Streamwise variation in vessel geometry is a feature of many multiphase flows of practical interest, ranging from natural porous media flows to man-made lab-on-the-chip applications. The variable streamwise geometry typically exerts a dominant influence on bubble motion, and can lead to undesirable phenomena such as clogging of the vessel. Here, we study clogging in a fundamental configuration, where a tube of square cross-section is suddenly expanded over a short streamwise distance. The extent to which a bubble driven by constant flux flow broadens to partially fill the expansion depends on the balance between viscous and surface tension stresses, measured by the capillary number $\mathit{Ca}$. This broadening is accompanied by the slowing and momentary arrest of the bubble as $\mathit{Ca}$ is reduced towards its critical value for trapping. For $\mathit{Ca}\lt {\mathit{Ca}}_{c} $ the pressure drag forces on the quasi-arrested bubble are insufficient to force the bubble out of the expanded region so it remains trapped. We examine the conditions for trapping by varying bubble volume, flow rate of the carrier fluid, relative influence of gravity and length of expanded region. We find specifically that ${\mathit{Ca}}_{c} $ depends non-monotonically on the size of the bubble. We verify, with experiments and a capillary static model, that a bubble is released if the work of the pressure forces over the length of the trap exceeds the surface energy required for the trapped bubble to reenter the constricted square tube.

A computational model of the pseudo-incompressible equations is used to probe the range of validity of an extended Wentzel–Kramers–Brillouin theory (XWKB), previously derived through a distinguished limit of a multiple-scale asymptotic analysis of the Euler or pseudo-incompressible equations of motion, for gravity-wave packets at large amplitudes. The governing parameter of this analysis had been the scale-separation ratio $\varepsilon $ between the gravity wave and both the large-scale potential-temperature stratification and the large-scale wave-induced mean flow. A novel feature of the theory had been the non-resonant forcing of higher harmonics of an initial wave packet, predominantly by the large-scale gradients in the gravity-wave fluxes. In the test cases considered a gravity-wave packet is propagating upwards in a uniformly stratified atmosphere. Large-scale winds are induced by the wave packet, and possibly exert a feedback on the latter. In the limit $\varepsilon \ll 1$ all predictions of the theory can be validated. The larger $\varepsilon $ is the more the transfer of wave energy to the mean flow is underestimated by the theory. The numerical results quantify this behaviour but also show that, qualitatively, XWKB remains valid even when the gravity-wave wavelength approaches the spatial scale of the wave-packet amplitude. This includes the prevalence of first and second harmonics and the smallness of harmonics with wave number higher than two. Furthermore, XWKB predicts for the vertical momentum balance an additional leading-order buoyancy term in Euler and pseudo-incompressible theory, compared with the anelastic theory. Numerical tests show that this term is relatively large with up to $30\hspace{0.167em} \% $ of the total balance. The practical relevance of this deviation remains to be assessed in future work.

We study mixing in the presence of convective flow in a porous medium. Convection is characterized by the formation of vortices and stagnation points, where the fluid interface is stretched and compressed enhancing mixing. We analyse the behaviour of the mixing dynamics in different scenarios using an interface deformation model. We show that the scalar dissipation rate, which is related to the dissolution fluxes, is controlled by interfacial processes, specifically the equilibrium between interface compression and diffusion, which depends on the flow field configuration. We consider different scenarios of increasing complexity. First, we analyse a double-gyre synthetic velocity field. Second, a Rayleigh–Bénard instability (the Horton–Rogers–Lapwood problem), in which stagnation points are located at a fixed interface. This system experiences a transition from a diffusion controlled mixing to a chaotic convection as the Rayleigh number increases. Finally, a Rayleigh–Taylor instability with a moving interface, in which mixing undergoes three different regimes: diffusive, convection dominated and convection shutdown. The interface compression model correctly predicts the behaviour of the systems. It shows how the dependency of the compression rate on diffusion explains the change in the scaling behaviour of the scalar dissipation rate. The model indicates that the interaction between stagnation points and the correlation structure of the velocity field is also responsible for the transition between regimes. We also show the difference in behaviour between the dissolution fluxes and the mixing state of the systems. We observe that while the dissolution flux decreases with the Rayleigh number, the system becomes more homogeneous. That is, mixing is enhanced by reducing diffusion. This observation is explained by the effect of the instability patterns.

