Mechanisms for the degeneration of large-scale interfacial gravity waves are identified for lakes in which the effects of the Earth's rotation can be neglected. By assuming a simple two-layer model and comparing the timescales over which each of these degeneration mechanisms act, regimes are defined in which particular processes are expected to dominate. The boundaries of these regimes are expressed in terms of two lengthscale ratios: the ratio of the amplitude of the initial wave to the depth of the thermocline, and the ratio of the depth of the thermocline to the overall depth of the lake. Comparison of the predictions of this timescale analysis with the results from both laboratory experiments and field observations confirms its applicability. The results suggest that, for small to medium sized lakes subject to a relatively uniform windstress, an important mechanism for the degeneration of large-scale internal waves is the generation of solitons by nonlinear steepening. Since solitons are likely to break at the sloping boundaries, leading to localized turbulent mixing and enhanced dissipation, the transfer of energy from an initial basin-scale seiche to shorter solitons has important implications for the lake ecology.

The wakes behind square cylinders with variation in incidence angle are computed over a range of Reynolds numbers to elucidate the three-dimensional stability and dynamics up to a Reynolds number of Re = 300, based on the projected height of the inclined square cylinder. Three-dimensional instability modes are predicted and computed using a linear stability analysis technique and three-dimensional simulations, respectively. Depending on the incidence angle, the flow is found to transition to three-dimensional flow through either a mode A instability, or a subharmonic mode C instability. The mode A instability is predicted as the first-occurring instability at incidence angles smaller than 12° and greater than 26°, with the mode C instability predicted between these incidence angles. At a zero-degree angle of incidence, the wake instabilities closely match modes A, B and a quasi-periodic mode predicted in earlier studies behind square and circular cylinders. With increasing angle of incidence, the three-dimensional wake transition Reynolds number first increases from Re = 164 as the mode A instability weakens, before decreasing again beyond an incidence angle of 12° as the wake becomes increasingly unstable to the mode C instability, and then again to the mode A instability as the incidence angle approaches 45°. A spanwise autocorrelation analysis from computations over a cylinder span 20 times the square cross-section side length reveals that beyond the onset of three-dimensional instabilities, the vortex street breaks down with patterns consistent with spatio-temporal chaos. This effect was more pronounced at higher incidence angles.

Boundary conditions are developed for rapid granular flows in which the rheology is dominated by grain–grain collisions. These conditions are $\overline{v}_0 = {\rm const}\,{\rm d}\overline{v}_0/{\rm d}y$ and u0 = const du0/dy, where $\overline{v}$ and u are the thermal (fluctuation) and flow velocities respectively, and the subscript indicates that these quantities and their derivatives are to be evaluated at the wall These boundary conditions are derived from the nature of individual grain–wall collisions, so that the proportionality constants involve the appropriate coefficient of restitution ew for the thermal velocity equation, and the fraction of diffuse (i.e. non-specular) collisions in the case of the flow-velocity equation. Direct application of these boundary conditions to the problem of Couette-flow shows that as long as the channel width h is very large compared with a grain diameter d it is permissible to set $\overline{v} = 0$ at the wall and to adopt the no-slip condition. Exceptions occur where d/h is not very small, when the wall is not rough, and when the grain–wall collisions are very elastic. Similar insight into other flows can be obtained qualitatively by a dimensional analysis treatment of the boundary conditions. Finally, the more difficult problem of self-bounding fluids is discussed qualitatively.

The rise and deformation of a gas bubble in an otherwise stationary liquid contained in a closed, right vertical cylinder is investigated using a modified volume-of-fluid (VOF) method incorporating surface tension stresses. Starting from a perfectly spherical bubble which is initially at rest, the upward motion of the bubble in a gravitational field is studied by tracking the liquid–gas interface. The gas in the bubble can be treated as incompressible. The problem is simulated using primitive variables in a control-volume formulation in conjunction with a pressure–velocity coupling based on the SIMPLE algorithm. The modified VOF method used in this study is able to identify and physically treat features such as bubble deformation, cusp formation, breakup and joining. Results in a two-dimensional as well as a three-dimensional coordinate framework are presented. The bubble deformation and its motion are characterized by the Reynolds number, the Bond number, the density ratio, and the viscosity ratio. The effects of these parameters on the bubble rise are demonstrated. Physical mechanisms are discussed for the computational results obtained, especially the formation of a toroidal bubble. The results agree with experiments reported in the literature.

