We consider the enhanced mixing due to multiple cylinders organised in schools moving synchronously in a potential flow. Here simple interactions between cylinders are modelled by the method of image doublets. This is an extension to Thiffeault & Childress’s work (Phys. Lett. A, vol. 374, 2010, pp. 3487–3490) where fluid particle displacements due to non-interacting swimmers were analysed to produce an effective diffusivity that may have a significant impact in ocean mixing. Our results show that schools of two cylinders induce nonlinearly boosted diffusivity compared with the non-interacting case for general configuration parameters, except when they move along a straight line with small separation. We attribute this phenomenon to two different physical mechanisms via which interacting cylinders cooperate to generate long particle drifts depending on their formation. Finally, the effective diffusivity of schools of three or more cylinders in various configurations are also discussed.

The phenomenon of lock-in in vortex-induced vibration of a circular cylinder is investigated in the laminar flow regime ($20\leqslant Re\leqslant 100$). Direct time integration (DTI) and linear stability analysis (LSA) of the governing equations are carried out via a stabilized finite element method. Using the metrics that have been proposed in earlier studies, the lock-in regime is identified from the results of DTI. The LSA yields the eigenmodes of the coupled fluid–structure system, the associated frequencies ($F_{LSA}$) and the stability of the steady state. A linearly unstable system, in the absence of nonlinear effects, achieves large oscillation amplitude at sufficiently large times. However, the nonlinear terms saturate the response of the system to a limit cycle. For subcritical $Re$, the occurrence of lock-in coincides with the linear instability of the fluid–structure system. The critical $Re$ is the Reynolds number beyond which vortex shedding ensues for a stationary cylinder. For supercritical $Re$, even though the aeroelastic system is unstable for all reduced velocities ($U^{\ast }$) lock-in occurs only for a finite range of $U^{\ast }$. We present a method to estimate the time beyond which the nonlinear effects are expected to be significant. It is observed that much of the growth in the amplitude of cylinder oscillation takes place in the linear regime. The response of the cylinder at the end of the linear regime is found to depend on the energy ratio, $E_{r}$, of the unstable eigenmode. $E_{r}$ is defined as the fraction of the total energy of the eigenmode that is associated with the kinetic and potential energy of the structure. DTI initiated from eigenmodes that are linearly unstable and whose energy ratio is above a certain threshold value lead to lock-in. Interestingly, during lock-in, the oscillation frequency of the fluid–structure system drifts from $F_{LSA}$ towards a value that is closer to the natural frequency of the oscillator in vacuum ($F_{N}$). In the event of more than one eigenmode being linearly unstable, we investigate which one is responsible for lock-in. The concept of phase angle between the cylinder displacement and lift is extended for an eigenmode. The phase angle controls the direction of energy transfer between the fluid and the structure. For zero structural damping, if the phase angle of all unstable eigenmodes is less than 90°, the phase angle obtained via DTI evolves to a value that is close to 0°. If, on the other hand, the phase angle of any unstable eigenmode is more than 90°, it settles to 180°, approximately in the limit cycle. A new approach towards classification of modes is presented. The eigenvalues are tracked over a wide range of $U^{\ast }$ while keeping $Re$ and mass ratio ($m^{\ast }$) fixed. In general, for large values of $m^{\ast }$, the eigenmodes corresponding to the two leading eigenvalues exhibit a decoupled behaviour with respect to $U^{\ast }$. They are classified as the fluid and elastic modes. However, for relatively low $m^{\ast }$ such a classification is not possible. The two leading modes are coupled and are referred to as fluid–elastic modes. The regime of such occurrence is shown on the $Re{-}m^{\ast }$ parameter space.

We model the unsteady evolution of turbulent buoyant plumes following temporal changes to the source conditions. The integral model is derived from radial integration of the governing equations expressing the evolution of mass, axial momentum and buoyancy in the plume. The non-uniform radial profiles of the axial velocity and density deficit in the plume are explicitly captured by shape factors in the integral equations; the commonly assumed top-hat profiles lead to shape factors equal to unity. The resultant model for unsteady plumes is hyperbolic when the momentum shape factor, determined from the radial profile of the mean axial velocity in the plume, differs from unity. The solutions of the model when source conditions are maintained at constant values are shown to retain the form of the well-established steady plume solutions. We demonstrate through a linear stability analysis of these steady solutions that the inclusion of a momentum shape factor in the governing equations that differs from unity leads to a well-posed integral model. Therefore, our model does not exhibit the mathematical pathologies that appear in previously proposed unsteady integral models of turbulent plumes. A stability threshold for the value of the shape factor is also identified, resulting in a range of its values where the amplitudes of small perturbations to the steady solutions decay with distance from the source. The hyperbolic character of the system of equations allows the formation of discontinuities in the fields describing the plume properties during the unsteady evolution, and we compute numerical solutions to illustrate the transient development of a plume following an abrupt change in the source conditions. The adjustment of the plume to the new source conditions occurs through the propagation of a pulse of fluid through the plume. The dynamics of this pulse is described by a similarity solution and, through the construction of this new similarity solution, we identify three regimes in which the evolution of the transient pulse following adjustment of the source qualitatively differs.

