Granular jumps commonly develop during granular flows over complex topographies or when hitting retaining structures. While this process has been well-studied for hydraulic flows, in granular flows such jumps remain to be fully explored, given the role of interparticle friction. Predicting the length of granular jumps is a challenging question, relevant to the design of protection dams against avalanches. In this study, we investigate the canonical case of standing jumps formed in granular flows down smooth inclines using extensive numerical simulations based on the discrete element method. We consider both two- and three-dimensional configurations and vary the chute bottom friction to account for the crucial interplay between the sliding along the smooth bottom and the shearing across the granular bulk above. By doing so, we derived a robust scaling law for the jump length that is valid over a wide range of Froude numbers and takes into account the influence of the packing density. The findings have potential implications on a number of situations encountered in industry as well as problems associated with natural hazards.

Previous mathematical models of quasi-steady turbulent plumes and fountains have described the flow that results from a prescribed input of buoyancy. We offer a new perspective by asking, what input of buoyancy would give rise to a plume, or fountain, with given properties? Addressing this question by means of an analytical framework, we take a first step toward enabling a plume with specific characteristics, i.e. a synthetic plume, to be designed. We develop analytical solutions to the conservation equations that describe four kinds of turbulent flow: axisymmetric plume, starting fountain, line plume and wall plume. Crucially, our solutions do not require the buoyancy distribution to be specified, whether this be the source or off-source distribution. Key to our approach, we specify a function for the volume flux, $Q=f(z)$, and take advantage of the weak coupling between the conservation equations to uniquely express general solutions in terms of $f$. We show that any analytic function $f$ can form the basis for a set of solutions for the fluxes, local variables and local Richardson number, though $f/({\rm d}f/{\rm d}z)>0$ is a necessary condition for physically realistic solutions. As an example of plume synthesis, we show that an axisymmetric plume can have an invariant radius if there is an exponentially increasing input of buoyancy to the plume centreline. We also consider how plume synthesis could be achieved practically.

This article presents a theoretical modelling framework for the previously unconsidered case of turbulent wall plumes that detrain continually with height in stably stratified environments. Built upon the classic turbulent plume model, our approach incorporates turbulent detrainment with a variable coefficient of detrainment. Based on a linear constitutive relationship between the ratio of the detrainment to entrainment coefficients and the ambient buoyancy gradient, it is found that for linear ambient stratifications, a dynamic quasi-equilibrium region, characterised by a near invariant local plume Richardson number, is achieved, downstream of which this equilibrium rapidly breaks down. With increasing ambient buoyancy gradient, while the plume becomes increasingly slender with weaker vertical motions, the level at which the plume breaks down to form a horizontal intrusion first decreases and then increases. Moreover, distinct from classic purely entraining plumes, a detraining wall plume can swell within the pre-equilibrium adjustment stage provided the local Richardson number is sufficiently low $({Ri\ll 6})$, behaviour which is in accordance with observations made in filling-box experiments.

Many geological layers include cross bedding, which leads to different values for permeability along and across the bedding planes. We explore how such cross bedding impacts buoyancy-driven flow through an inclined aquifer. For each bedding angle and ratio of the permeability along and across the bedding, a free buoyancy-driven plume rises at a particular angle to the horizontal. If the angle of inclination of the aquifer to the horizontal is smaller than this angle, then the plume rises along the upper boundary, otherwise, somewhat surprisingly, the buoyant plume rises along the lower boundary of the aquifer. We present new laboratory experiments to support these predictions. We also test a model for the effective permeabilities which control the speed and the rate of spread of the plume along one or other boundary of the aquifer. We consider the impact of our results for modelling geological storage of ${\rm CO}_2$ or aquifer thermal energy storage.

