The growth and collapse of a cavitation bubble forming within the core of a line vortex was examined experimentally to determine how the dynamics and noise emission of the elongated cavitation bubble is influenced by the underlying non-cavitating vortex properties. A steady line vortex was formed downstream of a hydrofoil mounted in the test section of a recirculating water channel. A focused pulse of laser light was used to initiate a nucleus in the core of a vortex, allowing for the detailed examination of the growth, splitting and collapse of individual cavitation bubbles as they experience a reduction and recovery of the local static pressure. Images of single-bubble dynamics were captured with two pulse-synchronized high-speed video cameras. The shape and dynamics of single vortex cavitation bubbles are compared to the original vortex properties and the local static pressure in the vortex core, and an analysis was performed to understand the relationship between the non-cavitating vortex properties and the diameter of the elongated cavitation bubble. Acoustic emissions from the bubbles were detected during growing, splitting and collapse, revealing that the acoustic impulse created during collapse was four orders of magnitude higher than the noise emission due to growth and splitting. The dynamics and noise generation of the elongated bubbles are compared to that of spherical cavitation bubbles in quiescent flow. These data indicate that the core size and circulation are insufficient to scale the developed vortex cavitation. The non-cavitating vortex circulation and core size are not sufficient to scale the bubble dynamics, even though the single-phase pressure field is uniquely scaled by these parameters. A simple analytical model of the equilibrium state of the elongated cavitation bubble suggests that there are multiple possible equilibrium values of the elongated bubble radius, each with varying tangential velocities at the bubble interface. Thus, the details of the bubble dynamics and bubble–flow interactions will set the final bubble dimensions.

We study the reflection of acoustic shock waves grazing at a small angle over a rigid surface. Depending on the incidence angle and the Mach number, the reflection patterns are mainly categorized into two types, namely regular reflection and irregular reflection. In the present work, using the nonlinear KZ equation, this reflection problem is investigated for extremely weak shocks as encountered in acoustics. A critical parameter, defined as the ratio of the sine of the incidence angle and the square root of the acoustic Mach number, is introduced in a natural way. For step shocks, we recover the self-similar (pseudo-steady) nature of the reflection, which is well known from von Neumann's work. Four types of reflection as a function of the critical parameter can be categorized. Thus, we describe the continuous but nonlinear and non-monotonic transition from linear reflection (according to the Snell–Descartes laws) to the weak von-Neumann-type reflection observed for almost perfectly grazing incidence. This last regime is a new, one-shock regime, in contrast with the other, already known, two-shock (regular reflection) or three-shock (von Neumann-type reflection) regimes. Hence, the transition also resolves another paradox on acoustic shock waves addressed by von Neumann in his classical paper. However, step shocks are quite unrealistic in acoustics. Therefore, we investigate the generalization of this transition for N-waves or periodic sawtooth waves, which are more appropriate for acoustics. Our results show an unsteady reflection effect necessarily associated with the energy decay of the incident wave. This effect is the counterpart of step-shock propagation over a concave surface. For a given value of the critical parameter, all the patterns categorized for the step shock may successively appear when the shock is propagating along the surface, starting from weak von-Neumann-type reflection, then gradually turning to von Neumann reflection and finally evolving into nonlinear regular reflection. This last one will asymptotically result in linear regular reflection (Snell–Descartes). The transition back to regular reflection is one of two types, depending on whether a secondary reflected shock is observed. The latter case, here described for the first time, appears to be related to the non-constant state behind the incident shock, which prevents secondary reflection.

Static magnetic fields are known to be suitable for damping mean flow and turbulent motion in an electrically conducting liquid. In this paper, an experimental study is presented considering the influence of a horizontal magnetic field on a bubble-driven flow of a liquid metal. The investigation is focused on the liquid circulation inside a liquid metal column driven by a central jet produced by gas injection. The fluid vessel has a circular cross-section and electrically insulating walls. Low gas flow rates were applied, resulting in a plume of separated bubbles rising inside a spot around the cylinder axis. This axisymmetric configuration is exposed to a horizontal magnetic field. We present detailed experimental data describing the spatial as well as the temporal structure of the velocity field. Measurements of the vertical and the radial velocity component, respectively, were performed using the ultrasound Doppler velocimetry (UDV), allowing for the first time a complete mapping of the liquid velocity distribution for a bubble-driven liquid metal flow. The magnetic field considerably modified the global and local properties of the flow field compared to an ordinary bubble plume. In the parameter range considered here we did not find a prior flow suppression, but, in fact, a restructuring of the convective motion. The original axisymmetric flow field became anisotropic with respect to the direction of the magnetic field lines. An upwards flow dominated in a plane parallel to the magnetic field, whereas the recirculating motion was enforced in the orthogonal plane. Contrary to usual expectations, the application of a moderate magnetic field (100 < Ha < 400, 1 ≲ N ≲ 10) destabilizes the global flow and gives rise to transient, oscillating flow patterns with predominant frequencies.

