Direct numerical simulations of Taylor–Couette flow, i.e. the flow between two coaxial and independently rotating cylinders, were performed. Shear Reynolds numbers of up to $3\times 10^{5}$, corresponding to Taylor numbers of $\mathit{Ta}=4.6\times 10^{10}$, were reached. Effective scaling laws for the torque are presented. The transition to the ultimate regime, in which asymptotic scaling laws (with logarithmic corrections) for the torque are expected to hold up to arbitrarily high driving, is analysed for different radius ratios, different aspect ratios and different rotation ratios. It is shown that the transition is approximately independent of the aspect and rotation ratios, but depends significantly on the radius ratio. We furthermore calculate the local angular velocity profiles and visualize different flow regimes that depend both on the shearing of the flow, and the Coriolis force originating from the outer cylinder rotation. Two main regimes are distinguished, based on the magnitude of the Coriolis force, namely the co-rotating and weakly counter-rotating regime dominated by Rayleigh-unstable regions, and the strongly counter-rotating regime where a mixture of Rayleigh-stable and Rayleigh-unstable regions exist. Furthermore, an analogy between radius ratio and outer-cylinder rotation is revealed, namely that smaller gaps behave like a wider gap with co-rotating cylinders, and that wider gaps behave like smaller gaps with weakly counter-rotating cylinders. Finally, the effect of the aspect ratio on the effective torque versus Taylor number scaling is analysed and it is shown that different branches of the torque-versus-Taylor relationships associated to different aspect ratios are found to cross within 15 % of the Reynolds number associated to the transition to the ultimate regime. The paper culminates in phase diagram in the inner versus outer Reynolds number parameter space and in the Taylor versus inverse Rossby number parameter space, which can be seen as the extension of the Andereck et al. (J. Fluid Mech., vol. 164, 1986, pp. 155–183) phase diagram towards the ultimate regime.

Recent experimental studies have shown rich transition behaviour in rotating plane Couette flow (RPCF). In this paper we study the transition in supercritical RPCF theoretically by determination of equilibrium and periodic orbit tertiary states via Floquet analysis on secondary Taylor vortex solutions. Two new tertiary states are discovered which we name oscillatory wavy vortex flow (oWVF) and skewed vortex flow (SVF). We present the bifurcation routes and stability properties of these new tertiary states and, in addition, we describe a bifurcation procedure whereby a set of defected wavy twist vortices is approached. Further to this, transition scenarios at flow parameters relevant to experimental works are investigated by computation of the set of stable attractors which exist on a large domain. The physically observed flow states are shown to share features with states in our set of attractors.

In the present treatise, the stability of the boundary layer under solitary waves is analysed by means of the parabolized stability equation. We investigate both surface solitary waves and internal solitary waves. The main result is that the stability of the flow is not of parametric nature as has been assumed in the literature so far. Not only does linear stability analysis highlight this misunderstanding, it also gives an explanation why Sumer et al. (J. Fluid Mech., vol. 646, 2010, pp. 207–231), Vittori & Blondeaux (Coastal Engng, vol. 58, 2011, pp. 206–213) and Ozdemir et al. (J. Fluid Mech., vol. 731, 2013, pp. 545–578) each obtained different critical Reynolds numbers in their experiments and simulations. We find that linear instability is possible in the acceleration region of the flow, leading to the question of how this relates to the observation of transition in the acceleration region in the experiments by Sumer et al. or to the conjecture of a nonlinear instability mechanism in this region by Ozdemir et al. The key concept for assessment of instabilities is the integrated amplification which has not been employed for this kind of flow before. In addition, the present analysis is not based on a uniformization of the flow but instead uses a fully nonlinear description including non-parallel effects, weakly or fully. This allows for an analysis of the sensitivity with respect to these effects. Thanks to this thorough analysis, quantitative agreement between model results and direct numerical simulation has been obtained for the problem in question. The use of a high-order accurate Navier–Stokes solver is primordial in order to obtain agreement for the accumulated amplifications of the Tollmien–Schlichting waves as revealed in this analysis. An elaborate discussion on the effects of amplitudes and water depths on the stability of the flow is presented.

