Hot-wire measurements are carried out in grid-generated turbulence at moderate to low Taylor microscale Reynolds number $Re_{{\it\lambda}}$ to assess the appropriateness of the commonly used power-law decay for the mean turbulent kinetic energy (e.g. $k\sim x^{n}$, with $n\leqslant -1$). It is found that in the region outside the initial and final periods of decay, which we designate a transition region, a power law with a constant exponent $n$ cannot describe adequately the decay of turbulence from its initial to final stages. One is forced to use a family of power laws of the form $x^{n_{i}}$, where $n_{i}$ is a different constant over a portion $i$ of the decay time during the decay period. Accordingly, it is currently not possible to determine whether any grid-generated turbulence reported in the literature decays according to Saffman or Batchelor because the reported data fall in the transition period where $n$ differs from its initial and final values. It is suggested that a power law of the form $k\sim x^{n_{init}+m(x)}$, where $m(x)$ is a continuous function of $x$, could be used to describe the decay from the initial period to the final stage. The present results, which corroborate the numerical simulations of decaying homogeneous isotropic turbulence of Orlandi & Antonia (J. Turbul., vol. 5, 2004, doi:10.1088/1468-5248/5/1/009) and Meldi & Sagaut (J. Turbul., vol. 14, 2013, pp. 24–53), show that the values of $n$ reported in the literature, and which fall in the transition region, have been mistakenly assigned to the initial stage of decay.

The swash of solitary waves on a plane beach is studied using large-scale experiments. Ten wave cases are examined which range from non-breaking waves to plunging breakers. The focus of this study is on the influence of breaker type on flow evolution, spatiotemporal variations of bed shear stresses and run-up. Measurements are made of the local water depths, flow velocities and bed shear stresses (using a shear plate sensor) at various locations in the swash zone. The bed shear stress is significant near the tip of the swash during uprush and in the shallow flow during the later stages of downrush. In between, the flow evolution is dominated by gravity and follows an explicit solution to the nonlinear shallow water equations, i.e. the flow due to a dam break on a slope. The controlling scale of the flow evolution is the initial velocity of the shoreline immediately following waveform collapse, which can be predicted by measurements of wave height prior to breaking, but also shows an additional dependence on breaker type. The maximum onshore-directed bed shear stress increases significantly onshore of the stillwater shoreline for non-breaking waves and onshore of the waveform collapse point for breaking waves. A new normalization for the bed shear stress which uses the initial shoreline velocity is presented. Under this normalization, the variation of the maximum magnitudes of the bed shear stress with distance along the beach, which is normalized using the run-up, follows the same trend for different breaker types. For the uprush, the maximum dimensionless bed shear stress is approximately 0.01, whereas for the downrush, it is approximately 0.002.

In this article we investigate the importance of mass transfer effects in the effective acoustic properties of diluted bubbly liquids. The classical theory for wave propagation in bubbly liquids for pure gas bubbles is extended to capture the influence of mass transfer on the effective phase speed and attenuation of the system. The vaporization flux is shown to be important for systems close to saturation conditions and at low frequencies. We derive a general expression for the transfer function that relates bubble radius and pressure changes, solving the linear version of the conservation equations inside, outside and at the bubble interface. Simplified expressions for various limiting situations are derived in order to get further insight about the validity of the common assumptions typically applied in bubble dynamic models. The relevance of the vapour content, the mass transfer flux across the interface and the effect of variations of the bubble interface temperature is discussed in terms of characteristic non-dimensional numbers. Finally we derive the various conditions that must be satisfied in order to reach the low-frequency limit solutions where the phase speed no longer depends on the forcing frequency.

When a mixture of particles, which differ in both their size and their density, avalanches downslope, the grains can either segregate into layers or remain mixed, dependent on the balance between particle-size and particle-density segregation. In this paper, binary mixture theory is used to generalize models for particle-size segregation to include density differences between the grains. This adds considerable complexity to the theory, since the bulk velocity is compressible and does not uncouple from the evolving concentration fields. For prescribed lateral velocities, a parabolic equation for the segregation is derived which automatically accounts for bulk compressibility. It is similar to theories for particle-size segregation, but has modified segregation and diffusion rates. For zero diffusion, the theory reduces to a quasilinear first-order hyperbolic equation that admits solutions with discontinuous shocks, expansion fans and one-sided semi-shocks. The distance for complete segregation is investigated for different inflow concentrations, particle-size segregation rates and particle-density ratios. There is a significant region of parameter space where the grains do not separate completely, but remain partially mixed at the critical concentration at which size and density segregation are in exact balance. Within this region, a particle may rise or fall dependent on the overall composition. Outside this region of parameter space, either size segregation or density segregation dominates and particles rise or fall dependent on which physical mechanism has the upper hand. Two-dimensional steady-state solutions that include particle diffusion are computed numerically using a standard Galerkin solver. These simulations show that it is possible to define a Péclet number for segregation that accounts for both size and density differences between the grains. When this Péclet number exceeds 10 the simple hyperbolic solutions provide a very useful approximation for the segregation distance and the height of rapid concentration changes in the full diffusive solution. Exact one-dimensional solutions with diffusion are derived for the steady-state far-field concentration.

