A new method valid for highly dispersive and highly nonlinear water waves is presented. It combines a time-stepping of the exact surface boundary conditions with an approximate series expansion solution to the Laplace equation in the interior domain. The starting point is an exact solution to the Laplace equation given in terms of infinite series expansions from an arbitrary z-level. We replace the infinite series operators by finite series (Boussinesq-type) approximations involving up to fifth-derivative operators. The finite series are manipulated to incorporate Padé approximants providing the highest possible accuracy for a given number of terms. As a result, linear and nonlinear wave characteristics become very accurate up to wavenumbers as high as kh = 40, while the vertical variation of the velocity field becomes applicable for kh up to 12. These results represent a major improvement over existing Boussinesq-type formulations in the literature. A numerical model is developed in a single horizontal dimension and it is used to study phenomena such as solitary waves and their impact on vertical walls, modulational instability in deep water involving recurrence or frequency downshift, and shoaling of regular waves up to breaking in shallow water.

The formation of compound vortices from a ‘shielded’ vortex monopole embedded in a two-dimensional background straining flow is studied numerically. The computations are performed in plane polar coordinates using an infinite radial domain Navier–Stokes solver. The straining flow excites an azimuthal instability, leading to vortex tripole formation in the case of small strain rates. For larger strain rates, tripole formation proceeds briefly before the structure is destroyed by the dominant background straining flow.

Detailed three-dimensional measurements of the first vortex pairing of a large plane mixing layer reveal excitation of several three-dimensional instability modes. Time evolution in three-dimensional space (x, y, z, t) shows how the two-dimensional rollers become three-dimensional as they approach each other and that the linear growth of at least two instability waves leads to a spanwise periodic pairing. The results are based on phase-locked measurements made in three-dimensional spatial grids, with a mesh spacing of 8.5% of the fundamental instability wavelength. Spanwise-uniform, periodic acoustic excitation stabilizes the most probable two-dimensional natural features – roll-up and first pairing. The second subharmonic is added to study the effect of alternate streamwise pairing locations on the three-dimensional characteristics of vortex pairing. Velocities are measured using hot-wire anemometry, and the coherent structures are reconstructed from the ensemble-averaged vorticity field.

Vortex pairing is shown to initiate through local ‘bridging’ at the maxima of periodic spanwise undulations. The undulations result from linear amplification of various instability modes on pairing rollers having different strengths. Bridging results from the change of the relative phase between the spanwise undulations of the pairing rollers from in-phase (due to the initial translative mode) to out-of-phase (due to the amplification of bulging-like and non-axisymmetric modes). It is found that when pairing occurs sufficiently far upstream, only axisymmetric waves are amplified and the evolution results in axisymmetric merging. In contrast, when pairing occurs sufficiently far downstream, both axisymmetric and non-axisymmetric waves are amplified and the evolution results in non-axisymmetric merging.

The results indicate that vortex pairing is accompanied by the counter-rotating pairs of secondary structures (‘streamwise vortices’ or ‘ribs’) located in the mixing-layer braids and residing in the valleys of the spanwise-roller waves. Time evolution of these secondary structures shows that they move in the transverse direction, following the rollers.

This paper investigates the stability and large-displacement post-buckling behaviour of liquid-lined elastic rings. The fluid flow and the wall deformation are described by the free-surface Navier–Stokes equations and by geometrically nonlinear shell theory, respectively. The fluid–structure interaction problem is solved numerically by a finite element method. The compressive load on the ring is a combination of the external pressure and the effect of surface tension. Once this combined load exceeds a critical value, the subsequent non-axisymmetric collapse of the ring is controlled by the dynamics of the surface-tension-driven redistribution of fluid in the liquid lining. It is shown that, for sufficiently large surface tension, the ring can undergo a catastrophic collapse which leads to a complete occlusion of its lumen. A novel lubrication theory model, which ensures exact volume conservation for flows on strongly curved substrates, is developed and found to be capable of accurately describing the motion of the air–liquid interface and the fluid–structure interaction in the large-displacement regime, even in cases where the film thickness is large.

The findings have important implications for the occurrence of airway closure in lung diseases (such as oedema) which cause an increase in the thickness of the airways' liquid lining. It is shown that under such conditions, airways can become occluded even if the volume of fluid in their liquid lining is much smaller than that required to occlude them in their axisymmetric state.