Electrohydrodynamic (EHD) flow around a charged spherical colloid near an electrode was studied theoretically and experimentally to understand the nature of long-range particle–particle attraction near electrodes. Numerical computations for finite double-layer thicknesses confirmed the validity of an asymptotic methodology for thin layers. Then the electric potential around the particle was computed analytically in the limit of zero Péclet number and thin double layers for oscillatory electric fields at frequencies where Faradaic reactions are negligible. Streamfunctions for the steady component of the EHD flow were determined with an electro-osmotic slip boundary condition on the electrode surface. Accordingly, it was established how the axisymmetric flow along the electrode is related to the dipole coefficient of the colloidal particle. Under certain conditions, the flow is directed toward the particle and decays as r−4, in accord with observations of long-range particle aggregation. To test the theory, particle-tracking experiments were performed with fluorescent 300 nm particles around 50μm particles over a wide range of electric field strengths and frequencies. Treating the particle surface conductivity as a fitting parameter yields velocities in excellent agreement with the theoretical predictions. The observed frequency dependence, however, differs from the model predictions, suggesting that the effect of convection on the charge distribution is not negligible as assumed in the zero Péclet number limit.

This paper examines the consistency of the exact scaling laws for isotropic magnetohydrodynamic (MHD) turbulence in numerical simulations with large magnetic Prandtl numbers Pm and with Pm = 1. The exact laws are used to elucidate the structure of the magnetic and velocity fields. Despite the linear scaling of certain third-order correlation functions, the situation is not analogous to the case of Kolmogorov turbulence. The magnetic field is adequately described by a model of a stripy (folded) field with direction reversals at the resistive scale. At currently available resolutions, the cascade of kinetic energy is short-circuited by the direct exchange of energy between the forcing-scale motions and the stripy magnetic fields. This non-local interaction is the defining feature of isotropic MHD turbulence.

For moderate Reynolds numbers, a sphere colliding with a wall in the normal direction will lead to a trailing recirculating wake, threading over the sphere after impact and developing into a complex vortex-ring system as it interacts with vorticity generated at the wall. The primary vortex ring, consisting of the vorticity from the wake of the sphere prior to impact, persists and convects, relatively slowly, outwards away from the sphere owing to the motion induced from its image. The outward motion is arrested only a short distance from the axis because of the strong interaction with the secondary vorticity. In this paper, the structure and evolution of this combined vortex system, consisting of a strong compact primary vortex ring surrounded by and interacting with the secondary vorticity, is quantified through a combined experimental and numerical study. The Reynolds-number range investigated is (100 < Re < 2000). At Reynolds numbers higher than about 1000, a non-axisymmetric instability develops, leading to rapid distortion of the ring system. The growth of the instability does not continue indefinitely, because of the dissipative nature of the flow system; it appears to reach a peak when the wake vorticity first forms a clean primary vortex ring. A comparison of the wavelength, growth rate and perturbation fields predicted from both linear stability theory and direct simulations, together with theoretical predictions, indicates that the dominant physical mechanism for the observed non-axisymmetric instability is centrifugal in nature. The maximum growth occurs at the edge of the primary vortex core, where the vorticity changes sign. Notably, this is a physical mechanism different from that proposed previously to explain the development of the three-dimensional flow of an isolated vortex ring striking a wall.

The evolution of disturbances in an annular duct with a non-isentropic radially varying mean flow is studied. Linear and nonlinear analyses are carried out to examine how the mean velocity and density gradients affect the stability and coupling between the disturbances. To isolate the effect of the mean-velocity gradients from that of the mean-density gradients two mean flows are considered, one with a Gaussian density profile and a uniform axial velocity and the other with Gaussian density and Gaussian axial-velocity distributions. For small-amplitude disturbances with the former mean flow profile, the vortical disturbances convect with the mean flow and density fluctuations grow linearly in space as a result of the interaction of the mean-density gradient with the disturbance radial velocity. Eigenmode analysis of the latter profile shows that unstable modes with exponential growth occur owing to the inflection point in the mean-velocity profile. These modes are almost independent of the mean-density profile and are most unstable for low azimuthal wavenumbers. Nonlinear solutions support the linear results and show an algebraic growth of the density for a range of azimuthal wavenumbers and both uniform and non-uniform mean-velocity profiles. The growth of the velocity fluctuations, however, is strongly dependent on the azimuthal wavenumber of the incident disturbance and the mean-velocity profile. The largest growth in the disturbance is observed at radial locations where the largest mean-flow gradients exist. Owing to the growth of the density fluctuations, coupled vorticity–entropy waves are observed downstream of a forced harmonic excitation in a non-isentropic flow. The forcing amplitudes of the incident waves were varied to see how the solutions change with amplitude. As the amplitude is increased, the waves continue to grow and a steepening of the gradients is observed as they propagate downstream until eventually very sharp density and velocity fronts form. These results show that the mean-flow and density profiles play an important role in the evolution of low-azimuthal-wavenumber disturbances which can couple strongly to the duct acoustic modes during combustion instabilities.