Efficient mixing, typically characterised by chaotic advection, is hard to achieve in low Reynolds number conditions because of the linear nature of the Stokes equation that governs the motion. Here we show that low Reynolds number swimmers moving in quasi-periodic orbits can result in considerable stretching and folding of fluid elements. We accurately follow packets of tracers within the fluid domain and show that their trajectories become chaotic as the swimmer’s trajectory densely fills its invariant torus. The mixing process is demonstrated in two dimensions using the Quadroar swimmer that autonomously propels and tumbles along quasi-periodic orbits with multi-loop turning trajectories. We demonstrate and discuss that the streamlines of the flow induced by the Quadroar closely resemble the oscillatory flow field of the green alga Chlamydomonas reinhardtii. Our findings can thus be utilized to understand the interactions of microorganisms with their environments, and to design autonomous robotic mixers that can sweep and mix an entire volume of complex geometry containers.

The coupled motion is investigated for a mechanical system consisting of water and a body freely floating in it. Water occupies either a half-space or a layer of constant depth into which an infinitely long surface-piercing cylinder is immersed, thus allowing us to study two-dimensional modes. Under the assumption that the motion is of small amplitude near equilibrium, a linear setting is applicable, and for the time-harmonic oscillations it reduces to a spectral problem with the frequency of oscillations as the spectral parameter. Within this framework, it is shown that the total energy of the water motion is finite and the equipartition of energy holds for the whole system. On this basis two results are obtained. First, the so-called semi-inverse procedure is applied for the construction of a family of two-dimensional bodies trapping the heave mode. Second, it is proved that no wave modes can be trapped provided that their frequencies exceed a bound depending on the cylinder properties, whereas its geometry is subject to some restrictions and, in some cases, certain restrictions are imposed on the type of mode.

A dual-plane snapshot proper orthogonal decomposition (POD) analysis of turbulent pipe flow at a Reynolds number of 104 000 is presented. The high-speed particle image velocimetry data were simultaneously acquired in two planes, a cross-stream plane (2D–3C) and a streamwise plane (2D–2C) on the pipe centreline. The cross-stream plane analysis revealed large structures with a spatio-temporal extent of $1{-}2R$, where $R$ is the pipe radius. The temporal evolution of these large-scale structures is examined using the time-shifted correlation of the cross-stream snapshot POD coefficients, identifying the low-energy intermediate modes responsible for the transition between the large-scale modes. By conditionally averaging based on the occurrence/intensity of a given cross-stream snapshot POD mode, a complex structure consisting of wall-attached and -detached large-scale structures is shown to be associated with the most energetic modes. There is a pseudo-alignment of these large structures, which together create structures with a spatio-temporal extent of approximately $6R$, which appears to explain the formation of the very-large-scale motions previously observed in pipe flow.

This paper describes linear and nonlinear interactions between forced axial acoustic oscillations and the global mode of the reacting wake. This work is motivated by the problem of combustion instabilities, where acoustic oscillations associated with natural combustor modes excite hydrodynamic instabilities of the flow that, in turn, induce heat release oscillations. Wake flows with density stratification can be globally stable or unstable at high Reynolds numbers, and so the density change across the flame has significant influence on the natural flame and flow dynamics. Measurements were obtained in a facility in which flame density ratio, lip velocity and forcing frequency are independently varied using 5 kHz particle image velocimetry and Mie scattering measurements. By varying the density ratio, the hydrodynamic global mode growth rate can be systematically varied. In addition, measurements and analyses were performed where the forcing frequency is varied relative to the global mode frequency. While axial forcing excites a varicose response of the shear layers, the sinuous mode is the most rapidly growing. As expected, forcing at a frequency near the wake’s global mode frequency leads to rapid growth in vortical disturbance amplitude, and the symmetric vortices quickly stagger as they convect downstream leading to a large scale, sinuous flapping of the wake and flame. A linear, local stability analysis, together with a nonlinear analysis, help elucidate the physics that govern the vortex staggering. The study concludes with an analysis of the heat release dynamics. Significantly, the study shows that the heat release exhibits quite different sensitivities than the fluid dynamics; e.g. axial forcing of the flow near its global mode frequency leads to a reduction in heat release oscillations. This is true even though this forcing frequency maximizes the local degree of vortically induced flame flapping. Thus, the results of this study show some phenomena that contradict conventional notions, namely that conditions which align the frequency of a hydrodynamic global mode with that of an acoustic mode may lead to diminished forced heat release oscillations in bluff body combustors.

