Numerical simulations of the evolution of gravity wave spectra of fairly narrow bandwidth have been performed both for two and three dimensions. Simulations using the nonlinear Schrödinger (NLS) equation approximately verify the stability criteria of Alber (1978) in the two-dimensional but not in the three-dimensional case. Using a modified NLS equation (Trulsen et al. 2000) the spectra ‘relax’ towards a quasi-stationary state on a timescale (ε2ω0)−1. In this state the low-frequency face is steepened and the spectral peak is downshifted. The three-dimensional simulations show a power-law behaviour ω−4 on the high-frequency side of the (angularly integrated) spectrum.

In this paper we present results from a numerical investigation of turbulent channel flow in the presence of a compliant wall. The compliant wall is modelled as a homogeneous spring-supported plate. The simulation code is validated both by comparison with an alternative code and by reproducing results of linear stability theory. Our results demonstrate that with the wall compliance we used in the simulation there is little change in the very long-time behaviour of the turbulent skin friction drag and little modification to the near-wall turbulent coherent structures. The values of pertinent statistical quantities of the turbulence near the compliant walls converge to those near a rigid wall and the statistical effect of the wall compliance on the turbulent channel flow is small.

Stereoscopic particle image velocimetry (PIV) was used to measure all three instantaneous components of the velocity field in streamwise–spanwise planes of a turbulent boundary layer at Reτ=1060 (Reθ=2500). Datasets were obtained in the logarithmic layer and beyond. The vector fields in the log layer (z+=92 and 150) revealed signatures of vortex packets similar to those proposed by Adrian and co-workers in their PIV experiments. Groups of legs of hairpin vortices appeared to be coherently arranged in the streamwise direction. These regions also generated substantial Reynolds shear stress, sometimes as high as 40 times −uw. A feature extraction algorithm was developed to automate the identification and characterization of these packets of hairpin vortices. Identified patches contributed 28% to −uw while occupying only 4% of the total area at z+=92. At z+=150, these patches occupied 4.5% of the total area while contributing 25% to −uw. Beyond the log layer (z+=198 and 530), the spatial organization into packets is seen to break down.

We consider the steady propagation of an air finger into a buckled elastic tube initially filled with viscous fluid. This study is motivated by the physiological problem of pulmonary airway reopening. The system is modelled using geometrically nonlinear Kirchhoff–Love shell theory coupled to the free-surface Stokes equations. The resulting three-dimensional fluid–structure-interaction problem is solved numerically by a fully coupled finite element method.

The system is governed by three dimensionless parameters: (i) the capillary number, Ca=μU/σ*, represents the ratio of viscous to surface-tension forces, where μ is the fluid viscosity, U is the finger's propagation speed and σ* is the surface tension at the air–liquid interface; (ii) σ=σ*/(RK) represents the ratio of surface tension to elastic forces, where R is the undeformed radius of the tube and K its bending modulus; and (iii) A∞=A*∞/(4R2), characterizes the initial degree of tube collapse, where A*∞ is the cross-sectional area of the tube far ahead of the bubble.

The generic behaviour of the system is found to be very similar to that observed in previous two-dimensional models (Gaver et al. 1996; Heil 2000). In particular, we find a two-branch behaviour in the relationship between dimensionless propagation speed, Ca, and dimensionless bubble pressure, p*b/(σ*/R). At low Ca, a decrease in p*b is required to increase the propagation speed. We present a simple model that explains this behaviour and why it occurs in both two and three dimensions. At high Ca, p*b increases monotonically with propagation speed and p*b/(σ*/R) ∝ Ca for sufficiently large values of σ and Ca. In a frame of reference moving with the finger velocity, an open vortex develops ahead of the bubble tip at low Ca, but as Ca increases, the flow topology changes and the vortex disappears.

An increase in dimensional surface tension, σ*, causes an increase in the bubble pressure required to drive the air finger at a given speed; p*b also increases with A*∞ and higher bubble pressures are required to open less strongly buckled tubes. This unexpected finding could have important physiological ramifications. If σ* is sufficiently small, steady airway reopening can occur when the bubble pressure is lower than the external (pleural) pressure, in which case the airway remains buckled (non-axisymmetric) after the passage of the air finger. Furthermore, we find that the maximum wall shear stresses exerted on the airways during reopening may be large enough to damage the lung tissue.

A filament of an incompressible highly viscous fluid that is supported at its ends sags under the influence of gravity. Its instantaneous shape resembles that of a catenary, but evolves with time. At short times, the shape is dominated by bending deformations. At intermediate times, the effects of stretching become dominant everywhere except near the clamping boundaries where bending boundary layers persist. Finally, the filament breaks off in finite time via strain localization and pinch-off.

