The evolution of two oblique wave packets trailing, a transitional spot in a laminar boundary layer was investigated in order to determine the extent of the interaction between the packets and the spot. The experimental investigation, carried out on two slightly different laminar boundary layers characterized by Falkner-Skan constants of β = 0 and β = 0.2, revealed that very small pressure gradients can have significant effects on the stability of the laminar boundary layer and the rate at which it is contaminated by a turbulent spot. Some simple, novel statistical procedures for treating the data were developed and were used to accentuate the understanding of the physical processes governing transition to turbulence.

A pattern-recognition procedure designed to extract footprints of organized structures from turbulent signals is developed and used to analyse the large-eddy organization of several turbulent wake flows. The pattern-recognition technique is intended to be a general-purpose analytical tool that makes no use of specific flow characteristics, and that can be implemented as a computer code independent of the types of signals to be processed. The technique is applied to analyse the wake generated by a single cylinder at downstream positions ranging from x/D = 10 to x/D = 220. Also the structural features of the wakes behind a rotating cylinder, two cylinders of unequal diameters and two cylinders of equal diameter, one rotating, are examined at x/D = 140. In the near wake the large-scale motions detected are Kármán vortices, whose periodic activity persists up to 60 diameters. Further downstream the most significant coherent structures detected are single and double rollers with shear-aligned vorticity, whose dimensions and velocity intensities are properly scaled by the half-width of the wake and the local r.m.s. values, respectively. The similarities observed in the organized motions identified in the different wakes at x/D = 140, suggest that the roller organization may be an intrinsic characteristic of fully developed turbulent plane wake flows, irrespective of initial conditions.

The temperature signals measured in a fully turbulent, thermally contaminated wake behind a heated cylinder at x/D = 140, are analysed using a pattern-recognition procedure. The temperature patterns detected show that the wake flow is dominated by shear-aligned structures with a strong three-dimensional character. The temperature footprints have a limited extent in the spanwise and streamwise directions, while their dimension is of the order of the wake width in the vertical coordinate. The footprints are characterized by a steep temperature gradient at their back edge, which appears as a sharp hot-to-cold transition. The topological features inferred from the temperature signals suggest that the entrainment process is accomplished mainly by engulfing external fluid. The large-scale organized structures are interpreted as the temperature footprints of the double-rollers characterizing the large-eddy motion in far wakes. The analysis of present results within the context of previous studies about intermittency in heated wakes, brings forward the idea that the intermittent bulges are the emerging double rollers at the three-dimensional turbulent/non-turbulent interface.

An important phase of turbulence production in the flow past a wall occurs with the intermittent inflexional instability of the streamwise current. But according to a linearized inviscid calculation this instability (Kelvin-Helmholtz) can be reduced or eliminated by the presence of a (slippery) wall. Attention is therefore directed to the temporal evolution of a finite-amplitude patch of inflected fluid, i.e. one that is localized in the downstream direction. The model has piecewise uniform vorticity, and the contour-dynamical method is used. Numerical integrations show that sufficiently wide initial patches will eject slow fluid near the wall until it comes into close contact with the free stream, whereupon the ejection is deflected around a large eddy which is surrounded by a stable shear flow. The parametric regime in which this kind of finite instability occurs is sketched, and the Reynolds stress is computed. The initial condition assumed in this calculation depends on the prior existence and intensification of a local spanwise circulation, and this process is briefly discussed using a separate two-dimensional calculation. This shows that widely separated vorticity isopleths tend to completely merge, implying that such fronts in a real fluid may only be viscously limited. The analogous process of potential vorticity frontogenesis may be important in oceanic coastal currents.

The stability to three dimensional disturbances of bubbles rising rectilinearly in an inviscid fluid is studied numerically. It is found, in contrast with earlier work, that the interaction of hydrodynamic pressure forces and surface tension does not lead to linear instability of the bubble path.

The steady motion of a flat surfboard propelled by a solitary wave is considered. The shape of the free surface and the flow of the fluid are determined numerically by series truncation for flows without spray or splash. These flows all bifurcate from the uniform horizontal flow at the critical value of the Froude number. Various limiting cases of these special flows are described analytically. Flows past submerged hydrofoils are discussed also.