We study the occurrence of the multiple steady states that flows in a collapsible tube can develop under the effect of: (i) geometrical alterations (e.g. stenosis), (ii) variations of the mechanical properties of the tube wall, or (iii) variations of the external pressure acting on the conduit. Specifically, if the approaching flow is supercritical, two steady flow states are possible in a restricted region of the parameter space: one of these flow states is wholly supercritical while the other produces an elastic jump that is located upstream of the variation. In the latter case the flow undergoes a transition through critical conditions in the modified segment of the conduit. Both states being possible, the actual state is determined by the past history of the system, and the parameter values show a hysteretic behaviour when shifting from one state to the other. First we set up the problem in a theoretical framework assuming stationary conditions, and then we analyse the dynamics numerically in a one-dimensional framework. Theoretical considerations suggest that the existence of multiple states is associated with non-uniqueness of the steady-state solution, which is confirmed by numerical simulations of the fully unsteady problem.

The nonlinear instability of the density-inverted granular Leidenfrost state and the resulting convective motion in strongly shaken granular matter are analysed via a weakly nonlinear analysis of the hydrodynamic equations. The base state is assumed to be quasi-steady and the effect of harmonic shaking is incorporated by specifying a constant granular temperature at the vibrating plate. Under these mean-field assumptions, the base-state temperature decreases with increasing height away from the vibrating plate, but the density profile consists of three distinct regions: (i) a collisional dilute layer at the bottom, (ii) a levitated dense layer at some intermediate height and (iii) a ballistic dilute layer at the top of the granular bed. For the nonlinear stability analysis (Shukla & Alam, J. Fluid Mech., vol. 672, 2011b, pp. 147–195), the nonlinearities up to cubic order in the perturbation amplitude are retained, leading to the Landau equation, and the related adjoint stability problem is formulated taking into account appropriate boundary conditions. The first Landau coefficient and the related modal eigenfunctions (the fundamental mode and its adjoint, the second harmonic and the base-flow distortion, and the third harmonic and the cubic-order distortion to the fundamental mode) are calculated using a spectral-based numerical method. The genesis of granular convection is shown to be tied to a supercritical pitchfork bifurcation from the density-inverted Leidenfrost state. Near the bifurcation point the equilibrium amplitude ($A_{e}$) is found to follow a square-root scaling law, $A_{e}\sim \sqrt{{\it\Delta}}$, with the distance ${\it\Delta}$ from the bifurcation point. We show that the strength of convection (measured in terms of velocity circulation) is maximal at some intermediate value of the shaking strength, with weaker convection at both weaker and stronger shaking. Our theory predicts that at very strong shaking the convective motion remains concentrated only near the top surface, with the bulk of the expanded granular bed resembling the conduction state of a granular gas, dubbed as a floating-convection state. The linear and nonlinear patterns of the density and velocity fields are analysed and compared with experiments qualitatively. Evidence of 2:1 resonance is shown for certain parameter combinations. The influences of bulk viscosity, effective Prandtl number, shear work and free-surface boundary conditions on nonlinear equilibrium states are critically assessed.

Magnetohydrodynamic (MHD) turbulence at low magnetic Reynolds number is experimentally investigated by studying a liquid metal flow in a cubic domain. We focus on the mechanisms that determine whether the flow is quasi-two-dimensional, three-dimensional or in any intermediate state. To this end, forcing is applied by injecting a DC current $I$ through one wall of the cube only, to drive vortices spinning along the magnetic field. Depending on the intensity of the externally applied magnetic field, these vortices extend part or all of the way through the cube. Driving the flow in this way allows us to precisely control not only the forcing intensity but also its dimensionality. A comparison with the theoretical analysis of this configuration singles out the influences of the walls and of the forcing on the flow dimensionality. Flow dimensionality is characterised in several ways. First, we show that when inertia drives three-dimensionality, the velocity near the wall where current is injected scales as $U_{b}\sim I^{2/3}$. Second, we show that when the distance $l_{z}$ over which momentum diffuses under the action of the Lorentz force (Sommeria & Moreau, J. Fluid Mech., vol. 118, 1982, pp. 507–518) reaches the channel width $h$, the velocity near the opposite wall $U_{t}$ follows a similar law with a correction factor $(1-h/l_{z})$ that measures three-dimensionality. When $l_{z}<h$, by contrast, the opposite wall has less influence on the flow and $U_{t}\sim I^{1/2}$. The central role played by the ratio $l_{z}/h$ is confirmed by experimentally verifying the scaling $l_{z}\sim N^{1/2}$ put forward by Sommeria & Moreau ($N$ is the interaction parameter) and, finally, the nature of the three-dimensionality involved is further clarified by distinguishing weak and strong three-dimensionalities previously introduced by Klein & Pothérat (Phys. Rev. Lett., vol. 104 (3), 2010, 034502). It is found that both types vanish only asymptotically in the limit $N\rightarrow \infty$. This provides evidence that because of the no-slip walls, (i) the transition between quasi-two-dimensional and three-dimensional turbulence does not result from a global instability of the flow, unlike in domains with non-dissipative boundaries (Boeck et al. Phys. Rev. Lett., vol. 101, 2008, 244501), and (ii) it does not occur simultaneously at all scales.