Shoaling solitary internal waves are ubiquitous features in the coastal regions of the world's oceans where waves with a core of recirculating fluid (trapped cores) can provide an effective transport mechanism. Here, numerical evidence is presented which suggests that there is a close connection between the limiting behaviour of large-amplitude solitary waves and the formation of such waves via shoaling. For some background states, large-amplitude waves are broad, having a nearly horizontal flow in their centre. The flow in the centre of such waves is called a conjugate flow. For other background states, large-amplitude waves can reach the breaking limit, at which the maximum current in the wave is equal to the wave's propagation speed. The presence of a background current with near-surface vorticity of the same sign as that induced by the wave can change the limiting behaviour from the conjugate-flow limit to the breaking limit. Numerical evidence is presented here which suggests that if large solitary waves cannot reach the breaking limit in the shallow water, that is if the background flow has a conjugate flow, then waves with trapped cores will not be formed via shoaling. It is also shown that, due to a change in the limiting behaviour of large waves, an appropriate background current can enable the formation of waves with trapped cores in stratifications for which such waves are not formed in the absence of a background current.

We derive analytic solutions for the forced linear shallow water equation of the following form:

<?TeX \partial^2 Y \over \partial t^2}-b{\partial \over \partial x}\left(x {\partial Y \over \partial x}\right)={\partial^2 f\over \partial t^2 ?>

for x>0, in which Y(x,t) denotes an unknown variable, f(x,t) a prescribed forcing function and b a positive constant. This equation has been used to describe landslide-generated tsunamis and also long waves induced by moving atmospheric pressure distributions. We discuss particular and general solutions. We then compare our results with numerical solutions of the same equation and with the corresponding solutions of the nonlinear depth-integrated equations and discuss them in terms of landslide-generated tsunamis.

This paper deals with a specially designed slender ship moving steadily at a chosen supercritical speed in a suitable narrow shallow channel such that the wave resistance becomes zero. It is a sequel to Chen & Sharma (1997), which predicted theoretically the existence of a ship–channel configuration of zero wave resistance. It was derived from a two-soliton solution of the Kadomtsev–Petviashvili equation. The present theory comprises an improved shallow-water wave equation of Boussinesq type for the far field and an enhanced slender body approximation in the near field. Moreover, a physical model experiment performed in the Duisburg shallow water towing tank (VBD) is reported. It confirms that at the exact design condition the ship's reflected bow wave cancels the stern wave so completely that there are almost no waves behind the ship and the measured resistance is reduced so much that the wave resistance component is practically zero.

When a drop impacts on a liquid surface it entraps a small amount of air under its centre as the two liquid surfaces meet. The contact occurs along a ring enclosing a thin disk of air. We use the next-generation ultra-high-speed video camera, capable of 1 million f.p.s. (Etoh et al. 2002), to study the dynamics of this air sheet as it contracts due to surface tension, to form a bubble or, more frequently, splits into two bubbles. During the contraction of the air disk an azimuthal undulation, resembling a pearl necklace, develops along its edge. The contraction speed of the sheet is accurately described by a balance between inertia and surface tension. The average initial thickness of the air sheet decreases with higher impact Reynolds numbers, becoming less than one micron. The total volume of air entrapped depends strongly on the bottom curvature of the drop at impact. A sheet of micro-bubbles is often observed along the original interface. Oguz–Prosperetti bubble rings are also observed. For low Weber numbers (We<20) a variety of other entrapment phenomena appear.

The paper is devoted to an experimental and numerical investigation of the problem of excitation of three-dimensional Tollmien–Schlichting (TS) waves in a boundary layer on an airfoil owing to scattering of an acoustic wave on localized microscopic surface non-uniformities. The experiments were performed at controlled disturbance conditions on a symmetric airfoil section at zero angle of attack. In each set of measurements, the acoustic wave had a fixed frequency fac, in the range of unstable TS-waves. The three-dimensional surface non-uniformity was positioned close to the neutral stability point at branch I for the two-dimensional perturbations. To avoid experimental difficulties in the distinction of the hot-wire signals measured at the same (acoustic) frequency but having a different physical nature, the surface roughness was simulated by a quasi-stationary surface non-uniformity (a vibrator) oscillating with a low frequency fv. This led to the generation of TS-wavetrains at combination frequencies f1,2=fac ∓ fv. The spatial behaviour of these wavetrains has been studied in detail for three different values of the acoustic frequency. The disturbances were decomposed into normal oblique TS-modes. The initial amplitudes and phases of these modes (i.e. at the position of the vibrator) were determined by means of an upstream extrapolation of the experimental data. The shape of the vibrator oscillations was measured by means of a laser triangulation device and mapped onto the Fourier space.