We report an experimental study of confinement effects in quasi-2-D turbulent Rayleigh–Bénard convection. The experiments were conducted in five rectangular cells with their height $H$ and length $L$ being the same and fixed, while the width $W$ was different for each cell to produce lateral aspect ratios (${\it\Gamma}=W/H$) of 0.6, 0.3, 0.2, 0.15 and 0.1. Direct flow field measurements reveal that the large-scale flow slows down as ${\it\Gamma}$ decreases and there are more plumes travelling through the bulk region. Moreover, the reversal frequency of the large-scale flow is found to increase drastically in smaller ${\it\Gamma}$ cells, by more than 1000-fold for the highest value of Rayleigh number reached in the experiment. The reversal frequency can be well described by a stochastic model developed by Ni et al. (J. Fluid Mech., vol. 778, 2015, R5) and the probability density functions (PDF) of the time interval between successive reversals are found to follow Poisson statistics as in the 3-D system. It is further observed that the bulk temperature fluctuation increases significantly and its PDF changes from exponential to Gaussian as ${\it\Gamma}$ decreases. The influences of geometric confinement on the global heat transport are also investigated. The measured Nu–Ra relationship suggests that, as the lateral aspect ratio decreases, the relative weight of the boundary layer contribution in the global heat transport increases compared to that from the bulk. These results demonstrate that in the quasi-2-D geometry, geometric confinement has strong effects on both the global and local properties in turbulent convective flows, which are very different from the previous findings in 3-D and true 2-D systems.

We report joint Lagrangian velocity and temperature measurements in turbulent thermal convection. Measurements are performed using an improved version (extended autonomy) of the neutrally buoyant instrumented particle (Shew et al., Rev. Sci. Instrum., vol. 78, 2007, 065105) that was used by Gasteuil et al. (Phys. Rev. Lett., vol. 99, 2007, 234302) to performed experiments in a parallelepipedic Rayleigh–Bénard cell. The temperature signal is obtained from a radiofrequency transmitter. Simultaneously, we determine a particle’s position and velocity with one camera, which grants access to the Lagrangian heat flux. Due to the extended autonomy of the present particle, we obtain well-converged temperature and velocity statistics, as well as pseudo-Eulerian maps of velocity and heat flux. Present experimental results have also been compared with the results obtained by a corresponding campaign of direct numerical simulations and Lagrangian tracking of massless tracers. The comparison between experimental and numerical results shows the accuracy and reliability of our experimental measurements and points also out the finite-size effects of the particle. Finally, the analysis of Lagrangian velocity and temperature frequency spectra is shown and discussed. In particular, we observe that temperature spectra exhibit an anomalous $f^{-2.5}$ frequency scaling, likely representing the ubiquitous passive and active scalar behaviour of temperature.

A free falling, absorbing liquid drop hit by a nanosecond laser pulse experiences a strong recoil pressure kick. As a consequence, the drop propels forward and deforms into a thin sheet which eventually fragments. We study how the drop deformation depends on the pulse shape and drop properties. We first derive the velocity field inside the drop on the time scale of the pressure pulse, when the drop is still spherical. This yields the kinetic energy partition inside the drop, which precisely measures the deformation rate with respect to the propulsion rate, before surface tension comes into play. On the time scale where surface tension is important, the drop has evolved into a thin sheet. Its expansion dynamics is described with a slender-slope model, which uses the impulsive energy partition as an initial condition. Completed with boundary integral simulations, this two-stage model explains the entire drop dynamics and its dependence on the pulse shape: for a given propulsion, a tightly focused pulse results in a thin curved sheet which maximizes the lateral expansion, while a uniform illumination yields a smaller expansion but a flat symmetric sheet, in good agreement with experimental observations.

A new, exact Floquet theory is presented for linear waves in two-layer fluids over a periodic bottom of arbitrary shape and amplitude. A method of conformal transformation is adapted. The solutions are given, in essentially analytical form, for the dispersion relation between wave frequency and generalized wavenumber (Floquet exponent), and for the waveforms of free wave modes. These are the analogues of the classical Lamb’s solutions for two-layer fluids over a flat bottom. For internal modes the interfacial wave shows rapid modulation at the scale of its own wavelength that is comparable to the bottom wavelength, whereas for surface modes it becomes a long wave carrier for modulating short waves of the bottom wavelength. The approximation using a rigid lid is given. Sample calculations are shown, including the solutions that are inside the forbidden bands (i.e. Bragg resonated).