The work, addressing subsonic, supersonic and hypersonic boundary-layer instability, is motivated by the need for more understanding of compressible transition at high global Reynolds numbers Re. In the supersonic case, the so-called ‘first modes’ of instability found/suggested by previous Orr-Sommerfeld computations can be identified as triple-deck oblique ones, directed outside the local wave-Mach-cone directions. Less oblique instability modes inside are not of Orr-Sommerfeld form since they are substantially affected by non-parallel flow effects. The maximum linear growth rates are determined for a range of supersonic and subsonic free-stream Mach numbers M∞, and comparisons are made with previous computations, showing fairly good agreement at moderate Mach numbers. A second mound of unstable wavenumbers and frequencies is also evident. In addition, the nonlinear version is set up and emphasized (with attention drawn to a recent paper by the author (1988a) showing the possibility of nonlinear break-up), and certain extremes are examined including those of transonic and hypersonic boundary layers. In the hypersonic limit a new regime is found (for many conditions, including the insulated plate), namely $M_{\infty \sim Re^{\frac{1}{10}}$, in which non-parallel-flow effects enter to control the main disturbances, and it is concluded that the restriction $M_{\infty \ll Re^{\frac{1}{10}}$ applies to the Orr-Sommerfeld approach. This is a very severe restriction in practice.

We consider the centrifugal separation of an initially homogeneous mixture in an asymmetric geometry. The mixture is shown to acquire a uniform and negative relative vorticity, manifested as a retrograde circulation, as the heavier phase is forced outwards by the centrifugal acceleration. The perturbation theory formulated accounts for inertial effects and endwall boundary layers as well as slight deviations of endwall shape from a level plane. The time-dependent separation process is described and the flow field, including kinematic shocks, is calculated in some cases of technological interest.

The Benjamin (1968) analysis of a two-fluid density current is extended to include the effect of vorticity within the current. In the case of constant vorticity, the density-current depth is shown to lie between the limits of half and two-thirds of the channel depth. More general vorticity distributions are also considered, namely those that have: (i) a maximum in the upper and lower regions of the density current; and (ii) a maximum in the middle of the density current. In the former, as in the case of constant vorticity, density-current structures exist, whereas in the latter, deep overturning circulations predominate which can cause a ‘blocking’ of the upstream inflow. A generalized propagation formula which includes the effects of finite depth and rear inflow into the density current is established and the uniqueness issue is considered.

The analysis is further extended to a three-fluid system, composed of physically distinct component flows, namely, a density current, an overturning updraught region in upper levels ahead of the density current and an updraught in which the fluid ascends without overturning to its outflow level. Two types of behaviour are identified. First, a symmetric mode in which the density current and the overturning updraught have the same depth and, second, an asymmetric mode with solutions restricted to a certain parameter range. A special case in which the fluids have the same density illustrates the basic dynamics of the problem and also the nature of the vertical transport of momentum.

High-resolution, high-Reynolds-number numerical solutions of fully three-dimensional, decaying, geostrophic turbulence are examined. The results include the demonstration of a substantial degree of similarity between geostrophic and two-dimensional turbulence: transfer of energy to larger scales; transfer of potential enstrophy to smaller scales; vanishing energy dissipation as the Reynolds number increases; the emergence and growth to dominance of isolated, coherent vortices; and a competition between the vortices and Rossby waves, with an associated horizontal anisotropy when the latter are dominant. Properties that are distinct to geostrophic turbulence include the following: approximate three-dimensional wavenumber isotropy, with significant departures on large scales due to boundedness of the domain and on smaller scales due to anisotropic spectrum transfer rates; insensitivity of solution properties to anisotropy or vertical inhomogeneity in the dissipation; persistence of vertical inhomogeneity; development of inhomogeneity due to solid vertical boundaries; and the processes of alignment, attachment, and vertical straining associated with the finite vertical extent of the coherent vortices.

When fluid is withdrawn from a body of stratified fluid the surfaces of constant density are deformed towards the region of withdrawal. The equations describing the flow caused by withdrawal through a point sink in a two-layer unbounded system in which viscous forces dominate are formulated using the boundary-integral representation of Stokes flow. It is shown by dimensional and analytic arguments that surface tension between the layers is a necessary condition for the stability of an interfacial equilibrium in which only one fluid is withdrawn. The critical flow rate above which both fluids are withdrawn is determined numerically as a function of the capillary number. When the flow is supercritical a small adaptation of the numerical scheme allows the proportion of fluid withdrawn from each layer to be found. The various analyses and conclusions further our understanding of the physical processes that determine the compositional output of volcanic eruptions that tap an underlying stratified reservoir of magma.