Using a model experiment, we explore the dynamics of inertial liquid jets arising from a gravitational cavity collapse. The focus of the study is to elucidate the link between both the dynamical and kinematical properties of the jet and the initial cavity geometry, for a wide range of physical parameters. We demonstrate that the jets exhibit shape similarity and reveal a robust relationship between the jet tip velocity and the initial cavity geometry, regardless of the details of the collapse process. We argue that this relation reflects a flow focusing mechanism, and we propose a simple model capturing the key features of the erupting jet velocity scaling. Finally, the relevance of these results to other jets occurring in e.g. large bubble detachment or wave impact on walls is discussed.

The dynamics of an axisymmetric liquid bridge of a fluid of density ${\it\rho}$, viscosity ${\it\mu}$ and surface tension ${\it\sigma}$ held between two co-axial disks of equal radius $e$ is studied when one disk is slowly moved with a velocity $U(t)$. The analysis is performed using a one-dimensional model for thin bridges. We consider attached boundary conditions (the contact line is fixed to the boundary of the disk), neglect gravity and limit our analysis to short bridges such that there exists a stable equilibrium shape. This equilibrium is a Delaunay curve characterized by the two geometric parameters $\ell _{0}=V_{0}/({\rm\pi}e^{3})$ and $S=\ell ({\rm\pi}e^{2})/V_{0}$, where $V_{0}$ is the volume of fluid and $\ell$ the length of the bridge. Our objective is to analyse the departure of the dynamical solution from the static Delaunay shape as a function of $\ell _{0}$, $S$, the Ohnesorge number $Oh={\it\mu}/\sqrt{{\it\rho}{\it\sigma}e}$, and the instantaneous velocity $U(t)$ and acceleration $\partial _{t}U$ (non-dimensionalized using $e$ and ${\it\sigma}/{\it\rho}$) of the disk. Using a perturbation theory for small velocity and acceleration, we show that (i) a non-homogeneous velocity field proportional to $U(t)$ and independent of $Oh$ is present within the bridge; (ii) the area correction to the equilibrium shape can be written as $U^{2}A_{i}+U\,Oh\,A_{v}+\partial _{t}UA_{a}$ where $A_{i}$, $A_{v}$ and $A_{a}$ are functions of $\ell _{0}$ and $S$ only. The characteristics of the velocity field and the shape corrections are analysed in detail. For the case of a cylinder ($S=1$), explicit expressions are derived and used to provide some insight into the break-up that the deformation would induce. The asymptotic results are validated and tested by direct numerical simulations when the velocity is constant and when it oscillates. For the constant velocity case, we demonstrate that the theory provides a very good estimate of the dynamics for a large range of parameters. However, a systematic departure is observed for very small $Oh$ due to the persistence of free eigenmodes excited during the transient. These same eigenmodes also limit the applicability of the theory to oscillating bridges with large oscillating periods. Finally, the perturbation theory is applied to the cylindrical solution of Frankel & Weihs (J. Fluid Mech., vol. 155 (1985), pp. 289–307) obtained for constant velocity $U$ when the contact lines are allowed to move. We show that it can be used to compute the correction associated with acceleration. Finally, the effect of gravity is discussed and shown to modify the equilibrium shape but not the main results obtained from the perturbation theory.

