Turbulence locally created in laminar pipe flows shows sudden decay or splitting after a stochastic waiting time. In laboratory experiments, the mean waiting time was observed to increase double-exponentially as the Reynolds number ($Re$) approaches its critical value. To understand the origin of this double-exponential increase, we perform many pipe flow direct numerical simulations of the Navier–Stokes equations, and measure the cumulative histogram of the maximum axial vorticity field over the pipe (turbulence intensity). In the domain where the turbulence intensity is not small, we observe that the histogram is well-approximated by the Gumbel extreme value distribution. The smallest turbulence intensity in this domain roughly corresponds to the transition value between the locally stable turbulence and a metastable (edge) state. Studying the $Re$ dependence of the fitting parameters in this distribution, we derive that the time scale of the transition between these two states increases double-exponentially as $Re$ approaches its critical value. On the contrary, in smaller turbulence intensities below this domain, we observe that the distribution is not sensible to the change of $Re$. This means that the decay time of the metastable state (to the laminar state) is stochastic but $Re$-independent in average. Our observation suggests that the conjecture made by Goldenfeld et al. to derive the double-exponential increase of turbulence decay time is approximately satisfied in the range of $Re$ we studied. We also discuss using another extreme value distribution, Fréchet distribution, instead of the Gumbel distribution to approximate the histogram of the turbulence intensify, which reveals interesting perspectives.

Two-point statistical equations of the liquid-phase indicator function are used to appraise the physics of liquid–gas shear flows. The contribution of the different processes in the combined scale/physical space is quantified by means of direct numerical simulations of a temporally liquid–gas shear layer. Light is first shed onto the relationship between two-point statistics of the phase indicator and the geometrical properties of the liquid/gas interface, namely its surface density, mean and Gaussian curvatures. Then, the theory is shown to be adequate for highlighting the preferential direction of liquid transport in either scale or flow position space. A direct cascade process, i.e. from large to small scales, is observed for the total phase indicator field, while the opposite applies for the randomly fluctuating part, suggesting a transfer of ‘energy’ from the mean to the fluctuating component. In the space of positions within the flow, the flux tends to redistribute energy from the centreline to the edge of the shear layer. The influence of the mean shear rate and statistical inhomogeneities on the different scales of the liquid field are revealed.

A numerical study of the $d$-dimensional eddy damped quasi-normal Markovian equations is performed to investigate the dependence on spatial dimension of homogeneous isotropic fluid turbulence. Relationships between structure functions and energy and transfer spectra are derived for the $d$-dimensional case. Additionally, an equation for the $d$-dimensional enstrophy analogue is derived and related to the velocity derivative skewness. Comparisons are made to recent four-dimensional direct numerical simulation results. Measured energy spectra show a magnified bottleneck effect which grows with dimension whilst transfer spectra show a varying peak in the nonlinear energy transfer as the dimension is increased. These results are consistent with an increased forward energy transfer at higher dimensions, further evidenced by measurements of a larger asymptotic dissipation rate with growing dimension. The enstrophy production term, related to the velocity derivative skewness, is seen to reach a maximum at around five dimensions and may reach zero in the limit of infinite dimensions, raising interesting questions about the nature of turbulence in this limit.