We have utilized a computational model of semi-infinite air bubble progression in a surfactant-doped, fluid-filled rigid capillary to investigate the continual interfacial expansion dynamics that occur during the opening of collapsed pulmonary airways. This model simulates mixed-kinetic conditions with nonlinear surfactant equations of state similar to those of pulmonary surfactant. Several dimensionless parameters govern the system responses: the capillary number (Ca) that relates viscous to surface tension forces; the elasticity number (El), a measure of the ability of surfactant to modify the surface tension; the bulk Péclet number (Pe), relating bulk convection rates to diffusion rates; the adsorption and desorption Stanton numbers (Sta and Std) that relate the adsorption/desorption rates to surface convective rates; and finally the adsorption depth (λ), a dimensionless bulk surfactant concentration parameter. We investigated this model by performing detailed parameter variation studies at fixed and variable equilibrium concentrations. We find that the surfactant properties can strongly influence the interfacial pressure drop through modification of the surface tension and the creation of Marangoni stresses that influence the viscous stresses along the interface. In addition, these studies demonstrate that, depending upon the range of parameters, either film thickening or film thinning responses are possible. In particular, we find that when Pe[Gt ]1 (as with pulmonary surfactant) or when sorption rates are low, concentration profiles can substantially differ from near-equilibrium approximations and can result in film thinning. These responses may influence stresses on epithelial cells that line pulmonary airways and the stability of these airways, and may be important to the delivery of exogenous surfactant to deep regions of the lung.

The flow in a rectangular cavity driven by the sinusoidal motion of the floor in its own plane has been studied both experimentally and computationally over a broad range of parameters. The stability limits of the time-periodic two-dimensional base state are of primary interest in the present study, as it is within these limits that the flow can be used as a viable surface viscometer (as outlined theoretically in Lopez & Hirsa 2001). Three flow regimes have been found experimentally in the parameter space considered: an essentially two-dimensional time-periodic flow, a time-periodic three-dimensional flow with a cellular structure in the spanwise direction, and a three-dimensional irregular (in both space and time) flow. The system poses a space–time symmetry that consists of a reflection about the vertical mid-plane together with a half-period translation in time (RT symmetry); the two-dimensional base state is invariant to this symmetry. Computations of the two-dimensional Navier–Stokes equations agree with experimentally measured velocity and vorticity to within experimental uncertainty in parameter regimes where the flow is essentially uniform in the spanwise direction, indicating that in this cavity with large spanwise aspect ratio, endwall effects are small and localized for these cases. Two classes of flows have been investigated, one with a rigid no-slip top and the other with a free surface. The basic states of these two cases are quite similar, but the free-surface case breaks RT symmetry at lower forcing amplitudes, and the structure of the three-dimensional states also differs significantly between the two classes.

Dissipation-range intermittency was first observed by Batchelor & Townsend (1949) in high Reynolds number turbulent flows. It typically manifests itself in spatio-temporal binary behaviour which is characterized by long, quiescent periods in the signal which are interrupted by short, active ‘events’ during which there are large excursions away from the average. It is shown that Leray's weak solutions of the three-dimensional incompressible Navier–Stokes equations can have this binary character in time. An estimate is given for the widths of the short, active time intervals, which decreases with the Reynolds number. In these ‘bad’ intervals singularities are still possible. However, the average width of a ‘good’ interval, where no singularities are possible, increases with the Reynolds number relative to the average width of a bad interval.

The production of vorticity or circulation production in shock wave diffraction over sharp convex corners has been numerically simulated and quantified. The corner angle is varied from 5° to 180°. Total vorticity is represented by the circulation, which is evaluated by integrating the velocity along a path enclosing the perturbed region behind a diffracting shock wave. The increase of circulation in unit time, or the rate of circulation production, depends on the shock strength and wall angle if the effects of viscosity and heat conductivity are neglected. The rate of vorticity production is determined by using a solution-adaptive code, which solves the Euler equations. It is shown that the rate of vorticity production is independent of the computational mesh and numerical scheme by comparing solutions from two different codes. It is found that larger wall angles always enhance the vorticity production. The vorticity production increases sharply when the corner angle is varied from 15° to 45°. However, for corner angles over 90°, the rate of vorticity production hardly increases and reaches to a constant value. Strong shock waves produce vorticity faster in general, except when the slipstream originating from the shallow corner attaches to the downstream wall. It is found that the vorticity produced by the slipstream represents a large proportion of the total vorticity. The slipstream is therefore a more important source of vorticity than baroclinic effects in shock diffraction.