The influence of viscous heating on the stability of Taylor–Couette flow is investigated theoretically. Based on a linear stability analysis it is shown that viscous heating leads to significant destabilization of the Taylor–Couette flow. Specifically, it is shown that in the presence of viscous dissipation the most dangerous disturbances are axisymmetric and that the temporal characteristic of the secondary flow is very sensitive to the thermal boundary conditions. If the temperature difference between the two cylinders is small, the secondary flow is stationary as in the case of isothermal Taylor–Couette flow. However, when the temperature difference between the two cylinders is large, time-dependent secondary states are predicted. These linear stability predictions are in agreement with the experimental observations of White & Muller (2000) in terms of onset conditions as well as the spatiotemporal characteristics of the secondary flow. Nonlinear stability analysis has revealed that over a broad range of operating conditions, the bifurcation to the time-dependent secondary state is subcritical, while stationary states result as a consequence of supercritical bifurcation. Moreover, the supercritically bifurcated stationary state undergoes a secondary bifurcation to a time-dependent flow. Overall, the structure of the time-dependent state predicted by the analysis compares very well with the experimental observations of White & Muller (2000) that correspond to slowly moving vortices parallel to the cylinder axis. The significant destabilization observed in the presence of viscous heating arises as the result of the coupling of the perturbation velocity and the base-state temperature gradient that gives rise to fluctuations in the radial temperature distribution. Due to the thermal sensitivity of the fluid these fluctuations greatly modify the fluid viscosity and reduce the dissipation of disturbances provided by the viscous stress terms in the momentum equation.

The dramatic effects of viscous dissipation on the stability of Newtonian Taylor–Couette (TC) flows are studied experimentally using flow visualization techniques. Viscous heating, parameterized by the Nahme–Griffith number Na, drives a transition to a new, oscillatory mode of instability when coupled with the effects of centrifugal destabilization. This instability, consisting of travelling axisymmetric vortices, only occurs when viscous heating and centrifugal destabilization are both present. Step tests in cylinder velocity show that the time following initiation of shearing required for onset of instability scales well with the time for the fluid to reach a steady temperature profile under the action of viscous heating. The onset time can be dramatically reduced at fixed Na by increasing the centrifugal destabilization through the addition of co-rotation of the outer cylinder. The onset time can also be reduced while holding the centrifugal destabilization constant by increasing the amount of viscous heating (i.e. holding Reynolds number Re constant while increasing Na). The effects of viscous heating on the critical conditions of Newtonian TC flows are also quantified using ramp tests in cylinder velocity. These tests reveal the large extent to which viscous heating is destabilizing; at Na ≈ 2, a transition occurs at a critical Re that is less than 5% of the isothermal value.

Double-diffusive finger convection is studied experimentally in a transparent Hele-Shaw cell for a two-solute system. A less dense sucrose solution is layered on top of a more dense salt solution using a laminar flow technique, and convective motion is followed photographically from the static state. We systematically increase solute concentrations from dilute to the solubility limit of the salt solution while maintaining a fixed buoyancy ratio of approximately 1.08. Across the 14 experiments conducted, the convective motion shows considerable variation in both structure and time scale. We find that new finger pairs form continuously within a finger generation zone where complexity increases with Rayleigh number, reaches a peak, and then decreases for highly concentrated solutions. The vertical fnger length scale grows linearly in time across the full concentration range. The vertical finger velocity also increases linearly with Rayleigh number, but as the concentrations increase, deviation from linearity and asymmetrical convection occur. The horizontal length scale grows as a power law in time with the exponent constant over most of the range; again, deviations are observed for highly concentrated solutions. The observed deviations at high concentrations are attributed to the increasing nonlinearity in the governing equations as the solutions approach their solubility limits. There, the fluid properties become functions of solute concentration and vary significantly within the experimental fields suppressing structural complexity, imparting asymmetry to the convective motion, and influencing emergent vertical and horizontal length scales and their growth.

Fully developed flows are often used to describe fluid motion in complex geometrical systems, including the human macrocirculation. In fact they may frequently be quite inappropriate even for geometrically simple pipes, owing to the unfeasibly large viscous entry lengths required. Inviscid adjustment to changes in geometry, however, occurs on the lengthscale of the pipe diameter. Inviscid idealizations are therefore more likely to apply in relatively short arterial sections. We aim to quantify the distances involved by calculating the rates of spatial decay for a general disturbance superimposed on an idealized base flow. Both irrotational and rotational base flows are examined, although in the latter case there can exist non-decaying inertial waves, so that an arbitrary inflow need not attain an inviscid state independent of the downstream coordinate. In the rotational case, we therefore restrict attention to those flows which settle down to perturbations of such a state, whereas the potential flows can be regarded as developing from an arbitrary input.