When surface wetting drives liquids to invade porous media or microstructured materials with uniform channels, the penetration distance is known to increase as the square root of time. We demonstrate, experimentally and theoretically, that shape variations of the channel, in the flow direction, modify this ‘diffusive’ response. At short times, the shape variations are not significant and the imbibition is still diffusive. However, at long times, different power-law responses occur, and their exponents are uniquely connected to the details of the geometry. Experiments performed with conical tubes clearly show the two theoretical limits. Several extensions of these ideas are described.

This paper describes experiments on small solid particle settling behaviour in stationary homogeneous isotropic air turbulence. We present here a new methodology using a recently developed cruciform apparatus: a large horizontal cylindrical vessel equipped with a pair of counter-rotating fans and perforated plates at each end is used to generate stationary near-isotropic turbulence in the core region between the two perforated plates and a long vertical vessel is used to supply heavy descending particles from its top. This novel experimental design, without the unwanted influences from the injection of particles, the mean flow, and the decay of turbulence, allows direct imaging and velocity measurements of the two-way interaction between heavy particles and homogeneous isotropic turbulence. Consequently, the spatiotemporal responses of both fluid turbulence and particle settling can be determined by high-speed digital particle image velocimetry and accelerometry, together with the wavelet transform analysis for the first time. Hence, experimental information on and thereby understanding of the particle settling rate, preferential accumulation, and turbulence modification due to the presence of the particles is obtained.

We found that the particle settling velocity (${V}_{s})$ is much greater than the terminal velocity (${V}_{t})$ in still fluid for which the value of (${V}_{s}\,{-}\, {V}_{t})$ reaches a maximum of 0.13$u^\prime $ when the Stokes number $\hbox{\it St}\,{ =}\,\tau_{p}/\tau_{k}\,{\approx}\,$1 and ${V}_{t}/u^\prime \,{\approx}\,$0.5 at $\hbox{\it Re}_{\lambda }\,{=}\,$120 and $\hbox{\it Re}_{p} \,{<}\,$1, in good agreement with previous numerical results, where $\tau _{p}$ is the particle's relaxation time, $\tau _{k}$ is the Kolmogorov time scale, $u^\prime$ is the energy-weighted r.m.s. turbulent intensity, and $\hbox{\it Re}_{\lambda}$ and $\hbox{\it Re}_{p}$ are the Reynolds numbers based on the Taylor microscale ($\lambda$) and the mean diameter of particles, respectively. Non-uniform particle concentration fields are observed and most significant when $\hbox{\it St}\,{\approx}\,$1.0, at which the particle clusters accumulate preferentially around the outer perimeter of small intense banana-shaped vortical structures. These clusters can turn and stretch banana-shaped vortical structures toward the gravitational direction and thus significantly increase the mean settling rate especially when $\hbox{\it St}\,{ =}\,1$. From spatiotemporal analysis of the flatness factor, it is found that the characteristic length and time scales of these preferential particle clusters are related to the spacing between the adjacent intense vorticity structures of the order $\lambda$ and the time passage of these clustering structures of the order $\tau _{k}$, respectively. By comparing the average frequency spectra between laden (heavy particle) and unladen (neutral particle) turbulent flows over the measurement field at a fixed $\hbox{\it Re}_{\lambda }\,{=}\,$120, turbulence augmentation is found for most frequencies in the gravitational direction, especially for $\hbox{\it St}\,{\ge}\,$1. In the transverse direction, augmentation occurs only at higher frequencies beyond the Taylor microscale for all values of $\hbox{\it St}$ studied varying from 0.36 to 1.9. The increase in the size of energy spectra (turbulence augmentation) due to the presence of heavy particles is greatest at $\tau_{k}^{ - 1 }$ when $\hbox{\it St}\,{\approx}\,$1.0. Furthermore, the slip velocities between fluid turbulence and heavy particles can stimulate the laden turbulent flow to become more intermittent in the dissipation range. Finally, a simple energy balance model for turbulence modification is given to explain these results and areas for further study identified.

