical simulation is based on a mixed Euler–Lagrangian quadratic boundary-element method and accounts for fully nonlinear wave–wave and wave–body interactions in the context of potential flow. We find that the progressive gravity–capillary ring waves (with frequency $2\unicode[STIX]{x1D714}_{0}$) become unstable to (small-amplitude) radial cross-wave disturbances when the body-motion parameter $k_{0}a$ exceeds the threshold value $\unicode[STIX]{x1D700}_{c}$, where $a$ is the amplitude of body oscillation and $k_{0}$ is the wavenumber of the ring wave at subharmonic frequency $\unicode[STIX]{x1D714}_{0}$. The predicted $\unicode[STIX]{x1D700}_{c}$ from nonlinear simulations under the assumption of ideal fluid, which decreases with increasing $k_{0}r_{1}$, is generally smaller than the experimental measurement of Tatsuno et al. (Rep. Res. Inst. Appl. Mech. Kyushu University, vol. 17, 1969, pp. 195–215) by approximately 50 %. When the viscous effects in body-surface and free-surface boundary layers are taken into account, the predicted $\unicode[STIX]{x1D700}_{c}$ matches the experimental data excellently. The unstable modes are characterized as the progressive radial cross-waves at the subharmonic frequency ($\unicode[STIX]{x1D714}_{0}$) with the growth rates generally increasing with $k_{0}a$. The maximum growth rate is achieved for the cross-wave mode with the azimuthal wavenumber $m^{\ast }\sim 1.2k_{0}r_{1}$. These distinctive features of instability obtained in numerical simulations are consistent with the experimental observations. From the comparison with the weakly nonlinear analysis of Shen & Liu (J. Fluid Mech., vol. 869, 2019, pp. 439–467), it is found that inclusion of finite-amplitude ring wave effects generally reduces the growth rate of unstable modes but has an insignificant influence on the shape of unstable modes and the value of $\unicode[STIX]{x1D700}_{c}$. Moreover, for moderately steep ring waves, nonlinear interactions of a few unstable modes can excite broadbanded unstable subharmonic cross-wave modes, leading to the formation/observation of distinctive non-axisymmetric wave patterns during long-time evolutions.

It is widely acknowledged that streamwise ridges on the bed of open-channel flows generate secondary currents (SCs). A recent discovery of meandering long streamwise counter-rotating vortices in open-channel flows, known as very-large-scale motions (VLSMs), raises a question regarding the interrelations between VLSMs and SCs in flows over ridge-covered fully rough beds. To address it, we conducted long-duration experiments using stereoscopic particle image velocimetry, covering a range of ridge spacings ($s$) from ${\approx}0.4$ to ${\approx}4$ flow depths ($H$). For a benchmark no-ridge case, the flow is quasi-two-dimensional in the central part of the channel, exhibiting a strong spectral signature of VLSMs, as expected. With ridges on the bed at $s\lessapprox 2H$, two SC cells are formed between neighbouring ridges and VLSMs are entirely suppressed, suggesting that ridge-induced SCs prevent the formation of VLSMs by absorbing their energy or overpowering their formation. At the same time, velocity auto- and cross-spectra reveal a new feature that can be explained by low-amplitude meandering of the alternating low- and high-momentum flow regions associated with instantaneous manifestations of SCs. Two-point velocity correlations and smooth velocity field reconstructions using proper orthogonal decomposition further support the validity of this effect. Its origin is probably due to the instability related to the presence of inflection points in the spanwise distribution of the streamwise velocity within the SC cells. These results have implications for bed friction in open channels, where the friction factor may increase if depth-scale SCs are present, or decrease under conditions of sub-depth-scale SCs and suppressed VLSMs.

We perform direct numerical simulations of spiral turbulent Taylor–Couette (TC) flow for $400\leqslant Re_{i}\leqslant 1200$ and $-2000\leqslant Re_{o}\leqslant -1000$, i.e. counter-rotation. The aspect ratio $\unicode[STIX]{x1D6E4}=\text{height}/\text{gap width}$ of the domain is $42\leqslant \unicode[STIX]{x1D6E4}\leqslant 125$, with periodic boundary conditions in the axial direction, and the radius ratio $\unicode[STIX]{x1D702}=r_{i}/r_{o}=0.91$. We show that, with decreasing $Re_{i}$ or with decreasing $Re_{o}$, the formation of a turbulent spiral from an initially ‘featureless turbulent’ flow can be described by the phenomenology of the Ginzburg–Landau equations, similar as seen in the experimental findings of Prigent et al. (Phys. Rev. Lett., vol. 89, 2002, 014501) for TC flow at $\unicode[STIX]{x1D702}=0.98$ an $\unicode[STIX]{x1D6E4}=430$ and in numerical simulations of oblique turbulent bands in plane Couette flow by Rolland & Manneville (Eur. Phys. J., vol. 80, 2011, pp. 529–544). We therefore conclude that the Ginzburg–Landau description also holds when curvature effects play a role, and that the finite-wavelength instability is not a consequence of the no-slip boundary conditions at the upper and lower plates in the experiments. The most unstable axial wavelength $\unicode[STIX]{x1D706}_{z,c}/d\approx 41$ in our simulations differs from findings in Prigent et al., where $\unicode[STIX]{x1D706}_{z,c}/d\approx 32$, and so we conclude that $\unicode[STIX]{x1D706}_{z,c}$ depends on the radius ratio $\unicode[STIX]{x1D702}$. Furthermore, we find that the turbulent spiral is stationary in the reference frame of the mean velocity in the gap, rather than the mean velocity of the two rotating cylinders.

When a drop of fluid containing long, flexible polymers breaks up, it forms threads of almost constant thickness, whose size decreases exponentially in time. Using an Oldroyd-B fluid as a model, we show that the thread profile, rescaled by the thread thickness, converges to a similarity solution. Using the correspondence between viscoelastic fluids and nonlinear elasticity, we derive similarity equations for the full three-dimensional axisymmetric flow field in the limit that the viscosity of the solvent fluid can be neglected. Deriving a conservation law along the thread, we can calculate the stress inside the thread from a measurement of the thread thickness. The explicit form of the velocity and stress fields can be deduced from a solution of the similarity equations. Results are validated by detailed comparison with numerical simulations.