The nonlinear stability of a weakly supercritical shear flow with vertical temperature (density) stratification is investigated. It is shown that the usual Lin's rule of ‘indenting’ a singularity at the point of wave-flow resonance (the so-called critical layer, CL) is inapplicable for evaluating the nonlinear effects. To this end, a consistent procedure for deriving a nonlinear evolution equation is suggested and realized for the viscous critical-layer regime. The procedure takes into account the interaction of the fundamental harmonic with the second harmonic as well as with the zeroth one (i.e. with the mean-flow distortion). It is shown that the nonlinear factors both act in the same manner - at Prandtl number η ≤ 1 they limit the instability but at η > 1 they enhance it and convey a ‘burst-like’ character to it.

It is found that CL is the region of strongest interactions between the harmonics. Hence the nonlinear contribution does not actually depend on the type of original flow model chosen. A simple physical interpretation is given to illustrate the mechanism governing the nonlinearity effects on the stability in the viscous critical-layer regime.

A general method for computing the hydrodynamic interactions among N suspended particles, under the condition of vanishingly small particle Reynolds number, is presented. The method accounts for both near-field lubrication effects and the dominant many-body interactions. The many-body hydrodynamic interactions reproduce the screening characteristic of porous media and the ‘effective viscosity’ of free suspensions. The method is accurate and computationally efficient, permitting the dynamic simulation of arbitrarily configured many-particle systems. The hydrodynamic interactions calculated are shown to agree well with available exact calculations for small numbers of particles and to reproduce slender-body theory for linear chains of particles. The method can be used to determine static (i.e. configuration specific) and dynamic properties of suspended particles that interact through both hydrodynamic and non-hydrodynamic forces, where the latter may be any type of Brownian. colloidal, interparticle or external force. The method is also readily extended to dynamically simulate both unbounded and bounded suspensions.

A uniformly valid theory (for all wavelengths and angle of incidence) is derived, correct to second order in the slenderness parameter. The leading-order contribution is the unified theory, derived by Newman (1978) and Sclavounous (1982), and it is shown here that for an important class of bodies - those with an ‘almost uniform’ cross-section - the leading-order correction is of higher order than second, and can be disregarded.

In part 1 of this work the shallow-water limit is considered, and some numerical experiments show good agreement between the observed error and the predicted one. In part 2 (to follow) the theory is extended to the arbitrary-water-depth case, with similar results and accuracy. In both cases it is assumed that the mean forward velocity is zero.

For grid Reynolds numbers from 12800 to 81000, the Frenkiel-Klebanoff-Huang data for approximately isotropic homogeneous grid-generated turbulence shows that the longitudinal correlation function is given by the simple empirical expression f = [1 + (r/2L)]−3, where r(≥ 0.01M) is the separation distance between two points in the fluid flow and L = L(t) is the integral scale. It follows that the longitudinal velocity correlation 〈u1(x + re,t)u1(x,t)〉 = u2f with e = (1,0,0) is invariant under the separation-distance time-contraction transformations $r\rightarrow [r+(1 - \lambda)2L], t\rightarrow\lambda^{\frac{5}{2}}t$ for all positive parameter values λ ≤ 1. Conversely, if the longitudinal correlation function is prescribed to have the form f = J (r/L(t)), then the indicated transformation invariance holds if and only if [Fscr ](ξ) = (1 + ½ξ)−3. It is also shown that a Gaussian normal probability distribution at t = 0 and the Kármán-Howarth equation for all t > 0 are compatible with the transformation invariance and associated expression for f.

A conventional (non-density-weighted) averaging method is used to study variable-density turbulent flows, in particular, a helium-nitrogen mixing layer. A careful order-of-magnitude analysis is carried out, first in relative density fluctuations, and then in the ratio of cross-stream to streamwise lengthscales. In this way it is shown that, to lowest order, in jets and shear layers, the turbulence is unafffected by the density fluctuations, and conventional models can be used. The non-uniform density distribution influences only the mean-continuity and mean-momentum equations. Calculations (using a new form for the scalar-dissipation equation based on relaxation to an equilibrium timescale ratio) show good agreement with experiment. Calculation with a less truncated system indicates that neglected terms have little effect. We use a modified Patankar-Spalding method that overcomes numerical stability difficulties.

Streamline, streakline, and material-line flow-visualization techniques have been numerically simulated in the vortex-shedding flow field from a finite-element simulation of the two-dimensional Navier-Stokes equations at a Reynolds number of 110. The results have been used (i) to characterize the wake in terms of its critical-point trajectories, and (ii) to verify that the two-dimensional Navier-Stokes model predicts the mechanism of vortex shedding experimentally observed by Gerrard (1978). A technique for determining vorticity balances in the flow field is also presented.