Experimental observations of the collapse of initially static columns of sand in axisymmetric and two-dimensional geometries are presented. The experiments were carried out using cylinders and rectangular tanks 30 to 60 cm tall, and cover aspect ratios between 0.5 and 20, where the aspect ratio is defined as the ratio of the initial height to horizontal extent of the column. The final positions of sand grains from different points initially on the outer surface of the columns are mapped. For all axisymmetric columns the point of maximum runout is found to originate from a point at fractional height 0.74 ± 0.03 of the initial vertical height of the column, independent of the aspect ratio. For two-dimensional columns the corresponding point is 0.65 ± 0.07. Collapses of columns of water-saturated sand into water display a different form of flow, which leads to there being no such well-defined point. In this case, grains from all but the innermost, basal areas of the initial column can end up in the outermost region of the final deposit. For collapses in air and aspect ratios greater than 1, the detail of the initial geometry is relatively insignificant in determining the shape of the final deposit. The results of this and previous studies thus have general applicability, even to situations with less initial symmetry. Movies are available with the online version of the paper.

This paper deals with the propagation of nearly resonant gravity waves in two-layer flows over a bottom topography assuming that both fluids are incompressible and inviscid. Evolution equations are derived for weakly nonlinear surface-layer and internal-layer waves in the hydraulic limit of infinite wavelength. Special emphasis is placed on the flow regime where the quadratic nonlinear parameter associated with internal-layer waves is small or vanishes. For example, this is the case for all possible density ratios if the velocities in both layers are equal and if the interface height is close to one-half the total fluid-layer height. The waves then exhibit so-called mixed nonlinearity leading in turn to the formation of positive and negative hydraulic jumps. Considerations based on a model equation for the internal dissipative–dispersive structure of hydraulic jumps indicate that the admissibility of discontinuities in this regime depends strongly on the relative magnitudes of dispersion and dissipation. Surprisingly, these admissible hydraulic jumps may violate the wave-speed-ordering relationship which requires that the upstream wave speed does not exceed the propagation speed of the discontinuity. An important example is provided by the inviscid hydraulic jump, which has been known for some time, although its non-classical nature, in that it transmits rather than absorbs waves, has apparently not been recognized before.

An integral method is presented for determining effects of buoyancy due to heat release on the properties of reacting jets and plumes. This method avoids the Morton entrainment hypothesis entirely, and thus removes the ad hoc ‘entrainment modelling’ required in most other integral approaches. We develop the integral equation for the local centreline velocity uc(x), which allows modelling in terms of the local flow width δ (x). In both the momentum-dominated jet limit and buoyancy-dominated plume limit, dimensional arguments show δ (x) ≈ x, and experimental data show the proportionality factor cδ to remain constant between these limits. The entrainment modelling required in traditional integral methods is thus replaced by the observed constant cδ value in the present method. In non-reacting buoyant jets, this new integral approach provides an exact solution for uc(x) that shows excellent agreement with experimental data, and gives simple expressions for the virtual origins of jets, plumes and buoyant jets. In the exothermically reacting case, the constant cδ value gives an expression for the buoyancy flux B(x) that allows the integral equation for uc(x) to be solved for arbitrary exit conditions. The resulting uc(x) determines the local mass, momentum and buoyancy fluxes throughout the flow, as well as the centreline mixture fraction ζc(x) and thus the flame length L. The latter provides the proper parameters Ω andΛ that determine buoyancy effects on the flame, and provides power-law scalings in the momentum-dominated and buoyancy-dominated limits. Comparisons with buoyant flame data show excellent agreement over a wide range of conditions.