We report an experimental and numerical study on Rayleigh–Bénard convection in a slender rectangular geometry with the aspect ratio $\varGamma$ varying from 0.05 to 0.3 and a Rayleigh number range of $10^5\leqslant Ra\leqslant 3\times 10^9$. The Prandtl number is fixed at $Pr=4.38$. It is found that the onset of convection is postponed when the convection domain approaches the quasi-one-dimensional limit. The onset Rayleigh number shows a $Ra_c=328\varGamma ^{-4.18}$ scaling for the experiment and a $Ra_c=810\varGamma ^{-3.95}$ scaling for the simulation, both consistent with a theoretical prediction of $Ra_c\sim \varGamma ^{-4}$. Moreover, the effective Nusselt–Rayleigh scaling exponent $\beta =\partial (\log Nu)/\partial (\log Ra)$ near the onset of convection also shows a rapid increase with decreasing $\varGamma$. Power-law fits to the experimental and numerical data yield $\beta =0.290\varGamma ^{-0.90}$ and $\beta =0.564\varGamma ^{-0.92}$, respectively. Near onset, the flow shows a stretched cell structure. In this regime, the velocity and temperature variations in a horizontal cross-section are found to be almost invariant with height in the core region of a slender domain. As the Rayleigh number increases, the system evolves from the viscous dominant regime to a plume-controlled one, a feature of which is enhancement in the heat transport efficiency. Upon further increase of $Ra$, the flow comes back to the classical boundary-layer-controlled regime, in which the quasi-one-dimensional geometry has no apparent effect on the global heat transfer.

Based on a generalized local Kolmogorov–Hill equation expressing the evolution of kinetic energy integrated over spheres of size $\ell$ in the inertial range of fluid turbulence, we examine a possible definition of entropy and entropy generation for turbulence. Its measurement from direct numerical simulations in isotropic turbulence leads to confirmation of the validity of the fluctuation relation (FR) from non-equilibrium thermodynamics in the inertial range of turbulent flows. Specifically, the ratio of probability densities of forward and inverse cascade at scale $\ell$ is shown to follow exponential behaviour with the entropy generation rate if the latter is defined by including an appropriately defined notion of ‘temperature of turbulence’ proportional to the kinetic energy at scale $\ell$.

This paper revisits the study by Bailey et al. (J. Fluid Mech., vol. 615, 2008, pp. 121–138), adopting a higher-fidelity calibration approach to reveal subtle flow variations with Reynolds numbers that were not discernible previously. The paper aims therefore to provide insights into the characteristics of azimuthal and streamwise pipe flow structures adopting two-point joint statistics and spectral analysis for shear Reynolds numbers in the range $2\times {10^3}\le {Re_\tau }\le {16\times {10^3}}$, where ${Re_\tau }$ is based on the wall friction velocity $u_{\tau }$, the pipe radius $R$, and the fluid kinematic viscosity $\nu$. The streamwise velocity fluctuations were measured at four wall-normal locations, $0.1\le {x_{{2}}/R}\le {0.7}$, covering the logarithmic and core regions of fully developed turbulent pipe flow based on 35–41 azimuthal probe separations using, simultaneously, two single hot-wire probes. A unique in situ calibration approach for both probes was adopted where a potential flow was insured, resulting in consistent and precise pipe flow data. The azimuthal velocity correlation, the cross-power spectral density and the coherence function of the streamwise velocity fluctuations are discussed, revealing a clear dependence of the azimuthal scales of the large and very large flow motions on the wall-normal location, the azimuthal separation, the streamwise wavenumber and the Reynolds number. Along the logarithmic region, a linear growth of the azimuthal scales of the large- and very-large-scale structures was observed; however, they do scale nonlinearly and reach their maximum sizes in the core region, i.e. near the centreline of the pipe. Additionally, the streamwise very-large- and large-scale motions were evaluated using the premultiplied energy spectra, showing wavelengths ${\approx }{18R}$ and ${\approx }{3R}$ for ${Re_{\tau }}\approx {16\times {10^3}}$ at half of the pipe radius, respectively.

We present a theoretical study of viscous slug motion inside a microscopically rough capillary tube, where pronounced stick–slip motion can emerge at slow displacement rates. The mathematical description of this intermittent motion can be reduced to a system of ordinary differential equations, which also describe the motion of a pendulum inside a fluid-filled rotating drum. We use this analogy to show that the stick–slip motion transitions to steady sliding at high displacement rates. We characterize this crossover with a simple scaling relation and show that the crossover is accompanied by a shift in the dominant energy dissipation mechanisms within the system.

