Numerical simulations investigating the formation and stability of quasi-two-dimensional coherent vortices in rotating homogeneous three-dimensional flow are described. In a numerical study of shear flows Lesieur, Yanase & Métais (1991) found that cyclones (respectively anticyclones) with |ω2D| ∼ O(2Ω), where ω2D is the vorticity and Ω is the rotation rate, are stabilized (respectively destabilized) by the rotation. A study of triply periodic pseudo-spectral simulations (643) was undertaken in order to investigate the vorticity asymmetry in homogeneous turbulence. Specifically, we examine (i) the possible three-dimensionalization of initially two-dimensional vortices and (ii) the emergence of quasi-two-dimensional structures in initially-isotropic three-dimensional turbulence. Direct numerical simulations of the Navier—Stokes equations are compared with large-eddy simulations employing a subgridscale model based on the second-order velocity structure function evaluated at the grid separation and with simulations employing hyperviscosity.

Isolated coherent two-dimensional vortices, obtained from a two-dimensional decay simulation, were superposed with a low-amplitude three-dimensional perturbation, and used to initialize the first set of simulations. With Ω = 0, a three-dimensionalization of all vortices was observed. This occurred first in the small scales in conjunction with the formation of longitudinal hairpin vortices with vorticity perpendicular to that of the initial quasi-two-dimensional flow. In agreement with centrifugal stability arguments, when 2Ω = [ω2D]rms a rapid destabilization of anticyclones was observed to occur, whereas the initial two-dimensional cyclonic vortices persisted throughout the simulation. At larger Ω, both cyclones and anticyclones remained two-dimensional, consistent with the Taylor—Proudman theorem. A second set of simulations starting from isotropic three-dimensional fields was initialized by allowing a random velocity field to evolve (Ω = 0) until maximum energy dissipation. When the simulations were continued with 2Ω = [ω · Ω]rms/Ω, the three-dimensional flow was observed to organize into two-dimensional cyclonic vortices. At larger Ω, two-dimensional anticyclones also emerged from the initially-isotropic flow. The consequences for a variety of industrial and geophysical applications are clear. For quasi-two-dimensional eddies whose characteristic circulation times are of the order ofder of Ω−1, rotation induces a complete disruption of anticyclonic vortices, while stabilizing cyclonic ones.

The evolution of three-dimensional disturbances in an incompressible mixing layer in an inviscid fluid is investigated as an initial-value problem. A Green's function approach is used to obtain a general space–time solution to the problem using a piecewise linear model for the basic flow, thereby making it possible to determine complete and closed-form analytical expressions for the variables with arbitrary input. Structure, kinetic energy, vorticity, and the evolution of material particles can be ascertained in detail. Moreover, these solutions represent the full three-dimensional disturbances that can grow exponentially or algebraically in time. For large time, the behaviour of these disturbances is dominated by the exponentially increasing discrete modes. For the early time, the behaviour is controlled by the algebraic variation due to the continuous spectrum. Contrary to Squire's theorem for normal mode analysis, the early-time behaviour indicates growth at comparable rates for all values of the wavenumbers and the initial growth of these disturbances is shown to rapidly increase. In particular, the disturbance kinetic energy can rise to a level approximately ten times its initial value before the exponentially growing normal mode prevails. As a result, the transient behaviour can trigger the roll-up of the mixing layer and its development into the well-known pattern that has been observed experimentally.

We present a new model to describe the thermal and compositional evolution of a binary alloy which is cooled from above. Explicit account is taken of the nucleation of crystals in the cold upper thermal boundary layer, the growth of crystals in the turbulently convecting interior, and their subsequent gravitational settling to the floor of the chamber. The crystallization of one solid phase only is considered. When the residence time of a typical crystal within the convecting bulk is short compared with the overall cooling time of the fluid, the crystal size distribution loses memory of earlier conditions in the fluid and the number density simply decays exponentially with the cube of the crystal size. A quasi-steady state exists in which the rate of crystal production is balanced by the rate of sedimentation at the floor, allowing the volume fraction of suspended crystals to remain small until convection ceases to be vigorous.

We focus on the situation in which the latent heat released by solidification would far exceed the heat flux extracted through convection if the melt undercooling were maintained equal to the initial temperature difference applied at the cold upper boundary. In this case, either the growth or nucleation of crystals must be limited in order that the fluid continues to cool. Both the growth-limited and nucleation-limited regimes may develop during the cooling of an individual fluid body, depending upon the thermal boundary condition at the upper boundary of the convecting portion of the fluid.

