Polymers in a turbulent flow are subject to intense strain, which can cause their scission and thereby limit the experimental study and application of phenomena such as turbulent drag reduction and elastic turbulence. In this paper, we study polymer scission in homogeneous isotropic turbulence, through a combination of stochastic modelling, based on a Gaussian time-decorrelated random flow, and direct numerical simulations (DNS) with both one-way (passive) and two-way (active) coupling of the polymers, modelled as bead-spring chains, and the flow. For the first scission of passive polymers, the stochastic model yields analytical predictions which are found to be in good agreement with results from the DNS, for the temporal evolution of the fraction of unbroken polymers and the statistics of the survival of polymers. The impact of scission on the dynamics of a turbulent polymer solution is investigated through DNS with two-way coupling (active polymers). Our results indicate that the reduction of kinetic energy dissipation due to feedback from stretched polymers is an inherently transient effect, which is lost as the polymers break up. Thus, the overall dissipation reduction is maximized by an intermediate polymer relaxation time, for which polymers stretch significantly but without breaking too quickly. We also study the dynamics of the polymer fragments which form after scission; these daughter polymers can themselves undergo subsequent, repeated, breakups to produce a hierarchical population of polymers with a range of relaxation times and scission rates.

This empirical study aims to characterize the dynamical behaviour of sprays using time-series analysis of the size–velocity data acquired using a phase Doppler particle analyser. The prime motivation of this analysis is to capture the spatio-temporal correlations using time-series modelling paradigms that provide valuable new insights into spray dynamics. As a first step, we study long-held assumptions, especially on stationarity and time unsteadiness. We show that air-blast sprays have increased drop size as well as velocity ordering near the edge of the spray. Analysis of the inter-particle time of the droplets shows non-Poisson behaviour where droplets that are closely spaced in time are also closely spaced in the size and velocity coordinates. Temporal auto-correlation and partial auto-correlation calculations reveal the presence of inherent correlated features in the spray. This correlation is stronger and short lived in an air-blast spray and weaker but more persistent in a pressure swirl spray. These correlations render the probability density function (p.d.f.) estimate obtained from standard methods inaccurate; therefore, we propose a technically correct way of estimating the p.d.f. using a suitable downsampling and averaging method. Statistical analysis of residuals (from appropriate autoregressive integrated moving average time-series models) uncovers an interesting feature of spray data pertaining to heteroskedasticity (stochastically changing variance) of the diameter series. In order to account for heteroskedasticity, appropriate generalized autoregressive conditional heteroskedasticity models are developed. Finally, we present a utilitarian view of these results as an empirically consistent boundary condition implementation tool for computational fluid dynamics (CFD).

An oscillating spanwise pressure gradient is imposed numerically to control the flow separation and reduce the drag of a turbulent flow in a channel with square bars. The transverse flow produces a maximum drag reduction of 25 %, due to lower pressure and skin-friction drag forces. The pressure drag reduction reaches a maximum of 22 % and is due to a decrease of the positive high pressure in front of the bars and an increase of the low negative pressure behind the bars. The skin-friction drag reduction is caused by a lower wall-shear stress along the cavity between the bars where the flow is fully attached, while the wall-shear stress on the crest of the bars and in the separated region behind the bars is unaffected. The spanwise laminar flow obtained by neglecting the nonlinear terms involving the turbulent velocity fluctuations is used to compute the power spent for oscillating the fluid along the spanwise direction and an excellent agreement is found with the power spent obtained by the averaged turbulent flow. A marginal or negative net power saved is found by subtracting the power employed for controlling the flow from the power saved thanks to the transverse flow. The control reduces the total drag as the integral of the Reynolds stresses along the horizontal line connecting the corners of two consecutive bars is decreased, which in turn impacts the pressure and wall-shear stress reductions.

Direct numerical simulations are conducted to study the receptivity and transition mechanisms in a solitary wave boundary layer developing over randomly organized wave-like bottom topography. The boundary layer flow shows a selective response to broadband perturbations from the bottom, and develops streamwise-elongated streaks. When the streaks reach high amplitudes, they indirectly amplify streamwise-elongated vortices through modulating small-scale fluctuations and pressure fields. These stronger vortices in turn stir the boundary layer more effectively and further intensify streaks via the lift-up mechanism. This nonlinear feedback loop increases the sensitivity of the boundary layer to the roughness level and yields dramatic variations among cases sharing the same Reynolds number with differing roughness height. Three different local breakdown scenarios are observed depending on the amplitude of the streaks: (i) two-dimensional wave instabilities in the regions with weak streaks; (ii) inner shear-layer instabilities in the regions with moderate-amplitude streaks; and (iii) rapidly growing outer shear-layer instabilities in the regions with highly elevated high-amplitude streaks. Inner instabilities have the slowest growth rate among all transition paths, which confirms the previous predictions on the stabilising role of moderate-amplitude streaks (Önder & Liu, J. Fluid. Mech., vol. 896, 2020, A20).

