In a previous paper, Phillips (1960) showed that two or three trains of gravity waves may interact so as to produce a fourth (tertiary) wave whose wave-number is different from any of three primary wave-numbers k1, k2, k3, and whose amplitude grows in time. Such resonant interactions may produce an appreciable modification of the spectrum of ocean waves within a few hours. In this paper, by a slightly different method, the interaction is calculated in detail for the simplest possible case: when two of the three primary wave-numbers are equal (k3 = k1).

It is found that, when k1 and k2 are parallel or antiparallel, the interaction vanishes unless k1 = k2. Generally, if θ denotes the angle between k1 and k2, the rate of growth of the tertiary wave with time is a maximum when θ [eDot ] 17°; the rate of growth with horizontal distance is a maximum when θ [eDot ] 24°. The calculations show that it should be possible to detect the tertiary wave in the laboratory.

Surface-renewal motions in the interfacial region below a gas-liquid interface were experimentally investigated in relation to bursting motions in the wall region. To estimate the frequency of the appearance of surface-renewal eddies, mass-transport experiments with methylene-blue solution, together with velocity measurements, were done in an open-channel flow. The instantaneous concentration of methylene-blue tracer emitted from a point source positioned in the buffer layer was measured at the free surface downstream from the source by an optical probe. Instantaneous streamwise velocity was measured using a laser-Doppler velocimeter at a position in the buffer region. Frequencies of both surface-renewal and bursting events were computed from these concentration and velocity signals using a conditional-averaging method. In order to clarify whether the surface-renewal eddies actually dominate mass transfer across the gas-liquid interface, gas-absorption experiments were added. Carbon dioxide was absorbed into the water flow across the calm free surface and its mass-transfer coefficient on the liquid side was measured under the same flow conditions as used in the above mass-transport experiments. The results show that the surface-renewal motions originate in the bursting motions which vigorously occur in the buffer region. That is, the decelerated fluid which is strongly lifted towards the outer layer by bursting almost always arrives at the free surface and renews the free surface. The frequency of the surface renewal, as well as the bursting frequency, is uniquely determined by the wall variables or the outer-flow variables and the Reynolds number. Mass transfer across the gas-liquid interface is dominated by the large-scale surface-renewal eddies, and the mass-transfer coefficient on the liquid side is proportional to the square-root of the surface-renewal frequency.

This paper presents a study of the convection in acetone due jointly to the thermocapillary (Marangoni) and thermogravitational effects. The liquid (acetone) is submitted to a horizontal temperature difference. Experiments and numerical simulations both show the existence of three different states: monocellular steady states, multicellular steady states and spatio-temporal structures. The results are discussed and compared with the linear stability analysis of Smith & Davis (1983).

An experimental investigation was designed to test the hypothesis that all axisymmetric turbulent free jets become asymptotically independent of the source conditions and may be described by classical similarity analysis. Effects of initial conditions were studied by varying jet exit boundary conditions and the global density ratio. The exit velocity profile and turbulence level was changed by using both pipe and nozzle flow hardware. Initial density differences were imposed by using three gases: helium, methane, and propane. The scalar field (concentration) in the momentum-dominated regime of the far field (10 to 60 jet exit diameters downstream) of turbulent free jets was characterized using Rayleigh light scattering as the diagnostic. The results show that regardless of the initial conditions axisymmetric turbulent free jets decay at the same rate, spread at the same angle, and both the mean and r.m.s. values collapse in a form consistent with full self-preservation. The means and fluctuations follow a law of full self-preservation in which two virtual origins must be specified. The two displacements are required to account for the effects of a finite source of momentum and different development of the velocity and mass distributions in the near fields of the jets. The memory of the jet is embodied in these two virtual origins.

The time-space evolution of a steep water wave reaching the breaking stage is observed by means of a visualization technique. In particular, the asymmetry of the wave profile in the near-breaking region is displayed. Measurements at breaking onset on a sample of breaking waves show a relation between the rate of asymmetry growth and the breaker type. The shape evolution of a plunging crest after breaking has started, and the related splash-up phenomenon and its part in the air-entrainment process are also observed.