We calculate how the mean crystal size within the sedimented crystal pile evolves as the fluid cools. During the growth-limited regime, the mean crystal size in the crystal pile typically decreases with height, owing to the decrease in the extracted heat flux and the greater efficiency of crystal settling as the fluid layer becomes shallower. In contrast, during the nucleation-limited regime, the fluid undercooling may increase significantly as the fluid cools, and inverse grading (large crystals over small) is possible. We discuss the possible application of our theory to the cooling of large igneous intrusions.

Steady two-dimensional flows past a parabolic obstacle lying on the free surface in water of finite depth are considered. The fluid is treated as inviscid and incompressible and the flow is assumed to be irrotational. Gravity is included in the free-surface condition. The problem is solved numerically by using boundary integral equation techniques. It is shown that there are solutions for which the flow is supercritical both upstream and downstream and others for which the flow is subcritical both upstream and downstream. These flows have continuous tangents at both ends of the obstacle at which separation occurs. For supercritical flows, there are up to three solutions corresponding to the same value of the Froude number when the obstacle is concave and up to two solutions when the obstacle is convex. For subcritical flows, there are solutions with waves behind the obstacle. As the Froude number decreases, these waves become steeper and the numerical calculations suggest that they, ultimately, reach limiting configurations with a sharp crest forming a 120° angle.

We study the effect of surface roughness and coatings on fluid flow over a solid surface. In the limit of small-amplitude roughness and thin lubricating films we are able to derive asymptotically an effective slip boundary condition to replace the no-slip condition over the surface. When the film is absent, the result is a Navier slip condition in which the slip coefficient equals the average amplitude of the roughness. When a layer of a second fluid covers the surface and acts as a lubricating film, the slip coefficient contains a term which is proportional to the viscosity ratio of the two fluids and which depends on the dynamic interaction between the film and the fluid. Limiting cases are identified in which the film dynamics can be decoupled from the outer flow.

Modelling of the complete second-order structure of homogeneous, neutrally stratified atmospheric boundary-layer turbulence, including spectra of all velocity components and cross-spectra of any combination of velocity components at two arbitrarily chosen points, is attempted. Two models based on Rapid Distortion Theory (RDT) are investigated. Both models assume the velocity profile in the height interval of interest to be approximately linear. The linearized Navier–Stokes equation together with considerations of ‘eddy’ lifetimes are then used to modify the spatial second-order structure of the turbulence. The second model differs from the first by modelling the blocking by the surface in addition to the shear. The resulting models of the spectral velocity tensor contain only three adjustable parameters: a lengthscale describing the size of the largest energy-containing eddies, a non-dimensional number used in the parametrization of ‘eddy’ lifetime, and the third parameter is a measure of the energy dissipation.

Two atmospheric experiments, both designed to investigate the spatial structure of turbulence and both running for approximately one year, are used to test and calibrate the models. Even though the approximations leading to the models are very crude they are capable of predicting well the two-point second-order statistics such as cross-spectra, coherences and phases, on the basis of measurements carried out at one point. The two models give very similar predictions, the largest difference being in the coherences involving vertical velocity fluctuations, where the blocking by the surface seems to have a significant effect.

Modulation of the growth rate of short capillary–gravity surface wind waves in the presence of a long wave with steepness much smaller than the maximum is studied theoretically. The Miles (1962) mechanism taking into account the viscous wave stresses in the air flow is considered to be the main process of short-wave generation. The short-wave growth rate is defined by the wind velocity gradient in the viscous sublayer of the logarithmic boundary layer. The long wave propagating on the wave surface induces an additional component of the wind velocity gradient oscillating with the length and time periods of the long wave, which results in modulation, with the same period, of the growth rate of the short wave riding on the long one. The growthrate modulation amplitude depends on the parameter M being of the order of the relation between the oscillating and the mean wind velocity gradients in the viscous sublayer \[M=\frac{2kac}{u^2_*}(ckv_{\alpha})^{1/2} \] (where c, k, a are the phase velocity, the wavenumber and the elevation amplitude of the long wave; va is the viscosity coefficient in the air; u* is the wind friction velocity). When M = O(1) (weak winds and long waves) the oscillating component of the shortwave growth rate is of the same order as the mean one. If M is much smaller than unity, then the relative amplitude of the growth rate is of the same order as the steepness of the long wave.

The magnetic moment of individual living magnetic bacteria was determined by motion analysis in a time-dependent magnetic field. For this purpose we had to estimate the drag exerted on the moving bacterium by the surrounding liquid. First, the bacterium was approximated by an ellipsoid. In order to determine drag coefficients for more complicated (and realistic) forms, a model experiment was built. In this experiment enlarged models of bacteria were rotated in a viscous liquid and the torque acting upon them was measured. Computing algorithms were developed in order to calculate drag coefficients of magnetic bacteria and to simulate their motion in magnetic fields. The experimental and numerical determination of the drag coefficients agree within their error bounds. Besides the determination by motion analysis, the bacterial magnetic moment was also calculated from the number and size of magnetic particles contained in the bacterium as seen in an electron microscope. The results of both calculations agree well.