The Orr-Sommerfeld equation admits two solution modes for the two-dimensional plane wake. These are the sinuous mode with antisymmetric streamwise fluctuations and the varicose mode with symmetric streamwise fluctuations. The varicose mode is often ignored because its amplification rates are considerably less than those of the sinuous mode. An experimental investigation of the varicose mode in a two-dimensional turbulent wake was undertaken to determine if this mode of instability agrees as well with linear stability theory, as did the sinuous mode in previous experiments (Wygnanski, Champagne & Marasli 1986). The experiments demonstrated that, although it is possible to generate a nearly pure symmetric disturbance wave, it is very difficult to do as the flow is very sensitive to the slightest asymmetries which might be present in the experiments. These asymmetries are preferentially amplified, resulting in the eventual distortion of an initially prominent symmetric wave. It was therefore necessary to decompose phase-averaged measurements of the streamwise component of the velocity fluctuations into their symmetric and antisymmetric parts, and the results were compared with the appropriate theoretical eigenfunctions from linear stability theory. The lateral distribution of the amplitude and the phase of each mode agree reasonably well with their theoretical counterparts from the Orr-Sommerfeld equation. Slowly diverging linear theory predicts the streamwise variation of the sinuous mode quite well, but fails to do so for the varicose mode. An eddy-viscosity model, coupled with the slowly diverging linear equations, predicts the streamwise variation of both modes reasonably well and describes the transverse distributions of the perturbation amplitudes for both modes, but it fails to predict the distribution of phase for the varicose mode.

A formulation, previously employed to find exact Navier-Stokes solutions for planar disturbances in two- and three-dimensional flows with spatially uniform rates of strain, is here adapted to incorporate the contribution of various types of body force. In the absence of body forces, it is known that unbounded flows with constant vorticity and elliptical streamlines are unstable to certain planar disturbances, which are amplified by a Floquet mechanism. The influence of a Coriolis force upon this instability mechanism is here described in detail, as an illustration of the general formulation. The results are likely to be of geophysical interest and may also have relevance to the breakdown of closed-eddy structures in turbulence. The final section of the paper reviews other systems for which analogous exact solutions may be obtained.

We analyse the melting and/or freezing that can occur when a very large layer of hot fluid begins to flow turbulently over a cold solid retaining boundary. This is a form of Stefan problem and the response is determined by the balance between the turbulent heat flux from the fluid, H, and the (initially infinite) conductive flux into the solid. We show that solidification of the flow at the boundary must always occur initially, unless the freezing temperature of the fluid, Tf, is less than the initially uniform temperature, T0, of the semi-infinite solid. We determine the evolution of the solidified region and show that with time it will be totally remelted. Melting and ablation of the solid retaining boundary will then generally follow, unless its melting temperature exceeds that of the turbulent flow. The maximum thickness of the solidified crust is shown to scale with k2(Tf − T0)2/ρkHL and its evolution takes place on a timescale of k2(Tf − T0)2/kH2, where k is the thermal conductivity, k the thermal diffusivity, ρ the density and L the latent heat, with all these material properties assumed to be equal for fluid and solid.

The spectral balances involved in shaping the short gravity wave region of the ocean wave-height spectrum have been the subject of recent physical models. In terms of the wind friction velocity u*, gravitational acceleration g and local wavenumber k, these models predict a wavenumber dependence of $k^{-\frac{7}{2}}$, where k = |k|, and a linear dependence on u* for the equilibrium range of gravity waves above the spectral peak. In this paper we present the results of an experimental determination of the wavenumber spectrum for the wavelength range of 0.2−1.6 m, based on stereophotogrammetric determinations from an oil platform under open ocean conditions.

From our observations, for this wavenumber range, the one-dimensional equilibrium wavenumber spectrum was determined as \[ \phi (k_i) \sim \left(\frac{u^2_*k}{g}\right)^{\gamma} k^{-3}_{i}\;\;\;\;\;\;\;(i=1,2 \;\;\; K = (k_1,k_2)) \] where γ = 0.09±0.09 at the 95% confidence level. These limits embrace wind-independent approximations to the observed one-dimensional and two-dimensional wavenumber spectra of the form \[ \phi (k_i) \sim B k^{-3}_i \;\;\; (i = 1,2), \] and \[ \psi(k_i) \sim A k^{-4}, \] respectively, with B ∼ 10−4 and A ∼ 0.3 × 10−4 for $(u^2_*k_i/g)=10^{-2}$ and k = |k| is expressed in cycles/metre.