Neither the phenomenological nor the structural approach to the determination of constitutive equations has yet shown itself to be capable of producing useful and predictive descriptions of the majority of technologically important complex fluids. In the present paper we explore the suggestion that significant progress can be made when these two complementary approaches to rheology are combined. For this initial study we restrict our attention to materials which can be modelled as a suspension of particles in a Newtonian fluid, thereby including most polymer solutions while excluding polymer melts. By applying phenomenological techniques to the basic formulation of suspension mechanics we are able to deduce a common simplified constitutive model for all suspension-like materials and to reveal its physical origin. The present analysis demonstrates that the constitutive model of Hand (1962), involving a single second-order tensor, is not sufficiently general for a rigorous description of the majority of suspension-like materials. Consideration of the constitutive forms for the limiting cases of near-equilibrium and strongly non-equilibrium microstructure suggests, however, that Hand's model may provide a reasonable approximation to the exact constitutive behaviour which is useful over the whole range of flow strengths.

The problem of the slow viscous flow of a fluid through a random porous medium is considered. The macroscopic Darcy's law, which defines the fluid permeability k, is first derived in an ensemble-average formulation using the method of homogenization. The fluid permeability is given explicitly in terms of a random boundary-value problem. General variational principles, different to ones suggested earlier, are then formulated in order to obtain rigorous upper and lower bounds on k. These variational principles are applied by evaluating them for four different types of admissible fields. Each bound is generally given in terms of various kinds of correlation functions which statistically characterize the microstructure of the medium. The upper and lower bounds are computed for flow interior and exterior to distributions of spheres.

The velocity measuring technique described in this paper consists of measuring the time of flight of a tracer of heated air from an electrically pulsed wire to one of two sensor wires which are operated as resistance thermometers. These sensor wires are at right angles to the pulsed wire and are placed one on either side of the pulsed wire. The instrument may be used in highly turbulent flows including regions in which flow reversals occur. The paper discusses the theoretical behaviour of the probe and the results of some calibration experiments.

We investigate the in-line and cross-flow vortex-induced vibrations of a long cylindrical tensioned beam, with length to diameter ratio L/D = 200, placed within a linearly sheared oncoming flow, using three-dimensional direct numerical simulation. The study is conducted at three Reynolds numbers, from 110 to 1100 based on maximum velocity, so as to include the transition to turbulence in the wake. The selected tension and bending stiffness lead to high-wavenumber vibrations, similar to those encountered in long ocean structures. The resulting vortex-induced vibrations consist of a mixture of standing and travelling wave patterns in both the in-line and cross-flow directions; the travelling wave component is preferentially oriented from high to low velocity regions. The in-line and cross-flow vibrations have a frequency ratio approximately equal to 2. Lock-in, the phenomenon of self-excited vibrations accompanied by synchronization between the vortex shedding and cross-flow vibration frequencies, occurs in the high-velocity region, extending across 30% or more of the beam length. The occurrence of lock-in disrupts the spanwise regularity of the cellular patterns observed in the wake of stationary cylinders in shear flow. The wake exhibits an oblique vortex shedding pattern, inclined in the direction of the travelling wave component of the cylinder vibrations. Vortex splittings occur between spanwise cells of constant vortex shedding frequency. The flow excites the cylinder under the lock-in condition with a preferential in-line versus cross-flow motion phase difference corresponding to counter-clockwise, figure-eight orbits; but it damps cylinder vibrations in the non-lock-in region. Both mono-frequency and multi-frequency responses may be excited. In the case of multi-frequency response and within the lock-in region, the wake can lock in to different frequencies at various spanwise locations; however, lock-in is a locally mono-frequency event, and hence the flow supplies energy to the structure mainly at the local lock-in frequency.