This paper reports on a theoretical analysis of the rich variety of spatio-temporal patterns observed recently in inclined layer convection at medium Prandtl number when varying the inclination angle ${\it\gamma}$ and the Rayleigh number $R$. The present numerical investigation of the inclined layer convection system is based on the standard Oberbeck–Boussinesq equations. The patterns are shown to originate from a complicated competition of buoyancy driven and shear-flow driven pattern forming mechanisms. The former are expressed as longitudinal convection rolls with their axes oriented parallel to the incline, the latter as perpendicular transverse rolls. Along with conventional methods to study roll patterns and their stability, we employ direct numerical simulations in large spatial domains, comparable with the experimental ones. As a result, we determine the phase diagram of the characteristic complex 3-D convection patterns above onset of convection in the ${\it\gamma}{-}R$ plane, and find that it compares very well with the experiments. In particular we demonstrate that interactions of specific Fourier modes, characterized by a resonant interaction of their wavevectors in the layer plane, are key to understanding the pattern morphologies.

We present direct numerical simulations of Taylor–Couette flow with grooved walls at a fixed radius ratio ${\it\eta}=r_{i}/r_{o}=0.714$ with inner cylinder Reynolds number up to $Re_{i}=3.76\times 10^{4}$, corresponding to Taylor number up to $Ta=2.15\times 10^{9}$. The grooves are axisymmetric V-shaped obstacles attached to the wall with a tip angle of 90°. Results are compared to the smooth wall case in order to investigate the effects of grooves on Taylor–Couette flow. We focus on the effective scaling laws for the torque, flow structures, and boundary layers. It is found that, when the groove height is smaller than the boundary layer thickness, the torque is the same as that of the smooth wall cases. With increasing $Ta$, the boundary layer thickness becomes smaller than the groove height. Plumes are ejected from the tips of the grooves and secondary circulations between the latter are formed. This is associated with a sharp increase of the torque, and thus the effective scaling law for the torque versus $Ta$ becomes much steeper. Further increasing $Ta$ does not result in an additional slope increase. Instead, the effective scaling law saturates to the ‘ultimate’ regime effective exponents seen for smooth walls. It is found that even though after saturation the slope is the same as for the smooth wall case, the absolute value of torque is increased, and more so with the larger size of the grooves.

The horizontal gradient of potential vorticity (PV) across the tropopause typically declines with lead time in global numerical weather forecasts and tends towards a steady value dependent on model resolution. This paper examines how spreading the tropopause PV contrast over a broader frontal zone affects the propagation of Rossby waves. The approach taken is to analyse Rossby waves on a PV front of finite width in a simple single-layer model. The dispersion relation for linear Rossby waves on a PV front of infinitesimal width is well known; here, an approximate correction is derived for the case of a finite-width front, valid in the limit that the front is narrow compared to the zonal wavelength. Broadening the front causes a decrease in both the jet speed and the ability of waves to propagate upstream. The contribution of these changes to Rossby wave phase speeds cancel at leading order. At second order the decrease in jet speed dominates, meaning phase speeds are slower on broader PV fronts. This asymptotic phase speed result is shown to hold for a wide class of single-layer dynamics with a varying range of PV inversion operators. The phase speed dependence on frontal width is verified by numerical simulations and also shown to be robust at finite wave amplitude, and estimates are made for the error in Rossby wave propagation speeds due to the PV gradient error present in numerical weather forecast models.

Characteristics of laboratory-scale bubble-driven buoyant plumes in a stably stratified quiescent fluid are studied using large-eddy simulation (LES). As a bubble plume entrains stratified ambient water, its net buoyancy decreases due to the increasing density difference between the entrained and ambient fluids. A large fraction of the entrained fluid eventually detrains and falls along an annular outer plume from a height of maximum rise (peel height) to a neutral buoyancy level (trap height), during which less buoyant scalars (e.g. small droplets) are trapped and dispersed horizontally, forming quasi-horizontal intrusion layers. The inner/outer double-plume structure and the peel/intrusion process are found to be more distinct for cases with small bubble rise velocity, while weak and unstable when the slip velocity is large. LES results are averaged to generate distributions of mean velocity and turbulent fluxes. These distributions provide data for assessing the performance of previously developed closures used in one-dimensional integral plume models. In particular, the various LES cases considered in this study yield consistent behaviour for the entrainment coefficients for various plume cases. Furthermore, a new continuous peeling model is derived based on the insights obtained from LES results. Comparing to previous peeling models, the new model behaves in a more self-consistent manner, and it is expected to provide more reliable performance when applied in integral plume models.

