We report here on the results of a series of experiments carried out on a turbulent spot in a distorted duct to study the effects of a divergence with straight streamlines preceded by a short stretch of transverse streamline curvature, both in the absence of any pressure gradient. It is found that the distortion produces substantial asymmetry in the spot: the angles at which the spot cuts across the local streamlines are altered dramatically (in contradiction of a hypothesis commonly made in transition zone modelling), and the Tollmien–Schlichting waves that accompany the wing tips of the spot are much stronger on the outside of the bend than on the inside. However there is no strong effect on the internal structure of the spot and the eddies therein, or on such propagation characteristics as overall spread rate and the celerities of the leading and trailing edges. Both lateral streamline curvature and non-homogeneity of the laminar boundary layer into which the spot propagates are shown to be strong factors responsible for the observed asymmetry. It is concluded that these factors produce chiefly a geometric distortion of the coherent structure in the spot, but do not otherwise affect its dynamics in any significant way.

Thermocapillary convection in three-dimensional rectangular finite containers with rigid lateral walls is studied. The upper surface of the fluid layer is assumed to be flat and non-deformable but is submitted to a temperature-dependent surface tension. The realistic ‘no-slip’ condition at the sidewalls makes the method of separation of variables inapplicable for the linear problem. A spectral Tau method is used to determine the critical Marangoni number and the convective pattern at the threshold as functions of the aspect ratios of the container. The influence on the critical parameters of a non-vanishing gravity and a non-zero Biot number at the upper surface is also examined. The nonlinear regime for pure Marangoni convection (Ra = 0) and for Pr = 104, Bi = 0 is studied by reducing the dynamics of the system to the dynamics of the most unstable modes of convection. Owing to the presence of rigid walls, it is shown that the convective pattern above the threshold may be quite different from that predicted by the linear approach. The theoretical predictions of the present study are in very good agreement with the experiments of Koschmieder & Prahl (1990) and agree also with most of Dijkstra's (1995a, b) numerical results. Important differences with the analysis of Rosenblat, Homsy & Davis (1982b) on slippery walls containers are emphasized.

A numerical study is performed for time-varying natural convection of an incompressible Boussinesq fluid in a sidewall-heated square cavity. The temperature at the cold sidewall Tc is constant, but at the hot sidewall a time-varying temperature condition is prescribed, $ T_H = \overline{T_H} + \Delta T^{\prime} \sin ft $. Comprehensive numerical solutions are found for the time-dependent Navier–Stokes equations. The numerical results are analysed in detail to show the existence of resonance, which is characterized by maximal amplification of the fluctuations of heat transfer in the interior. Plots of the dependence of the amplification of heat transfer fluctuations on the non-dimensional forcing frequency ω are presented. The failure of Kazmierczak & Chinoda (1992) to identify resonance is shown to be attributable to the limitations of the parameter values they used. The present results illustrate that resonance becomes more distinctive for large Ra and Pr ∼ 0(1). The physical mechanism of resonance is delineated by examining the evolution of oscillating components of flow and temperature fields. Specific comparisons are conducted for the resonance frequency ωr between the present results and several other previous predictions based on the scaling arguments.

A reappraisal is made of the Brown & Michael (1954, 1955) equation that is widely used to model high-Reynolds-number vortex shedding in two dimensions by rectilinear vortices of time-dependent circulations. It is concluded that the equation introduces an unbalanced and unacceptable surface force that can significantly influence predicted flow characteristics. A corrected equation is derived which removes this force, and is applied to determine the sound generated at low Mach numbers when a line vortex translates around the edge of a rigid half-plane. The solution of this problem in the absence of vortex shedding (Crighton 1972) is extended by permitting shedding to occur at the edge in accordance with the unsteady Kutta condition. The shed vorticity is assumed to roll-up into a concentrated core whose motion is calculated by both the emended and original Brown & Michael equations. The two models exhibit large qualitative differences in the predicted wake flow near the edge; both predict significant reductions in the radiated sound, but the reduction is smaller by about 4 dB for the emended Brown & Michael equation.

This paper deals with the initial region (x/d = 1 to 20) of a turbulent wake generated by a mesh strip. The purpose of the study is to identify the formation mechanism of the large-scale vortices which have developed by the end of this zone. The region is characterized by the evolution and interaction of two parallel shear layers generated at the edges of the mesh. Multi-point hot-wire velocity measurements establish that the merging and interaction of the small-scale eddies account for the formation of the large-scale vortices. A numerical simulation based on a discrete vortex method was carried out to gain further insight into the interaction between the two shear layers. Both the experimental and numerical results suggest that low-frequency perturbation plays an important role in the communication between the two shear layers.