Using hot-wire-anemometer dynamic-calibration methods, fully developed pipe-flow turbulence measurements have been taken in the Reynolds-number range 80 × 103 to 260 × 103. Comparisons are made with the results of previous workers, obtained using static-calibration methods. From the dynamic-calibration results, a consistent and systematic correlation for the distribution of turbulence quantities becomes evident, the resulting correlation scheme being similar to that which has previously been established for the mean flow. The correlations reported have been partly conjectured in the past by many workers but convincing experimental evidence has always been masked by the scatter in the results, no doubt caused by the difficulties associated with static-calibration methods, particularly the earlier ones. As for the mean flow, the turbulence intensity measurements appear to collapse to an inner and outer law with a region of overlap, from which deductions can be made using dimensional arguments. The long-suspected similarity of the turbulence structure and its consistency with the established mean-flow similarity appears to be confirmed by the measurements reported here.

A mathematical model of convection, obtained by truncation from the two-dimensional Boussinesq equations, is shown to exhibit a bifurcation from symmetrical cells to tilted non-symmetrical ones. A subsequent bifurcation leads to time-dependent flow with similarly tilted transient plumes and a large-scale Lagrangian mean flow. This change of symmetry is similar to that occurring with the advent of a large-scale flow and transient tilted plumes seen in laboratory experiments on turbulent convection at high Rayleigh number. Though not intended as a description of turbulent convection, the model does bring out in a theoretically tractable context the possibility of the spontaneous change of symmetry suggested by the experiments.

Further bifurcations of the model lead to stable chaotic phenomena as well. These are numerically found to occur in association with heteroclinic orbits. Some mathematical results clarifying this association are also presented.

This study provides a detailed exploration of the near-field shear-layer instabilities associated with a gaseous jet injected normally into crossflow, also known as the transverse jet. Jet injection from nozzles which are flush as well as elevated with respect to the tunnel wall are explored experimentally in this study, for jet-to-crossflow velocity ratios R in the range 1 ≲ R ≤ 10 and with jet Reynolds numbers of 2000 and 3000. The results indicate that the nature of the transverse jet instability is significantly different from that of the free jet, and that the instability changes in character as the crossflow velocity is increased. Dominant instability modes are observed to be strengthened, to move closer to the jet orifice, and to increase in frequency as crossflow velocity increases for the regime 3.5 < R ≤ 10. The instabilities also exhibit mode shifting downstream along the jet shear layer for either nozzle configuration at these moderately high values of R. When R is reduced below 3.5 in the flush injection experiments, single-mode instabilities are dramatically strengthened, forming almost immediately within the shear layer in addition to harmonic and subharmonic modes, without any evidence of mode shifting. Under these conditions, the dominant and initial mode frequencies tend to decrease with increasing crossflow. In contrast, the instabilities in the elevated jet experiments are weakened as R is reduced below about 4, probably owing to an increase in the vertical coflow magnitude exterior to the elevated nozzle, until R falls below 1.25, at which point the elevated jet instabilities become remarkably similar to those for the flush injected jet. Low-level jet forcing has no appreciable influence on the shear-layer response when these strong modes are present, in contrast to the significant influence of low-level forcing otherwise. These studies suggest profound differences in transverse-jet shear-layer instabilities, depending on the flow regime, and help to explain differences previously observed in transverse jets controlled by strong forcing.

Using molecular dynamic simulations, we investigate the characteristics of dense flows of model cohesive grains. We describe their rheological behaviour and its origin at the scale of the grains and of their organization. Homogeneous plane shear flows give access to the constitutive law of cohesive grains which can be expressed by a simple friction law similar to the case of cohesionless grains, but intergranular cohesive forces strongly enhance the resistance to the shear. Then we show the consequence on flows down a slope: a plugged region develops at the free surface where the cohesion intensity is the strongest. Moreover, we measure various indicators of the microstructure within flows which evidence the aggregation of grains owing to cohesion and we analyse the properties of the contact network (force distributions and anisotropy). This provides new insights into the interplay between the local contact law, the microstructure and the macroscopic behavior of cohesive grains. Movies are available with the online version of the paper.

An investigation has been made of the flow over spiked cone-cylinders mainly at a Mach number of 10. Five distinct types of flow were discovered. The occurrence of each is defined in terms of spike length and cone angle.

Following a discussion of the flow mechanism, a semi-empirical separation relation is derived. This can be used to predict the flow pattern for the most common type of separated flow.

