We consider the evolution under the action of surface tension of wedges and cones of viscous fluid whose initial semi-angles are close to π/2. A short time after the fluid is released from rest, there is an inner region, where surface tension and viscosity dominate, and an outer region, where inertia and viscosity dominate. We also find that the velocity of the tip of the wedge or cone is singular, of O(log(1/t)), as time, t, tends to zero. After a long time, the free surface asymptotes to a similarity form where deformations are of O(t2/3), and capillary waves propagate away from the tip. However, a distance of O(t3/4) away from the tip, viscosity acts to damp out the capillary waves.

We solve the linearized governing equations using double integral transforms, which we calculate numerically, and use asymptotic techniques to approximate the solutions for small and large times. We also compare the asymptotic solution for the inviscid fat wedge with a numerical solution of the nonlinear inviscid problem for wedges of arbitrary semi-angle.

A new mechanism is proposed for the generation of Tollmien–Schlichting (T–S) waves by free-stream turbulence. For definiteness and self-consistency, the mechanism is described mathematically by using a triple-deck formalism. The free-stream turbulence is represented by convecting gusts consisting of the so-called vortical and entropy waves of small amplitude. We show that suitable convecting gusts can interact with sound waves in the free stream to produce a forcing that has the same time and length scales as those of the T–S waves, thereby exciting such waves in the vicinity of the lower branch of the neutral stability curve. The T–S waves so produced have the order of magnitude of ε2R5/16, where ε is the amplitude of the free-stream disturbance and R the global Reynolds number. The scale conversion is achieved without resorting to any non-homogeneity on the wall, and hence the mechanism operates in a flat boundary layer. Furthermore, the T–S waves so generated do not undergo any immediate decay, as they may do in some other receptivity processes. For homogeneous isotropic free-stream turbulence, the spectrum of the T–S waves is obtained. The efficiency of the receptivity mechanism is assessed by parametric studies.

An experimental study was conducted in a turbulent spray flame in which droplets were produced ultrasonically at low velocity relative to the host gas. In this fashion, injector-specific effects on the two-phase flow were minimized and a scenario generally characteristic of the far field of practical spray systems could be simulated. Close to the burner exit, the spray flame appeared as a dense column of drops burning with an envelope flame. Further downstream, it opened up slowly in the radial direction and developed a turbulent ‘brush’ appearance. Measurements of the size, velocity and concentration of the droplets, and of gas-phase velocity and temperature were made by combining a Phase-Doppler interferometric technique with Stokes/anti-Stokes Raman thermometry. The experimental data were used to derive scaling and self-similarity for the Reynolds-averaged continuity and momentum equations using suitable transformations.

Results showed three distinct regions, on the basis of the behaviour of the gas axial velocity in the spray flame. In the lower part of the flame, the gas momentum increased because of vaporization. In the intermediate region of the spray flame, the axial velocity decayed along the centreline as an inverse power of the distance from the virtual origin, with exponents smaller than unity. In the upper part of the spray flame, the flow field recovered the axial velocity decay that is typical of incompressible jets, namely as an inverse of the axial distance. Self-similar behaviour held for the axial velocity throughout the intermediate region. The vapour source term in the gas continuity equation scaled approximately as an inverse power of axial distance, and exhibited self-similarity throughout the spray flame. As a result, a simple model of the Reynolds stress term could be formulated, in which two competing contributions appear: one, that is due to turbulent transport, tends to increase the value of the velocity correlation; another, that is due to the vaporization term, tends to reduce the value of the velocity correlation and can be construed as a vaporization-induced tendency towards relaminarization. The first term is modelled by a classic gradient-transport approach yielding an empirical mixing length relating the velocity correlation to the average velocity gradient. Model and experiments are found to be in good agreement, especially sufficiently far from the injector, where one-way coupling between the two phases holds.

In Part 1 of this work we presented a new mathematical formulation for numerical investigation of capillary wave dynamics. This permitted calculations to be readily performed on a desktop computer. Applications given were primarily to waves in one surface dimension. In Part 2 we describe further applications to waves in two surface dimensions and also to waves in shallow water (not, however, shallow for the capillary waves). Wavenumber–frequency spectra for wind waves are calculated. As was observed in tank experiments by Hara et al. (1997), our calculations show both a frequency spread and a frequency up-shifting which suggests that capillary waves are ‘dragged along’ by longer waves. Fine-scale roughness near wind wave crests shows a transitory nature, changing with the wave pattern. We discuss implications of this for microwave remote sensing. The propagation of short gravity waves in shallow water is studied. As these waves develop bore-like forward faces, generation of parasitic capillary waves is observed. Generation rates for these are substantially greater than observed for waves in deep water.

