Nonlinear diffraction of low-amplitude gravity waves in deep water due to a slightly submerged obstacle is studied experimentally in a wave channel and theoretically. The obstacle is either a circular cylinder or a rectangular shelf. The incoming waves (with wavelength λ) undergo strong nonlinear deformations at the obstacle when the wave amplitude is finite. An infinite number of superharmonic waves are then introduced to the flow. Their wavelengths far away from the obstacle are λ/4, λ/9, λ/16,…, due to the dispersion relation being quadratic in the wave frequency. The superharmonic wave amplitudes grow with increasing incoming wave amplitude up to saturation values. They are found to be prominent at the obstacle's lee side and vanishingly small at the weather side. The second- and third-harmonic wave amplitudes are, surprisingly, in several examples found to be comparable to the incoming wave amplitude. Up to 25% of the incoming energy flux may be transferred to the shorter waves. The theoretical model accounts for nonlinearity by the Boussinesq equations in the shallow region above the obstacle, with patching to linearized potential theory in the deep water. The theory explains both qualitatively and quantitatively the trends observed in the experiments up to breaking.

A Monte Carlo simulation is developed for the study of rapidly deforming, steady, simple shear flows of inelastic disks or spheres. The simulation is based on the theoretical framework of the kinetic theory of dense gases. In the simulation, space is discarded in an explicit sense and replaced by an isotropic, homogeneous, and uncorrelated space based on the assumption of a state of simple shear, a uniform concentration field, and molecular chaos. The simulation generates a distribution of particle velocities which corresponds to the parameters of the flow. The velocity distribution is a numerical solution to the Boltzmann equation under these conditions. The Monte Carlo simulation defines the limits to the accuracy of analytical granular flow theories based on the kinetic theory and the assumption of molecular chaos.

Moore's asymptotic analysis of vortex-sheet motion predicts that the Kelvin–Helmholtz instability leads to the formation of a weak singularity in the sheet profile at a finite time. The numerical studies of Meiron. Baker & Orszag, and of Krasny, provide only a partial validation of his analysis. In this work, the motion of periodic vortex sheets is computed using a new, spectrally accurate approximation to the Birkhoff–Rott integral. As advocated by Krasny, the catastrophic effect of round-off error is suppressed by application of a Fourier filter, which itself operates near the level of the round-off. It is found that to capture the correct asymptotic behaviour of the spectrum, the calculations must be performed in very high precision, and second-order terms must be included in the Ansatz to the spectrum. The numerical calculations proceed from the initial conditions first considered by Meiron, Baker & Orszag. For the range of amplitudes considered here, the results indicate that Moore's analysis is valid only at times well before the singularity time. Near the singularity time the form of the singularity departs away from that predicted by Moore, with the real and imaginary parts of the solutions becoming differentiated in their behaviour; the real part behaves in accordance with Moore's prediction, while the singularity in the imaginary part weakens. In addition, the form of the singularity apparently depends upon the initial amplitude of the disturbance, with the results suggesting that either Moore's analysis gives the complete form of the singularity only in the zero amplitude limit, or that the initial data considered here is not yet sufficiently small for the behaviour to be properly described by the asymptotic analysis. Convergence of the numerical solution beyond the singularity time is not observed.

The bulk motion of a granular material affects its apparent thermal as well as its apparent mechanical properties. This paper presents the simultaneous measurements of the apparent viscosity and thermal conductivity for a dry granular material undergoing shear in an annular shear cell. Both properties are seen to vary linearly with the shear rate. As such, it can be argued that both the apparent conductivity and viscosity are proportional to the square root of the granular temperature in exactly the same way as the kinetic theory of gases predicts that the conductivity and viscosity of a perfect gas vary as the square root of the thermodynamic temperature. Thus, analogies can be drawn between the mechanical and thermal behaviour of a granular flow that share much with similar — a.k.a. Reynolds — analogies for both laminar and turbulent flows of simple fluids. However, the results do indicate fundamental differences in the internal transport of heat and momentum. In particular, heat may only be transmitted by the streaming motion of the particles, while momentum may also be exchanged during interparticle collisions.