An experiment on a flat rectangular plate facing a uniform flow at $Re=264\,000$ shows the importance of the base pressure loading on the asymmetric static modes of the turbulent wake. The plate is free to rotate around its short symmetry axis. For plates with aspect ratio ${\it\kappa}<6$, the angular position exhibits strong random discontinuities between steady states of non-zero angles. The steady states have long time durations, more than one order of magnitude greater than the convective time scale. The discontinuities, comparable to rare and violent events, are due to strong fluid forces associated with a drastic global change of the three-dimensional wake – mainly the switching between the static asymmetric modes. A clear transition occurs at ${\it\kappa}=6$, for which the angular fluctuations are minimum, leading for ${\it\kappa}>6$ to a classical fluid structure interaction with periodic fluctuations. The transition is supported by a recent global stability analysis of rectangular fixed plates in the laminar regime.

The transitional regime of incompressible pressure-driven flows inside an annular pipe is investigated using accurate direct numerical simulation in long computational domains. At marginally low friction Reynolds number $Re_{{\it\tau}}$, turbulence occurs in the form of intermittent localised structures. Different types of localisation are identified as the aspect ratio is varied from ${\it\eta}=0.8$ to $0.1$. These coherent structures vary from helical turbulence at ${\it\eta}=0.8$ to streamwise-localised puffs at ${\it\eta}=0.1$. They are respectively analogous to the stripe patterns and puffs formerly identified in plane channel flow and cylindrical pipe flow. Helical turbulence has been tracked down to ${\it\eta}=0.3$, accompanied by a monotonic reduction of the pitch angle. For ${\it\eta}=0.3$ and marginally low $Re_{{\it\tau}}$, these turbulence structures localise in the streamwise direction, giving rise to a new regime of helical puffs with chirality. The present results suggest that helical puffs mediate the transition from globally axisymmetric puffs to helical stripe patterns.

We consider mixing of the density field in stratified turbulence and argue that, at sufficiently high Reynolds numbers, stationary turbulence will have a mixing efficiency and closely related mixing coefficient described solely by the turbulent Froude number $Fr={\it\epsilon}_{k}/(Nu^{2})$, where ${\it\epsilon}_{k}$ is the kinetic energy dissipation, $u$ is a turbulent horizontal velocity scale and $N$ is the Brunt–Väisälä frequency. For $Fr\gg 1$, in the limit of weakly stratified turbulence, we show through a simple scaling analysis that the mixing coefficient scales as ${\it\Gamma}\propto Fr^{-2}$, where ${\it\Gamma}={\it\epsilon}_{p}/{\it\epsilon}_{k}$ and ${\it\epsilon}_{p}$ is the potential energy dissipation. In the opposite limit of strongly stratified turbulence with $Fr\ll 1$, we argue that ${\it\Gamma}$ should reach a constant value of order unity. We carry out direct numerical simulations of forced stratified turbulence across a range of $Fr$ and confirm that at high $Fr$, ${\it\Gamma}\propto Fr^{-2}$, while at low $Fr$ it approaches a constant value close to ${\it\Gamma}=0.33$. The parametrization of ${\it\Gamma}$ based on $Re_{b}$ due to Shih et al. (J. Fluid Mech., vol. 525, 2005, pp. 193–214) can be reinterpreted in this light because the observed variation of ${\it\Gamma}$ in their study as well as in datasets from recent oceanic and atmospheric measurements occurs at a Froude number of order unity, close to the transition value $Fr=0.3$ found in our simulations.

Oscillatory Stokes flows, with zero mean, are subjected to subcritical transition to turbulence. The maximal energy growth of perturbations is computed in the subcritical regime through an optimisation method. The results show strong amplifications during half a period. The energy transfer from the base flow involves an Orr mechanism with two-dimensional vorticity waves, and the maximum energy scales exponentially with the Reynolds number. Nonlinear simulations show that low-energy perturbations are sufficient to trigger turbulent flow.

What is the turbulent drag force experienced by an object moving in a rotating fluid? This open and fundamental question can be addressed by measuring the torque needed to drive an impeller at a constant angular velocity ${\it\omega}$ in a water tank mounted on a platform rotating at a rate ${\it\Omega}$. We report a dramatic reduction in drag as ${\it\Omega}$ increases, down to values as low as 12 % of the non-rotating drag. At small Rossby number $Ro={\it\omega}/{\it\Omega}$, the decrease in the drag coefficient $K$ follows the approximate scaling law $K\sim Ro$, which is predicted in the framework of nonlinear inertial-wave interactions and weak-turbulence theory. However, stereoscopic particle image velocimetry measurements indicate that this drag reduction instead originates from a weakening of the turbulence intensity in line with the two-dimensionalization of the large-scale flow.