An experimental study of Rayleigh-Bénard convection in helium gas at roughly 5 K is performed in a cell with aspect ratio 1. Data are analysed in a ‘hard turbulence’ region (4 × 107 < Ra < 6 × 1012) in which the Prandtl number remains between 0.65 and 1.5. The main observation is a simple scaling behaviour over this entire range of Ra. However the results are not the same as in previous theories. For example, a classical result gives the dimensionless heat flux, Nu, proportional to $Ra^{\frac{1}{3}}$ while experiment gives an index much closer to $\frac{2}{7}$. A new scaling theory is described. This new approach suggests scaling indices very close to the observed ones. The new approach is based upon the assumption that the boundary layer remains in existence even though its Rayleigh number is considerably greater than unity and is, in fact, diverging. A stability analysis of the boundary layer is performed which indicates that the boundary layer may be stabilized by the interaction of buoyancy driven effects and a fluctuating wind.

We study the behaviour of shallow (of order δ [Lt ] 1) water waves excited by a small (of order ε [Lt ] 1) amplitude bottom disturbance in the presence of a uniform oncoming flow with either constant or slowly varying Froude number F. When F* ≡ |F − 1|ε−½ [Gt ] 1, the speed and free surface perturbations are of order ν = O(ε); these grow to become of order ε½ if F* = O(1). Therefore, the asymptotic expansions of the solution for ε → 0 depend on the order of F*. These expansions are constructed in a form which remains valid for times of order ν−1; they are then matched to provide results which are also valid for all F. The analytic results exhibit the interesting effects of weak nonlinearities including the steepening of waves and eventual formation of bores if δν−½ [Lt ] 1, the surface rippling due to dispersion if δν−½ = O(1), the strong interaction of waves and the periodic generation of upstreampropagating solitary waves if F* = O(1), etc. All these results are confirmed by numerical integration of the governing equations.

A high resolution, two component laser-Doppler anemometer has been used for turbulence measurements at a high data rate in a channel flow of water. Measurements of the velocity components in the stream direction and in a direction normal to the wall are reported over the Reynolds number range of 3000–40000. The combination of high spatial resolution and high data rates enabled accurate reconstruction of time dependent velocity traces. Long-time statistical averages of these signals clearly show that profiles of the dimensionless turbulence quantities such as turbulence intensities and Reynolds stress are strongly Reynolds-number dependent over a large part of the channel flow. For instance, in the Reynolds-number range of this investigation, it is shown that the fluctuating turbulence quantities do not scale with wall variables even as close as 15 viscous lengths from the wall. The velocity traces and associated power spectra exposed two phenomena which may explain the Reynolds number dependencies.

Generalized Taylor dispersion theory extends the basic long-time, asymptotic scheme of Taylor and Aris greatly beyond the class of rectilinear duct and channel flow dispersion problems originally addressed by them. This feature has rendered it indispensable for studying flow and dispersion phenomena in porous media, chromatographic separation processes, heat transfer in cellular media, sedimentation of non-spherical Brownian particles, and transport of flexible clusters of interacting Brownian particles, to mention just a few examples of the broad class of non-unidirectional transport phenomena encompassed by this scheme. Moreover, generalized Taylor dispersion theory enjoys the attractive feature of conferring a unified paradigmatic structure upon the analysis of such apparently disparate physical problems. For each of the problems thus treated it provides an asymptotic, macroscale description of the original microscale transport process, being based upon a convective-diffusive ‘model’ problem characterized by a set of constant (position- and time-independent) phenomenological coefficients.

The present contribution formally substantiates the scheme. This is accomplished by demonstrating that the coarse-grained (macroscale) transport ‘model’ equation leads to a solution which accords asymptotically with the leading-order behaviour of the comparable solution of the exact (microscale) convective–diffusive problem underlying the transport process. It is also shown, contrary to current belief, that no systematic improvement in the asymptotic order of approximation is possible through the incorporation of higher-order gradient terms into the model constitutive equation for the coarse-grained flux. Moreover, the inherent difference between the present rigorous asymptotic scheme and the dispersion models resulting from Gill–Subramanian moment-gradient expansions is illuminated, thereby conclusively resolving a long-standing puzzle in longitudinal dispersion theory.

We investigate the stability of the one-dimensional solitary waves solutions of the equations proposed by McKenzie to model the ascent of melts in the Earth interior. We show that for small porosity and two-dimensional horizontal disturbances with long wavelength, these solitary waves are unstable. We also exhibit two- and three-dimensional solitary-wave solutions of the McKenzie equations.