We present the results of a combined experimental and theoretical investigation of the dynamics of water-walking insects and spiders. Using high-speed videography, we describe their numerous gaits, some analogous to those of their terrestrial counterparts, others specialized for life at the interface. The critical role of the rough surface of these water walkers in both floatation and propulsion is demonstrated. Their waxy, hairy surface ensures that their legs remain in a water-repellent state, that the bulk of their leg is not wetted, but rather contact with the water arises exclusively through individual hairs. Maintaining this water-repellent state requires that the speed of their driving legs does not exceed a critical wetting speed. Flow visualization reveals that the wakes of most water walkers are characterized by a series of coherent subsurface vortices shed by the driving stroke. A theoretical framework is developed in order to describe the propulsion in terms of the transfer of forces and momentum between the creature and its environment. The application of the conservation of momentum to biolocomotion at the interface confirms that the propulsion of water walkers may be rationalized in terms of the subsurface flows generated by their driving stroke. The two principal modes of propulsion available to small water walkers are elucidated. At driving leg speeds in excess of the capillary wave speed, macroscopic curvature forces are generated by deforming the meniscus, and the surface behaves effectively as a trampoline. For slower speeds, the driving legs need not substantially deform the surface but may instead simply brush it: the resulting contact or viscous forces acting on the leg hairs crossing the interface serve to propel the creature forward.

Supersonic boundary-layer receptivity to different types of free-stream disturbance is studied for a Mach 4.5 boundary-layer flow over a flat plate by using the approaches of both direct numerical simulation and linear stability theory. This paper is Part 3 of a three-part study of the receptivity of supersonic boundary layers to free-stream disturbances. The present paper investigates receptivity to four types of different free-stream disturbances, i.e. slow and fast acoustic waves, entropy waves, and vorticity waves. A high-order shock-fitting scheme is used in the numerical simulation in order to accurately account for the effects of interactions between free-stream disturbance waves and the oblique shock wave. Numerical results on the generation of fast acoustic waves by free-stream entropy waves or vorticity waves are compared with those of a linear theory. Good agreement is obtained in both wave angles and amplitudes immediately behind the bow shock. It is found that the second-mode receptivity to free-stream slow acoustic waves is several times stronger than that to free-stream fast acoustic waves. This is because free-stream slow acoustic waves can directly induce and interact with the first and second Mack modes, while free-stream fast acoustic waves cannot. Instead, the free-stream fast acoustic waves can only induce and interact with stable mode I waves, which in turn induce unstable Mack modes. In the cases of receptivity to free-stream entropy waves and vorticity waves, it is found that the oblique shock wave created by the displacement of the boundary layer plays an important role because boundary-layer disturbances are mainly induced by fast acoustic waves generated behind the shock by free-stream forcing waves. As a result, mechanisms of the receptivity to free-stream entropy and vorticity waves are very similar to those of the receptivity to free-stream fast acoustic waves.