The flow of a planar jet (sheet) impinging onto a solid flat plate with slip is examined theoretically. The jet is assumed to spread out in a thin layer bounded by a hydraulic jump, and draining at the edge of the plate. In contrast to an adhering jet, a slipping jet does not admit a similarity solution. Taking advantage of the different scaling in each region, series expansions are used in the developing and fully viscous layers, which are matched at the transition point. We show that a slipping film exhibits a singularity in the normal stress at the leading edge of the boundary layer, as opposed to the singularity in velocity and shear stress for an adhering film. The boundary-layer and film heights are both found to decrease with slip relative to a smooth substrate, roughly like $\sqrt{30x/Re}-2S$, whereas the slip velocity intensifies like $S\sqrt{Re/30x}$ with slip. Here, $x$ is the distance along the plate, $S$ is the slip length and $Re$ is the Reynolds number (in units of the jet width). The transition is delayed by slip. Guided by the measurements of Duchesne et al. (Europhys. Lett., vol. 107, 2014, p. 54002) for a circular adhering jet, the hydraulic-jump height and location are determined for a planar jet, and are found to increase with the Froude number (flow rate) like $Fr^{1/4}$ and $Fr^{5/8}$ respectively, essentially independently of slip length.

A weak, laminar shear flow of a monodisperse suspension of high-Reynolds-number, low-Weber-number bubbles is studied in a novel experimental configuration. Nitrogen bubbles are formed through an array of small capillaries at the base of a tall channel with a small inclination from the vertical. The bubbles generate a unidirectional shear flow, in which the denser suspension near the bottom wall falls and the lighter suspension near the top wall rises. Profiles of the bubble and liquid velocities and the bubble volume fraction are obtained using hot-film and dual impedance probes. To our knowledge, measurements of the laminar shear properties of a nearly homogeneous bubble suspension have not previously been reported.

A steady shear flow is observed in which the bubble velocity variation across the channel is typically less than 20% of the mean bubble velocity. The velocity and volume fraction gradients increase with channel inclination and exhibit little or no dependence on the mean gas volume fraction. To explain the magnitude of the volume fraction gradients, it is necessary to consider the effects of both the lift force and the effective bubble-phase diffusivity in balancing the segregating tendency of the cross-channel component of the buoyancy force. The bubble velocity gradient can be understood in terms of a balance of the component of the buoyancy force parallel to the channel walls and an effective viscosity associated with the Reynolds stresses produced by bubble-induced liquid velocity fluctuations. Theories for bubbles rising with potential-flow hydrodynamic interactions predict an instability of the homogeneous state due to a negative Maxwell pressure. However, the hydrodynamic diffusivity inferred from our experiments is large enough to mitigate the clustering effects of the Maxwell pressure. Consistent with this, a vigorous instability of the homogeneous state of the bubble suspension is only observed at volume fractions larger than 5%–20% with the critical volume fraction depending on the angle of inclination.

A characteristic feature of axisymmetric jets, and turbulent shear flows in general, is the entrainment of mass across the turbulent/non-turbulent interface (TNTI). The multi-scale nature of the TNTI surface area was recently observed to exhibit power-law scaling with a fractal dimension, $D_{f}$, between $D_{f}=2.3{-}2.4$, inferred from two-dimensional data, in both high Reynolds number boundary layers and the far field of axisymmetric jets. In this paper, we show that the fractal scaling previously observed in the far field of an axisymmetric jet is established at the end of the potential core. Simultaneous measurements of the velocity and scalar fields were obtained and coarse grain filtering was applied over two decades of scale separation, showing that $D_{f}$ evolves to ${\approx}2.35$ at $x/d=4.6$, which is similar to $D_{f}$ found in the far field between $x/d=40{-}60$. This is evidence that scale separation becomes sufficiently developed to achieve scale invariance of the TNTI surface area in the near field of the jet well before self-similarity is established. We also observe that the onset of this geometric scale invariance coincides with the onset of radial homogeneity shown by two-point velocity correlations. Finally, we present a simple theoretical basis for these results using an exact fractal construction based on the Koch curve and applying a coarse-grain filtering analysis.