It is shown that the linear and nonlinear exciting force coefficients, the added mass and radiation damping matrices, and the far-field wave amplitude (reflection and transmission coefficients in the two-dimensional case) can all be written as stationary values of well-defined functionals. As a consequence these quantities can be accurately determined with relatively crude approximations for the diffraction and radiation potentials. Numerical experiments confirm this feature: by inverting a 4 × 4 real symmetric matrix the results obtained by Vugts (1968), who computed the added mass and radiation damping matrices for several different geometries, were recovered over the whole range of frequencies.

Two integral theorems are proved which are applicable to the motion of an incompressible fluid in three dimensions. From either of these theorems one can derive the monopole component of the pressure fluctuation at infinity when a bubble undergoes non-spherical oscillations. The results confirm and generalize some recent calculations of this effect (Longuet-Higgins 1989a). They also provide a basis for a physical discussion of the origin of the monopole terms.

Statistical properties of the position of the turbulent/non-turbulent interface in the near wake of stationary and spinning cylinders have been measured for Re = 2000, using image processing techniques. The results include the first statistical moments, intermittency and burst-rate profiles, auto- and cross-correlation functions, auto- and cross-spectra and phase spectra. They indicate that the rotation of the cylinder causes substantial changes in the structure of the flow.

We use the methods of statistical mechanics to develop a theoretical relationship between the observed oceanic spectra and the probability distributions usually studied in statistical mechanics. We also find that the assumption that in terms of Lagrangian variables the oceanic internal wave field is near canonical equilibrium (i.e. the internal wave modes are populated in accordance with a Maxwell–Boltzmann-type distribution) yields expressions for the various marginal or reduced Eulerian spectra associated with both moored and towed measurements which are in striking qualitative agreement with experiment. In developing this theory it is important to distinguish carefully between Lagrangian and Eulerian variables. The important difference between the two sets of variables is due to the advective nonlinearity (i.e. (v∇) v where v is the Eulerian velocity) which is present only in the Eulerian frame. Our method treats the dynamics within the Lagrangian frame, where because of the absence of the advective nonlinearity it is fundamentally simpler, and then transforms to the Eulerian or measurement frame. We find that at small wavenumbers the four-dimensional Eulerian frequency wavenumber spectrum is approximately equal to the corresponding Lagrangian frequency wavenumber spectrum. At large wavenumbers, however, advective contributions become important and the two types of spectra are significantly different. While from a Lagrangian frame point of view the system is entirely wavelike, at large wavenumbers the Eulerian spectrum is not confined to the dispersion surface and the system, from an Eulerian frame point of view, is not wavelike. Further, the three-dimensional Eulerian wavenumber spectrum exhibits a large-wavenumber advective tail which decays as a power law and results in one-dimensional marginal spectra which are in excellent qualitative agreement with experiment. The above features are exhibited independent of the detailed nature of the underlying Lagrangian frequency wavenumber spectrum.

In this study the flow through and around a parallel-walled channel with an obstruction (flat plate) placed at the channel inlet is investigated. Depending on the position of the obstruction, the flow inside the channel is in a direction opposite to that outside, stagnant or in the same direction as outside but with reduced magnitude. Flow visualization in water is used to examine the fluid motion, although some wind tunnel measurements have been made and are also reported. The parameters that have been varied are the gap between the obstruction and the entry to the channel, the length of the channel and the Reynolds number. The maximum value of the reverse flow velocity is found to be about 20% of that of the flow outside. The maximum forward velocity inside the channel (when it occurs) is only about 65% of the outside velocity even for very large gaps between the obstruction and the channel entrance. A tentative explanation is offered for the observed features.

This paper describes the stabilization of compressor surge by an active method. It is known that surge follows when small disturbances grow in an unstable compression system, and that small growth can be modelled through a linear stability analysis. An active element is here introduced to counter any tendency to instability and the control law governing the active stabilizer is determined from linear theory. We follow precisely the suggestion put forward by Epstein et al. (1986) and verify that their theory conforms to practice. The theory is verified in an experiment on a compression system whose plenum volume is controlled. Suppression of the flow instability was achieved by switching on the controller and the compressor was made to operate stably on a part of its characteristic beyond the nature stall line. Furthermore the controlled compressor is much more resilient to external disturbances than is the natural case. The controller is even effective on deep surge – a feature of great interest but hardly predictable from the Epstein et al. initiative for this kind of study.