Slightly heated swirling jets have been investigated experimentally in order to determine the effect of swirl degree on the mean and turbulent flow pattern. In particular, it has been shown that the swirl action strongly intensifies the turbulent momentum and heat transport processes in the close vicinity of the jet outlet and tends to quench them further downstream.

The vortex excitation of rectangular cylinders having side ratios of 0.2, 0.4 and 0.6, with a long side normal to the flow, in a mode of lateral translation, is investigated experimentally in a wind tunnel using free- and forced-oscillation methods. The range of reduced wind speeds investigated, 3-12, includes the vortex-resonance regime. The forced-oscillation experiment includes measurements of the fluctuating lift force at amplitudes up to 10% of the length of a long side. The experiments were also performed on cylinders with a long fixed splitter plate. The results of the measurements show that vortex excitation of a rectangular cylinder is strongly dependent on the side ratio. It is suggested that the critical change of the mean base pressure of an oscillating rectangular cylinder with increasing side ratio is closely correlated with the vortex-excitation characteristics. The concept of vortex excitation as aeroelastic flutter occurring in a fluid-body coupled system is proposed on the basis of the experimental results. The most essential feature of vortex excitation as a coupled flutter is that the fluid subsystem can be set into resonance by the body motion. A rapid phase-angle change in the lift force through vortex resonance produces a large out-of-phase force component which excites the motion of the body subsystem.

Large-scale organized structures in the turbulent plane wake of a circular cylinder are investigated in air up to a downstream distance of 40d at a Reynolds number of Red = 1.3 × 104; d is the cylinder diameter. Velocity signals from a linear transverse rake of 8 X-wires are sampled simultaneously to calculate the instantaneous span wise vorticity. We have appropriately smoothed the temporal traces of vorticity to obtain time evolutions (including the transverse displacement, sign, strength and size distributions) of organized structures identified by vorticity contour maps. The periodicity of the initial structures is rapidly lost: dispersion in streamwise spacing, transverse displacement, strength and size of structures increases with increasing downstream distance.

Particular emphasis is placed on examining alternative general schemes for educing coherent structures in natural or unexcited turbulent shear flows, especially in their fully developed states. The optimal eduction scheme employed involves centring the rake at the most probable transverse location of centres of advected structures and accepting those structures that: (i) are centred at the midpoint of the rake, (ii) have a peak value of (smoothed) vorticity of a given sign above a specified level, and (iii) have streamwise and transverse extents of the (smoothed) vorticity contours above a specified size. From successive accepted structure signatures the instants of occurrence of structure centres (i.e. smoothed vorticity peaks) are identified. Un-smoothed signals are then time-aligned with respect to these instants and ensemble averaged to educe coherent structure and incoherent turbulence characteristics. Further enhancement is achieved by iteratively improving the time-alignment by maximizing the cross-correlation of individual structure vorticity with the ensemble-averaged vorticity and by discarding structures that require excessive time shifts or that produce significantly weak peak correlation values.

Following this optimal scheme, large-scale coherent structures have been educed in the fully turbulent wake. The average structure centre is found to be closer to the wake centreline than the half-width location, and the structure size does not increase in proportion to the wake width, suggesting that transverse wandering of structures (including their three-dimensionality) increases significantly with increasing downstream distance. The various flow properties associated with coherent and incoherent turbulence, and the coherent structure dynamics, in particular the role of vortex stretching (at the saddle) in turbulence production and mixing, are discussed.

This paper describes an experimental study of turbulent boundary layers over two-dimensional spanwise groove and three-dimensional sandgrain roughnesses in the ‘transition regime’ between hydraulically smooth and fully rough conditions. Mean-flow measurements show that a state of kinematic near-self-preservation is also reached by sandgrain roughness and not just by d-type grooved roughness alone as commonly believed; sandgrain roughness simply requires an order-of-magnitude-longer length to reach such a state. The two roughness Reynolds numbers demarcating the boundaries of the transition regime of k-type roughnesses are found to decrease with increasing roughness-element spanwise aspect ratio (span/height). A more important role of the upper-Reynolds-number limit of the transition regime in the drag behaviour is indicated. The two Reynolds-number limits of the transition regime correlate with the two critical Reynolds numbers that describe the stability of the vortex-shedding process existing behind a similar but isolated roughness element lying submerged in an otherwise laminar boundary layer. The results provide a guideline for reducing k-type rough-wall drag by lowering the spanwise aspect ratio of the roughness elements. The vortex-shedding process in rough-wall turbulent boundary layers is described by the stability parameter $U\tau (\overline{T}/\nu)^{\frac{1}{2}}$ whose value is the same for all roughnesses examined herein; here Uτ is the friction velocity, $\overline{T}$ is the mean time period of vortex shedding and v is the kinematic viscosity of the fluid.