The present findings do not support the wavenumber dependence predicted by the recent models in this wavenumber range and are at variance with their predicted dependence on the friction velocity. However, our observations are generally consistent with the radar reflectivity dependence on wind direction and wind speed under Bragg scattering conditions within our wavenumber range. The experimental observations also point out the potentially important role of wave-breaking of longer wave components in influencing the spectral levels of short gravity wave components.

Steady three-dimensional convection flows induced by the knot instability of two-dimensional convection rolls are studied numerically for various Prandtl numbers. The Galerkin method is used to obtain the three-dimensional solutions of the basic equations in the case of rigid, infinitely conducting boundaries. These solutions exhibit the typical knot-like structure superimposed onto the basic rolls. The Nusselt number and kinetic energy of motion do not differ much for two- and three-dimensional solutions and the toroidal part of the kinetic energy associated with vertical vorticity always remains a small fraction of the total in the case of the knot solution. The analysis of the steady solutions is complemented by a stability analysis with respect to disturbances that fit the same horizontal periodicity interval as the knot solution. All instabilities correspond to Hopf bifurcations. Some example of finite-amplitude oscillatory knot convection are presented.

This paper discusses the refraction of plane shock waves in media with arbitrary equations of state. Previous work is reviewed briefly, then a rigorous definition of wave impedance is formulated. Earlier definitions are shown to be unsatisfactory. The impedance is combined with the boundary conditions at the media interface to study both head-on and oblique shock incidence. The impedance determines the nature of the reflected and transmitted waves, their intensities, and the fractions of energy and power that are reflected and transmitted. The refractive index is also defined and determines whether or not a wave will be refracted, and also helps determine whether the wave system will be regular or irregular. The fundamental law of refraction is derived and shown to be a consequence of the fact that an arbitrary point on a shock or an expansion wave follows a ray path of minimum time between any two points on the path. This is a generalization of Fermat's Principle to media that are deformed and convected by the waves propagating through them.

The envelope equation of Dysthe (1979), which provides an extension of the nonlinear Schrödinger equation (NLS) to fourth order in wave steepness, is used to discuss higher-order modulation effects on the long-time evolution of solitary wave envelopes in deep water. The Dysthe equation admits solitary-wave solutions, similar to those of the NLS. Using perturbation methods, it is shown that an initial disturbance in the form of a solitary wave group of the NLS evolves to a solitary wave of the Dysthe equation having lower peak amplitude and moving with higher speed than the original wave; the increase in wave speed is caused by a downshift in wavefrequency. Asymptotic expressions are derived for this amplitude decrease and frequency downshift, which are consistent with numerical and experimental results.

In this paper we examine some general features of the time-dependent dynamics of drop deformation and breakup at low Reynolds numbers. The first aspect of our study is a detailed numerical investigation of the ‘end-pinching’ behaviour reported in a previous experimental study. The numerics illustrate the effects of viscosity ratio and initial drop shape on the relaxation and/or breakup of highly elongated droplets in an otherwise quiescent fluid. In addition, the numerical procedure is used to study the simultaneous development of capillary-wave instabilities at the fluid-fluid interface of a very long, cylindrically shaped droplet with bulbous ends. Initially small disturbances evolve to finite amplitude and produce very regular drop breakup. The formation of satellite droplets, a nonlinear phenomenon, is also observed.

Measurements of velocity and temperature fluctuations are made in a turbulent boundary layer with nominally zero pressure gradient for two different slightly heated wall conditions: impermeable and porous surfaces. The temperature fluctuations are measured at three points in the flow to permit the identification of two spatially coherent events: coolings and heatings. Conditional velocity vectors in the plane of mean shear are viewed in a reference frame which translates at a constant velocity. Conditioning is on coolings, heatings or a combination of these events. Sectional streamlines, derived from the velocity vector data, show a succession of critical points: saddles and unstable foci. Coolings are aligned with diverging separatrices through the saddles whereas heatings are identified with the foci. Coolings are associated with a large strain rate and also a large spanwise vorticity: this result seems consistent with the presence of hairpin vortices which extend to different distances from the wall. In contrast, the strain rate and spanwise vorticity are small at the foci. The stabilizing influence of suction is observed in the topology of the organized motion and in the contribution from this motion to the conventional stresses, temperature variance and heat fluxes.