The present investigation deals with the development stage (x/d = 20–100) of the porous-body wake, generated by a mesh strip. Pattern-recognition analysis was applied to the velocity signals from hot-wire measurements in the vertical centreplane and various horizontal planes in the flow field. Spanwise coherent structures (with ωz) and lateral coherent structures (ωy) have been identified. An examination of the fine-scale turbulence is used to show that the two kinds of structures are indeed connected. In the near region of the wake, the lateral structures are ‘ribs’ attached to the spanwise vortices. In the intermediate region, the interaction of the two structures changes the original spanwise vortices into three-dimensional structures. Beyond this zone, in the far region, the’ legs’ of the original spanwise structure merge with the’ ribs’, and it can be inferred that the coherent structures within the flow are essentially randomly distributed hairpin-like vortices.

The influence of different boundary conditions applied in the contact line region on the outer meniscus shape is analysed by means of a finite-element numerical simulation of the steady movement of a liquid-gas meniscus in a capillary tube. The free-surface steady shape is obtained by solving the unsteady creeping-flow approximation of the Navier–Stokes equations starting from some initial shape. Comparisons of the outer solutions obtained using two different inner models, together with that published by Lowndes (1980), indicate the relative insensitivity of the outer solution to the type of model utilized in the contact line region.

Turbulent air flow over a surface gravity wave of small amplitude is studied analytically on the basis of a family of rapid-distortion turbulence models. Results for the wave growth rate do not depend sensitively on the specific choice of these models. However, the agreement with results based on a so-called truncated mixing-length model (Belcher & Hunt 1993) is poor, despite physical similarity of the models. The present analysis also shows that the use of turbulence models based on rapid-distortion theory leads to significant underestimation of observed growth rates of high-frequency waves.

Using techniques developed in our previous publication (Mackaplow et al. 1994), we complete a comprehensive set of numerical simulations of the volume-averaged stress tensor in a suspension of rigid, non-Brownian slender fibres at zero Reynolds number. In our problem formulation, we use slender-body theory to develop a set of integral equations to describe the interfibre hydrodynamic interactions at all orders. These integral equations are solved for a large number of interacting fibres in a periodically extended box. The simulations thus developed are an accurate representation of the suspensions at concentrations up to and including the semidilute regime. Thus, large changes in the suspensions properties can be obtained. The Theological properties of suspensions with concentrations ranging from the dilute regime, through the dilute/semi-dilute transition, and into the semi-dilute regime, are surprisingly well predicted by a dilute theory that takes into account two-body interactions. The accuracy of our simulations is verified by their ability to reproduce published suspension extensional and shear viscosity data for a variety of fibre aspect ratios and orientation distributions, as well as a wide range of suspension concentrations.

The techniques developed in Part 1 of the present series are here applied to two-dimensional solutions of the equations governing the magnetohydrodynamics of ideal incompressible fluids. We first demonstrate an isomorphism between such flows and the flow of a stratified fluid subjected to a field of force that we describe as ‘pseudo-gravitational’. We then construct a general Casimir as an integral of an arbitrary function of two conserved fields, namely the vector potential of the magnetic field, and the analogous potential of the ‘modified vorticity field’, the additional frozen field introduced in Part 1. Using this Casimir, a linear stability criterion is obtained by standard techniques. In §4, the (Arnold) techniques of nonlinear stability are developed, and bounds are placed on the second variation of the sum of the energy and the Casimir of the problem. This leads to criteria for nonlinear (Lyapunov) stability of the MHD flows considered. The appropriate norm is a sum of the magnetic and kinetic energies and the mean-square vector potential of the magnetic field.