We have measured the probability density functions (PDFs) of density fluctuations and of density-gradient fluctuations in decaying stratified turbulence, using a thermally stratified wind tunnel. The turbulence was generated by passing the flow through a biplanar grid at the entrance to the test section. The linear mean vertical temperature gradient could be adjusted to produce different stratification strengths. The PDFs of the density-gradient fluctuations exhibit extended exponential tails, while those for the density fluctuations are nearly Gaussian. As the turbulence decays away from the grid the exponential tails of the density gradient PDFs become steeper and the central rounded part of the distribution widens. The tail steepness scales approximately as $Re \lambda{-\frac{1}{2}}$. Buoyancy forces are not the cause of the exponential tails, since when normalized in r.m.s. units the behaviour of the tails is independent of stratification strength. The vertical temperature gradients ∂θ/∂z (measured using two cold wires) show a strong positive skewness close to the grid where the turbulence is most vigorous. This skewness is not caused by non-Boussinesq effects and is present for all stratification strengths. We propose a simple phenomenological model (similar to that of Budwig et al. 1985), based on stirring of fluid parcels advected in the mean gradient, to explain the presence of this skewness. The skewness observed by other researchers and their interpretations are discussed in the context of this model. The buoyancy flux PDF also shows strong exponential tails and is very strongly skewed. Both of these properties are consistent with joint-Gaussian statistics of the vertical velocity and temperature fluctuations.

A perturbation scheme is developed to analyse the disturbance field produced by acoustic forcing over a flat plate with non-localized surface irregularities. Both the amplitude of the forcing and the height of the irregularity are assumed to be small. At first order, two modes are calculated: a Stokes mode resulting from the acoustic forcing, and a wall mode resulting from the surface irregularity. These modes interact at second order to produce a forced travelling wave with the frequency of the acoustic wave and a wavenumber associated with the surface irregularity. Streamwise variations in the mean flow mediate a distiibuted energy transfer between the forced mode and the eigenmode. Sufficiently far downstream, the forced-mode amplitude becomes small and the total disturbance is dominated by the resulting eigenmode. Receptivity amplitudes, expressed in terms of effective branch I values, are O(10) for a broad range of surface wavenumbers.

We consider the problem of three interacting resonant waves with arbitrary (non-conservative) nonlinear coupling. Such coupling arises naturally in the interaction of waves on shear flows, and in interactions between interfacial and gravity waves. We focus on the case where two modes are damped and have identical properties, and the third is linearly unstable. When the damping rates dominate the growth rate, the dynamics evolves on two disparate timescales and it is then possible to reduce the system to a multi-modal one-dimensional map, thus revealing clearly the complex sequence of bifurcations that occurs as the parameters are varied. We also investigate the effect on the equations of small additive noise; this can be simply modelled by a (deterministic) perturbation to the map. It is shown that even at very low levels, the effect of noise can be extremely important in determining the period and amplitude of the oscillations.

Asymptotic methods are used to determine the dispersion equation for disturbances of rotating parallel flows in shallow water. From this equation the unstable modes and their growth rates are determined. The solution involves seven asymptotic expansions which are matched together. The results supplement and extend those which have been obtained previously using numerical methods by Griffiths, Killworth & Stern (1982) and by Hayashi & Young (1987).

The dynamics of a thin film of Newtonian fluid coating the inner surface of an elastic circular tube is analysed. This problem is motivated by an interest in the closure of small airways of the lungs either by formation of a liquid bridge, the collapse of the airway wall or a combination of both processes. Liquid bridge formation is due to the destabilization of the liquid film that coats the inner surface of airways, while wall collapse can be due to either the high surface tension of the air–liquid interface or the flexibility of the wall.

Nonlinear evolution equations for the film thickness and wall position are derived using lubrication theory, but an accurate representation of the curvatures of both the liquid and wall interfaces is employed which is valid for thick films. These approximations allow closure to be predicted. In addition, these approximations are justified by comparison with rigid-wall results obtained by solving the full Navier–Stokes equations and because fluid inertia only becomes important in the very late stages of closure. The linear stability of these equations is examined using normal-mode analysis for infinitesimal disturbances and the nonlinear stability is investigated by solving the governing equations numerically using the method of lines. Solutions show that there is a critical film thickness, strongly dependent on fluid and wall properties, above which unstable waves grow to form liquid bridges. The critical film thickness decreases with increasing surface tension or wall compliance since waves grow faster. Even for relatively stiff airways, the volume of fluid in the liquid lining required for closure can be approximately 70% of the volume for the rigid-tube case. Wall damping is an important effect only when the airway is sufficiently compliant. Airway closure occurs more rapidly with increasing unperturbed film thickness, surface tension and wall flexibility and decreasing wall damping.