Mixing and transport of a stratifying scalar are investigated at a density interface imbedded in a turbulent shear flow. Steady-state interfacial shear flows are generated in a laboratory water channel for layer Richardson numbers, Ri, between about 1 and 10. The flow field is made optically homogeneous, enabling the use of laser-induced fluorescence with photodiode array imaging to measure the concentration field at high resolution. False-colour images of the concentration field provide valuable insight into interfacial dynamics: when the local mean shear Richardson number, Ris, is less than about 0.40–0.45, interfacial mixing appears to be dominated by Kelvin–Helmholtz (K–H) instabilities; when Ris is somewhat larger than this, interfacial mixing appears to be dominated by shear-driven wave breaking. In both cases, vertical transport of mixed fluid from the interfacial region into adjacent turbulent layers is accomplished by large-scale turbulent eddies which impinge on the interface and scour fluid from its outer edges.

Motivated by the experimental findings, a model for interfacial mixing and entrainment is developed. A local equilibrium is assumed in which the rate of loss of interfacial fluid by eddy scouring is balanced by the rate of production (local mixing) by interfacial instabilities and molecular diffusion. When a single layer is turbulent and entraining, the model results are as follows: in the molecular-diffusion-dominated regime, δ/h ∼ Pe−1/2 and E ∼ Ri−1Pe−1/2; in the wave-breaking-dominated regime, δ/h ∼ Ri−1/2 and E ∼ Ri−3/2; and in the K–H-dominated regime, δ/h ∼ Ri−1 and E ∼ Ri−2, where δ is the interface thickness, h is the boundary-layer thickness, Pe is the Péclet number, and E is the normalized entrainment velocity. In all three regimes, the maximum concentration anomaly, [gcy ]m ∼ Ri−1. When both layers are turbulent and entraining, E and δ depend on combinations of parameters from both layers.

In this paper we analyse in detail, and for the first time, the rôle of torsion in the dynamics of twisted vortex filaments. We demonstrate that torsion may influence considerably the motion of helical vortex filaments in an incompressible perfect fluid. The binormal component of the induced velocity, asymptotically responsible for the displacement of the vortex filament in the fluid, is closely analysed. The study is performed by applying the prescription of Moore & Saffman (1972) to helices of any pitch and a new asymptotic integral formula is derived. We give a rigorous proof that the Kelvin régime and its limit behaviour are obtained as a limit form of that integral asymptotic formula. The results are compared with new calculations based on the re-elaboration of Hardin's (1982) approach and with results obtained by Levy & Forsdyke (1928) and Widnall (1972) for helices of small pitch, here also re-elaborated for the purpose.

Boundary layers arising from flows which oscillate parallel to a permeable bed, and are subject to oscillating percolation of the same frequency as the bed parallel flow, referred to here as ‘ventilated oscillatory boundary layers’, are the subject of this laboratory study. These boundary layers are intended to approximate naturally occurring wave boundary layers over permeable beds. Measurements of boundary-layer velocities, bed stress and turbulent flow properties are presented. It is observed that suction (flow into the bed) enhances the near-bed velocities and bed stress while injection (flow out of the bed) leads to a reduction in these quantities. As the ventilated oscillatory boundary layer experiences both these phenomenon in one full cycle, the result is a net stress and a net boundary-layer velocity in an otherwise symmetric flow. While production of turbulence attributable to injection is enhanced, the finite time required for this to occur leads to a greater vertically averaged turbulence in the suction half-cycle. Turbulence generated in the suction half-cycle is maintained in a compact layer much closer to the bed. These effects appear to hold for $\widetilde{Re}$ ranging from 105 to 106 and for oscillations other than sinusoidal.

A study of the effect of external straining and shearing flows on the evolution and form of breakup of vortex rings has been performed. Two orientations each of straining and shearing flows are considered. A theoretical analysis of the ring motion for small strain and shear rates is performed, and it is found that for shearing and straining flows in the plane of the ring, the ring may oscillate periodically. For a straining flow with compression normal to the initial plane of the ring, the linear theory predicts that the ring radius will expand with time. For shearing flow normal to the initial plane of the ring, the linear theory predicts tilting of the ring in the direction of the shear flow rotation.