Process industries often involve handling non-cohesive solid materials which vary in size and shape. A comprehensive understanding of such systems helps in effective handling of industrial operations. Here, we studied heterogeneous mixtures of dumbbells and discs flowing out of a two-dimensional silo using discrete element method simulations. We analysed discharge dynamics of the mixtures in two regimes, namely the free-flow regime ($W/d>=15$) and the interrupted flow regime ($W/d<=10$), where $W$ and $d$ are the orifice width and diameter of each of the circles of a dumbbell. One of the intriguing results is a decrease in the flow rate $Q$ of mixtures with an increase in the fraction of dumbbells $X_{db}$ in both of the regimes analysed. This can be attributed to the geometrical interlocking among the particles and a hindrance to the rotation of dumbbells. The time-averaged (coarse-grained) flow fields reveal an increase in the size of the stagnant zone beside the orifice with an increase in $X_{db}$. The stagnant zone hinders the particles flowing next to it, which is another reason for a decrease in $Q$ with an increase in $X_{db}$. In the interrupted flow regime, we investigated clogged states of the mixtures using arch morphology, the fraction of dumbbells and number of particles in an arch, and avalanche sizes.

The stability of sliding spreading films of Herschel–Bulkley fluid is investigated theoretically, motivated by a dramatic fingering pattern observed experimentally and proposed theoretically to originate from an extensional flow instability of shear-thinning fluids. Considering the thin-film limit, we construct axisymmetric base states and then test their stability towards non-axisymmetric perturbations by numerically solving the initial-value problem. We complement the numerics with analytical solutions for early and late times. The stability analysis demonstrates that spreading thinning films are unstable. At late times, where the spreading of the base state becomes self-similar, non-axisymmetric patterns can develop strongly if the fluid has a yield stress or is sufficiently shear thinning.

In this paper we experimentally study the transitional range of Reynolds numbers in plane Couette–Poiseuille flow, focusing our attention on the localized turbulent structures triggered by a strong impulsive jet and the large-scale flow generated around these structures. We present a detailed investigation of the large-scale flow and show how its amplitude depends on Reynolds number and amplitude perturbation. In addition, we characterize the initial dynamics of the localized turbulent spot, which includes the coupling between the small and large scales, as well as the dependence of the advection speed on the large-scale flow generated around the spot. Finally, we provide the first experimental measurements of the large-scale flow around an oblique turbulent band.

We report on experiments of the quasi-static growth and detachment of air bubbles in water from a superhydrophobic substrate, overcoming the maximum size limitation of conventional injectors due to the Rayleigh–Taylor instability. The observations are in good agreement with a hydrostatic model, demonstrating that bubbles grow through a sequence of quasi-equilibrium states. Our experiments corroborate the theoretical prediction of a maximum bubble volume of approximately $6.04 {\rm \pi}$ and a critical base radius of $3.22$, both numbers in units of the capillary length (Michael & Williams, Proc. R. Soc. Lond. A. vol. 351, 1976, pp. 117–127). This maximum is also reached when bubbles grow in an unbounded, ideally non-wetting surface, establishing the ultimate size limit of quasi-static bubble formation.

The behaviour of a fluid at the interface with a solid boundary is affected, to a large extent, by the potential landscape imposed on the fluid by the solid. Fluid slip at the interface with a solid boundary is modelled here as forced Brownian motion in a periodic potential landscape. The resulting model goes beyond simple transition-state-theory approaches and uses well-defined atomistic parameters to capture the salient features of the slip process in both the linear and nonlinear forcing regimes, yielding excellent agreement with molecular dynamics simulation results, as well as previous modelling results. An explicit expression for the Navier slip coefficient in terms of molecular-level system parameters is derived.

We consider extensional flows of a dense layer of spheres in a viscous fluid and employ force and torque balances to determine the trajectory of particle pairs that contribute to the stress. In doing this, we use Stokesian dynamics simulations to guide the choice of the near-contacting pairs that follow such a trajectory. We specify the boundary conditions on the representative trajectory, and determine the distribution of particles along it and how the stress depends on the microstructure and strain rate. We test the resulting predictions using the numerical simulations. Also, we show that the relation between the tensors of stress and strain rate involves the second and fourth moments of the particle distribution function.