We address – motivated in part by the findings of Gong et al. (1996) and Miller (1995) – the role of streamwise-oriented large-scale structures in a developed flow between a sinusoidal bottom wall and a flat top wall. Particle image velocimetry (PIV) is used to examine the spatial variation of the velocity in different planes of the flow through a water channel with an aspect ratio of 12:1. The wave amplitude is equal to one tenth of the wall wavelength, Λ, and Reynolds numbers between 500 and 7300, defined with the bulk velocity and the half-height of the channel, are considered. To examine streamwise-oriented structures, the spanwise variation of the velocity field is studied in a plane parallel to the top wall, and in one that intersects the wavy surface at an uphill location. From a proper orthogonal decomposition (POD) of the streamwise velocity fluctuations, we obtain the dominant eigenfunctions with a characteristic spanwise scale of O(1.5Λ), which agrees with the scale of perturbations for the streamwise velocity at laminar conditions. A decomposition of the turbulent velocity field close to the uphill section of the wavy surface reveals smaller structures at a location that coincides with the Reynolds shear stress maximum.

Experimental studies of laminar axisymmetric starting plumes are performed to investigate the dependence of the flow on the Prandtl number, focusing on large Prandtl numbers. Thermal plumes are generated by a small electric heater in a glass tank filled with viscous oils. Prandtl numbers in the range of 7–104 were investigated. Experimental conditions are such that viscosity variations due to temperature differences are negligible. Plumes ascend in two different regimes as a function of distance to source. At short distances, the plumes accelerate owing to the development of the viscous boundary layer. At distances larger than about five times the heater size, the ascent velocity is constant and increases as a function of the Prandtl number, as predicted by theory for steady plumes. This velocity is, within experimental error, proportional to the steady plume centreline velocity.

Particle image velocimetry (PIV) measurements and flow visualization in a water tunnel show that vortex shedding at the leading and trailing edges of rectangular cylinders can be simultaneously phase-locked to transverse velocity perturbations when the applied perturbation Stp is close to an impinging leading-edge vortex/trailing-edge vortex shedding (ILEV/TEVS) frequency. The transverse perturbations, analogous to β-mode duct acoustic resonances, are generated through harmonic oscillations of the sidewalls. When this occurs, the leading-edge vortices are found always to pass the trailing edge at the same phase in the perturbation cycle regardless of the chord-to-thickness (c/t) ratio. Applying perturbations at an Stp not equal to the natural global frequency also results in phase-locked vortex shedding from the leading edge, and a near wake with a frequency equal to the perturbation frequency. This is consistent with previous experimental findings. However, vortex shedding at the trailing edge is either weaker or non-existent. PIV results and flow visualization showed trailing-edge vortex growth was weaker because leading-edge vortices arrive at the trailing edge at a phase in the perturbation cycle where they interfere with trailing-edge shedding. The frequencies at which trailing-edge vortices form for different c/t ratios correspond to the natural ILEV/TEVS frequencies. As in the case of natural shedding, peaks in base suction occur when the leading-edge vortices pass the trailing edge at the phase in the perturbation cycle (and thus in the leading-edge shedding cycle) that allows strong trailing-edge shedding. This is the reason for the similarity in the Stvs.c/t relationship for three seemingly different sets of experiments.

Measurements in a laboratory flume have shown unexpected wave-induced changes in the vertical profile of the mean horizontal velocity. Two theoretical explanations for these changes have been proposed so far. One is based on a local force balance in the longitudinal direction and the other relies on secondary circulations in the cross-sectional plane. In this study, a two-dimensional (2DV lateral) model based on the so-called generalized Lagrangian mean (GLM) formulation has been developed to investigate the three-dimensional effect of waves on the steady current and in particular to investigate the validity of the two fundamentally different explanations. Formulations for the three-dimensional wave-induced driving force have been implemented in an existing 2DV non-hydrostatic numerical flow model. Computations for regular waves following and opposing a turbulent current over a horizontal bed have been carried out and the results are compared with both experimental results and results from an existing numerical model. The results clearly indicate predominance of the longitudinal component of the wave-induced driving force over the cross-sectional components. Although the 2DV model has only been applied to and verified with measurements in wave–current systems in a laboratory flume, the approach followed here is relevant for a wider class of problems of wave–current interactions.