Numerical calculations are performed with both single vortex filaments and with a three-dimensional discrete vortex element method. The numerical calculations confirm the predictions of the linear theory for values of strain and shear rates below a certain critical value (which depends on the ratio R/σ0 of initial ring to core radii), whereas for strain and shear rates above this value the ring becomes very elongated with time and eventually pinches off. Three distinct regimes of long-time behaviour of the ring have been identified. Regime selection depends on initial ring geometry and orientation and on values of strain and shear rates. These regimes include (i) periodic oscillations with no pinching off, (ii) pinching off at the ring centre, and (iii) development of an elongated vortex pair at the ring centre and wider ‘heads’ near the ends (with pinching off just behind the heads). The boundaries of these regimes and theoretical reasons for the vortex behaviour in each case are described. It is also shown that the breakup of stretched vortex rings exhibits a self-similar behaviour, in which the number and size of ‘offspring’ vortices, at the point of pinching-off the ring, may be scaled by the product of the strain rate e (or shear rate s) and the oscillation period τ of a slightly elliptical ring with mean radius R.

A similarity solution is used to analyse the flow in the thin gap between a floating disk and an air table, the similarity solution being that for an axisymmetric stagnation point but with an upper boundary condition within the boundary layer. Viscous effects increase the height of the floating disk by 20% even at a Reynolds number of 50. Theoretical predictions are compared with experimental observations. The effect of the pressure distribution under the disk on the air flux through the table is examined.

Laminar–turbulent transition mechanisms for a supersonic boundary layer are examined by numerically solving the governing partial differential equations. It is shown that the dominant mechanism for transition at low supersonic Mach numbers is associated with the breakdown of oblique first-mode waves. The first stage in this breakdown process involves nonlinear interaction of a pair of oblique waves with equal but opposite angles resulting in the evolution of a streamwise vortex. This stage can be described by a wave–vortex triad consisting of the oblique waves and a streamwise vortex whereby the oblique waves grow linearly while nonlinear forcing results in the rapid growth of the vortex mode. In the second stage, the mutual and self-interaction of the streamwise vortex and the oblique modes results in the rapid growth of other harmonic waves and transition soon follows. Our calculations are carried all the way into the transition region which is characterized by rapid spectrum broadening, localized (unsteady) flow separation and the emergence of small-scale streamwise structures. The r.m.s. amplitude of the streamwise velocity component is found to be on the order of 4–5 % at the transition onset location marked by the rise in mean wall shear. When the boundary-layer flow is initially forced with multiple (frequency) oblique modes, transition occurs earlier than for a single (frequency) pair of oblique modes. Depending upon the disturbance frequencies, the oblique mode breakdown can occur for very low initial disturbance amplitudes (on the order of 0.001% or even lower) near the lower branch. In contrast, the subharmonic secondary instability mechanism for a two-dimensional primary disturbance requires an initial amplitude on the order of about 0.5% for the primary wave. An in-depth discussion of the oblique-mode breakdown as well as the secondary instability mechanism (both subharmonic and fundamental) is given for a Mach 1.6 flat-plate boundary layer.

Dilute aqueous solutions of long alcohol chains were recently found to cause a quadratic dependence of surface tension on the temperature without affecting other bulk properties of the liquid: σ = σ0 + αQ(T − T0)2, αQ > 0. The impact of such Marangoni instability on the behaviour of a thin liquid layer is studied in this work. We derive an equation describing a nonlinear spatiotemporal evolution of a thin film. The behaviour of the perturbed film in the absence of gravity, critically depends on whether the temperature T0, yielding a minimal surface tension, is attained on the surface of the film. When this is the case, a qualitatively new behaviour is observed: perturbations of the film interface may evolve into continuous steady patterns that do not rupture. Otherwise, the observed patterns due to the linear and quadratic Marangoni effects are qualitatively similar and result in the rupture of the film into separate drops.

Fluctuating wall-pressure measurements have been made on the centreline upstream of a blunt fin in a Mach 5 turbulent boundary layer. By examining the ensemble-averaged wall-pressure distributions for different separation shock foot positions, it has been shown that local fluctuating wall-pressure measurements are due to a distinct pressure distribution, [weierp ]i, which undergoes a stretching and flattening effect as its upstream boundary translates aperiodically between the upstream-influence and separation lines. The locations of the maxima and minima in the wall-pressure standard deviation can be accurately predicted using this distribution, providing quantitative confirmation of the model. This model also explains the observed cross-correlations and ensemble-average measurements within the interaction. Using the [weierp ]i model, wall-pressure signals from under the separated flow region were used to reproduce the position–time history of the separation shock foot. The unsteady behaviour of the primary horseshoe vortex and its relation to the unsteady separation shock is also described. The practical implications are that it may be possible to predict some of the unsteady aspects of the flowfield using mean wall-pressure distributions obtained from either computations or experiments; also, to minimize the fluctuating loads caused by the unsteadiness, flow control methods should focus on reducing the magnitude of the [weierp ]i gradient (∂[weierp ]i/∂x).