Three-dimensional simulations are performed to investigate swimming and prey capture by a paddling jellyfish. First, the three-dimensional vortex–vortex and vortex–body interactions are revealed, as the jellyfish swims forwards through several cycles of active muscle contraction followed by passive energy recapture via shape recovery. For varied transient paddling force and paddling frequency, we analyse the resultant changes of a jellyfish's swimming speed, interactive power, cost of transport and prey clearance rate. The pressure field around the periodically deformed elastic bell and the circulation generated by starting and stopping vortex rings are presented in greater detail to better understand the biophysical interactions that support swimming. Second, to reveal prey-specific interception and feeding behaviour, using a dynamical-system-based approach and modified Maxey–Riley equation, we compute the trajectories of the surrounding infinitesimal, inertial, opposite and normally escaping prey or plankton that hover around the medusa and are swept differently via the paddling-created velocity field. Accordingly, the diverse prey trajectories are obtained with varied paddling force, resonant driving of the elastic bell and for two different bell fineness ratios. These trajectories are then used to compute the finite-time Lyapunov exponent fields and identify particle Lagrangian coherent structures for various motile/strategically evasive prey, for five swimming cycles. The detected geometric separatrices unambiguously map and demarcate differently driven upstream fluid regions of a medusa and illustrate precisely from where an intercepted prey can be brought into the jellyfish bell, or safely stored in a capture region for ingestion, and from where a prey will surely escape. Hereby, for the first time, the prey-specific target regions, the physically well-defined three-dimensional capture surfaces and the generated cycle-to-cycle prey clearance rate are presented/analysed like never before, which provide a significantly advanced understanding on diverse predator–prey interactions and resultant success rate in prey capture. Several supplementary movies that show detailed fluid–structure interactions, transient entrainment of the floating prey and eventual prey confinement inside a secured capture surface are provided for two different jellyfish morphologies (fineness ratios 0.3 and 0.5) that help to better comprehend the natural prey encounter and hunting processes.

Breaking waves entrain gas beneath the surface. The wave-breaking process energizes turbulent fluctuations that break bubbles in quick succession to generate a wide range of bubble sizes. Understanding this generation mechanism paves the way towards the development of predictive models for large-scale maritime and climate simulations. Garrett et al. (J. Phys. Oceanogr., vol. 30, 2000, pp. 2163–2171) suggested that super-Hinze-scale turbulent break-up transfers entrained gas from large- to small-bubble sizes in the manner of a cascade. We provide a theoretical basis for this bubble-mass cascade by appealing to how energy is transferred from large to small scales in the energy cascade central to single-phase turbulence theories. A bubble break-up cascade requires that break-up events predominantly transfer bubble mass from a certain bubble size to a slightly smaller size on average. This property is called locality. In this paper, we analytically quantify locality by extending the population balance equation in conservative form to derive the bubble-mass-transfer rate from large to small sizes. Using our proposed measures of locality, we show that scalings relevant to turbulent bubbly flows, including those postulated by Garrett et al. (J. Phys. Oceanogr., vol. 30, 2000, pp. 2163–2171) and observed in breaking-wave experiments and simulations, are consistent with a strongly local transfer rate, where the influence of non-local contributions decays in a power-law fashion. These theoretical predictions are confirmed using numerical simulations in Part 2 (Chan et al., J. Fluid. Mech. vol. 912, 2021, A43), revealing key physical aspects of the bubble break-up cascade phenomenology. Locality supports the universality of turbulent small-bubble break-up, which simplifies the development of cascade-based subgrid-scale models to predict oceanic small-bubble statistics of practical importance.

Breaking waves generate a distribution of bubble sizes that evolves over time. Knowledge of how this distribution evolves is of practical importance for maritime and climate studies. The analytical framework developed in Part 1 (Chan, Johnson & Moin, J. Fluid Mech., vol. 912, 2021, A42) examined how this evolution is governed by the bubble-mass flux from large- to small-bubble sizes which depends on the rate of break-up events and the distribution of child bubble sizes. These statistics are measured in Part 2 as ensemble-averaged functions of time by simulating ensembles of breaking waves, and identifying and tracking individual bubbles and their break-up events. The large-scale break-up dynamics is seen to be statistically unsteady, and two intervals with distinct characteristics were identified. In the first interval, the dissipation rate and bubble-mass flux are quasi-steady, and the theoretical analysis of Part 1 is supported by all observed statistics, including the expected $-10/3$ power-law exponent for the super-Hinze-scale size distribution. Strong locality is observed in the corresponding bubble-mass flux, supporting the presence of a super-Hinze-scale break-up cascade. In the second interval, the dissipation rate decays, and the bubble-mass flux increases as small- and intermediate-sized bubbles become more populous. This flux remains strongly local with cascade-like behaviour, but the dominant power-law exponent for the size distribution increases to $-8/3$ as small bubbles are also depleted more quickly. This suggests the emergence of different physical mechanisms during different phases of the breaking-wave evolution, although size-local break-up remains a dominant theme. Parts 1 and 2 present an analytical toolkit for population balance analysis in two-phase flows.