This paper is the first in a series of investigations having the overall objective of developing a new technique for treating the slow viscous motion past finite assemblages of particles of arbitrary shape. The new method, termed the multi-pole representation technique, is based on the theory that any object conforming to a natural co-ordinate system in a particle assemblage can be approximated by a truncated series of multi-lobular disturbances in which the accuracy of the representation is systematically improved by the addition of higher order multipoles. The essential elements of this theory are illustrated by examining the flows past finite line arrays of axisymmetric bodies such as spheres and spheroids which conform to special natural co-ordinate systems. It is demonstrated that this new procedure converges more rapidly and is simpler to use than the method of reflexions and represents the desired boundaries more precisely than the point-force approximation even when the objects are touching one another. Comparison of these solutions with the exact solutions of Stimson & Jeffery (1926) for the two sphere problem demonstrates the rapidity of convergence of this multipole procedure even when the spheres are touching. Drag results are also presented for flows past chains containing up to 101 spheres as well as for chains containing up to 15 prolate or oblate spheroids. The potential value of the technique is suggested by the rapidity with which the drag calculations were made, the 101 sphere problem requiring about 10 seconds on an IBM 360–65 computer to determine both the fluid flow and the drag coefficient.

We investigate the possibility of using proper orthogonal decomposition (POD) in reconstructing complete flow fields from gappy data. The incomplete fields are created from DNS snapshots of flow past a circular cylinder by randomly ommiting data points. We first examine the effectiveness of an existing method and subsequently introduce modifications that make the method robust and lead to the maximum possible resolution at a certain level of spatio-temporal gappiness. We simulate three levels of gappiness at approximately 20%, 50% and 80% in order to investigate the limits of applicability of the new procedure. We find that for the two lower levels of gappiness both the temporal and spatial POD modes can be recovered accurately leading to a very accurate representation of the velocity field. The resulting resolution is improved by more than five times compared to the existing method. However, for 80% gappiness only a few temporal modes are captured accurately while the corresponding spatial modes are noisy. We explain this breakdown of the method in terms of a simple perturbation analysis. This new methodology can be a building block in an effort to develop effective data assimilation techniques in fluid mechanics applications.

The Benjamin–Feir instability is a modulational instability in which a uniform train of oscillatory waves of moderate amplitude loses energy to a small perturbation of other waves with nearly the same frequency and direction. The concept is well established in water waves, in plasmas and in optics. In each of these applications, the nonlinear Schrödinger equation is also well established as an approximate model based on the same assumptions as required for the derivation of the Benjamin–Feir theory: a narrow-banded spectrum of waves of moderate amplitude, propagating primarily in one direction in a dispersive medium with little or no dissipation. In this paper, we show that for waves with narrow bandwidth and moderate amplitude, any amount of dissipation (of a certain type) stabilizes the instability. We arrive at this stability result first by proving it rigorously for a damped version of the nonlinear Schrödinger equation, and then by confirming our theoretical predictions with laboratory experiments on waves of moderate amplitude in deep water. The Benjamin–Feir instability is often cited as the first step in a nonlinear process that spreads energy from an initially narrow bandwidth to a broader bandwidth. In this process, sidebands grow exponentially until nonlinear interactions eventually bound their growth. In the presence of damping, this process might still occur, but our work identifies another possibility: damping can stop the growth of perturbations before nonlinear interactions become important. In this case, if the perturbations are small enough initially, then they never grow large enough for nonlinear interactions to become important.

By considering the entrainment effect on the intermittency in the free boundary of shear layers, a set of turbulence model equations for the turbulent kinetic energy k, the dissipation rate ε, and the intermittency factor γ is proposed. This enables us to incorporate explicitly the intermittency effect in the conventional K–ε turbulence model equations. The eddy viscosity νt is estimated by a function of K, ε and γ. In contrast to the closure schemes of previous intermittency modelling which employ conditional zone averaged moments, the present model equations are based on the conventional Reynolds averaged moments. This method is more economical in the sense that it halves the number of partial differential equations to be solved. The proposed K–ε–γ model has been applied to compute a plane jet, a round jet, a plane far wake and a plane mixing layer. The computational results of the model show considerable improvement over previous models for all these shear flows. In particular, the spreading rate, the centreline mean velocity and the profiles of Reynolds stresses and turbulent kinetic energy are calculated with significantly improved accuracy.