A magnetic field is shown to be asymptotically (t → ∞) decaying in a flow of finite conductivity with v = Cr, where C = Cζ(t) is a random matrix. The decay is exponential, and its rate does not depend on the conductivity. However, the magnetic energy increases exponentially owing to growth of the domain occupied by the field. The spatial distribution of the magnetic field is a set of thin ropes and (or) layers.

Instability waves, commonly called T-S waves, can be introduced in a laminar boundary layer by periodic heating of flush-mounted heating elements. Experiments have demonstrated that nearly complete cancellation of a T-S wave excited in this way can be achieved by using a second downstream heating element with a suitable phase shift. As one application of the technique, a single element together with a feedback loop activated by measured wall shear stress has been used to reduce the amplitude of naturally occurring laminar instability waves. A significant increase in the transition Reynolds number has been achieved.

Two modal decomposition techniques are employed to analyse the stability of wind turbine wakes. A numerical study on a single wind turbine wake is carried out focusing on the instability onset of the trailing tip vortices shed from the turbine blades. The numerical model is based on large-eddy simulations (LES) of the Navier–Stokes equations using the actuator line (ACL) method to simulate the wake behind the Tjæreborg wind turbine. The wake is perturbed by low-amplitude excitation sources located in the neighbourhood of the tip spirals. The amplification of the waves travelling along the spiral triggers instabilities, leading to breakdown of the wake. Based on the grid configurations and the type of excitations, two basic flow cases, symmetric and asymmetric, are identified. In the symmetric setup, we impose a 120° symmetry condition in the dynamics of the flow and in the asymmetric setup we calculate the full 360° wake. Different cases are subsequently analysed using dynamic mode decomposition (DMD) and proper orthogonal decomposition (POD). The results reveal that the main instability mechanism is dispersive and that the modal growth in the symmetric setup arises only for some specific frequencies and spatial structures, e.g. two dominant groups of modes with positive growth (spatial structures) are identified, while breaking the symmetry reveals that almost all the modes have positive growth rate. In both setups, the most unstable modes have a non-dimensional spatial growth rate close to $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\pi /2$ and they are characterized by an out-of-phase displacement of successive helix turns leading to local vortex pairing. The present results indicate that the asymmetric case is crucial to study, as the stability characteristics of the flow change significantly compared to the symmetric configurations. Based on the constant non-dimensional growth rate of disturbances, we derive a new analytical relationship between the length of the wake up to the turbulent breakdown and the operating conditions of a wind turbine.

When a circular cylinder oscillates transversely in water at rest, a three-dimensional streaming flow streaked with the chains of separated dye sheets is produced over a certain range of amplitude of the oscillation. This paper reports observations of this three-dimensional flow instability.

Friction factor data from two recent pipe flow experiments are combined to provide a comprehensive picture of the friction factor variation for Reynolds numbers from 10 to 36,000,000.

Boundary conditions are developed for rapid granular flows in which the rheology is dominated by grain–grain collisions. These conditions are $\overline{v}_0 = {\rm const}\,{\rm d}\overline{v}_0/{\rm d}y$ and u0 = const du0/dy, where $\overline{v}$ and u are the thermal (fluctuation) and flow velocities respectively, and the subscript indicates that these quantities and their derivatives are to be evaluated at the wall These boundary conditions are derived from the nature of individual grain–wall collisions, so that the proportionality constants involve the appropriate coefficient of restitution ew for the thermal velocity equation, and the fraction of diffuse (i.e. non-specular) collisions in the case of the flow-velocity equation. Direct application of these boundary conditions to the problem of Couette-flow shows that as long as the channel width h is very large compared with a grain diameter d it is permissible to set $\overline{v} = 0$ at the wall and to adopt the no-slip condition. Exceptions occur where d/h is not very small, when the wall is not rough, and when the grain–wall collisions are very elastic. Similar insight into other flows can be obtained qualitatively by a dimensional analysis treatment of the boundary conditions. Finally, the more difficult problem of self-bounding fluids is discussed qualitatively.