In order to model the evolution of a solitary wave near an obstacle or over an uneven bottom, the long-wave equations including curvature effects are introduced to describe the deformation and fission of a barotropic solitary wave passing over a shelf or an obstacle. The numerical results obtained from these equations are shown to be in good agreement with an analytical model derived by Germain (1984) in the framework of a generalized shallow-water theory, and with experimental results collected in a large channel equipped with a wave generator. Given the initial conditions, i.e. amplitude of the incident solitary wave, water depth in the deep region, and height of the shelf or the barrier, it is possible to predict the amplitude and number of the transmitted solitary waves as well as the amplitude of the reflected wave, and to describe the shape of the free surface at any time.

This article provides theoretical conditions for the use and meaning of a stability analysis around a mean flow. As such, it may be considered as an extension of the works by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to non-parallel flows and by Turton et al. (Phys. Rev. E, vol. 91 (4), 2015, 043009) to broadband flows. Considering a Reynolds decomposition of the flow field, the spectral (or temporal Fourier) mode of the fluctuation field is found to be equal to the action on a turbulent forcing term by the resolvent operator arising from linearisation about the mean flow. The main result of the article states that if, at a particular frequency, the dominant singular value of the resolvent is much larger than all others and if the turbulent forcing at this frequency does not display any preferential direction toward one of the suboptimal forcings, then the spectral mode is directly proportional to the dominant optimal response mode of the resolvent at this frequency. Such conditions are generally met in the case of weakly non-parallel open flows exhibiting a convectively unstable mean flow. The spatial structure of the singular mode may in these cases be approximated by a local spatial stability analysis based on parabolised stability equations (PSE). We have also shown that the frequency spectrum of the flow field at any arbitrary location of the domain may be predicted from the frequency evolution of the dominant optimal response mode and the knowledge of the frequency spectrum at one or more points. Results are illustrated in the case of a high Reynolds number turbulent backward facing step flow.

The sources of error associated with the use of particle-tracking techniques in the measurement of velocity and vorticity fields in moderately three-dimensional turbulent flows are analysed. The two dominant sources of error are the visualisation error, resulting from the limited resolution of the optical data acquisition system, and the sampling error, due to limited particle concentration. Their relative importance is discussed.

The performance of the interpolation methods used to translate the measurements from the positions of the particles to an arbitrary point is discussed, and a non-parametric algorithm is given to estimate the errors that arise, using only the available data. The smoothing of the results to produce flow maps of a given statistical significance is also discussed. Finally, the method is validated using simultaneous laser-Doppler velocity measurements.

The system is applied to measurements of the near wake of a circular cylinder. Velocity and vorticity maps are provided which throw light on the process by which the large eddies form and relax to their final equilibrium configurations.

A computational fluid dynamics cavitation model based on the Eulerian–Lagrangian approach and suitable for hole-type diesel injector nozzles is presented and discussed. The model accounts for a number of primary physical processes pertinent to cavitation bubbles, which are integrated into the stochastic framework of the model. Its predictive capability has been assessed through comparison of the calculated onset and development of cavitation inside diesel nozzle holes against experimental data obtained in real-size and enlarged models of single- and multi-hole nozzles. For the real-size nozzle geometry, high-speed cavitation images obtained under realistic injection pressures are compared against model predictions, whereas for the large-scale nozzle, validation data include images from a charge-coupled device (CCD) camera, computed tomography (CT) measurements of the liquid volume fraction and laser Doppler velocimetry (LDV) measurements of the liquid mean and root mean square (r.m.s.) velocities at different cavitation numbers (CN) and two needle lifts, corresponding to different cavitation regimes inside the injection hole. Overall, and on the basis of this validation exercise, it can be argued that cavitation modelling has reached a stage of maturity, where it can usefully identify many of the cavitation structures present in internal nozzle flows and their dependence on nozzle design and flow conditions.

An inviscid model of two-dimensional vortex shedding behind a square-based section is developed. The model uses a discrete-vortex approximation for the free shear layers. The motion of the shear layers is computed from the velocities of the discrete vortices, which in turn are derived through a Schwartz-Christoffel transformation of the section. The flow round the body is impulsively started from rest and initially develops symmetrically. The introduction of a small asymmetric disturbance results in asymmetric interaction of the shear layers amplifying into steady vortex-shedding motion.

The model is shown to predict the form of vortex shedding, the Strouhal number and some other flow quantities to a good degree of agreement with experimental results.