A direct numerical simulation is presented of an elliptical instability observed in the laboratory within an elliptically distorted, rapidly rotating, fluid-filled cylinder (Malkus 1989). Generically, the instability manifests itself as the pairwise resonance of two different inertial modes with the underlying elliptical flow. We study in detail the simplest ‘subharmonic’ form of the instability where the waves are a complex conjugate pair and which at weakly supercritical elliptical distortion should ultimately saturate at some finite amplitude (Waleffe 1989; Kerswell 1992). Such states have yet to be experimentally identified since the flow invariably breaks down to small-scale disorder. Evidence is presented here to support the argument that such weakly nonlinear states are never seen because they are either unstable to secondary instabilities at observable amplitudes or neighbouring competitor elliptical instabilities grow to ultimately disrupt them. The former scenario confirms earlier work (Kerswell 1999) which highlights the generic instability of inertial waves even at very small amplitudes. The latter represents a first numerical demonstration of two competing elliptical instabilities co-existing in a bounded system.

Model equations that govern the evolution of internal gravity waves at the interface of two immiscible inviscid fluids are derived. These models follow from the original Euler equations under the sole assumption that the waves are long compared to the undisturbed thickness of one of the fluid layers. No smallness assumption on the wave amplitude is made. Both shallow and deep water configurations are considered, depending on whether the waves are assumed to be long with respect to the total undisturbed thickness of the fluids or long with respect to just one of the two layers, respectively. The removal of the traditional weak nonlinearity assumption is aimed at improving the agreement with the dynamics of Euler equations for large-amplitude waves. This is obtained without compromising much of the simplicity of the previously known weakly nonlinear models. Compared to these, the fully nonlinear models' most prominent feature is the presence of additional nonlinear dispersive terms, which coexist with the typical linear dispersive terms of the weakly nonlinear models. The fully nonlinear models contain the Korteweg–de Vries (KdV) equation and the Intermediate Long Wave (ILW) equation, for shallow and deep water configurations respectively, as special cases in the limit of weak nonlinearity and unidirectional wave propagation. In particular, for a solitary wave of given amplitude, the new models show that the characteristic wavelength is larger and the wave speed is smaller than their counterparts for solitary wave solutions of the weakly nonlinear equations. These features are compared and found in overall good agreement with available experimental data for solitary waves of large amplitude in two-fluid systems.

Numerical studies of two-dimensional, transonic flows of dense gases of retrograde type, known as BZT gases, around thin airfoils are presented. The computations are guided by a recent asymptotic theory of Rusak & Wang (1997). It provides a uniformly valid solution of the flow around the entire airfoil surface which is composed of outer and inner solutions. A new transonic small-disturbance (TSD) equation solver is developed to compute the nonlinear BZT gas flow in the outer region around most of the airfoil. The flow in the inner region near the nose of the airfoil is computed by solving the problem of a sonic flow around a parabola. Numerical results of the composite solutions calculated from the asymptotic formula are compared with the solutions of the Euler equations. The comparison demonstrates that, in the leading order, the TSD solutions of BZT gas flows represent the essence of the flow character around the airfoil as computed from the Euler equations. Furthermore, guided by the asymptotic formula, the computational results demonstrate the similarity rules for transonic flows of BZT gases. There are differences between the self-similar cases that may be related to the error associated with the accuracy of the asymptotic solution. A discussion on the flow patterns around an airfoil at transonic speeds and at various upstream thermodynamic conditions is also presented. The paper provides important guidelines for future studies on this subject.

Unsteady flow in collapsible tubes has been widely studied for a number of different physiological applications; the principal motivation for the work of this paper is the study of blood flow in the jugular vein of an upright, long-necked subject (a giraffe). The one-dimensional equations governing gravity- or pressure-driven flow in collapsible tubes have been solved in the past using finite-difference (MacCormack) methods. Such schemes, however, produce numerical artifacts near discontinuities such as elastic jumps. This paper describes a numerical scheme developed to solve the one-dimensional equations using a more accurate upwind finite volume (Godunov) scheme that has been used successfully in gas dynamics and shallow water wave problems. The adapatation of the Godunov method to the present application is non-trivial due to the highly nonlinear nature of the pressure–area relation for collapsible tubes.

The code is tested by comparing both unsteady and converged solutions with analytical solutions where available. Further tests include comparison with solutions obtained from MacCormack methods which illustrate the accuracy of the present method.