The existence of fast dynamos caused by steady motion of an electrically conducting fluid is established by consideration of a two-dimensional spatially periodic flow: the velocity, which is independent of the vertical coordinate z, is finite and continuous everywhere but the vorticity is infinite at the X-type stagnation points. A mean-field model is developed using boundary-layer methods valid in the limit of large magnetic Reynolds number R. The magnetic field is confined to sheets, width of order R−½. The mean magnetic field lies and is uniform on horizontal planes: its direction is independent of time but rotates once about the vertical axis over a short distance 2πl, where l−1 = R½β and β is a vertical stretched wavenumber independent of R. Its alternating direction gives it a rope-like structure within the sheets. An α-effect is calculated for the model, whose strength for a given flow is a function of β and R. Two sources of α-effect are isolated whose relative importance depends critically on the size of β. When the vorticity is finite everywhere and β [Lt ] 1, the dynamo is ‘almost’ fast with growth rates of order (ln R)−1. The maximum growth rate ln (ln R)/ln R occurs when, correct to leading order, β is (ln R)−½. The asymptotic results valid for large R compare excellently with Roberts (1972) modal analysis for finite R.

This paper extends previous theoretical work on waves in floating ice plates to take account of the following effects: (i) compressive stress in the plane of the plate, (ii) uniform flow in the underlying water, and (iii) stratification of the underlying water. The first two effects are unlikely to be important in practice, causing respectively a slight decrease in phase speed and mainly a re-orientation of the wave pattern due to a steadily moving source. A two-layer model is used to describe stratification, which introduces a new system of slow internal waves associated with the layer interface, while the surface flexural waves are only slightly modified. In the case of unstratified water there is a minimum speed cmin such that more slowly moving sources excite a static rather than a wavelike response in the ice. With stratified water there remains a variety of steady wave patterns due to the internal waves, at source speeds below cmin. Another important effect of stratification is to greatly increase wave drag. For certain source load distributions, internal-wave amplitudes may grow until linear theory is no longer applicable.

An acoustic line source is placed in the neighbourhood of a circular cylinder. Acoustic streaming is generated in a Stokes shear-wave layer, at the surface of the cylinder, beyond which it persists. It is the streaming outside the Stokes layer, for large values of a streaming Reynolds number, that is the subject of the present paper.

An initial state consisting of sugar solution lying above a denser salt solution in a Hele-Shaw cell is unstable to disturbances that evolve into long, slender fingers. An analysis of the structure of fully evolved (infinitely long) fingers that are independent of the vertical coordinate concludes that fingers with a width of the order of the buoyancy-layer thickness have maximum growth rate. Since effective gravity can be altered by inclining the Hele-Shaw cell toward the horizontal, fingers of different preferred widths can be established. An abrupt change of the angle of inclination changes the preferred width. A stability analysis of the resulting initial-value problem shows that perturbations with a vertical scale of the order of the buoyancy-layer thickness grow, and fluid from each finger penetrates laterally into the two adjacent fingers. The unstable modes resemble those observed experimentally by Taylor & Veronis (1986). It turns out that all vertically uniform fingers, even ones with the preferred width of the basic state, are unstable to a non-oscillatory peturbation that changes straight fingers to ones that have a vertically wavy structure. In all cases the vertical scale of the most unstable disturbance is of the order of the buoyancy-layer thickness. Also included is a discussion of the need for a model describing the transient evolution of fingers and particularly one that contains an analysis of the role of the transition region between the salt-finger zone and the reservoirs above and below.

The spiralling flow within a curved tube of rectangular cross-section normally forms two distinct cells in the plane of the cross-section, but a transition to a different, four-cell flow is known to occur at a critical value of the axial pressure gradient. This paper shows that, for a square cross-section, the transition is a result of a complex structure of multiple, symmetric and asymmetric solutions. The structure is revealed by solving extended systems of equations for steady-state, fully developed, laminar flow, in a finite-element approximation, to locate exactly the positions of singular points. Continuation methods are used to trace the paths of these points as the aspect ratio and radius of curvature vary. For a square cross-section, multiple, symmetric flows of both two and four cells are found in two distinct ranges of axial pressure gradient q. In addition, a pair of asymmetric solutions arise from a symmetry-breaking bifurcation point on the primary flow branch. The paths of limit points and symmetry-breaking bifurcation points are obtained as the aspect ratio γ varies, and a transcritical bifurcation point is found at γ = 1.43. For larger aspect ratios the secondary four-cell branch is disconnected from that of the primary two-cell flow. The bifurcation set in the two parameters q and γ has a complex structure, with a number of higher-order singularities; in particular, the path of limit points is found to cross in the manner of an unfolded swallowtail catastrophe. A stability analysis shows that all multiple solutions except the two-cell type are unstable with respect to either symmetric or antisymmetric perturbations. For values of the aspect ratio less than 1.43 there is a range of axial pressure gradients for which there is no stable flow. The critical Dean numbers of the singular points are found to vary strongly at small values of radius of curvature β, but the bifurcation set remains qualitatively the same. Previous work is interpreted in the light of the present results.