A technique has been developed to measure the instantaneous shear stress at the boundary over which a liquid is flowing. It is being used to study turbulence in the immediate vicinity of a pipe wall. A reaction is conducted on an electrode mounted flush with a solid wall at high enough voltages to reduce the concentration of the reacting species to zero at the surface. Under these conditions, the rate of reaction is controlled by the rate of mass transfer. The electrode is analogous to a constant-temperature hot-wire anemometer in that the surface concentration is kept constant and the current flowing in the circuit is related to the surface shear stress. For fully developed turbulence the limiting velocity intensity, based on the local average velocity, is 0·32. Some of the velocity fluctuations are as large as the local average velocity and their distribution is nearly symmetric about the average. The ratio of the longitudinal to circumferential scale is about 30 : 1, because the circumferential scale is very small. There is some indication that close to a wall the velocity fluctuations in the circumferential direction are much smaller than the longitudinal fluctuations.

The stability of a horizontal layer of Maxwellian fluid heated from below is considered. Critical Rayleigh numbers, wave-numbers, and frequencies for overstability are determined for both free and rigid boundaries. Elasticity is found to destabilize the fluid, and the presence of rigid boundaries is found to be slightly stabilizing.

The linear stability of two superimposed layers of fluid, heated from below and separated by a thin conducting plate, is investigated. It is shown that when the ratio of the depths of the layers is close to 1:2, two distinct modes of convection can occur with preferred horizontal wavenumber in the ratio 1:2. The nonlinear evolution of a disturbance consisting of both modes is considered, and it is shown that travelling waves are the preferred mode of nonlinear convection for a wide range of parameter values. Other possible types of behaviour, including modulated waves and an attracting homoclinic trajectory, are also described in detail.

Extensive schlieren studies and yawmeter traverses of the wake behind slender cone-cylinders at large angles of incidence have shown that the flow pattern is generally steady. Under certain flow conditions, however, the wake exhibits an instability which is not understood. For cross-flow Reynolds numbers in the subcritical region the wake can be described in terms of a cross-flow Strouhal number which has a constant value of 0·2 for cross-flow Mach number components (Mc) up to 0·7 and then increases steadily to a value of 0·6 at Mc = 1·6. The strength of the wake vortices varies substantially with Mc, increasing to a maximum at Mc ≈ 0·7 and then decreasing rapidly for higher values of Mc. Schlieren photographs of the wake have been analysed by means of the impulse flow analogy and also by considering the vortices to be part of a yawed infinite vortex street. The impulse flow analogy is shown to be of use in determining the cross-flow Strouhal number but estimates of vortex strength are too high. The Kármán vortex street theory combined with the sweepback principle leads to reliable estimates of vortex strength up to Mc = 1·0.

Information is given on the spacing, path and strength of the vortices shed from the body for flow conditions varying from incompressible speeds up to Mc = 1·0. Finally this information is used to determine the vortex drag of a two-dimensional circular cylinder below Mc = 1·0.

The formation of turbulent patterns in plane Couette flow is investigated near the onset of transition, using numerical simulation in a very large domain of size 800 h × 2 h × 356 h. Based on a maximum observation time of 20 000 inertial units, the threshold for the appearance of sustained turbulent motion is Rec = 324 ± 1. For Rec < Re ≤ 380, turbulent-banded patterns form, irrespective of whether the initial perturbation is a noise or localized disturbance. Measurements of the turbulent fraction versus Re show evidence for a discontinuous phase transition scenario where turbulent spots play the role of the nuclei. Using a smaller computational box, the angle selection of the turbulent bands in the early stages of their development is shown to be related to the amplitude of the initial perturbation.

At any free surface at which the tangential stress τns vanishes there must be a surface vorticity ω = −2κq, where κ is the curvature and q the tangential velocity. In a surface wave on water, this condition produces a (Stokes) boundary layer with thickness of order δ = (2ν/σ)½, where ν is the kinemetic viscosity and σ the radian frequency of the wave. To first order in the wave steepness parameter ak, the vorticity remains within the boundary layer, but at second order some escapes through the Stokes layer. The mean vorticity $\overline{\omega}$ at the outer edge of the Stokes layer is of order 2(ak)2σ, twice the mean vorticity generated at the free surface.

These results are applied to steep capillary waves, particularly the parasitic capillaries often seen on the forward face of short gravity waves. Because of the high value of σ for the capillaries, the vorticity they generate is much larger than that generated by the gravity wave itself. Hence the capillaries contribute significantly to the vortex (roller) often found at the crest of short gravity waves, when capillaries are present. It is argued that the crest roller and the capillaries form a cooperative system, a ‘capillary roller’ in which each supports the other, with the aid of surface tension and viscosity. Energy is supplied by the gravity wave.

A capillary roller is one instance of a more general phenomenon: a ‘capillary bore’, which is a noticeable feature of many disturbed water surfaces.