Finally the possibility of roll waves occurring in collapsible tubes is also considered, both as a test case for the scheme and as an interesting phenomenon in its own right, arising out of the similarity of the collapsible tube equations to those governing shallow water flow.

The breaking of internal waves propagating in a stratified fluid of constant buoyancy frequency on a sloping boundary was investigated numerically. It was found that at the boundary, nonlinear non-resonant interactions between the incident and reflected waves produced higher-mode waves. These modes had frequencies greater than the local buoyancy frequency and so could not radiate from the interaction region. The energy level of trapped waves increased with time and subsequently led to overturning of the density field. At the critical frequency, when the reflected wave propagated in a direction parallel to the slope, wave overturning occurred near the wall, but the point of overturning moved off the bottom as the propagation angle changed away from that of the bottom slope as the waves became increasingly supercritical. The internal wave reflection coefficient generally increased as the effects of nonlinearity and viscosity decreased, but depended strongly on the forcing frequency and the angle of the sloping boundary.

The intermittency of pressure and pressure gradient in stationary isotropic turbulence at low to moderate Reynolds numbers is studied by direct numerical simulation (DNS) and theoretically. The energy spectra scale in Kolmogorov units as required by the universal-equilibrium hypothesis, but the pressure spectra do not. It is found that the variances of the pressure and pressure gradient are larger than those computed using the Gaussian approximation for the fourth-order moments of velocity, and that the variance of the pressure gradient, normalized by Kolmogorov units, increases roughly as [Rscr ]1/2λ, where [Rscr ]λ is the Taylor microscale Reynolds number. A theoretical explanation of the Reynolds number dependence is presented which assumes that the small-scale pressure field is driven by coherent small-scale vorticity–strain domains. The variance of the pressure gradient given by the model is the product of the variance of ui,juj,i, the source term of the Poisson equation for pressure, and the square of an effective length of the small-scale coherent vorticity–strain structures. This length can be expressed in terms of the Taylor and Kolmogorov microscales, and the ratio between them gives the observed Reynolds number dependence. Formal asymptotic matching of the spectral scaling observed at small scales in the DNS with the classical scaling at large scales suggests that at high Reynolds numbers the pressure spectrum in these forced flows consists of three scaling ranges which are joined by two inertial ranges, the classical k−7/3 range and a k−5/3 range at smaller scale. It is not possible, within the classical Kolmogorov theory, to determine the length scale at which the inertial range transition occurs because information beyond the energy dissipation rate is required.

The film flow down an inclined plane has several features that make it an interesting prototype for studying transition in a shear flow: the basic parallel state is an exact explicit solution of the Navier–Stokes equations; the experimentally observed transition of this flow shows many properties in common with boundary-layer transition; and it has a free surface, leading to more than one class of modes. In this paper, unstable wavepackets – associated with the full Navier–Stokes equations with viscous free-surface boundary conditions – are analysed by using the formalism of absolute and convective instabilities based on the exact Briggs collision criterion for multiple k-roots of D(k, ω) = 0; where k is a wavenumber, ω is a frequency and D(k, ω) is the dispersion relation function.

The main results of this paper are threefold. First, we work with the full Navier–Stokes equations with viscous free-surface boundary conditions, rather than a model partial differential equation, and, guided by experiments, explore a large region of the parameter space to see if absolute instability – as predicted by some model equations – is possible. Secondly, our numerical results find only convective instability, in complete agreement with experiments. Thirdly, we find a curious saddle-point bifurcation which affects dramatically the interpretation of the convective instability. This is the first finding of this type of bifurcation in a fluids problem and it may have implications for the analysis of wavepackets in other flows, in particular for three-dimensional instabilities. The numerical results of the wavepacket analysis compare well with the available experimental data, confirming the importance of convective instability for this problem.

The numerical results on the position of a dominant saddle point obtained by using the exact collision criterion are also compared to the results based on a steepest-descent method coupled with a continuation procedure for tracking convective instability that until now was considered as reliable. While for two-dimensional instabilities a numerical implementation of the collision criterion is readily available, the only existing numerical procedure for studying three-dimensional wavepackets is based on the tracking technique. For the present flow, the comparison shows a failure of the tracking treatment to recover a subinterval of the interval of unstable ray velocities V whose length constitutes 29% of the length of the entire unstable interval of V. The failure occurs due to a bifurcation of the saddle point, where V is a bifurcation parameter. We argue that this bifurcation of unstable ray velocities should be observable in experiments because of the abrupt increase by a factor of about 5.3 of the wavelength across the wavepacket associated with the appearance of the bifurcating branch. Further implications for experiments including the effect on spatial amplification rate are also discussed.