In this paper, the nonlinear dynamics of an axisymmetric liquid bridge held captive between two coaxial, circular, solid disks that are separated at a constant velocity are considered. As the disks are continuously pulled apart, the bridge deforms and ultimately breaks when its length attains a limiting value, producing two drops that are supported on the two disks. The evolution in time of the bridge shape and the rupture of the interface are investigated theoretically and experimentally to quantitatively probe the influence of physical and geometrical parameters on the dynamics. In the computations, a one-dimensional model that is based on the slender jet approximation is used to simulate the dynamic response of the bridge to the continuous uniaxial stretching. The governing system of nonlinear, time-dependent equations is solved numerically by a method of lines that uses the Galerkin/finite element method for discretization in space and an adaptive, implicit finite difference technique for discretization in time. In order to verify the model and computational results, extensive experiments are performed by using an ultra-high-speed video system to monitor the dynamics of liquid bridges with a time resolution of 1/12 th of a millisecond. The computational and experimental results show that as the importance of the inertial force – most easily changed in experiments by changing the stretching velocity – relative to the surface tension force increases but does not become too large and the importance of the viscous force – most easily changed by changing liquid viscosity – relative to the surface tension force increases, the limiting length that a liquid bridge is able to attain before breaking increases. By contrast, increasing the gravitational force – most readily controlled by varying disk radius or liquid density – relative to the surface tension force reduces the limiting bridge length at breakup. Moreover, the manner in which the bridge volume is partitioned between the pendant and sessile drops that result upon breakup is strongly influenced by the magnitudes of viscous, inertial, and gravitational forces relative to surface tension ones. Attention is also paid here to the dynamics of the liquid thread that connects the two portions of the bridge liquid that are pendant from the top moving rod and sessile on the lower stationary rod because the manner in which the thread evolves in time and breaks has important implications for the closely related problem of drop formation from a capillary. Reassuringly, the computations and the experimental measurements are shown to agree well with one another.

We use numerical and asymptotic techniques to study the stability of a two-phase air/water flow above a flat porous plate. This flow is a model of the boundary layer which forms on a yawed cylinder and can be used as a useful approximation to the air flow over swept wings. The air and water form an immiscible interface which can destabilize the flow, leading to travelling wave disturbances which move along the attachment line. This instability occurs for lower Reynolds numbers than is the case in the absence of a water layer. The two-fluid flow can be used as a crude model of the effect of heavy rain on the leading edge of a swept wing.

We also investigate the instability of inviscid stationary modes. We calculate the effective wavenumber and orientation of the stationary disturbance when the fluids have identical physical properties. Using perturbation methods we obtain corrections due to a small stratification in viscosity, thus quantifying the interfacial effects. Our analytical results are in agreement with the numerical solution which we obtain for arbitrary fluid properties.

Mild to steep standing waves of the fundamental mode are generated in a narrow rectangular cylinder undergoing vertical oscillation with forcing frequencies of 3.15 Hz to 3.34 Hz. A precise, non-intrusive optical wave profile measurement system is used along with a wave probe to accurately quantify the spatial and temporal surface elevations. These standing waves are also simulated by a two-dimensional spectral Cauchy integral code. Experiments show that contact-line effects increase the viscous natural frequency and alter the neutral stability curves. Hence, as expected, the addition of the wetting agent Photo Flo significantly changes the stability curve and the hysteresis in the response diagram. Experimentally, we find strong modulations in the wave amplitude for some forcing frequencies higher than 3.30 Hz. Reducing contact-line effects by Photo-Flo addition suppresses these modulations. Perturbation analysis predicts that some of this modulation is caused by noise in the forcing signal through ‘sideband resonance’, i.e. the introduction of small sideband forcing can generate large modulations of the Faraday waves. The analysis is verified by our numerical simulations and physical experiments. Finally, we observe experimentally a new form of steep standing wave with a large symmetric double-peaked crest, while simulation of the same forcing condition results in a sharper crest than seen previously. Both standing wave forms appear at a finite wave steepness far smaller than the maximum steepness for the classical standing wave and a surface tension far smaller than that for a Wilton ripple. In both physical and numerical experiments, a stronger second harmonic (in time) and temporal asymmetry in the wave forms suggest a 1:2 resonance due to a non-conventional quartet interaction. Increasing wave steepness leads to a new form of breaking standing waves in physical experiments.

Breaking waves generated by a two-dimensional hydrofoil moving near a free surface at constant speed (U∞), angle of attack and depth of submergence were studied experimentally. The measurements included the mean and fluctuating shape of the breaking wave, the surface ripples downstream of the breaker and the vertical distribution of vertical and horizontal velocity fluctuations at a single station behind the breaking waves. The spectrum of the ripples is highly peaked and shows little variation in both its peak frequency and its shape over the first three wavelengths of the wavetrain following the breaker. For a given speed, as the breaker strength is increased, the high-frequency ends of the spectra are nearly identical but the spectral peaks move to lower frequencies. A numerical instability model, in conjunction with the experimental data, shows that the ripples are generated by the shear flow developed at the breaking region. The spectrum of the vertical velocity fluctuations was also found to be highly peaked with the same peak frequency as the ripples, while the corresponding spectrum of the horizontal velocity fluctuations was found not to be highly peaked. The root-mean-square (r.m.s.) amplitude of the ripples (νrms) increases with increasing speed and with decreasing depth of submergence of the hydrofoil, and decreases as x-1/2 with increasing distance x behind the breaker. The quantity (gνrms)/(U∞Vrms) (where Vrms is the maximum r.m.s. vertical velocity fluctuation and g is the gravitational acceleration) was found to be nearly constant for all of the measurements.