We numerically investigate the flow structure of periodic steady water waves of fixed relative mass flux propagating on rotational flows with piece-wise constant vorticity. We show that, for wave solutions along the global bifurcation diagram, the stagnation point can first occur internally or at the bottom, and then again occurs at the crest with further increase in wave height. We observe that the bifurcation diagram has a new branch which is not connected to the trivial solution. Furthermore, we present an in-depth discussion of the results on pressure distributions and particle trajectories beneath large-amplitude steady waves near stagnation. We also expand on previous results concerning the amplitude and mass flux of steady water waves travelling on rotational flows with discontinuous vorticity.

This paper discusses a hitherto unexplored flow phenomenon, namely the internal hydraulic jump in thin films during co-current and counter-current two-layer flow between parallel plates. The problem corresponds to a special case of plane Poiseuille flow where the velocity profile changes continuously in the streamwise distance. Since an exact solution of Navier–Stokes equations is not possible, we reformulate the approximate shallow water theory, conventionally adopted to analyse viscous jumps in single-layer laminar flow. The standalone theory has been extensively validated with experimental data for coflow of the two phases and numerical simulations for both co- and counter-current flow. In the limit of zero viscosity ratio, the theoretical results reduce to the expression proposed by Dhar et al. (J. Fluid Mech., vol. 884, no. A11, 2020, pp. 1–26) for single-layer viscous jumps. For a holistic understanding, numerical simulations are used to unravel the physics at the jump, where the analysis displays singularity. The theory in conjunction with simulation reveals recirculation zones even in co-current laminar flow and delineates wavy, smooth and submerged jumps, displayed as a phase diagram. We thus demonstrate the efficacy of shallow water theory which, despite the approximations involved, can be used as a reliable tool for a priori prediction of viscous jumps in two-layer flow with much less effort and resources compared to numerical simulations. Use of an approximate analysis to obtain multifaceted results for a complex flow phenomenon has rarely been explored previously. This paper is also the first study reporting experiments on viscous jumps for two-layer flow in a shallow water analogue.

Motivated by understanding the dynamics of stellar and planetary interiors, we have performed a set of direct numerical simulations of Boussinesq convection in a rotating full sphere. The domain is internally heated with fixed temperature and stress-free boundary conditions, but fixed heat flux and no-slip boundary conditions are also briefly considered. We particularly focus on the large-scale coherent structures and the mean zonal flows that can develop in the system. At Prandtl number of unity, as the thermal forcing (measured by the Rayleigh number) is increased above the value for the onset of convection, we find a relaxation oscillation regime, followed by a geostrophic turbulence regime. Beyond this we see for the first time the existence of large-scale coherent vortices that form on the rotation axis. All regime boundaries are well described by critical values of the convective Rossby number $Ro_c$, with transitions from oscillatory to geostrophic turbulence, and then to the large-scale vortex regime at values $Ro_c\approx 0.2$ and $Ro_c\approx 1.5$, respectively. The zonal flow is controlled by the convective Rossby number and changes its direction when the flow transitions from the geostrophic turbulence regime to the large-scale vortex regime. While the non-zonal flow speed and heat transfer can be described by the so-called inertial scaling in the geostrophic turbulence regime, the formation of large-scale vortices appears to reduce both the non-zonal flow speed and the efficiency of convective heat transfer.

A specific family of spanwise-localised invariant solutions of plane Couette flow exhibits homoclinic snaking, a process by which spatially localised invariant solutions of a nonlinear partial differential equation smoothly grow additional structure at their fronts while undergoing a sequence of saddle-node bifurcations. Homoclinic snaking is well understood in the context of simpler pattern-forming systems such as the one-dimensional Swift–Hohenberg equation with cubic-quintic nonlinearity. The Swift–Hohenberg solutions closely resemble the snaking solutions of plane Couette flow, yet this remarkable resemblance and the mechanisms supporting homoclinic snaking within the three-dimensional Navier–Stokes equations remain to be fully understood. Studies of Swift–Hohenberg revealed the central importance of discrete symmetries for homoclinic snaking to be supported by an equation. We therefore study the structural stability of the characteristic snakes-and-ladders structure associated with homoclinic snaking in three-dimensional plane Couette flow for flow modifications that break symmetries of the flow. We demonstrate that wall-normal suction modifies the bifurcation structure of three-dimensional plane Couette solutions in the same way a symmetry-breaking quadratic term modifies solutions of the one-dimensional Swift–Hohenberg equation. These modifications are related to the breaking of the discrete rotational symmetry. At large amplitudes of the symmetry-breaking wall-normal suction the connected snakes-and-ladders structure is destroyed. Previously unknown solution branches are created and can be parametrically continued to vanishing suction. This yields new localised solutions of plane Couette flow that exist in a wide range of Reynolds numbers.