The effect of solute concentration on hindered diffusion of sphere-like colloidal solutes in stiff polymer hydrogels is examined theoretically and experimentally. In the theoretical development, it is shown that the presence of the gel fibres enhances the effect of concentration on the thermodynamic driving force for gradient diffusion, while simultaneously reducing the effect of concentration on the hydrodynamic drag. The result is that gradient diffusion depends more strongly on solute concentration in gels than it does in pure solution, by an amount that depends on the partition coefficient and hydraulic permeability of the gel–solute system. Quantitative calculations are made to determine the concentration-dependent diffusivity correct to first order in solute concentration. In order to compare the theoretical predictions with experimental data, rates of diffusion have been measured for nonionic micelles and globular proteins in solution and agarose hydrogels at two gel concentrations. The measurements were performed by using holographic interferometry, through which one monitors changes in refractive index as gradient diffusion takes place within a transparent gel. If the solutes are modelled as spheres with short-range repulsive interactions, then the experimentally measured concentration dependence of the diffusivities of both the protein and micelles is in good agreement with the theoretical predictions.

In this paper we consider the causal response of the inviscid shear-layer flow over an elastic surface to excitation by a time-harmonic line force. In the case of uniform flow, Brazier-Smith & Scott (1984) and Crighton & Oswell (1991) have analysed the long-time limit of the response. They find that the system is absolutely unstable for sufficiently high flow speeds, and that at lower speeds there exist certain anomalous neutral modes with group velocity directed towards the driver (in contradiction of the usual radiation condition of out-going disturbances). Our aim in this paper is to repeat their analysis for more realistic shear profiles, and in particular to determine whether or not the uniform-flow results can be regained in the limit in which the shear-layer thickness on a length scale based on the fluid loading, denoted ε, becomes small. For a simple broken-line linear shear profile we find that the results are qualitatively similar to those for uniform flow. However, for the more realistic Blasius profile very significant differences arise, essentially due to the presence of the critical layer. In particular, we find that as ε → 0 the minimum flow speed required for absolute instability is pushed to considerably higher values than was found for uniform flow, leading us to conclude that the uniform-flow problem is an unattainable singular limit of our more general problem. In contrast, we find that the uniform-flow anomalous modes (written as exp (ikx − iωt), say) do persist for non-zero shear over a wide range of ε, although now becoming non-neutral. Unlike the case of uniform flow, however, the k-loci of these modes can now change direction more than once as the imaginary part of ω is increased, and we describe the connection between this behaviour and local properties of the dispersion function. Finally, in order to investigate whether or not these anomalous modes might be realizable at a finite time after the driver is switched on, we evaluate the double Fourier inversion integrals for the unsteady flow numerically. We find that the anomalous mode is indeed present at finite time, once initial transients have propagated away, not only for impulsive start-up but also when the forcing amplitude is allowed to grow slowly from a small value at some initial instant. This behaviour has significant implications for the application of standard radiation conditions in wave problems with mean flow.

Surface waves superimposed upon a larger-scale flow are blocked and reflected at the points where the group velocities balance the convection by the larger-scale flow. In this study, we first extended the theory of Shyu & Phillips (1990) to the situation when short deep-water gravity waves propagate obliquely upon a steady unidirectional irrotational current and are reflected by it. In this case, the uniformly valid solution and the WKBJ solution of the short waves were derived from the Laplace equation and the kinematical and dynamical boundary conditions. These solutions in terms of some parameters (the expressions for which have also been deduced in this case) take the same forms as those derived by Shyu & Phillips, which by referring to Smith's (1975) theory can even be proved to be valid for gravity waves in an intermediate-depth region and near a curved moving caustic induced by an unsteady multidirectional irrotational current. In this general case, the expressions for certain parameters in these solutions cannot be obtained so that their values must be estimated in a numerical calculation. The algorithm for estimates of some of these parameters that are responsible for the amplitude of the reflected wave not being equal to that of the incident wave in the vicinity of the caustic and therefore are crucial for the computer calculation of the ray solution to be continued after reflection, was illustrated through numerical tests. This algorithm can avoid the error magnification phenomenon that occurred in the previous estimates of the reflected wave in the vicinity of the caustic using the action conservation principle directly. The forms of the solutions have also been utilized to clarify the wave profiles near caustics in a general situation, which indicate that in storm conditions freak waves characterized by a steeper forward face preceded by a deep trough will probably occur in the caustic regions.