We introduce a projection-based model reduction method that systematically accounts for nonlinear interactions between the resolved and unresolved scales of the flow in a low-dimensional dynamical systems model. The proposed method uses a separation of time scales between the resolved and subscale variables to derive a reduced-order model with cubic closure terms for the truncated modes, generalizing the classic Stuart–Landau equation. The leading-order cubic terms are determined by averaging out fast variables through a perturbation series approximation of the action of a stochastic Koopman operator. We show analytically that this multiscale closure model can capture both the effects of mean-flow deformation and the energy cascade before demonstrating improved stability and accuracy in models of chaotic lid-driven cavity flow and vortex pairing in a mixing layer. This approach to closure modelling establishes a general theory for the origin and role of cubic nonlinearities in low-dimensional models of incompressible flows.

We report the results of a theoretical investigation of the stability of a hydrodynamic analogue of Landau levels, specifically circular orbits arising when a millimetric droplet self-propels along the surface of a vibrating, rotating liquid bath. Our study elucidates the form of the stability diagram characterising the critical memory at which circular orbits destabilise, and the form of instability. Particular attention is given to rationalising observations reported in prior experimental works, including the prevalence of resonant wobbling instabilities, in which the instability frequency is approximately twice the orbital frequency. We also explore the physical mechanism responsible for the onset of instability. Specifically, we compare the efficacy of different heuristic arguments proposed in prior studies, including propositions that the most unstable orbits arise when their radii correspond to the zeros of Bessel functions or when their associated wave intensity is extremised. We establish a new relation between orbital stability and the mean wave field, which supersedes existing heuristic arguments and suggests a rationale for the alternate wobbling and monotonic instabilities arising at onset as the orbital radius is increased progressively.

The interaction of small-scale vortical structures with the surrounding fluid are studied using a fully resolved three-dimensional experimental data set of homogeneous turbulence measured at the centre of a von Kármán mixing flow facility and a direct numerical simulation (DNS) data set of forced isotropic turbulence. To identify the small-scale vortices and their boundaries, an objective observer-independent definition was implemented to avoid arbitrariness and is the first implementation applied to experimental measurements of small-scale turbulence. Volume-averaged and conditional statistics are presented to demonstrate consistency between the experimental and DNS data sets. To examine the interaction of the structures with the surrounding flow field, we examine the flow across the boundary of vortex structures by adopting a similar methodological approach to that used to investigate the local entrainment and detrainment across the turbulent–non-turbulent interface. The probability density function (p.d.f.) of entrainment velocity conditioned on the vortex boundary exhibited a non-Gaussian distribution that skewed slightly in favour of entrainment and is remarkably similar to the p.d.f.s of entrainment velocity observed in boundary layers and jets. We analyse the enstrophy transport equation conditioned on radial and axial coordinates of the vortices to quantify the inviscid and viscous components of the entrainment/detrainment process. A comparison with Burgers vortices is made and it is found that the Burgers vortex model captures the vortex structure average size and the mechanisms of enstrophy transport in the radial direction, but is unable to capture local statistics and describe the governing physics along the axes of the vortices.

We investigate the influence of rotation on the onset and saturation of the Faraday instability in a vertically oscillating two-layer miscible fluid using a theoretical model and direct numerical simulations (DNS). Our analytical approach utilizes Floquet analysis to solve a set of the Mathieu equations obtained from the linear stability analysis. The solution of the Mathieu equations comprises stable and harmonic, and subharmonic unstable regions in a three-dimensional stability diagram. We find that the Coriolis force delays the onset of the subharmonic instability responsible for the growth of the mixing zone size at lower forcing amplitudes. However, at higher forcing amplitudes, the flow is energetic enough to mitigate the instability delaying effect of rotation, and the evolution of the mixing zone size is similar in both rotating and non-rotating environments. These results are corroborated by DNS at different Coriolis frequencies and forcing amplitudes. We also observe that for $(\,f/\omega )^2<0.25$, where $f$ is the Coriolis frequency, and $\omega$ is the forcing frequency, the instability and the turbulent mixing zone size-$L$ saturates. When $(\,f/\omega )^2\geq 0.25$, the turbulent mixing zone size-$L$ never saturates and continues to grow.