The local implications of the well-known flux conservation equations of Morton et al. (Proc. R. Soc. Lond. A, vol. 234, 1956, p.1) for plumes and jets are considered. Given the vertical velocity distributions of a model plume or jet, the divergence-free radial velocity distributions are calculated. It is shown that in general the velocity of the plume boundary is not described by the local total fluid velocity in this way. A two-fluid model tracking the evolution of both ‘plume’ and ‘ambient’ fluid is proposed which resolves this apparent inconsistency and also provides a way of explicitly describing the mixing process within a model plume. The plume boundary acts as a phase boundary across which ambient fluid is entrained, and the plume boundary moves at the velocity of the plume fluid. The difference between the plume-fluid radial velocity and the total fluid velocity quantifies in a natural way the purely horizontal entrainment flux of ambient fluid into the plume across the phase boundary at the plume edge.

Transverse jets arise in many applications, including propulsion, effluent dispersion, oil field flows, and V/STOL aerodynamics. This study seeks a fundamental, mechanistic understanding of the structure and evolution of vorticity in the transverse jet. We develop a high-resolution three-dimensional vortex simulation of the transverse jet at large Reynolds number and consider jet-to-crossflow velocity ratios r ranging from 5 to 10. A new formulation of vorticity-flux boundary conditions accounts for the interaction of channel wall vorticity with the jet flow immediately around the orifice. We demonstrate that the nascent jet shear layer contains not only azimuthal vorticity generated in the jet pipe, but wall-normal and azimuthal perturbations resulting from the jet–crossflow interaction. This formulation also yields analytical expressions for vortex lines in the near field as a function of r.

Transformation of the cylindrical shear layer emanating from the orifice begins with axial elongation of its lee side to form sections of counter-rotating vorticity aligned with the jet trajectory. Periodic roll-up of the shear layer accompanies this deformation, creating complementary vortex arcs on the lee and windward sides of the jet. Counter-rotating vorticity then drives lee-side roll-ups in the windward direction, along the normal to the jet trajectory. Azimuthal vortex arcs of alternating sign thus approach each other on the windward boundary of the jet. Accordingly, initially planar material rings on the shear layer fold completely and assume an interlocking structure that persists for several diameters above the jet exit. Though the near field of the jet is dominated by deformation and periodic roll-up of the shear layer, the resulting counter-rotating vorticity is a pronounced feature of the mean field; in turn, the mean counter-rotation exerts a substantial influence on the deformation of the shear layer. Following the pronounced bending of the trajectory into the crossflow, we observe a sudden breakdown of near-field vortical structures into a dense distribution of smaller scales. Spatial filtering of this region reveals the persistence of counter-rotating streamwise vorticity initiated in the near field.

Observations of the flow structures formed by rigid fibres of high aspect ratio sedimenting within a viscous fluid at a Reynolds number of approximately 10−4 confirm the existence of an instability as reported in previous theories, experiments, and numerical simulations. Using data generated from particle image velocimetry measurements, we quantify the sedimentation structures over a wide range of parameters, which include the height of fluid, cross-section of the sedimentation cell, fibre dimensions, fluid properties, and volume fractions ranging from dilute to semi-dilute. Alternating structures of streamers and backflow regions which span the height of the sedimentation cell form at short times and transition from large wavelengths to smaller wavelength as the sedimentation proceeds. No simple dependence of the horizontal wavelength on the length scales and concentration was observed in the experiments, suggesting the need for additional analysis. We also report the mean velocities and velocity fluctuations; the strength of the velocity fluctuations strongly correlates with the size of the vertical component of the sedimentation structure. Measurements of the orientation distribution, using an efficient and newly employed technique, agree with previously published results. A movie is available with the online version of the paper.

The motion of a spherical particle near the interface between two immiscible viscous fluids undergoing simple shear flow is considered in the limit of small Reynolds and capillary numbers where the interface exhibits negligible deformation. Taking advantage of the rotational symmetry of the boundaries of the flow with respect to the axis that is normal to the interface and passes through the particle centre, the problem is formulated as a system of one-dimensional integral equations for the first Fourier coefficients of the unknown components of the traction and velocity along the particle and interface contours. The results document the particle translational and angular velocities, and reveal that the particle slips while rolling over the interface under the influence of a simple shear flow, for any viscosity ratio. In the second part of the investigation, the motion of an axisymmetric particle straddling a planar interface is considered. The results confirm a simple exact solution when a particle with top-down symmetry is immersed half-way in each fluid and translates parallel to the interface, reveal a similar simple solution for a particle that is held stationary in simple shear flow, and document the force and torque exerted on a spherical particle for more general arrangements. The onset of a non-integrable singularity of the traction at the contact line prohibits the computation of the translational and angular velocities of a freely suspended particle convected under the action of a shear flow.