The interaction between an 8:1 aspect ratio rectangular jet and a flat plate, placed parallel to the jet, is addressed in this study. At high subsonic conditions and for certain relative locations of the plate, a resonance takes place with accompanying audible tones. Even when the tone is not audible the sound pressure level spectra are often marked by conspicuous peaks. The frequencies of these peaks, as functions of the plate’s length, its location relative to the jet as well as jet Mach number, are studied in an effort to understand the flow mechanism. It is demonstrated that the tones are not due to a simple feedback between the nozzle exit and the plate’s trailing edge; the leading edge also comes into play in determining the frequency. With parametric variation, it is found that there is an order in the most energetic spectral peaks; their frequencies cluster in distinct bands. The lowest frequency band is explained by an acoustic feedback involving diffraction at the plate’s leading edge. Under the resonant condition, a periodic flapping motion of the jet column is seen when viewed in a direction parallel to the plate. Phase-averaged Mach number data on a cross-stream plane near the plate’s trailing edge illustrate that the jet cross-section goes through large contortions within the period of the tone. Farther downstream a clear ‘axis switching’ takes place for the time-averaged cross-section of the jet that does not occur otherwise for a non-resonant condition.

The lateral migration properties of a rigid spherical particle suspended in a pressure-driven flow through channels with square cross-sections were investigated numerically, in the range of Reynolds numbers ($Re$) from 20 to 1000. The flow field around the particle was computed by the immersed boundary method to calculate the lateral forces exerted on the particle and its trajectories, starting from various initial positions. The numerical simulation showed that eight equilibrium positions of the particle are present at the centres of the channel faces and near the corners of the channel cross-section. The equilibrium positions at the centres of the channel faces are always stable, whereas the equilibrium positions at the corners are unstable until $Re$ exceeds a certain critical value, $Re_{c}$. At $Re\approx Re_{c}$, additional equilibrium positions appear on a heteroclinic orbit that joins the channel face and corner equilibrium positions, and the lateral forces along the heteroclinic orbit are very small. As $Re$ increases, the channel face equilibrium positions are shifted towards the channel wall at first, and then shifted away from the channel wall. The channel corner equilibrium positions exhibit a monotonic shift towards the channel corner with increasing $Re$. Migration behaviours of the particle in the cross-section are also predicted for various values of $Re$. These numerical results account for the experimental observations of particle distributions in the cross-section of micro and millimetre scale channels, including the characteristic alignment and focusing of the particles, the absence of the corner equilibrium positions at low $Re$ and the progressive shift of the equilibrium positions with $Re$.

In light of the successes of the Navier–Stokes equations in the study of fluid flows, similar continuum treatment of granular materials is a long-standing ambition. This is due to their wide-ranging applications in the pharmaceutical and engineering industries as well as to geophysical phenomena such as avalanches and landslides. Historically this has been attempted through modification of the dissipation terms in the momentum balance equations, effectively introducing pressure and strain-rate dependence into the viscosity. Originally, a popular model for this granular viscosity, the Coulomb rheology, proposed rate-independent plastic behaviour scaled by a constant friction coefficient ${\it\mu}$. Unfortunately, the resultant equations are always ill-posed. Mathematically ill-posed problems suffer from unbounded growth of short-wavelength perturbations, which necessarily leads to grid-dependent numerical results that do not converge as the spatial resolution is enhanced. This is unrealistic as all physical systems are subject to noise and do not blow up catastrophically. It is therefore vital to seek well-posed equations to make realistic predictions. The recent ${\it\mu}(I)$-rheology is a major step forward, which allows granular flows in chutes and shear cells to be predicted. This is achieved by introducing a dependence on the non-dimensional inertial number $I$ in the friction coefficient ${\it\mu}$. In this paper it is shown that the ${\it\mu}(I)$-rheology is well-posed for intermediate values of $I$, but that it is ill-posed for both high and low inertial numbers. This result is not obvious from casual inspection of the equations, and suggests that additional physics, such as enduring force chains and binary collisions, becomes important in these limits. The theoretical results are validated numerically using two implicit schemes for non-Newtonian flows. In particular, it is shown explicitly that at a given resolution a standard numerical scheme used to compute steady-uniform Bagnold flow is stable in the well-posed region of parameter space, but is unstable to small perturbations, which grow exponentially quickly, in the ill-posed domain.