The properties of a turbulent flow are often described in terms of velocity correlations in space, in time, and in space-time. In this paper the interpretation of velocity correlation measurements which are made in a region of highintensity turbulence is considered in some detail. Under these conditions it is shown that some account must be taken of the effects of both mean and fluctuating shear stresses which are continuously modifying the turbulent structure. For an almost frozen pattern, for example, in the turbulence behind a grid, the turbulent convection velocity is amost equal to the mean flow velocity, while the space correlation and auto-correlation of the velocity fluctuations are simply related through this velocity. In contrast to this, when the intensity is high, the convection velocity may differ considerably from the mean velocity, while it is shown that different turbulent spectral components appear to travel at different speeds. This means that the turbulent spectrum and the turbulent space scales are no longer simply related. For example, the high-frequency spectral components may be ascribed to both the high-velocity eddies and the small wave-number components acting together.

Experimental results are presented which indicate the conditions under which the assumption of a frozen pattern leads to uncertainties in the subsequent interpretation of the measurements. The measurements also show that the observed difference between the mean and the convection velocity may be qualitatively explained in terms of the skewness of the velocity signals.

Direct numerical simulations of the flow past a fixed oblate spheroidal bubble are carried out to determine the range of parameters within which the flow may be unstable, and to gain some insight into the instability mechanism. The bubble aspect ratio χ (i.e. the ratio of the major axis length over the minor axis length) is varied from 2.0 to 2.5 while the Reynolds number (based on the upstream velocity and equivalent bubble diameter) is varied in the range 102 ≤ Re ≤ 3 × 103. As vorticity generation at the bubble surface is at the root of the instability, theoretical estimates for the maximum of the surface vorticity and the surface vorticity flux are first derived. It is shown that, for large aspect ratios and high Reynolds numbers, the former evolves as χ8/3 while the latter is proportional to χ7/2Re−1/2. Then it is found numerically that the flow first becomes unstable for χ = χc ≈ 2.21. As the surface vorticity becomes independent of Re for large enough Reynolds number, the flow is unstable only within a finite range of Re, this range being an increasing function of χ − χc. An empirical criterion based on the maximum of the vorticity generated at the body surface is built to determine whether the flow is stable or not. It is shown that this criterion also predicts the correct threshold for the wake instability past a rigid sphere, suggesting that the nature of the body surface does not really matter in the instability mechanism. Also the first two bifurcations of the flow are similar in nature to those found in flows past rigid axisymmetric bluff bodies, such as a sphere or a disk. Wake dynamics become more complex at higher Reynolds number, until the Re−1/2-dependency of the surface vorticity flux makes the flow recover its steadiness and eventually its axisymmetry. A qualitative analysis of the azimuthal vorticity field in the base flow at the rear of the bubble is finally carried out to make some progress in the understanding of the primary instability. It is suggested that the instability originates in a thin region of the flow where the vorticity gradients have to turn almost at right angle to satisfy two different constraints, one at the bubble surface, the other within the standing eddy.

The flow in the wake of a two-dimensional blunt-trailing-edge body was investigated in the Reynolds number range, Reynolds number being referred to base height, 1·3 × 104 to 4·1 × 104. The effects of splitter plates and base bleed on the vortex street were examined. Measurements were made of the longitudinal spacing between vortices and the velocity of the vortices, and compared with values predicted by von Kármán's potential vortex street model. The lateral spacing was estimated by using both the von Kármán and Kronauer stability criteria. A new universal wake Strouhal number is devised, using the value of lateral spacing predicted by the Kronauer stability condition as the length dimension. A correlation of bluff-body data was found when pressure drag coefficient times Strouhal number was plotted against base pressure.

The distribution of vorticity in the wake of a hovering bird or insect is considered. The wake is modelled by a chain of coaxial small-cored circular vortex rings stacked one upon another; each member of the chain is generated by a single wing-stroke. Circulation is determined by the animal's weight and the time for which a single ring must provide lift; ring size is calculated from the circulation distribution on the animal's wing. The theory is equally applicable to birds and insects, although the mechanism of ring formation differs. This approach avoids the use of lift and drag coefficients and is not bound by the constraints of steady-state aerodynamics; it gives a wake configuration in agreement with experimental observations. The classical momentum jet approach has steady momentum flux in the wake, and is difficult to relate to the wing motions of a hovering bird or insect; the vortex wake can be related to the momentum jet, but adjacent vortex elements are disjoint and momentum flux is periodic.

The evolution of the wake starting from rest is considered by releasing vortex rings at appropriate time intervals and allowing them to interact in their own velocity fields. The resulting configuration depends on the feathering parameter f (which depends on the animal's morphology); f increases with body size. At the lower end of the wake rings coalesce to form a single large vortex, which breaks away from the rest of the wake at intervals. Wake contraction depends on f; the minimum areal contraction of one-half (as in momentum-jet theory) occurs only in the limit f → 0, but values calculated for smaller insects of just over one-half suggest that the momentum jet may be a good approximation to the wake when f is small.