We report a knot cascade of magnetic field lines through the stepwise reconnection of a pair of orthogonal helical flux tubes with opposite chirality. The magnetic-surface field is developed to identify the evolution of flux tubes and pinpoint the reconnection region. We find that the incipient X-type magnetic reconnection generates antiparallel U-shaped field lines moving in opposite directions away from the plasmoid. This reconnection is characterized by the decay of the magnetic flux through the diagonal symmetric planes. Subsequently, overhand magnetic knots are tied via the secondary reconnection, coinciding with the notable conversion from magnetic energy to kinetic energy. The knotted field lines are then rotated and stretched in the plasmoid by the vortical-like local motion induced by the magnetic knots themselves via the Lorentz force. This nonlinear evolution triggers the tertiary reconnection to form double overhand knots, and then further reconnections to produce more complex knots. From field lines extracted at different times, the knot cascade via the sequence of reconnections in a finite time period is quantified by the increment of the minimum crossing number and the migration of the probability density function of the Alexander–Briggs notation. Finally, the knot cascade slows down and then terminates due to resistive and viscous dissipations.

The effect of turbulence on snow precipitation is not incorporated into present weather forecasting models. Here we show evidence that turbulence is in fact a key influence on both fall speed and spatial distribution of settling snow. We consider three snowfall events under vastly different levels of atmospheric turbulence. We characterize the size and morphology of the snow particles, and we simultaneously image their velocity, acceleration and relative concentration over vertical planes approximately $30\ \textrm {m}^2$ in area. We find that turbulence-driven settling enhancement explains otherwise contradictory trends between the particle size and velocity. The estimates of the Stokes number and the correlation between vertical velocity and local concentration are consistent with the view that the enhanced settling is rooted in the preferential sweeping mechanism. When the snow vertical velocity is large compared to the characteristic turbulence velocity, the crossing trajectories effect results in strong accelerations. When the conditions of preferential sweeping are met, the concentration field is highly non-uniform and clustering appears over a wide range of scales. These clusters, identified for the first time in a naturally occurring flow, display the signature features seen in canonical settings: power-law size distribution, fractal-like shape, vertical elongation and large fall speed that increases with the cluster size. These findings demonstrate that the fundamental phenomenology of particle-laden turbulence can be leveraged towards a better predictive understanding of snow precipitation and ground snow accumulation. They also demonstrate how environmental flows can be used to investigate dispersed multiphase flows at Reynolds numbers not accessible in laboratory experiments or numerical simulations.

We present an experimental investigation of the acceleration statistics and the added mass tensor of deformable gas bubbles in turbulence. By simultaneously tracking both bubbles and their surrounding flow in three dimensions, we find two independent ways of estimating the bubble acceleration: either directly measured from three-dimensional bubble trajectories or indirectly calculated from the bubble's equation of motion. When such an equation is projected onto the bubble frame, the added-mass coefficient becomes a diagonal tensor with three elements being linked to the standard deviation of bubble acceleration along three bubble principal axes. This constraint aids in experimentally determining the added mass coefficient tensor. The obtained trend of $C_A$ seems to agree with Lamb's potential flow solutions for spheroids, suggesting that the added-mass force on deformable bubbles can be modelled using spheroids with the same geometry and orientation. In addition, the probability density function of the relative orientation between the semi-major axis of deformed bubbles and the slip acceleration in turbulence is shown. A surprising finding is that the bubble orientation, indicated by the bubble's major axis, is not random in turbulence but rather is preferentially aligned with the slip acceleration. The degree of this alignment increases as bubbles deform more. Because accelerating along the major axis of a more deformed bubble entails reduced added mass, the acceleration standard deviation of deformable bubbles increases as a function of the bubble aspect ratio.