Self-similar plane solutions for the inertial stage of gravity currents are related to the initial parameters and a coefficient that is determined by the boundary condition at the front. Different relations have been proposed for the boundary condition in terms of a Froude number at the front, none of which have a sound theoretical or experimental basis. This paper focuses on considerations of the appropriate Froude number based on results of lock-exchange experiments in which extended inertial gravity currents are generated in a rectangular cross-section channel. We use ‘top-hat’ vertical density profiles of the currents to obtain an ‘equivalent’ depth, defined by profiles having the same buoyancy at every position as the real profiles. As in previous work, our experimental results show that in the initial constant-velocity phase the Froude number can be defined in terms of the lock depth. However, as the current enters the similarity phase when the initial release conditions are no longer relevant, we find that the Froude number is more appropriately defined in terms of the maximum height of the head. Strictly speaking, the self-similar solution to the shallow-water equations requires a front condition that uses the height at the rear of the head. We find that this rear Froude number is not constant and is a function of the head Reynolds number over the range 400–4500.

We describe a series of laboratory experiments to study convective structures in rotating fluids (distilled water) in ranges of Rayleigh flux number Raf from 106 to 2 × 1011 and of Taylor number Ta from 106 to 1012. An intermediate quasi-stationary ring pattern of convection was found to arise from the interaction of the onset of convection with the fluid spin-up, for which we determined the times of origin and destruction, the distances between the rings, and the diameter of the central ring in terms of Raf and Ta. The ring structure evolves into a vortex grid which can be regular or irregular. In terms of Raf and Ta the regular grid exists in the linear regime, when the number of vortices N is in accord with the linear theory, when $N \propto h^{-2} Ta^{\frac{1}{3}} \propto \Omega^{\frac{2}{3}}$, or in the nonlinear regime when N ∝ h−2Ta½Raf−⅙ ∝ where Ω is the angular velocity and h is the fluid depth. In the irregular regime we always have N ∝ Ω. The transition from the regular regime to the irregular one is rather gradual and is determined by the value of the ordinary Rayleigh number, which we found to be greater than the first critical number Ra ∝ Ta2/3 by a factor about 25–40. In the transition region vortex interactions are observed, which start with rotation of two adjacent vortices around a common axis, then the vortices come closer and rotation accelerates, following which the vortices form a double helix and then coalesce into one stronger vortex.

Some other qualitative experiments show that if the rotating vessel with the convective fluid is inclined to the horizontal, the vortex grid is formed along the rotation axis in accordance with the Proudman–Taylor theorem.

In this joint theoretical, numerical and experimental study, we investigate the phenomenon of forced generation of nonlinear waves by disturbances moving steadily with a transcritical velocity through a layer of shallow water. The plane motion considered here is modelled by the generalized Boussinesq equations and the forced Korteweg-de Vries (fKdV) equation, both of which admit two types of forcing agencies in the form of an external surface pressure and a bottom topography. Numerical results are obtained using both theoretical models for the two types of forcings. These results illustrate that within a transcritical speed range, a succession of solitary waves are generated, periodically and indefinitely, to form a procession advancing upstream of the disturbance, while a train of weakly nonlinear and weakly dispersive waves develops downstream of an ever elongating stretch of a uniformly depressed water surface immediately behind the disturbance. This is a beautiful example showing that the response of a dynamic system to steady forcing need not asymptotically tend to a steady state, but can be conspicuously periodic, after an impulsive start, when the system is being forced at resonance.