The formation of abnormal (extreme) waves in coastal areas can be triggered by wave–seabed interaction, in particular by steep bottom changes. As an incident equilibrium sea state passes over a submerged step or bar, non-equilibrium dynamics appears locally and forces the sea state to a new, finite-depth equilibrium along with strong non-Gaussian statistics and an intensified occurrence probability of large waves. In this study, the experimental case Run 3 reported by Trulsen et al. (J. Fluid Mech., vol. 882, 2020, R2) has been investigated numerically with a fully nonlinear model. Furthermore, as both shoaling and de-shoaling effects exist in the set-up with a bar-profile bottom, an additional simulation with a step-profile bottom is performed to isolate the de-shoaling effects. The model is proven excellent by the confrontation of the measurements and simulated results in both time and spectral domains. Strong non-Gaussian behaviour of the sea state is highlighted after the up-slope transition by combining spectral and bi-spectral analyses, and characteristic parameters. With a harmonic extraction approach, we show evidence that both second- and third-order effects triggered by the non-equilibrium dynamics significantly enhance the local kurtosis and occurrence of extreme waves. The statistics of kinematics shows the asymmetry of the wave field evolves somewhat independently in the horizontal and vertical directions. By comparing the simulations of bar- and step-profile cases, we find the de-shoaling process is responsible for the upstream modulation of nonlinear and dispersive parameters, and the enhancement of kurtosis of both horizontal and vertical velocities and horizontal acceleration over the down-slope area.

A Mach-10 hypersonic boundary layer of air overriding a cold, isothermal, non-catalytic flat wall, and with a stagnation enthalpy of $21.6\ \textrm {MJ}\ \textrm {kg}^{-1}$, is analysed using direct numerical simulations. The calculations include multicomponent transport, equilibrium vibrational excitation and chemical kinetics for air dissociation. The initially laminar boundary layer undergoes transition to turbulence by the resonance of a two-dimensional mode injected by a suction-and-blowing boundary condition imposed over a narrow spanwise porous strip. The ensuing turbulent boundary layer has a momentum Reynolds number of 3826 near the outflow of the computational domain. The relatively low temperature of the free stream renders the air chemically frozen there. However, the high temperatures generated within the boundary layer by viscous aerodynamic heating, peaking at a wall-normal distance $y^\star \simeq 10\text {--}20$ in semi-local viscous units, lead to air dissociation in under-equilibrium amounts equivalent to 4 %–7 % on a molar basis of atomic oxygen, along with smaller concentrations of nitric oxide, which is mainly produced by the Zel'dovich mechanism, and of atomic nitrogen, the latter being mostly in steady state. A statistical analysis of the results is provided, including the streamwise evolution of (a) the skin friction coefficient and dimensionless wall heat flux; (b) the mean profiles of temperature, velocity, density, molar fractions, chemical production rates and chemical heat-release rate; (c) the Reynolds stresses and root-mean-squares of the fluctuations of temperature, density, pressure, molar fractions and chemical heat-release rate; and (d) the temperature/velocity and mass-fraction/velocity correlations.

We report on turbulent thermal convection experiments in a rotating cylinder with a diameter ($D$) to height ($H$) aspect ratio of $\varGamma =D/H=0.5$. Using nitrogen and pressurised sulphur hexafluoride we cover Rayleigh numbers (Ra) from $8\times 10^{9}$ to $8\times 10^{14}$ at Prandtl numbers $0.72\lesssim {\textit {Pr}}\lesssim 0.94$. For these Ra we measure the global vertical heat flux (i.e. the Nusselt number – Nu), as well as statistical quantities of local temperature measurements, as a function of the rotation rate, i.e. the inverse Rossby number – 1/Ro. In contrast to measurements in fluids with a higher Pr we do not find a heat transport enhancement, but only a decrease of Nu with increasing 1/Ro. When normalised with Nu(0) for the non-rotating system, data for all different Ra collapse and, for sufficiently large 1/Ro, follow a power law ${\textit {Nu}}/{\textit {Nu}}_0\propto (1/{\textit {Ro}})^{-0.43}$. Furthermore, we find that both the heat transport and the temperature field qualitatively change at rotation rates $1/{\textit {Ro}}^*_1=0.8$ and $1/{\textit {Ro}}^*_2=4$. We interpret the first transition at $1/{\textit {Ro}}^*_1$ as change from a large-scale circulation roll to the recently discovered boundary zonal flow (BZF). The second transition at rotation rate $1/{\textit {Ro}}^*_2$ is not associated with a change of the flow morphology, but is rather the rotation rate for which the BZF is at its maximum. For faster rotation the vertical transport of warm and cold fluid near the sidewall within the BZF decreases and hence so does Nu.