Small differences in the densities of a river confluence's tributaries (i.e. 0.5 kg m$^{-3}$) have been proposed to cause coherent streamwise-oriented vortices (SOVs) in its mixing interface. These secondary flow structures are thought to result from density-driven gravity currents being laterally confined between the converging flows. However, empirical evidence for density SOVs and the confined gravity current mechanism is lacking. To this end, experiments are carried out in a laboratory confluence permitting a spectrum of thermal density differences between its tributaries. Particle image velocimetry and laser-induced fluorescence are used simultaneously to study the mixing interface's dynamics. The sensitivity of the mixing interface's secondary flow structure to the confluence's momentum ratio and the magnitude of the density difference is evaluated. Density SOVs are confirmed in the mixing interface and are caused by the gravity currents being confined laterally as the opposing flows merge and accelerate downstream. The SOVs are largest and most coherent when the momentum of the dense channel is greater than that of the light channel. The dynamics of these secondary flow structures is strongly coupled to periodic vertically orientated Kelvin–Helmholtz instabilities. The striking similarities between the empirically reproduced SOVs herein and those recently observed at the Coaticook-Massawippi confluence (Quebec, Canada), despite a two-order magnitude difference in physical scale, suggest density SOVs are a scale-independent flow structure at confluences when specific, yet relatively common, hydraulic and density conditions align.

This paper investigates the nonlinear evolution and acoustic radiation of coherent structures (CS) in the near-nozzle region of a subsonic turbulent circular jet. A CS is taken to be a wavepacket consisting of multiple ring/helical modes, which are considered to be inviscid instability waves supported by the mean-flow profile. As the three-dimensionality of helical modes is weak in the near-nozzle region, the ring and helical modes with the same frequency have nearly the same growth rates and critical levels. They coexist and interact with each other in their common critical layer at high Reynolds numbers. The self and mutual quadratic interactions generate a mean-flow distortion and streaks, which act back on the fundamental components through the cubic interaction. The amplitude of the CS is governed by an integro-partial-differential equation, a significant feature of which is that differentiations with respect to the azimuthal coordinate appear in the history-dependent nonlinear terms. The non-parallelism of the mean flow as well as the impact of fine-scale turbulence on CS are taken into account and found to affect the nonlinear terms. By solving the amplitude equation, the development of the constituting modes, streamwise vortices and streaks are described. For CS consisting of frequency sideband, low-frequency components are excited nonlinearly and amplify to reach a considerable level. By analysing the large-distance asymptote of the perturbation, the low-frequency acoustic waves are found to be emitted by the temporally–spatially varying mean-flow distortion and streaks generated by the nonlinear interactions of the CS, and are thereby determined on the basis of first principles. Interestingly, the energetic part of the streaky structure that contributes to the nonlinear dynamics does not radiate directly, and instead the Reynolds stresses driving the subdominant radiating components represent the true physical sources.

An equation for the evolution of mean kinetic energy, $E$, in a two-dimensional (2-D) or 3-D Rayleigh–Bénard system with domain height $L$ is derived. Assuming classical Nusselt-number scaling, $Nu \sim Ra^{1/3}$, and that mean enstrophy, in the absence of a downscale energy cascade, scales as $\sim E/L^2$, we find that the Reynolds number scales as $Re \sim Pr^{-1}Ra^{2/3}$ in the 2-D system, where $Ra$ is the Rayleigh number and $Pr$ the Prandtl number. Using the evolution equation and the Reynolds-number scaling, it is shown that $\tilde {\tau } \gtrsim Pr^{-1/2}Ra^{1/2}$, where $\tilde {\tau }$ is the non-dimensional convergence time scale. For the 3-D system, we make the estimate $\tilde {\tau } \gtrsim Ra^{1/6}$ for $Pr = 1$. It is estimated that the total computational cost of reaching the high $Ra$ limit in a simulation is comparable between two and three dimensions. The predictions are compared with data from direct numerical simulations.

The choking condition of a one-dimensional steady-state diabatic flow, with wall friction, of a perfect gas through a constant cross-section pipe has been analysed by applying a recent analytical solution. Such a choking condition can be achieved for an initially subsonic flow and a supersonic one, even when the heat flux goes from the fluid to the wall. Entropy–enthalpy diagrams have been analysed, and a universal choking condition for compressible diabatic flows with wall friction has been determined. The analytical solution of a supersonic flow, in which a normal shock occurs, has been obtained for a diabatic flow with wall friction and compared with the results of a numerical model.