In many engineering systems operating with a working fluid, the best efficiency is reached close to a condition of flow separation, which makes the prediction of this condition very important in industry. Provided that wall-based quantities can be measured, we know today how to obtain good predictions for two- and three-dimensional steady and periodic flows. In these flows, the separation is defined on a fixed line attached to a material surface. The last case to elucidate is the one where this line is no longer attached to the wall but on the contrary is contained within the flow. This moving separation is probably, however, the most common case of separation in natural flows and industrial applications. Since this case has received less attention during the past few years, we propose in this study to examine some properties of moving separation in two-dimensional, unsteady flows where the separation does not leave a signature on the wall. Since in this framework separation can be extracted by using a Lagrangian frame where the separation profile can be viewed as a hyperbolic unstable manifold, we propose a method to extract the separation point defined by the Lagrangian saddle point that belongs to this unstable manifold. In practice, the separation point and profile are initially extracted by detecting the most attracting Lagrangian coherent structure near the wall, and they can then be advected in time for subsequent instants. It is found that saddle points, which initially act as separation points in the viscous wall flow region, remarkably preserve their hyperbolicity even if they are ejected from the wall towards the inviscid region. Two test cases are studied: the creeping flow of a rotating and translating cylinder close to a wall, and the unsteady separation in the boundary layer generated by a planar jet impinging onto a plane wall.

The transformation of a time-dependent pressure pulse imposed on the front face of a rigid porous medium sample, mounted in a tunnel, through the sample and a fixed-volume air gap between the rear face of the sample and the end wall of a tunnel is studied both experimentally and analytically. In the experiments, rigid porous samples that are placed at various distances from a shock tube end wall are subjected to the impingement of shock waves. The pressure buildup behind the porous sample is monitored and compared with the pressure imposed at the front face of the porous sample. The shock tube is fitted with a short driver section in order to generate blast-like decaying pressure profiles, which continue to decay after the initial shock impingement. In this scenario, the measured pressure profile at the end wall, which is affected by the properties of the porous medium and the size of the air gap separating its rear face and the shock tube end wall, is significantly different from the pressure profile imposed on the front face of the porous sample. The mechanism governing the pressure transformation provided by the porous medium is attributed to a selective filtration process that attenuates the pressure changes associated with high frequencies. The results of the present study are also analysed in conjunction with previously published analytical and numerical models to achieve a broader understanding of the physical mechanisms affecting the pressure buildup.

We present the complete unfolding of streamwise localisation in a paradigm of extended shear flows, namely two-dimensional plane Poiseuille flow. Exact solutions of the Navier–Stokes equations are computed numerically and tracked in the streamwise wavenumber–Reynolds number parameter space to identify and describe the fundamental mechanism behind streamwise localisation, a ubiquitous feature of shear flow turbulence. Unlike shear flow spanwise localisation, streamwise localisation does not follow the snaking mechanism demonstrated for plane Couette flow.

The Riemann problem of planar shock waves is analysed for a dilute granular gas by solving Euler- and Navier–Stokes-order equations numerically. The density and temperature profiles are found to be asymmetric, with the maxima of both density and temperature occurring within the shock layer. The density peak increases with increasing Mach number and inelasticity, and is found to propagate at a steady speed at late times. The granular temperature at the upstream end of the shock decays according to Haff’s law (${\it\theta}(t)\sim t^{-2}$), but the downstream temperature decays faster than its upstream counterpart. Haff’s law seems to hold inside the shock up to a certain time for weak shocks, but deviations occur for strong shocks. The time at which the maximum temperature deviates from Haff’s law follows a power-law scaling with the upstream Mach number and the restitution coefficient. The origin of the continual build-up of density with time is discussed, and it is shown that the granular energy equation must be ‘regularized’ to arrest the maximum density.

We experimentally determine the phase diagram for impacting ethanol droplets on a smooth, sapphire surface in the parameter space of Weber number $\mathit{We}$ versus surface temperature $T$. We observe two transitions, namely the one towards splashing (disintegration of the droplet) with increasing $\mathit{We}$, and the one towards the Leidenfrost state (no contact between the droplet and the plate due to a lasting vapour film) with increasing $T$. Consequently, there are four regimes: contact and no splashing (deposition regime), contact and splashing (contact–splash regime), neither contact nor splashing (bounce regime), and finally no contact, but splashing (film–splash regime). While the transition temperature $T_{L}$ to the Leidenfrost state depends weakly, at most, on $\mathit{We}$ in the parameter regime of the present study, the transition Weber number $\mathit{We}_{C}$ towards splashing shows a strong dependence on $T$ and a discontinuity at $T_{L}$. We quantitatively explain the splashing transition for $T<T_{L}$ by incorporating the temperature dependence of the physical properties in the theory by Riboux & Gordillo (Phys. Rev. Lett., vol. 113(2), 2014, 024507; J. Fluid Mech., vol. 772, 2015, pp. 630–648).