Induced power in hovering is calculated as the limit of the mean rate of increase of wake kinetic energy as time progresses. It can be related to the classical momentum-jet induced power by a simple conversion factor. For an insect or hummingbird the usual momentum-jet estimate may be between 10 and 15% too low, but for a bird it may be as much as 50% too low. This suggests that few, if any, birds are able to sustain aerobic hovering, and that as small a value of f as possible would be necessary if the bird were to hover.

Tip losses (energy cost of the vortex-ring wake compared with the equivalent momentum jet) are negligible for insects, but can be in the range 15–20% for birds.

The effects of initial conditions on grid turbulence are investigated for low to moderate Reynolds numbers. Four grid geometries are used to yield variations in initial conditions and a secondary contraction is introduced to improve the isotropy of the turbulence. The hot-wire measurements, believed to be the most detailed to date for this flow, indicate that initial conditions have a persistent impact on the large-scale organization of the flow over the length of the tunnel. The power-law coefficients, determined via an improved method, also depend on the initial conditions. For example, the power-law exponent m is affected by the various levels of large-scale organization and anisotropy generated by the different grids and the shape of the energy spectrum at low wavenumbers. However, the results show that these effects are primarily related to deviations between the turbulence produced in the wind tunnel and true decaying homogenous isotropic turbulence (HIT). Indeed, when isotropy is improved and the intensity of the large-scale periodicity, which is primarily associated with round-rod grids, is decreased, the importance of initial conditions on both the character of the turbulence and m is diminished. However, even in the case where the turbulence is nearly perfectly isotropic, m is not equal to −1, nor does it show an asymptotic trend in x towards this value, as suggested by recent analysis. Furthermore, the evolution of the second- and third-order velocity structure functions satisfies equilibrium similarity only approximately.

The three-dimensional flow past two identical spherical bubbles moving side by side in a viscous fluid is studied numerically by solving the full Navier–Stokes equations. The bubble surface is assumed to be clean so that the outer flow obeys a zero-shear-stress condition. The present study describes the interaction between the two bubbles over a wide range of Reynolds number ($0.02\,{\le}\,Re\,{\le}\,500$, $Re$ being based on the bubble diameter and rise velocity), and separation $S$ ($2.25\,{\le}\,S\,{\le}\,20$, S being the distance between the bubble centres normalized by the bubble radius). The flow structure, the vorticity field, the sign of the interaction force and the magnitude of the drag and lift forces are analysed; in particular the latter are compared with analytical expressions available in the potential flow limit and in the limit of low-but-finite Reynolds number. This study sheds light on the role of the vorticity generated at the bubble surface in the interaction process. When vorticity remains confined in a boundary layer whose thickness is small compared to the distance between the two bubbles, the interaction is dominated by the irrotational mechanism that results in an attractive transverse force. In contrast, when viscous effects are sufficiently large, the vorticity field about each bubble interacts with that existing about the other bubble, resulting in a repulsive transverse force. Computational results combined with available high-Reynolds-number theory provide empirical expressions for the drag and lift forces in the moderate-to-large Reynolds number regime. They show that the transverse force changes sign for a critical Reynolds number whose value depends on the separation. Using these computational results it is shown that, depending on their initial separation, freely moving bubbles may either reach a stable equilibrium separation or move apart from each other up to infinity.

A model problem is analysed to study the microscopic flow near the surface of porous media. In the idealized system, we consider two-dimensional media consisting of infinite and semi-infinite periodic lattices of cylindrical inclusions. In Part 1, results for axial flow were presented. In this work results for transverse flow are presented and discussed in the context of macroscopic approaches such as slip coefficients and Brinkman's equation.

We present a single-layer feed-forward artificial neural network architecture trained through a supervised learning approach for the deconvolution of flow variables from their coarse-grained computations such as those encountered in large eddy simulations. We stress that the deconvolution procedure proposed in this investigation is blind, i.e. the deconvolved field is computed without any pre-existing information about the filtering procedure or kernel. This may be conceptually contrasted to the celebrated approximate deconvolution approaches where a filter shape is predefined for an iterative deconvolution process. We demonstrate that the proposed blind deconvolution network performs exceptionally well in the a priori testing of two-dimensional Kraichnan, three-dimensional Kolmogorov and compressible stratified turbulence test cases, and shows promise in forming the backbone of a physics-augmented data-driven closure for the Navier–Stokes equations.

The turbulence structure in zero-pressure-gradient boundary layers above smooth, rough and wavy surfaces was investigated. The mean flow, turbulence intensity and spectral data for both smooth and rough surfaces show support for the attached eddy hypothesis of Townsend (1976), the model for wall turbulence proposed by Perry & Chong (1982) and the extended version developed by Perry, Henbest & Chong (1986). Anomalies in hot-wire behaviour when measuring in the turbulent wall region of the flow were discovered and some of these have been resolved.