Heat and mass transport are generally closely correlated to momentum transport in shear flows. This so-called Reynolds analogy between advective heat or mass transport and momentum transport hinders efficiency improvements in engineering heat and mass transfer applications. I show through direct numerical simulations that in plane Couette and Taylor–Couette flow, rotation can strongly influence wall-to-wall passive tracer transport and make it much faster than momentum transport, clearly in violation of the Reynolds analogy. This difference between passive tracer transport, representative of heat/mass transport, and momentum transport is observed in steady flows with large counter-rotating vortices at low Reynolds numbers as well as in fully turbulent flows at higher Reynolds numbers. It is especially large near the neutral (Rayleigh's) stability limit. The rotation-induced Coriolis force strongly damps the streamwise/azimuthal velocity fluctuations when this limit is approached, while tracer fluctuations are much less affected. Accordingly, momentum transport is much more reduced than tracer transport, showing that the Coriolis force breaks the Reynolds analogy. At higher Reynolds numbers, this strong advective transport dissimilarity is accompanied by approximate limit cycle dynamics with intense low-frequency bursts of turbulence when approaching the neutral stability limit. The study demonstrates that simple body forces can cause clear dissimilarities between heat/mass and momentum transport in shear flows.

The energetically independent linear wave and geostrophic (vortex) solutions are shown to be a complete basis for velocity and density variables $(u,v,w,\rho )$ in a rotating non-hydrostatic Boussinesq fluid with arbitrary stratification and non-periodic vertical boundaries. This work extends the familiar wave-vortex decomposition for triply periodic domains with constant stratification. As a consequence of the decomposition, the fluid can be unambiguously separated into decoupled linear wave and geostrophic components at each instant in time, without the need for temporal filtering. The fluid can then be diagnosed for temporal changes in wave and geostrophic coefficients at each unique wavenumber and mode, including those that inevitably occur due to nonlinear interactions. We demonstrate that this methodology can be used to determine which physical interactions cause the transfer of energy between modes by projecting the nonlinear equations of motion onto the wave-vortex basis. In the particular example given, we show that an eddy in geostrophic balance superimposed with inertial oscillations at the surface transfers energy from the inertial oscillations to internal gravity wave modes. This approach can be applied more generally to determine which mechanisms are involved in energy transfers between wave and vortices, including their respective scales. Finally, we show that the nonlinear equations of motion expressed in a wave-vortex basis are computationally efficient for certain problems. In cases where stratification profiles vary strongly with depth, this approach may be an attractive alternative to traditional spectral models for rotating Boussinesq flow.

Bedload particle hops are defined as successive motions of a particle from start to stop, characterizing one of the most fundamental processes of bedload sediment transport in rivers. Although two transport regimes have been recently identified for short and long hops, respectively, there is still the lack of a theory explaining the mean hop distance–travel time scaling for particles performing short hops, which dominate the transport and may cover over 80 % of the total hop events. In this paper, we propose a velocity-variation-based formulation, the governing equation of which is intrinsically identical to that of Taylor dispersion for solute transport within shear flows. The key parameter, namely the diffusion coefficient, can be determined by hop distances and travel times, which are easier to measure and more accurate than particle accelerations. For the first time, we obtain an analytical solution for the mean hop distance–travel time relation valid for the entire range of travel times, which agrees well with the measured data. Regarding travel times, we identify three distinct regimes in terms of different scaling exponents: respectively, $\sim$1.5 for the initial regime and $\sim$5/3 for the transition regime, which define the short hops, and 1 for the Taylor dispersion regime defining long hops. The corresponding distribution of the hop distance is analytically obtained and experimentally verified. We also show that the conventionally used exponential distribution, as proposed by Einstein, is solely for long hops. Further validation of the present formulation is provided by comparing the simulated accelerations with measurements.