This paper concerns a relaxation of the assumption of uniform mixture composition in the interior of sonoluminescence bubbles. Intense temperature and pressure gradients within the bubble drive relative mass diffusion which overwhelms diffusion driven by concentration gradients. This thermal and pressure diffusion results in a robust compositional inhomogeneity in the bubble which lasts several orders of magnitude longer than the temperature peak or light pulse at the main collapse of the bubble. This effect has important consequences for control of sonoluminescence, gas dynamics, sonochemistry, and the physics of light production.

A finite-amplitude long-wave equation is derived to describe the effect of weak current shear on internal waves in a uniformly stratified fluid. This effect is manifested through the introduction of a nonlinear term into the amplitude evolution equation, representing a projection of the shear from physical space to amplitude space. For steadily propagating waves the evolution equation reduces to the steady version of the generalized Korteweg–de Vries equation. An analysis of this equation is presented for a wide range of possible shear profiles. The type of waves that occur is found to depend on the number and position of the inflection points of the representation of the shear profile in amplitude space. Up to three possible inflection points for this function are considered, resulting in solitary waves and kinks (dispersionless bores) which can have up to three characteristic lengthscales. The stability of these waves is generally found to decrease as the complexity of the waves increases. These solutions suggest that kinks and solitary waves with multiple lengthscales are only possible for shear profiles (in physical space) with a turning point, while instability is only possible if the shear profile has an inflection point. The unsteady evolution of a periodic initial condition is considered and again the solution is found to depend on the inflection points of the amplitude representation of the shear profile. Two characteristic types of solution occur, the first where the initial condition evolves into a train of rank-ordered solitary waves, analogous to those generated in the framework of the Korteweg–de Vries equation, and the second where two or more kinks connect regions of constant amplitude. The unsteady solutions demonstrate that finite-amplitude effects can act to halt the critical collapse of solitary waves which occurs in the context of the generalized Korteweg–de Vries equation. The two types of solution are then used to qualititatively relate previously reported observations of shock formation on the internal tide propagating onto the Australian North West Shelf to the observed background current shear.

The Darcy model with the Boussinesq approximations is used to study double-diffusive instability in a horizontal rectangular porous enclosure subject to two sources of buoyancy. The two vertical walls of the cavity are impermeable and adiabatic while Dirichlet or Neumann boundary conditions on temperature and solute are imposed on the horizontal walls. The onset and development of convection are first investigated using the linear and nonlinear perturbation theories. Depending on the governing parameters of the problem, four different regimes are found to exist, namely the stable diffusive, the subcritical convective, the oscillatory and the augmenting direct regimes. The governing parameters are the thermal Rayleigh number, RT, buoyancy ratio, N, Lewis number, Le, normalized porosity of the porous medium, ε, aspect ratio of the enclosure, A, and the thermal and solutal boundary condition type, κ, applied on the horizontal walls. On the basis of the nonlinear perturbation theory and the parallel flow approximation (for slender or shallow enclosures), analytical solutions are derived to predict the flow behaviour. A finite element numerical method is introduced to solve the full governing equations. The results indicate that steady convection can arise at Rayleigh numbers below the supercritical value, indicating the development of subcritical flows. At the vicinity of the threshold of supercritical convection the nonlinear perturbation theory and the parallel flow approximation results are found to agree well with the numerical solution. In the overstable regime, the existence of multiple solutions, for a given set of the governing parameters, is demonstrated. Also, numerical results indicate the possible occurrence of travelling waves in an infinite horizontal enclosure.

Analysis is used to show that a solution of the Navier–Stokes equations can be computed in terms of wave-like series, which are referred to as waves below. The mean flow is a wave of infinitely long wavelength and period; laminar flows contain only one wave, i.e. the mean flow. With a supercritical instability, there are a mean flow, a dominant wave and its harmonics. Under this scenario, the amplitude of the waves is determined by linear and nonlinear terms. The linear case is the target of flow-instability studies. The nonlinear case involves energy transfer among the waves satisfying resonance conditions so that the wavenumbers are discrete, form a denumerable set, and are homeomorphic to Cantor's set of rational numbers. Since an infinite number of these sets can exist over a finite real interval, nonlinear Navier–Stokes equations have multiple solutions and the initial conditions determine which particular set will be excited. Consequently, the influence of initial conditions can persist forever. This phenomenon has been observed for Couette–Taylor instability, turbulent mixing layers, wakes, jets, pipe flows, etc. This is a commonly known property of chaos.