The development of incompressible turbulent flow through a pipe of wavy cross-section was studied numerically by direct integration of the Navier–Stokes equations. Simulations were performed at Reynolds numbers of $4.5\times 10^{3}$ and $10^{4}$ based on the hydraulic diameter and the bulk velocity. Results for the pressure resistance coefficient ${\it\lambda}$ were found to be in excellent agreement with experimental data of Schiller (Z. Angew. Math. Mech., vol. 3, 1922, pp. 2–13). Of particular interest is the decrease in ${\it\lambda}$ below the level predicted from the Blasius correlation, which fits almost all experimental results for pipes and ducts of complex cross-sectional geometries. Simulation databases were used to evaluate turbulence anisotropy and provide insights into structural changes of turbulence leading to flow relaminarization. Anisotropy-invariant mapping of turbulence confirmed that suppression of turbulence is due to statistical axisymmetry in the turbulent stresses.

Microdroplet deposition is a technology that spans applications from tissue engineering to microelectronics. Our new high-speed imaging measurements reveal how sequential linear deposition of overlapping droplets on flat uniform substrates leads to striking non-uniform morphologies for moderate contact angles. We develop a simple physical model, which for the first time captures the post-impact dynamics drop-by-drop: surface-tension drives liquid redistribution, contact-angle hysteresis underlies initial non-uniformity, while viscous effects cause subsequent periodic variations.

We describe a model for the speed of an internal bore as a function of amplitude in continuous stratification of arbitrary form. The model is developed from the Dubreil-Jacotin–Long theory for nonlinear solitary waves in the conjugate flow limit, which represents an internal hydraulic jump, by allowing dissipation across the jump. The bore speeds predicted by the model are consistent in both the small- and large-amplitude limits with the waveguide intrinsic to the ambient stratification. The model therefore represents a significant advancement over previous theories limited to sharp two-layer stratification. The model shows good agreement with Navier–Stokes simulations of both undular and turbulent internal bores generated by dam break into a continuously stratified ambient with a finite pycnocline, predicting both the front speed as well as the velocity and density structure through the bore. A model is required for the structure of the energy dissipation, and we introduce a one-parameter closure that produces excellent agreement with numerical results, particularly in the parameter limit that maximizes the overall dissipation. By varying the dissipation parameter, the model reproduces previous two-layer theories in the thin-pycnocline limit, and suggests an improved two-layer front speed relationship. It is demonstrated that, even for the sharp two-layer limit, continuous stratification, and particularly the nonlinear waveguide, must be accounted for in order to accurately predict the bore speed and structure.

A laboratory experiment shows that fluid can be pumped from one end to another in a narrow channel whose sawtooth walls vibrate transversely opposite to each other. The phenomenon is referred to as ratcheting fluid. Inspired by this, we put forward here a theory describing the rectified steady flow, and the net directional pumping. In a conformally transformed plane, the induced steady streaming in the Stokes boundary layer of the oscillatory flow is analysed, elucidating the driving mechanism that is due to the nonlinearity and viscosity. The solution of the stream function is given, showing the complex spatial structure of the induced steady flow and its spatial average that is related to the directional pumping. Whereas the wall sawtooth shape is the primary source of asymmetry, the difference in entrance and exit flow conditions due to the geometries at channel ends is found to be a secondary source to break the left–right symmetry of the system. This latter can affect the net directional transport of fluid, in particular in short channels with a small number of sawteeth. Various influences on the net pumping rate are analysed.