When a water droplet is deposited within a sparsely miscible liquid medium, such as certain oils, the droplet surprisingly vanishes, even in a confined geometry. Such a phenomenon has crucial consequences for multiphase flows in which confined nano- and/or picolitre droplets are considered. We report here experiments of microdroplet dissolution in microchannels that reveal an enhancement of the shrinkage of confined water microdroplets in oil due to the permeability of the walls – made of polydimethilsiloxane (PDMS) – and a delay when collective effects are present. The system is first modelled assuming that the dissolution of the droplet in its surrounding liquid follows the Epstein–Plesset solution of the diffusion equation. The dissolution of small isolated droplets can indeed be described by this solution of the diffusion equation, while the vanishing of droplets larger than a certain critical value and those closer to other droplets requires numerical simulations. Experimental measurements and simulations compare well only when the boundary conditions of the confined system, the neighbouring droplets and, interestingly, the evaporative water vapour flux through the PDMS are all taken into account in the numerical model. Our results thus reveal the important role of the water solubility in oil and, most remarkably, of the water vapour transport through permeable walls.

Micro- and nanodroplets have many important applications such as in drug delivery, liquid–liquid extraction, nanomaterial synthesis and cosmetics. A commonly used method to generate a large number of micro- or nanodroplets in one simple step is solvent exchange (also called nanoprecipitation), in which a good solvent of the droplet phase is displaced by a poor one, generating an oversaturation pulse that leads to droplet nucleation. Despite its crucial importance, the droplet growth resulting from the oversaturation pulse in this ternary system is still poorly understood. We experimentally and theoretically study this growth in Hele-Shaw-like channels by measuring the total volume of the oil droplets that nucleate out of it. In order to prevent the oversaturated oil from exiting the channel, we decorated some of the channels with a porous region in the middle. Solvent exchange is performed with various solution compositions, flow rates and channel geometries, and the measured droplets volume is found to increase with the Péclet number, $Pe$, with an approximate effective power law $V\propto Pe^{0.50}$. A theoretical model is developed to account for this finding. With this model we can indeed explain the $V\propto Pe^{1/2}$ scaling, including the prefactor, which can collapse all data of the ‘porous’ channels onto one universal curve, irrespective of channel geometry and composition of the mixtures. Our work provides a macroscopic approach to this bottom-up method of droplet generation and may guide further studies on oversaturation and nucleation in ternary systems.

Interscale kinetic energy transfer in chemically reacting compressible isotropic turbulence is studied using numerical simulations at turbulent Mach numbers 0.2 and 0.8 for isothermal and exothermic reactions. At low turbulent Mach number $M_{t}=0.2$ for exothermic reaction, heat release greatly enhances expansion and compression motions, and induces the formation of shocklets which are not observed for isothermal reaction at the same turbulent Mach number. It is found that heat release through exothermic reactions enhances both the positive and the negative components of pressure–dilatation, as well as positive and negative components of subgrid-scale (SGS) kinetic energy flux, indicating an increase of both forward-scatter and backscatter of kinetic energy. The SGS flux of kinetic energy has a tendency to be from large scales to small scales. The solenoidal components of pressure–dilatation and SGS kinetic energy flux are minorly influenced by heat release of reaction, while the dilatational components of pressure–dilatation and SGS kinetic energy flux are significantly intensified by heat release at all length scales. Heat release enhances the kinetic energy of the dilatational mode through the pressure work, and leads to the increase of dilatational kinetic energy transfer from large scales to small scales through dilatational SGS flux, as well as the viscous dissipation of dilatational kinetic energy at small scales. Taylor Reynolds number has a minor influence on the qualitative statistical properties of interscale kinetic energy transfer terms.

Fish schools and bird flocks, which involve a variety of orderly formations, have fascinated people for a long time. The collective performance of two flapping foils in parallel, which can self-propel in both lateral and longitudinal directions, is numerically studied through two-dimensional simulations in this paper. For the first time it is numerically confirmed that two flapping swimmers can simultaneously converge to equilibrium distances in both lateral and longitudinal directions. Two types of stable formations have been observed and which type occurs depends on the phase difference between two foils. The staggered formation appears in the in-phase scenario, and the side-by-side formation occurs in the anti-phase scenario. Moreover, both types of the stable formations strongly depend on the flapping frequency and amplitude, but are independent of the perturbation of the initial distance between the two foils when the perturbation is small. In addition, considerable velocity enhancement can be achieved by two foils in all of the stable formations, as compared with that of a single foil. Moreover, the velocity of the two-foil system can be couched as a specific scaling law, which is determined by the type of the stable formations. Finally, the mechanism behind the stable formations has been analysed. The results obtained here may shed some light on the understanding of collective behaviour of fish schools and bird flocks.