This paper presents combined theoretical and experimental studies of the two-dimensional piston-like steady-state motions of a fluid in a moonpool formed by two rectangular hulls (e.g. a dual pontoon or catamaran). Vertical harmonic excitation of the partly submerged structure in calm water is assumed. A high-precision analytically oriented linear-potential-flow method, which captures the singular behaviour of the velocity potential at the corner points of the rectangular structure, is developed. The linear steady-state results are compared with new experimental data and show generally satisfactory agreement. The influence of vortex shedding has been evaluated by using the local discrete-vortex method of Graham (1980). It was shown to be small. Thus, the discrepancy between the theory and experiment may be related to the free-surface nonlinearity.

This paper presents numerical simulations for the interaction of an expansion wave with an incident shock wave of the opposite family, the specific aim being to study the resultant reflection of the now-perturbed shock wave from a solid surface. This problem is considered in the context of an incident flow entering a parallel duct, a situation that commonly arises in a range of flow-turning problems including supersonic intake flows. Once the incident shock conditions are such that Mach reflection must occur, it is shown that stabilization of a simple Mach reflection is only possible for a narrow range of Mach numbers and that this depends sensitively on the relative streamwise positioning of the origins of the shock wave and the expansion wave.

The efficiency of MEMS-based time-periodic micro chaotic mixers is experimentally and theoretically investigated in this study. A time-periodic flow perturbation was realized using digitally controlled solenoid valves to activate a source and sink alternately, acting together as a pair, with different driving frequencies. Working fluids with and without fluorescent dye were used in the micromixing experiments. The spatio-temporal variation of the mixing concentration during the mixing process was characterized at different Strouhal numbers, ranging from 0.03 to 0.74, using fluorescence microscopy. A simple kinematical model for the micromixer was used to demonstrate the presence of chaotic mixing. Specific stretching rate, Lyapunov exponent, and local bifurcation and Poincaré section analyses were used to identify the emergence of chaos. Two different numerical methods were employed to verify that the maximum Lyapunov exponent was positive in the proposed micromixer model. A simplified analytical analysis of the effect of Strouhal number is presented. Kolmogorov–Arnold–Mose (KAM) curves, which are mixing barriers, were also found in Poincaré sections. From a comparative study of the experimental results and theoretical analysis, a finite-time Lyapunov exponent (FTLE) was shown to be a more practical mixing index than the classical Lyapunov exponent because the time spent in mixing is the main concern in practical applications, such as bio-medical diagnosis. In addition, the FTLE takes into account both fluid stretching in terms of the stretching rate and fluid folding in terms of curvature.

Inertial convection in a fluid contained in a rotating cylinder heated uniformly from below is investigated on the basis of the assumption that convection at leading order can be represented by a single or several inertial wave modes which propagate either in the prograde or retrograde direction. Buoyancy forces appear at the next order to drive inertial convection against the effect of viscous damping. Asymptotic expressions for inertial convection for four different combinations of the sidewall boundary condition are derived for a cylinder of arbitrary aspect ratio. New convection patterns in rotating cylinders are revealed by the asymptotic analysis. A fully numerical solution of the same problem is also carried out, demonstrating a quantitative agreement between the asymptotic and numerical analysis.

We consider the scattering by a semi-infinite array of bodies of arbitrary geometry excited by an incident wave in the linear water-wave formulation (which reduces to the simpler case of Helmholtz scattering if the depth dependence can be removed). The theory presented here is extremely general, and we present example calculations for an array of floating elastic plates (a highly non-trivial scatterer). The solution method follows closely from the solution for point scatterers in a medium governed by Helmholtz's equation. We have made several extensions to this theory, considering water-wave scattering, allowing for bodies of arbitrary scattering geometry and showing how to include the effects of bound waves (called Rayleigh–Bloch waves in the water-wave context) in the formulation. We present results for scattering by arrays of cylinders that show the convergence of our methods and also some results for the case of scattering by floating elastic plates and fixed docks.

The conservation of mass, momentum and energy are not sufficient to close a system of jump relations for shocks propagating in a heterogeneous mixture of compressible fluids. We propose here a closed set of relations corresponding to a two-stage structure of shock fronts. At the first stage, microkinetic energy due to the relative motion of mixture components is produced at the shock front. At the second stage, this microkinetic energy disappears inducing strong variations in the thermodynamical states that reach mechanical equilibrium. The microkinetic energy produced at the shock front is estimated by using an idea developed earlier for turbulent shocks in compressible fluids. The relaxation zone between the shocked state and the equilibrium state is integrated over a thermodynamic path a justification of which is provided. Comparisons with experiments on shock propagation in a mixture of condensed materials confirm the proposed theory.