Motivated by understanding mass transport processes occurring in the vitreous chamber of the eye, we consider the steady streaming component of the flow generated in a viscoelastic fluid contained within a hollow, rigid sphere performing small-amplitude, periodic, torsional oscillations about an axis passing through its centre. The problem is solved semi-analytically, assuming that the amplitude of the oscillations is small. The paper extends the work by Repetto et al. (J. Fluid Mech., vol. 608, 2008, pp. 71–80), in which the case of a purely viscous fluid was analysed. However, in reality, in young and healthy subjects, the vitreous humour has complex rheological properties, and so here we model it as a viscoelastic fluid. A similar problem was studied by Nikolakis (Eine Theorie für stationäre Drifterscheinungen viskoelastischer Flüssigkeiten, 1992, VDI). In the present model, the steady streaming flow is governed by four dimensionless parameters. We show that, when we account for the viscoelasticity of the fluid, there is a considerably more complex set of possible flow regimes than was found in the purely viscous case, and the flows can be classified into five qualitatively different types. Whereas there was only one circulation cell in each hemisphere in the viscous case, accounting for viscoelasticity it is possible have either one, two or three circulation cells, with different senses of rotation, depending on the parameter values.

We study localised exact coherent structures in plane Poiseuille flow that are relative periodic orbits. They are obtained from extended states in smaller periodically continued domains, by increasing the length to obtain streamwise localisation and then by increasing the width to achieve spanwise localisation. The states maintain the travelling wave structure of the extended states, which is then modulated by a localised envelope on larger scales. In the streamwise direction, the envelope shows exponential localisation, with different exponents on the upstream and downstream sides. The upstream exponent increases linearly with Reynolds number $\mathit{Re}$, but the downstream exponent is essentially independent of $\mathit{Re}$. In the spanwise direction the decay is compatible with a power-law localisation. As the width increases the localised state undergoes further bifurcations which add additional unstable directions, so that the edge state, the relative attractor on the boundary between the laminar and turbulent motions, in the system becomes chaotic.

Using direct numerical simulations (DNS), we investigate the structure and energetics of breaking internal waves on slopes. We employ a Navier–Stokes code in an idealized three-dimensional domain where an internal solitary wave of depression impinges upon a sloping bottom. Seven cases with varying initial wave amplitude and bathymetric slope, but constant wave Reynolds number $\mathit{Re}_{w}$ are considered. Volume-integrated values of dissipation and irreversible mixing are related to the density and velocity structure of the wave throughout the breaking process. The majority of dissipation (63 %) occurs along the no-slip bottom boundary. Most of the remaining dissipation (35 %) and nearly all irreversible mixing occurs in the interior after breaking, when density overturns are present at the interface. Breaking introduces three-dimensionality to the flow field that is driven by the lateral breakdown of density overturns and the lobe–cleft instability typical of gravity currents. The resulting longitudinal rolls (streamwise vorticity) increase dissipation by roughly 8 % and decrease irreversible mixing by roughly 20 % when compared with a similar two-dimensional simulation. The bulk mixing efficiency is shown to increase for larger and smaller values of the internal Iribarren number ${\it\xi}$, with a minimum for intermediate values of ${\it\xi}$ and a peak near ${\it\xi}=0.8$ for plunging breakers. This trend is explained by the degree of two-dimensionality in the flow, and agrees with previous results in the literature after accounting for Reynolds number effects. Local turbulence quantities are also calculated at ‘virtual moorings’, and a location upslope of the breakpoint but downslope of the intersection of the pycnocline and the bottom is shown to provide a signal that is most representative of the volume-integrated dissipation and mixing results.

Thermoacoustic systems can oscillate self-excitedly, and often non-periodically, owing to coupling between unsteady heat release and acoustic waves. We study a slot-stabilized two-dimensional premixed flame in a duct via numerical simulations of a $G$-equation flame coupled with duct acoustics. We examine the bifurcations and routes to chaos for three control parameters: (i) the flame position in the duct, (ii) the length of the duct and (iii) the mean flow velocity. We observe period-1, period-2, quasi-periodic and chaotic oscillations. For certain parameter ranges, more than one stable state exists, so mode switching is possible. At intermediate times, the system is attracted to and repelled from unstable states, which are also identified. Two routes to chaos are established for this system: the period-doubling route and the Ruelle–Takens–Newhouse route. These are corroborated by analyses of the power spectra of the acoustic velocity. Instantaneous flame images reveal that the wrinkles on the flame surface and pinch-off of flame pockets are regular for periodic oscillations, while they are irregular and have multiple time and length scales for quasi-periodic and aperiodic oscillations. This study complements recent experiments by providing a reduced-order model of a system with approximately 5000 degrees of freedom that captures much of the elaborate nonlinear behaviour of ducted premixed flames observed in the laboratory.