We focus on the last surviving mode of decay in simple uniform pipe geometries, in particular a straight pipe, part of a torus, and a helical pipe. In this way we are able to assess the effects of curvature and torsion on the inviscid entry lengths.

Principally, it is shown that the rate of decay is fastest in a straight pipe and slowest in a toroidal pipe, with that in a helical pipe somewhere in between. Core vorticity tends to reduce the decay rate. If an idealized flow occurs in a geometrically simple arterial portion, our results determine its domain of validity.

The density change across premixed flames propagating in turbulent flows modifies the turbulence. The nature of that modification depends on the regime of turbulent combustion, the burner design, the orientation of the turbulent flame and the position within the flame. The present study addresses statistically stationary turbulent combustion in the flame-sheet regime, in which the laminar-flame thickness is less than the Kolmogorov scale, for flames stabilized on a vertically oriented cylindrical burner having fully developed upward turbulent pipe flow upstream from the exit. Under these conditions, rapidly moving wrinkled laminar flamelets form the axisymmetric turbulent flame brush that is attached to the burner exit. Predictions have been made of changes in turbulence properties across laminar flamelets in such situations, but very few measurements have been performed to test the predictions. The present work measures individual velocity changes and changes in turbulence across flamelets at different positions in the turbulent flame brush for three different equivalence ratios, for comparison with theory.

The measurements employ a three-element electrostatic probe (EP) and a two-component laser-Doppler velocimeter (LDV). The LDV measures axial and radial components of the local gas velocity, while the EP, whose three sensors are located in a vertical plane that passes through the burner axis, containing the plane of the LDV velocity components, measures arrival times of flamelets at three points in that plane. From the arrival times, the projection of flamelet orientation and velocity on the plane are obtained. All of the EP and LDV sensors are located within a fixed volume element of about 1 mm diameter to provide local, time-resolved information. The technique has the EP advantages of rapid response and good sensitivity and the EP disadvantages of intrusiveness and complexity of interpretation, but it is well suited to the type of data sought here.

Theory predicts that the component of velocity tangent to the surface of a locally planar flamelet remains constant in passing through the flamelet. The data are consistent with this prediction, within the accuracy of the measurement. The data also indicate that the component of velocity normal to the flamelet, measured with respect to the flamelet, tends to increase in passing through the flamelet, as expected. The flamelets thereby can generate anisotropy in initially isotropic turbulence. They also produce differences in turbulent spectra conditioned on unburnt or burnt gas. Local modifications of turbulence by flamelets thus are demonstrated experimentally. The modifications are quantitatively different at different locations in the turbulent flame brush but qualitatively similar in that the turbulence is enhanced more strongly in the radial direction than in the axial direction at all positions in these flames.

Area–volume properties of fluid interfaces are investigated to quantify the scale-local and cumulative structure. An area–volume density g3(λ) and ratio Ω3(λ) are introduced to examine the interfacial behaviour as a function of scale λ or across a range of scales, respectively. These measures are demonstrated on mixed-fluid interfaces from whole-field ∼10003 three-dimensional space–time concentration measurements in turbulent jets above the mixing transition, at Re ∼ 20000 and Sc ∼ 2000, recorded by laser-induced-fluorescence and digital-imaging techniques, with Taylor's hypothesis applied. The cumulative structure is scale dependent in Ω3(λ), with a dimension D3(λ) that increases with increasing scale. In contrast, the scale-local structure exhibits self-similarity in g3(λ) with an exponent αg ≈1.3 for these interfaces. The scale dependence in the cumulative structure arises from the large scales, while the self-similarity corresponds to the small-scale area–volume contributions. The small scales exhibit the largest area–volume density and provide the dominant contributions to the total area–volume ratio, which corresponds to ∼10 times the area of a purely large-scale interface for the present flow conditions. The self-similarity in the scale-local structure at small scales provides the key ingredient to extrapolate the area–volume behaviour to higher Reynolds numbers.

We decipher the rich and complex two-dimensional wave transition and evolution dynamics on a falling film both theoretically and numerically. Small-amplitude white noise at the inlet is filtered by the classical linear instability into a narrow Gaussian band of primary frequency harmonics centred about ωm. Weakly nonlinear zero-mode excitation and a secondary modulation instability then introduce a distinct characteristic modulation frequency Δ [Lt ] ωm. The primary wave field evolves into trains of solitary pulses with an average wave period of 2π/ωm. Abnormally large ‘excited’ pulses appear within this train at a relative frequency of Δ/ωm due to the modulation. The excited pulses travel faster than the equilibrium ones and eliminate them via coalescence to coarsen the pulse field downstream. The linear coarsening of wave period downstream is a universal (0.015/〈u〉) s cm−1 and the final wave frequency is the modulation frequency Δ for 0.1 < δ < 0.4 where 〈u〉 is the flat-film average Nusselt velocity, δ = (3R2/W)1/3/15 is a normalized Reynolds number, R is the flat-film Reynolds number and W the Weber number.