It is shown that certain similarity solutions relating to axisymmetric vortices in a viscous fluid over a plane wall can be associated with a point source or sink of vorticity at the origin and a line vortex along the symmetry axis of the system. The rotationality of the nonlinear terms in the momentum equation, due to the radial vorticity and the azimuthal flow field, induces a poloidal flow which relates to one-cell or two-cell configurations. It is shown that a small imput of radial vorticity into a strong line vortex can induce an intense up-draught. There are ranges of values of the parameters yielding two solutions. The extremeties of these ranges are associated with values that yield velocity breakdown.

Rayleigh (1883) observed that the frequency of parametrically excited capillary–gravity waves in a container of lateral dimensions large compared with the wavelength was only 75% of the frequency ω0 calculated from Kelvin's dispersion relation for waves of the observed length, and attributed this discrepancy to friction. A boundary-layer calculation on the assumption that the free surface acts as an inextensible film (as is typical for water in the laboratory) yields a 10% reduction from ω0. The remaining discrepancy may be plausibly attributed to a contamination-induced reduction of surface tension from the value assumed by Rayleigh, but the possibility remains that nonlinearity could account for a significant shift of the frequency from ω0. The solution of the weakly nonlinear problem for parametrically excited capillary–gravity waves reveals that this shift is positive for Rayleigh's data, whence the surface tension must have been even smaller than that inferred from Kelvin's dispersion relation. This solution also suggests quantitative errors in the solutions of Ezerskii et al. (1986) and Milner (1991) for the limiting case of deep-water capillary waves.

An analytical and computational study is presented on solution properties of strongly nonlinear vortex/wave interactions involving Tollmien–Schlichting waves, in boundary-layer transition. The longitudinal vortex part, i.e. the total mean flow, is governed by a three-dimensional vortex system but coupled, through an effective spanwise slip condition at the surface, with the accompanying wave part, so that both the vortex and the wave parts are unknowns. Terminal forms of the space-marching or time-marching problem are proposed first, yielding either a lift-off separation singularity or a strong-attachment singularity. Second, a similarity version of the complete system is addressed numerically and analytically. This leads to a number of interesting solution features as the typical wave pressure is increased into the strongly nonlinear regime. In particular, lift-off separation and attachment forms seem to emerge which are analogous with those proposed above. The flow developments beyond the terminal forms are discussed, together with the links of the work with recent computational results and, tentatively, with experimental observations including the creation of lambda vortices (as a form of lift-off separation).

The three-dimensional time-dependent turbulent flow in the stably stratified Ekman layer over a smooth surface is computed numerically by directly solving the Navier–Stokes equations, using the Boussinesq approximation to account for buoyancy effects. All relevant scales of motion are included in the simulation so that no turbulence model is needed. The Ekman layer is an idealization of the Earth's boundary layer and provides information concerning atmospheric turbulence models. We find that, when non-dimensionalized according to Nieuwstadt's local scaling scheme, some of the simulation data agree very well with atmospheric measurements. The results also suggest that Brost & Wyngaard's ‘constant Froude number’ and Hunt's ‘shearing length’ stable layer models for the dissipation rate of turbulent kinetic energy are both valid, when Reynolds number effects are accounted for. Simple gradient closures for the temperature variance and heat flux demonstrate the same variation with Richardson number as in Mason & Derbyshire's large-eddy simulation (LES) study, implying both that the models are relatively insensitive to Reynolds number and that local scaling should work well when applied to the stable atmospheric layer. In general we find good agreement between the direct numerical simulation (DNS) results reported here and Mason & Derbyshire's LES results.

The usual second-order two-point velocity correlations for homogeneous, isotropic turbulence in a non-divergent fluid are not applicable to tracer pairs in turbulent dispersion because on average the tracers separate as though in a divergent fluid. The present formulation accounts for the expansion that is associated with dispersion through a modification of the Karman–Howarth relations that includes the rate of expansion as an unspecified constant.