We investigate the statistical properties of Lagrangian tracers transported by a time-correlated compressible renewing flow. We show that the preferential sampling of the phase space performed by tracers yields significant differences between the Lagrangian statistics and its Eulerian counterpart. In particular, the effective compressibility experienced by tracers has a non-trivial dependence on the time correlation of the flow. We examine the consequence of this phenomenon on the clustering of tracers, focusing on the transition from the weak- to the strong-clustering regime. We find that the critical compressibility at which the transition occurs is minimum when the time correlation of the flow is of the order of the typical eddy turnover time. Further, we demonstrate that the clustering properties in time-correlated compressible flows are non-universal and are strongly influenced by the spatio-temporal structure of the velocity field.

The transition from laminar to complex spatio-temporal dynamics of plumes due to a localized buoyancy source is studied numerically. Several experiments have reported that this transition is sensitive to external perturbations. Therefore, a well-controlled set-up has been chosen for our numerical study, consisting of a localized heat source at the bottom of an enclosed cylinder whose sidewall is maintained at a fixed temperature which varies linearly up the wall. Restricting the dynamics to the axisymmetric subspace, the first instability is to a puffing state. However, for smaller Grashof numbers, the plume becomes unstable to three-dimensional perturbations and a swirling plume spontaneously appears. The next bifurcation, viewed in the rotating frame where the plume is stationary, also exhibits puffing and suggests a connection between the unstable axisymmetric puffing solution and the swirling plume. Further bifurcations result in quasi-periodic states with a very low-frequency modulation, and these eventually become spatio-temporally complex.

Liquid–bubble interaction, especially in complex two-phase bubbly flow under breaking waves, is still poorly understood. In the present study, we perform a large-eddy simulation using a Navier–Stokes solver extended to incorporate entrained bubble populations, using an Eulerian–Eulerian formulation for a polydisperse bubble phase. The volume-of-fluid method is used for free-surface tracking. We consider an isolated unsteady deep water breaking event generated by a focused wavepacket. Bubble contributions to dissipation and momentum transfer between the water and air phases are considered. The model is shown to predict free-surface evolution, mean and turbulent velocities, and integral properties of the entrained dispersed bubbles fairly well. We investigate turbulence modulation by dispersed bubbles as well as shear- and bubble-induced dissipation, in both spilling and plunging breakers. We find that the total bubble-induced dissipation accounts for more than 50 % of the total dissipation in the breaking region. The average dissipation rate per unit length of breaking crest is usually written as $b{\it\rho}g^{-1}c_{b}^{5}$, where ${\it\rho}$ is the water density, $g$ is the gravitational acceleration and $c_{b}$ is the phase speed of the breaking wave. The breaking parameter, $b$, has been poorly constrained by experiments and field measurements. We examine the time-dependent evolution of $b$ for both constant-steepness and constant-amplitude wavepackets. A scaling law for the averaged breaking parameter is obtained. The exact two-phase transport equation for turbulent kinetic energy (TKE) is compared with the conventional single-phase transport equation, and it is found that the former overpredicts the total subgrid-scale dissipation and turbulence production by mean shear during active breaking. All of the simulations are also repeated without the inclusion of a dispersed bubble phase, and it is shown that the integrated TKE in the breaking region is damped by the dispersed bubbles by approximately 20 % for a large plunging breaker to 50 % for spilling breakers. In the plunging breakers, the TKE is damped slightly or even enhanced during the initial stage of active breaking.

A new theory for the emergence of dispersion in shallow water hydrodynamics in two horizontal space dimensions is presented. Starting with the key properties of uniform flow in open-channel hydraulics, it is shown that criticality is the key mechanism for generating dispersion. Modulation of the uniform flow then leads to model equations. The coefficients in the model equations are related precisely to the derivatives of the mass flux, momentum flux and mass density. The theory gives a new perspective – from the viewpoint of hydraulics – on how and why key shallow water models like the Korteweg–de Vries equation and Kadomtsev–Petviashvili equations arise in the theory of water waves.