The motion of a continuously stratified fluid in a rotating annulus is investigated experimentally. The flow is driven by the differential rotation of a lid on the annulus in contact with the fluid. For small enough Rossby numbers the basic flow is found to be axisymmetric, but in contrast to the unstratified case the zonal flow is not independent of the depth, and there is significant vertical shear in the interior. As the Rossby number of the flow is increased the axisymmetric flow becomes unstable to non-axisymmetric baroclinic disturbances. Some features of these disturbances as well as the time response to the motion of the lid are examined.

The sedimentation velocity of colloidal dispersions is known from experiment and theory at dilute concentrations to be quite sensitive to the interparticle potential with attractions/repulsions increasing/decreasing the rate significantly at intermediate volume fractions. Since the differences necessarily disappear at close packing, this implies a substantial maximum in the rate for attractions. This paper describes the derivation of a robust upper bound on the velocity that reflects these trends quantitatively and motivates wider application of a simple theory formulated for hard spheres. The treatment pertains to sedimentation velocities slow enough that Brownian motion sustains an equilibrium microstructure without large-scale inhomogeneities in density.

There are many flows of practical importance where both Tollmien-Schlichting waves and Taylor-Görtler vortices are possible causes of transition to turbulence. In this paper, the effect of fully nonlinear Taylor-Görtler vortices on the growth of small-amplitude Tollmien-Schlichting waves is investigated. The basic state considered is the fully developed flow between concentric cylinders driven by an azimuthal pressure gradient. It is hoped that an investigation of this problem will shed light on the more complicated external-boundary-layer problem where again both modes of instability exist in the presence of concave curvature. The type of Tollmien-Schlichting waves considered have the asymptotic structure of lower-branch modes of plane Poiseuille flow. Whilst instabilities at lower Reynolds number are possible, the former modes are simpler to analyse and more relevant to the boundary-layer problem. The effect of fully nonlinear Taylor-Görtler vortices on both two-dimensional and three-dimensional waves is determined. It is shown that, whilst the maximum growth as a function of frequency is not greatly affected, there is a large destabilizing effect over a large range of frequencies.

We present a new solution for the steady boundary-layer flow over the rotating sphere that also accounts for the eruption of the boundary layer at the equator and other higher-order viscous effects. Non-parallel corrections to the local type I and type II convective instability modes of this flow are also computed as a function of spin rate. Our instability results are associated with the previously observed spiral vortices and remarkable agreement between our predictions of the number of vortices and experimental observations is found. Vortices travelling at 70 %–80 % of the local surface speed are found to be the most amplified for sufficient spin rates, also consistent with prior experimental observations.

We have studied the evolution of grid turbulence in a planar contraction by focusing on the flow at the centre symmetry plane. Measurements are carried out in water with inlet Taylor-microscale Reynolds number varying from 51 to 99. Detailed laser-Doppler anemometry measurements show that the streamwise fluctuating velocity component for contraction ratio $C\,{<}\,2.5$ closely follows the decay of grid turbulence in a straight channel. Furthermore, the turbulent kinetic energy reaches a minimum value in the range of contraction ratio $1.5\,{<}\,C\,{<}\,2.5$. Turbulent intensity, independent of contraction angle and Reynolds number, decays exponentially. The results show that the flow reaches its peak of anisotropy at $2.5\,{<}\,C\,{<}\,3.5$ and then returns to a nearly fully isotropic state inside the contraction. The return to isotropy within the contraction is attributed to the rapid part of the pressure–strain correlation term in the transport equation of the Reynolds-stress anisotropy tensor.