Thirty data sets of wavestaff measurements of a solitary feature in the Kelvin wake of the Coast Guard cutter Point Brower are analysed. The average characteristics of the feature between 1 and 4 km aft of the ship are shown to be consistent with those of an oblique nonlinear solitary wave packet. The ship speed is 7.7 m/s (Froude number 0.49) and the waves that comprise the packet have an average frequency of 3.28 rad/s. The ship speed and the wave frequency, together with Kelvin wake kinematics, imply that the feature appears at an average wake half-angle of 10.9°. The packet does not exhibit linear dispersion beyond 1 km aft of the ship and its average width is 8.9 m (measured at e−1 of the peak variance). The average amplitude is 1.1 times the theoretical amplitude of an oblique nonlinear solitary wave packet with the observed width. There is considerable variability from run to run, and there is evidence of dispersive spreading before 1 km aft of the ship. An aerial photograph shows a sinuous fluctuation of the feature; possible explanations for the fluctuation include small variations in initial conditions or a sinuous instability. The solitary feature is a possible explanation for the long bright lines observed in SEASAT SAR images in light to moderate winds and observed in sun glitter photos taken from the space shuttle.

This paper derives an equation for the potential-flow wave loading on a lattice-type offshore structure moving partially immersed in waves. It is for the limiting case of small lattice-member diameter, and deals entirely in member-centreline fluid properties, so that it can be applied computationally by a simple ‘stick model’ computer program. This field is currently served by a simple two-term semiempirical formula ‘Morison's equation’: the new equation is effectively a replacement for the Morison inertial term, allowing the Morison drag term (or some refinement of it) to describe exclusively the effects of vorticity, which can in principle be calculated to greater accuracy when isolated in this way.

The new equation calculates the potential-flow wave load accurate to second order in wave height, which is a great improvement on ‘Morison's equation’: such results can currently only be sought by very much more complicated and computationally intensive methods, of currently uncertain repeatability. Moreover the third-order error is localized at the free-surface intersection, so the equation remains attractive for fully nonlinear problems involving intermittent immersion of lattice members, which are currently beyond even the most sophisticated of these computationally intensive methods. It is shown that the primary reason for this large contrast in computational efficiency is that the loads are derived from energy considerations rather than direct integration of surface pressures, which requires a lower level of flow detail for a given level of load-calculation accuracy.

These improvements must of course be seen against the current levels of uncertainty over the calculation of vorticity-induced loads, which in many applications completely dwarf inaccuracies in potential-flow load calculation. The conditions are accordingly established under which the improvements are comparable to the total wave load predicted by the Morison drag and inertia terms in combination. They are that the lattice member diameter is greater than its length/10, or the relative fluid motion/5, or the structure's motion radius/20, or the wavelength/30: if any one of these conditions is satisfied, the new equation is worthwhile even when used in combination with simple vorticity-induced load calculations from a Morison drag term.

Techniques of bifurcation theory are used to study the porous-medium analogue of the classical Rayleigh-Bénard problem: Lapwood convection in a two-dimensional saturated porous cavity heated from below. Two particular aspects of the problem are focused upon: (i) the existence of multiple steady solutions and (ii) the influence of aspect ratio.

Convection begins only when the applied temperature difference (say) exceeds a critical value defined by linear stability theory. The resulting convective flow pattern depends both on the magnitude of the temperature difference and on the aspect ratio of the cavity. A weakly nonlinear analysis reveals the roles played by so-called secondary bifurcations in determining the formation of further, anomalous patterns at fixed aspect ratio. In addition to giving rise to alternative stable flows for identical operating conditions, the secondary bifurcations are required for the modal exchanges which take place as the aspect ratio varies, a process which causes an abrupt change in preferred flow pattern at certain critical values of the aspect ratio.

As a complement to and an extension of the weakly nonlinear analysis, numerical methods are used to determine the bifurcation processes and to elucidate the modal exchange mechanisms in both weakly and strongly convective flows. The effect of container size is studied by continuation methods to predict the variation of the critical Rayleigh number of the bifurcation points for aspect ratios in the range 0.5 to 2.0. In this way a stability map is obtained which shows the alternative patterns expected for particular operating conditions. The Nusselt number is computed and it is found that the alternative stable modes transfer significantly different amounts of heat through the medium.

The study has provided new information on the existence and characteristics of, and interactions between, alternative steady modes of two-dimensional Lapwood convection. The results have important ramifications for the modelling and design of physical systems in which convective flow in a saturated porous medium is stimulated by an imposed unstable temperature gradient.