The three-dimensional governing equations of the flow field that is developed when an elasto-plastic exible porous medium, capable of undergoing extremely large deformations, is struck head-on by a shock wave, are developed using a multi-phase approach. The one-dimensional version of these equations is solved numerically using an arbitrary Lagrangian–Eulerian (ALE) based numerical code. The numerical predictions are compared qualitatively to experimental results from various sources and good agreement is obtained. This study complements our earlier study in which we developed and solved, using a total variation diminishing (TVD) based numerical code, the governing equations of the flow field that is developed when an elastic rigid porous medium, capable of undergoing only very small deformations, is struck head-on by a shock wave.

We report on experiments with liquid sheets formed through the impact of a slender jet on a small disc at high Reynolds number. When the interaction with the surrounding air is negligible, the sheet spreads out radially and remains smooth. The study extends over the whole life of the sheet, considering the dynamics of its formation and destruction and paying special attention to the stationary regime, in which the transition from sheet to drops occurs.

A round liquid jet with density ρ, surface tension σ and diameter D0 impacting a solid circular surface at normal incidence with velocity U0 takes the form of a radially expanding sheet whose thickness decreases with distance from the impact point. When the sheet develops in a still environment with density ρa = αρ, it destabilizes, provided the impacting Weber number We = ρU20D0/σ is larger than about 40α−1/2, as a result of a shear instability with the surrounding medium, in a sinuous, flag-like motion. We show how the instability properties set both the radial extent of the liquid sheet and the drop formation process at its rim. The shear instability gives the liquid a flag-like motion, ultimately triggering a Rayleigh–Taylor instability at the rim of the sheet which disintegrates, at the radial location R, into disjointed droplets of size d such that

R/D0 ∼ α−2/3We−1/3 and d/D0 ∼ α−2/3We−1.

The features of the sheet instability, its radius and the droplet sizes are determined experimentally for a broad range of control parameters, using different liquids and ambient-medium densities.

Axisymmetric stability of viscous resistive magnetized Couette flow is re-examined, with the emphasis on flows that would be hydrodynamically stable according to Rayleigh's criterion: opposing gradients of angular velocity and specific angular momentum. In this regime, magnetorotational instabilities (MRI) may occur. Previous work has focused on the Rayleigh-unstable regime. To prepare for an experimental study of MRI, which is of intense astrophysical interest, we solve for global linear modes in a wide gap with realistic dissipation coefficients. Exchange of stability appears to occur through marginal modes. Velocity eigenfunctions of marginal modes are nearly singular at conducting boundaries, but magnetic eigenfunctions are smooth and obey a fourth-order differential equation in the inviscid limit. The viscous marginal system is of tenth order; an eighth-order approximation previously used for Rayleigh-unstable modes does not permit MRI. Peak growth rates are insensitive to boundary conditions. They are predicted with surprising accuracy by WKB methods even for the largest-scale mode. We conclude that MRI is achievable under plausible experimental conditions using easy-to-handle liquid metals such as gallium.

The flow in a completely filled rotating cylinder driven by the counter-rotation of the top endwall is investigated both numerically and experimentally. The basic state of this system is steady and axisymmetric, but has a rich structure in the radial and axial directions. The most striking feature, when the counter-rotation is sufficiently large, is the separation of the Ekman layer on the top endwall, producing a free shear layer that separates regions of flow with opposite senses of azimuthal velocity. This shear layer is unstable to azimuthal disturbances and a supercritical symmetry-breaking Hopf bifurcation to a rotating wave state results. For height-to-radius ratio of 0.5 and Reynolds number (based on cylinder radius and base rotation) of 1000, rotating waves with azimuthal wavenumbers 4 and 5 co-exist and are stable over an extensive range of the ratio of top to base rotation. Mixed modes and period doublings are also found, and a bifurcation diagram is determined. The agreement between the Navier–Stokes computations and the experimental measurements is excellent. The simulations not only capture the qualitative features of the multiple states observed in the laboratory, but also quantitatively replicate the parameter values over which they are stable, and produce accurate precession frequencies of the various rotating waves.