This study seeks to characterise the breakdown of the steady two-dimensional solution in the flow around a 180-degree sharp bend to infinitesimal three-dimensional disturbances using a linear stability analysis. The stability analysis predicts that three-dimensional transition is via a synchronous instability of the steady flows. A highly accurate global linear stability analysis of the flow was conducted with Reynolds number $\mathit{Re}<1150$ and bend opening ratio (ratio of bend width to inlet height) $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 5$. This range of $\mathit{Re}$ and $\unicode[STIX]{x1D6FD}$ captures both steady-state two-dimensional flow solutions and the inception of unsteady two-dimensional flow. For $0.2\leqslant \unicode[STIX]{x1D6FD}\leqslant 1$, the two-dimensional base flow transitions from steady to unsteady at higher Reynolds number as $\unicode[STIX]{x1D6FD}$ increases. The stability analysis shows that at the onset of instability, the base flow becomes three-dimensionally unstable in two different modes, namely a spanwise oscillating mode for $\unicode[STIX]{x1D6FD}=0.2$ and a spanwise synchronous mode for $\unicode[STIX]{x1D6FD}\geqslant 0.3$. The critical Reynolds number and the spanwise wavelength of perturbations increase as $\unicode[STIX]{x1D6FD}$ increases. For $1<\unicode[STIX]{x1D6FD}\leqslant 2$ both the critical Reynolds number for onset of unsteadiness and the spanwise wavelength decrease as $\unicode[STIX]{x1D6FD}$ increases. Finally, for $2<\unicode[STIX]{x1D6FD}\leqslant 5$, the critical Reynolds number and spanwise wavelength remain almost constant. The linear stability analysis also shows that the base flow becomes unstable to different three-dimensional modes depending on the opening ratio. The modes are found to be localised near the reattachment point of the first recirculation bubble.

The analysis of the linear stability of a planar Chapman–Jouguet detonation wave is reformulated for an arbitrary caloric (incomplete) equation of state in an attempt to better represent the stability properties of detonations in condensed-phase explosives. Calculations are performed on a ‘stiffened-gas’ equation of state which allows us to prescribe a finite detonation Mach number while simultaneously allowing for a detonation shock pressure that is substantially larger than the ambient pressure. We show that the effect of increasing the ambient sound speed in the material, for a given detonation speed, has a stabilizing effect on the detonation. We also show that the presence of the slow reaction stage, a feature of detonations in certain types of energetic materials, where the detonation structure is characterized by a fast reaction stage behind the detonation shock followed by a slow reaction stage, tends to have a destabilizing effect.

This paper describes a sampling technique which is non-periodic, but also non-random, and which allows a pulsed wire anemometer to be used to obtain time-domain information in separated flows. Data free from aliasing can be obtained at frequencies considerably higher than that associated with the maximum possible (periodic) sampling rate achievable with a pulsed-wire probe. Autocorrelation and spectral data obtained in three quite distinct types of flow are presented, as a demonstration of the technique. Some of the data serve to emphasize the special character of separated flows. It is shown that in some circumstances it is possible to obtain spectral data that are adequate for estimating the turbulent-energy dissipation rate (provided that an inertial subrange exists). The measurement of all the terms necessary to examine the balance of turbulent energy in separated flows – or, at least, those terms which can be obtained by standard hot-wire anemometry in low-intensity flows – is therefore a distinct possibility.

It has been found that ventilated cavities extending behind hydrofoils, plates, and other two-dimensional bodies, oscillate when the air supply rate is sufficient to reduce the cavitation number to about one-fifth of its natural value. As the rate increases further, higher modes of oscillation occur in which the cavity–water interface supports several waves that are convected downstream towards the wake, which, owing to a pinching-off action replacing the usual entrainment sink, consists of a sequence of large bubbles drifting downstream. A theory of such flows that allows both for the convected velocity fluctuations in the cavity, and for the transport of bubble volume down the wake, is given in this paper. Coupled with a rather simple phenomenological relation between the pressure fluctuations within the cavity and the departure of the pinched-off rear portion of the cavity—explained in terms of the action of the re-entrant jet—this theory successfully predicts the resonance frequencies obtained in experiments by Silberman & Song.

The theory also provides a solution of the more general problem of determining the fluctuations in the pressure distribution over the whole surface of the body, when it is in a prescribed unsteady motion along its axis of symmetry (the theory is confined to symmetrical bodies and flows). Thus the growth in drag due to a sudden increment in the upstream velocity can be predicted, and also the damping forces acting on the body when it is forced to oscillate at a given frequency. It is shown that in all cases the body is unstable.