The structure of turbulence and its role in the breaking wave dynamics within the surf zone have been investigated through laboratory experiments using several flow visualization techniques and a fibre-optic LDV system. The results indicate that there exists a characteristic structure of large-scale eddies referred to here as ‘horizontal eddies’ and ‘obliquely descending eddies’, which has a significant role in the generation of Reynolds stress and thus affects the deformation of the mean flow field. The experiments also reveal that these eddies caused by the wave breaking bring a large amount of vorticity (with non-zero average) into otherwise almost irrotational velocity fields, resulting in the generation of vorticity-related mean flow fields as well as turbulence (vorticity-containing velocity fluctuation). This means that the breaking waves in the surf zone can be regarded as pseudowaves which consist of irrotational velocity components as ‘wave motion’ and appreciable amounts of rotational mean velocity components as ‘eddying motion’ (with non-zero mean vorticity) together with turbulence. It is found that the generation of the mean rotational velocity component due to wave breaking causes considerable increase in mass and momentum transport, as compared with ordinary non-breaking waves, and thus a decrease in wave height.

The present work gives a comprehensive numerical study of the evolution and decay of cylindrical and spherical nonlinear acoustic waves generated by a sinusoidal source. Using pseudospectral and predictor–corrector implicit finite difference methods, we first reproduced the known analytic results of the plane harmonic problem to a high degree of accuracy. The non-planar harmonic problems, for which the amplitude decay is faster than that for the planar case, are then treated. The results are correlated with the known asymptotic results of Scott (1981) and Enflo (1985). The constant in the old-age formula for the cylindrical canonical problem is found to be 1.85 which is rather close to 2, ‘estimated’ analytically by Enflo. The old-age solutions exhibiting strict symmetry about the maximum are recovered; these provide an excellent analytic check on the numerical solutions. The evolution of the waves for different source geometries is depicted graphically.

The secondary instability of large-amplitude Görtler vortices in a growing boundary layer is discussed in the fully nonlinear regime. It is shown that the three-dimensional breakdown to a flow with wavy vortex boundaries, similar to that which occurs in the Taylor vortex problem takes place. However, the instability is confined to the thin shear layers which were shown by Hall & Lakin (1988) to trap the region of vortex activity. The disturbance eigenfunctions decay exponentially away from the centre of these layers, so that the upper and lower shear layers can support independent modes of instability. The structure of the instability, in particular its location and speed of downstream propagation, is found to be entirely consistent with recent experimental results. Furthermore, it is shown that the upper and lower layers support wavy vortex instabilities with quite different frequencies. This result is again consistent with the available experimental observations.

It is shown that the shear-induced distortion of the static structure factor (or equivalently, the pair distribution function) of colloidal fluids is a non-analytic function of the shear rate. The Smoluchowski equation shows boundary-layer behaviour at zero wave vector (large separations). The width of this boundary layer is proportional to the square-root of the shear rate. An explicit perturbation analysis of the two-particle Smoluchowski equation without hydrodynamic interactions is given. The onset of the non-Newtonian behaviour of the effective vicosity is due to the boundary-layer behaviour of the structure factor.

We derive several versions of the (complex) amplitude equation of an inviscid wave packet travelling on a slightly inhomogeneous (and possibly unsteady and viscous) unstable base flow. This is done with complete generality, without any reference to the dimensions of physical and propagation spaces, by using the usual high-frequency ansatz. The final results are extremely simple: volume integrals of a complex wave action density are conserved subject to an appropriate flux and a source term. The latter is expressible in a remarkably concise way in terms of the gradient of the base flow acceleration and vanishes when the base flow is inviscid. The simplicity of our results hinges on a transformation of the dependent variables and on a suitable decomposition of these in cross- and propagation spaces. Our results are also discussed with the help of three different Lagrangian densities and their associated kinematic wave theories which are based on a basic identity due to Hayes.

Measurements of the Reynolds stresses, integral lengthscales and Taylor microscales are reported for several cases of uniformly sheared turbulent flows with shear values in a range substantially wider than those of previous measurements. It is shown that such flows demonstrate a self-preserving structure, in which the dimensionless Reynolds stress ratios and the dissipation over production ratio, ε/P, remain essentially constant. Flows with sufficiently large $k_{\rm s} = (1/\overline{U_{\rm c}}){\rm d}\overline{U_1}dx_2$ have exponentially growing stresses and ε/P ≈ 0.68; a linear relationship between the coefficient in the exponentiallaw and ks is shown to be compatible with measurements having ks > 3. The possibility of a self-preserving structure with asymptotically constant stresses and ε/P ≈ 1.0 is also compatible with measurements, corresponding to flows with small values of ks. The integral lengthscales appear to grow according to a power law with an exponent of about 0.8, independent of the mean shear, while the Taylor microscales, in general, approach constant values. Various attempts to scale the stresses and to predict their evolution are discussed and the applicability of Hasen's theory is scrutinized. Finally, an ‘exact’ expression for the pressure-strain rate covariance is derived and compared to some popular models.