A series of laboratory experiments was conducted with a cambered bottom topography impulsively started from rest to a constant transcritical velocity U, the corresponding depth Froude number F = U/(gh0)½ (g being the gravitational constant and h0 the original uniform water depth) being nearly the critical value of unity. For the two types of forcing, the generalized Boussinesq model indicates that the surface pressure can be more effective in generating the precursor solitary waves than the submerged topography of the same normalized spatial distribution. However, according to the fKdV model, these two types of forcing are entirely equivalent. Besides these and some other rather refined differences, a broad agreement is found between theory and experiment, both in respect of the amplitudes and phases of the waves generated, when the speed is nearly critical (0.9 < F < 1.1) and when the forcing is sufficiently weak (the topography-height to water-depth ratio less than 0.15) to avoid breaking. Experimentally, wave breaking was observed to occur in the precursor solitary waves at low supercritical speeds (about 1.1 < F < 1.2) and in the first few trailing waves at high subcritical speeds (about 0.8 < F < 0.9), when sufficiently forced. For still lower subcritical speeds, the trailing waves behaved more like sinusoidal waves as found in the classical case and the forward-running solitary waves, while still experimentally discernible and numerically predicted for 0.6 > F > 0.2, finally disappear at F ≈ 0.2. In the other direction, as the Froude number is increased beyond F ≈ 1.2, the precursor soliton phenomenon was found also to evanesce as no finite-amplitude solitary waves can outrun, nor can any two-dimensional waves continue to follow, the rapidly moving disturbance. In this supercritical range and for asymptotically large times, all the effects remain only local to the disturbance. Thus, the criterion of the fascinating phenomenon of the generation of precursor solitons is ascertained.

Axisymmetric flows with a two-layer density stratification are produced by releasing either a constant flux of fluid from a point source or a constant volume of fluid into a rotating environment with a different density. In both experiments the density interface intersects one horizontal boundary, forming a front. Transition to non-axisym-metric flow is observed and can be described by two parameters: θ, the square of the ratio of the internal Rossby radius of deformation to the horizontal length scale of the flow, and δ, the fraction of the total fluid depth occupied by the layer inside the front. For θ [Lt ] 1 and δ > 10−1 unstable disturbances obtain most of their energy from the potential energy of the flow, whilst for δ < 10−1 extraction of kinetic energy from the basic shear becomes the dominant driving mechanism. When the front intersects the free surface, n = 2 is the minimum azimuthal wavenumber for an unstable disturbance. At large amplitude of the growing waves, baroclinic and barotropic processes combine to form n vortex dipole structures which entrain buoyant fluid from the original vortex and propagate radially over the free surface. Vortices are also produced by the continuous release of fluid from a confined source at its own density level in a region of constant density gradient. As in the two-layer case the axisymmetric vortex grows to a critical size and then becomes unstable to a disturbance with wavenumber n = 2, producing, at large amplitude, two vortex pairs.

The present paper reports an experimental study of the development of plane monochromatic waves in the boundary layer on a flat plate at Mach number M = 2.0. The wave characteristics of the plane waves are obtained. Three-dimensional disturbances with an angle of the wave vector to the flow χ = 50–70° are found to be the most unstable. It is shown that the disturbances, consisting of vortical and compressible modes, are engendered in a supersonic boundary layer by a local source of artificial disturbances. It is found that an increase in the bluntness of the leading edge of a plate stabilizes three-dimensional disturbances of a vortical mode.

A linear stability analysis is applied to a system consisting of a horizontal fluid layer overlying a layer of a porous medium saturated with the same fluid, with uniform heating from below. Surface-tension effects at a deformable upper surface are allowed for. The solution is obtained for constant-flux thermal boundary conditions.

The transient deformation of liquid capsules enclosed by elastic membranes subject to simple shear flow is studied numerically using a new implementation of the boundary element method. The numerical results for capsules with spherical unstressed shapes and varying degrees of surface elasticity are compared with the predictions of an asymptotic theory for small deformations due to Barthès-Biesel and coworkers, and the significance of nonlinear effects due to finite deformation is assessed. It is found that the capsules exhibit continuous elongation when the dimensionless shear rate becomes larger than a critical threshold, in agreement with recent experimental observations of capsules with polymerized interfaces. Membrane failure at large deformations is discussed with respect to membrane thinning and development of excessive elastic tensions, and it is argued that the location where the membrane is likely to rupture due to continued deformation is insensitive to the precise mechanism of rupture. The numerical results suggest that a dilute suspension of capsules behaves like shear-thinning medium with some elastic properties. Results of oblate spheroidal capsules suggest that the points of maximum membrane thinning and tension coincide but their location depends upon the unstressed capsule shape.