Direct simulations of flow in a channel with complex, time-dependent wall geometries facilitate an investigation of smart skin control in a turbulent wall layer (with skin friction drag reduction as the goal). The test bed is a minimal flow unit, containing one pair of coherent structures in the near-wall region: a high- and a low-speed streak. The controlling device consists of an actuator, Gaussian in shape and approximately twelve wall units in height, that emerges from one of the channel walls. Raising the actuator underneath a low-speed streak effects an increase in drag, raising it underneath a high-speed streak effects a reduction – indicating a mechanism for control. In the high-speed region, fast-moving fluid is lifted by the actuator away from the wall, allowing the adjacent low-speed region to expand and thereby lowering the average wall shear stress. Conversely, raising an actuator underneath a low-speed streak allows the adjacent high-speed region to expand, which increases skin drag.

When a curved pipe rotates about the centre of curvature, the fluid flowing in it is subjected to both Coriolis and centrifugal forces. Based on the analogy between laminar flows in stationary curved pipes and in orthogonally rotating pipes, the flow characteristics of fully developed laminar flow in rotating curved pipes are made clear and definite by similarity arguments, computational studies and using experimental data. Similarity arguments clarify that the flow characteristics in loosely coiled rotating pipes are governed by three parameters: the Dean number KLC, a body force ratio F and the Rossby number Ro. As the effect of Ro is negligible when Ro is large, computational results are presented for this case first, and then the effect of Ro is studied. Flow structure and friction factor are studied in detail. Variations of flow structure show secondary flow reversal at F ≈ −1, where the two body forces are of the same order but in opposite directions. It is also shown how the Taylor–Proudman effect dominates the flow structure when Ro is small. Computed curves of the friction factor for constant Dean number have their minimum at F ≈ −1. A composite parameter KL is introduced as a convenient governing parameter and used to correlate the characteristics. By applying KL to the analogy formula previously derived for two limiting flows, a semi-empirical formula for the friction factor is presented, which shows good agreement with the experimental data for a wide range of the parameters.

The flow field of a diffusion flame attached to a thick-rim injector between two coflowing streams of fuel and oxidiser is analysed in the Burke–Schumann limit of infinitely fast reaction rate. The length of the recirculation region immediately behind the injector and the velocity of the recirculating fluid are proportional to the shear stresses of the reactant streams on the wall of the injector for a range of rim thicknesses, and the structure of the flow in the wake depends then on three main non-dimensional parameters, measuring the gas thermal expansion due to the chemical heat release, the air-to-fuel stoichiometric ratio of the reaction, and the air-to-fuel ratio of wall shear stresses. The recirculation region shortens with increasing heat release, and the position of the flame in this region depends on the other two parameters. An asymptotic analysis is carried out for very exothermic reactions, showing that the region of high temperature around the flame is confined by neatly defined boundaries and the hot fluid moves like a high-velocity jet under a favourable self-induced pressure gradient. The immediate wake is surrounded by a triple-deck region where the interacting flow leads to an adverse pressure gradient and a reduced shear stress upstream of the injector rim for sufficiently exothermic reactions. Separation of the boundary layers on the wall of the injector, however, seems to be postponed to very large values of the gas thermal expansion.

In this paper we consider the development of shear dispersion following the introduction of a diffusing tracer substance into a tube or duct containing flowing fluid, with emphasis on the characterization of the temporal variation of concentration at a fixed axial position. Asymptotic results are derived by assuming that the distance downstream of the point of tracer introduction, appropriately non-dimensionalized, is large. First, we consider the central moments of the temporal concentration variation, including their dependence on transverse position and on the initial transverse distribution of tracer. The moments for finite Péclet number are expressed in terms of their infinite-Péclet-number counterparts, and the latter are given explicitly for Poiseuille flow. Then, assuming the Péclet number is infinite, we derive an approximate solution for the Green's function expressing tracer concentration following its introduction at an arbitrary point within the tube. The solution is expressed in terms of three numerically evaluated functions of a dimensionless time variable, with parametric dependence on the distance downstream of the point of tracer release. The method is illustrated by calculation of the approximate solution for dispersion in Poiseuille flow. Unlike previous approximations, the present solution is uniformly asymptotic and represents the tails of the concentration distribution as well as the approximately Gaussian central part; in these three regions, simpler analytic forms of the approximation are given. Comparison with previous computational solutions suggests the present approximation remains reasonably accurate even at quite short distances from the point where tracer is released.