One important feature of the mathematical model chosen is that it completely avoids the presence of a time-dependent sink at infinity—with its associated infinite pressures—by conserving total volume of wake and cavity in just the same way as vorticity is conserved in unsteady aerofoil theory.

We analyse two-dimensional clamped parallel elastic sheets which are partially immersed in liquid as a model for elasto-capillary coalescence. In the existing literature this problem is studied via minimal energy analysis of capillary and elastic energies of the post-coalescence state, yielding the maximal stable post-coalescence bundle size. Utilizing modal stability analysis and asymptotic analysis, we studied the stability of the configuration before the coalescence occurred. Our analysis revealed previously unreported relations between viscous forces, body forces, and the instability yielding the coalescence, thus undermining a common assumption that coalescence will occur as long as it will not create a bundle larger than the maximal stable post-coalesced size. A mathematical description of the process creating the hierarchical coalescence structure was obtained and yielded that the mean number of sheets per coalesced region is limited to the subset ${2}^{N} $ where $N$ is the set of natural numbers. Our theoretical results were illustrated by experiments and good agreement with the theoretical predictions was observed.

The present work reports some new aspects of non-axisymmetric gravity currents obtained from laboratory experiments, fully resolved simulations and box models. Following the earlier work of Zgheib et al. (Theor. Comput. Fluid Dyn., vol. 28, 2014, pp. 521–529) which demonstrated that gravity currents initiating from non-axisymmetric cross-sectional geometries do not become axisymmetric, nor do they retain their initial shape during the slumping and inertial phases of spreading, we show that such non-axisymmetric currents eventually reach a self-similar regime during which (i) the local front propagation scales as $t^{1/2}$ as in circular releases and (ii) the non-axisymmetric front has a self-similar shape that primarily depends on the aspect ratio of the initial release. Complementary experiments of non-Boussinesq currents and top-spreading currents suggest that this self-similar dynamics is independent of the density ratio, vertical aspect ratio, wall friction and Reynolds number $\mathit{Re}$, provided the last is large, i.e. $\mathit{Re}\geqslant O(10^{4})$. The local instantaneous front Froude number obtained from the fully resolved simulations is compared to existing models of Froude functions. The recently reported extended box model is capable of capturing the dynamics of such non-axisymmetric flows. Here we use the extended box model to propose a relation for the self-similar horizontal aspect ratio ${\it\chi}_{\infty }$ of the propagating front as a function of the initial horizontal aspect ratio ${\it\chi}_{0}$, namely ${\it\chi}_{\infty }=1+(\ln {\it\chi}_{0})/3$. The experimental and numerical results are in good agreement with the proposed relation.

We present an approach combining temporal dynamical systems methods with newly proposed spatial coupling measures – namely, coherence and cross-bicoherence – to identify and quantitatively describe low-dimensional dynamics in transitional open flows. The approach is used to describe a forced mixing layer as a low-dimensional temporal dynamical system and interpret its transitional vortex dynamics.

Experiments were performed in an initially laminar plane mixing layer inside an anechoic chamber using forcing of the fundamental instability only; the forcing frequency and amplitude are used as control parameters. Dynamical invariants calculated show that vortex roll-up and the feedback-driven first two pairing dynamics are well-described by one periodic and at least two low-dimensional chaotic attractors; a phase diagram delineating such dynamical states in the control parameter space is presented. The large spatial extents of these feedback-sustained states (verified using coherence and cross-bicoherence), spanning many instability wavelengths downstream, indicate spatial coupling; feedback has also been experimentally verified. At a fixed forcing frequency, as the forcing amplitude is decreased, the spatially coupled, periodic second pairing dynamics becomes chaotic and spatiotemporal (inferred from decay of coherence and cross-bicoherence); the dynamics in the domain that includes the first pairing, however, remains temporal. This loss of spatial coupling is accompanied by a sudden increase in the attractor dimension, and suggests spatiotemporal chaos. The combination of dynamical systems theory and spatial measures seems to be a promising approach to probe spatiotemporal dynamics in other open flows as well.