This study is focused on the prediction of coherent structures, propagating within a turbulent channel flow. We propose a derivation of the linearised problem based on a stochastic formulation of the Navier–Stokes equations. It consists in considering the transport of quantities by a resolved velocity (i.e. solution of the model) perturbed by a Brownian motion which models the unresolved turbulent fluctuations over the time-averaged field, here thought of as the underlying background turbulence. The associated linearised model, considering the mean velocity profile as given, predicts linear solutions evolving within a corrected mean velocity field and perturbed by modelled background turbulence. Two ways to define the statistics of the Brownian motion are proposed and compared: one based on full simulation data, and the second, data free, based on preliminary predictions from resolvent analysis. The technique is applied on turbulent channel flows at friction Reynolds numbers $Re_\tau =180$ and $Re_\tau =550$, and predictions are compared with direct numerical simulation results. We show that the principal components of an ensemble of solutions of this stochastic linearised system are able to represent the leading spectral proper orthogonal decomposition modes with a similar accuracy to optimal responses coming from resolvent analysis with an eddy-viscosity model at scales where strong production occurs. For the other scales, receiving energy by nonlinear redistribution, the present strategy improves the prediction. Moreover, the second mode is systematically well predicted over all scales. This behaviour is understood by the ability of the stochastic modelling to model positive and negative inter-scale energy transfers through stochastic diffusion and random stochastic transport, while the eddy-viscosity term in resolvent analysis is purely diffusive.

This study numerically investigates the flow characteristics past a solid and smooth sphere translating steadily along the axis of a cylindrical tube filled with wormlike micellar solutions in the creeping flow regime. The two-species Vasquez–Cook–McKinley and single-species Giesekus constitutive models are used to characterize the rheological behaviour of the micellar solutions. Once the Weissenberg number exceeds a critical value, an unsteady motion downstream of the sphere is observed in the case of the two-species model. We provide evidence that this unsteady motion downstream of the sphere is caused by the sudden rupture of long and stretched micelles in this region, resulting from an increase in the extensional flow strength. The corresponding single-species Giesekus model for the wormlike micellar solution, with no breakage and reformation, predicts a steady flow field under otherwise identical conditions. Therefore, it further provides evidence presented herein for the onset of this unsteady motion. Furthermore, we find that the onset of this unsteady motion downstream of the sphere is delayed as the ratio of sphere to tube diameter decreases. A similar kind of unsteady motion has also been observed in several earlier experiments for the problem of a sphere sedimenting in a tube filled with wormlike micellar solutions. We find a remarkable qualitative similarity in the flow characteristics between the present numerical results for a steadily translating sphere and prior experimental results for a falling sphere.

This paper presents an analytical solution of the Burnett equations for gaseous flow in a long microchannel. A non-dimensional analysis is first undertaken to reduce the governing equations into somewhat simplified differential equations, which are solved to obtain the pressure and velocity fields. The exact solution for pressure has been obtained by solving the cross-stream momentum equation, while the solution for velocity is obtained from the streamwise momentum equation. The required boundary conditions are obtained from the direct simulation Monte Carlo (DSMC) method. The obtained analytical solution is compared with the available DSMC and perturbation based solutions, and found to agree well. The work is particularly significant because analytical solutions of the Burnett equations are currently known in only a very few cases. The present analytical solution opens the possibility for further analysis by employing the expressions for pressure and velocity provided here.

Jumps in granular chute flow are obtained as continuous solutions of the properly adapted Saint–Venant equations. We elucidate their internal structure via a dynamical systems approach and show that the jumps in phase space manifest themselves as trajectories organized around the stable/unstable manifold of the fixed point representing the uniform flow. Based on this, we derive an analytic approximate expression for the jump length. The paper concludes with a numerical experiment confirming the stability of the jumps.