A weakly nonlinear Hamiltonian model for two-dimensional irrotational waves on water of finite depth is developed. The truncated model is used to study families of periodic travelling waves of permanent form. It is shown that non-symmetric periodic waves exist, which appear via spontaneous symmetry-breaking bifurcations from symmetric waves.

When the lengthscales and timescales on which a transport process occur are not much larger than the scales of variations in the velocity field experienced by a tracer particle, a description of the transport in terms of a local, averaged macroscale version of Fick's law is not applicable. Here, a non-local transport theory is developed in which the average mass flux is not simply proportional to the average local concentration gradient, but is given by a convolution integral over space and time of the average concentration gradient times a spatial- and temporal-wavelength-dependent diffusivity. The non-local theory is applied to the transport of a passive tracer in the advective field that arises in the bulk fluid of a porous medium, and the complete residence-time distribution - space-time response to a unit source input - of the tracer is determined. It is also shown how the method of moments may be simply recovered as a special case of the non-local theory. While developed in the context of and applied to tracer dispersion in porous media, the non-local theory presented here is applicable to the general problem of determining the average transport behaviour in advection-diffusion-type systems in which spatial and temporal variations are occurring on scales comparable with the scale of interest.

The present study of the pressure fluctuations in the interaction region oif a two-dimensionals compression flow established that the frequency of the shock-wave unsteadiness is of the same order as the bursting frequency of the upstream boundary layer and that this frequency is independent of the downstream separated flow. The conditional-sampling technique developed herein is capable of separating phenomena due to shock-wave oscillations from those due to transport phenomena of turbulence. The results show that turbulence as inferred from wall-pressure fluctuations may be significantly amplified approaching the shock.

This is an experimental investigation of the flow of an impulsively started linearly stratified, rotating fluid past obstacles located on either the right or left side of a channel of rectangular cross-section. The obstacle has plane sloping sides so as to represent a cape-like feature extending from a shoreline. Emphasis is given to the temporal flow development in the lee of the obstacle. The pertinent dimensionless parameters are the Rossby. Burger and Ekman numbers, the ratio of the fluid depth to obstacle width and the obstacle-to-channel-width ratio.

For sufficiently small Burger numbers and for a range of Rossby numbers the right-side obstacle produces fully attached flows at all levels. At larger Burger numbers attached lee anticyclones develop during a period of the order of ten advective timescales. At still larger Burger numbers the starting lee anticyclone is shed and followed by a weaker secondary anticyclone.

No fully attached regime could be found for the left-side obstacle for the range of parameters considered. For the smallest Burger numbers studied and for a range of Rossby numbers, a lee cyclone is formed on the order of one advective timescale; this eddy spins down at large times leaving a quiescent wake. At larger Burger numbers and for a range of Rossby numbers, cyclonic eddy shedding occurs.

Measurements of the streamwise location of the eddy centres, the streamwise extent of the eddies and the vorticities of the eddy cores are made as functions of a dimensionless time and the system parameters. It is found, for example, that the vorticity of the starting eddies at small times in the eddy-shedding regimes for both anticyclones and cyclones is strongest at the lowest observation levels. Ekman suction, however, tends to more strongly decrease the vortex strength in the lower levels compared to the middle levels so that at large times the vorticity in the middle levels exceeds that near the channel floor. Eddies with similar characteristics are found in the open ocean.

One-dimensional, nonlinear, shallow-water wave equations are derived from two-dimensional Boussinesq equations to investigate resonant reflections due to corrugated boundaries. Small, but short-wave undulations are introduced through water depth and channel width. Coupled nonlinear equations for the transmitted and reflected wave fields are derived and solved numerically. In the simple case where undulations are zero (the reflection is also zero), the governing equations are used to study the propagation of permanent shallow-water waves (cnoidal waves) and to examine the generation of higher harmonics in shallow-water waves. The present numerical results show that the nonlinear effects are very important in considering the resonant reflection of cnoidal waves from a rippled bed.