We calculate the hydrodynamic force on a small spherical, unsteady squirmer moving with a time-dependent velocity in a fluid at rest, taking into account convective and unsteady fluid inertia effects in perturbation theory. Our results generalise those of Lovalenti & Brady (J. Fluid Mech., vol. 256, 1993, pp. 561–605) from passive to active spherical particles. We find that convective inertia changes the history contribution to the hydrodynamic force, as it does for passive particles. We determine how the hydrodynamic force depends on the swimming gait of the unsteady squirmer. Since swimming breaks the spherical symmetry of the problem, the force is not determined completely by the outer solution of the asymptotic matching problem, as it is for passive spheres. There are additional contributions due to the inhomogeneous solution of the inner problem. We also compute the disturbance flow, illustrating convective and unsteady effects when the particle experiences a sudden start followed by a sudden stop.

A multi-agent deep reinforcement learning (DRL)-based model is presented in this study to reconstruct flow fields from noisy data. A combination of reinforcement learning with pixel-wise rewards, physical constraints represented by the momentum equation and the pressure Poisson equation, and the known boundary conditions is used to build a physics-constrained deep reinforcement learning (PCDRL) model that can be trained without the target training data. In the PCDRL model, each agent corresponds to a point in the flow field and learns an optimal strategy for choosing pre-defined actions. The proposed model is efficient considering the visualisation of the action map and the interpretation of the model operation. The performance of the model is tested by using direct numerical simulation-based synthetic noisy data and experimental data obtained by particle image velocimetry. Qualitative and quantitative results show that the model can reconstruct the flow fields and reproduce the statistics and the spectral content with commendable accuracy. Furthermore, the dominant coherent structures of the flow fields can be recovered by the flow fields obtained from the model when they are analysed using proper orthogonal decomposition and dynamic mode decomposition. This study demonstrates that the combination of DRL-based models and the known physics of the flow fields can potentially help solve complex flow reconstruction problems, which can result in a remarkable reduction in the experimental and computational costs.

Recent numerical studies have shown that forced, statistically isotropic turbulence develops a ‘thermal-equilibrium’ spectrum, $\mathcal {E}(k) \propto k^2$, at large scales. This behaviour presents a puzzle, as it appears to imply the growth of a non-zero Saffman integral, which would require the longitudinal velocity correlation function, $\chi (r)$, to satisfy $\chi (r\to \infty )\propto r^{-3}$. As is well known, the Saffman integral is an invariant of decaying turbulence, precisely because non-local interactions (i.e. interactions via exchange of pressure waves) are too weak to generate such correlations. Subject to certain restrictions on the nature of the forcing, we argue that the same should be true for forced turbulence. We show that long-range correlations and a $k^2$ spectrum arise as a result of the turbulent diffusion of linear momentum, and extend only up to a maximum scale that grows slowly with time. This picture has a number of interesting consequences. First, if the forcing generates eddies with significant linear momentum (as in so-called Saffman turbulence), a thermal spectrum is not reached – instead, a shallower spectrum develops. Secondly, the energy of turbulence that is forced for a while and then allowed to decay obeys Saffman's decay laws for a period that is much longer than the duration of the forcing stage.

This paper considers the initial stage of radiatively driven convection, when the perturbations from a quiescent but time-dependent background state are small. Radiation intensity is assumed to decay exponentially away from the surface, and we consider parameter regimes in which the depth of the water is greater than the decay scale of $e$ of the radiation intensity. Both time-independent and time-periodic radiation are considered. In both cases, the background temperature profile of the water column is time-dependent. A linear analysis of the system is performed based on these time-dependent profiles. We find that the perturbations grow in time according to $\exp [(\sigma (t) t)]$, where $\sigma (t)$ is a time-dependent growth rate. An appropriately defined Reynolds number is the primary dimensionless number characterising the system, determining the wavelength, vertical structure and growth rate of the perturbations. Simulations using a Boussinesq model (the Stratified Ocean Model with Adaptive Refinement) confirm the linear analysis.