A large-Reynolds-number asymptotic solution of the Navier–Stokes equations is sought for the motion of an axisymmetric vortex ring of small cross-section embedded in a viscous incompressible fluid. In order to take account of the influence of elliptical deformation of the core due to the self-induced strain, the method of matched of matched asymptotic expansions is extended to a higher order in a small parameter ε = (v/Γ)1/2, where v is the kinematic viscosity of fluid and Γ is the circulation. Alternatively, ε is regarded as a measure of the ratio of the core radius to the ring radius, and our scheme is applicable also to the steady inviscid dynamics.

We establish a general formula for the translation speed of the ring valid up to third order in ε. This is a natural extension of Fraenkel–Saffman's first-order formula, and reduces, if specialized to a particular distribution of vorticity in an inviscid fluid, to Dyson's third-order formula. Moreover, it is demonstrated, for a ring starting from an infinitely thin circular loop of radius R0, that viscosity acts, at third order, to expand the circles of stagnation points of radii Rs(t) and R˜s(t) relative to the laboratory frame and a comoving frame respectively, and that of peak vorticity of radius R˜p(t) as Rs ≈ R0 + [2 log(4R0/√vt) + 1.4743424] vt/R0, R˜s ≈ R0 + 2.5902739 vt/R0, and Rp ≈ R0 + 4.5902739 vt/R0. The growth of the radial centroid of vorticity, linear in time, is also deduced. The results are compatible with the experimental results of Sallet & Widmayer (1974) and Weigand & Gharib (1997).

The procedure of pursuing the higher-order asymptotics provides a clear picture of the dynamics of a curved vortex tube; a vortex ring may be locally regarded as a line of dipoles along the core centreline, with their axes in the propagating direction, subjected to the self-induced flow field. The strength of the dipole depends not only on the curvature but also on the location of the core centre, and therefore should be specified at the initial instant. This specification removes an indeterminacy of the first-order theory. We derive a new asymptotic development of the Biot-Savart law for an arbitrary distribution of vorticity, which makes the non-local induction velocity from the dipoles calculable at third order.

The primary objective of this work is to determine the detailed characteristics of the flow features induced in a boundary layer by suction through laminar flow control (LFC) perforations. An additional goal is to validate a predictive method for generic LFC suction surfaces and to apply this technique to typical flight condition configurations. Fundamental insights into the flow physics of LFC suction surfaces are obtained from a unique series of high-resolution three-component laser Doppler velocimetry (LDV) flow field measurements. The flow fields induced by isolated super-scale perforations under low-speed conditions are mapped and found to be strongly three-dimensional and profoundly different from the idealized concept of continuously distributed suction. Over a range of sub- and super-critical suction flow rates a variety of suction-dependent complex flow features are identified, including a pair of contra-rotating streamwise vortices, multiple co-rotating streamwise vortices, spanwise variations of the mean flow and inherently unstable boundary layer profiles. The measurements reveal that suction-induced transition commences with an instability of these attached vortices, resulting in the development of a pair of turbulent wedges downstream of the perforation. A finite-volume Navier–Stokes method is validated by simulating a variety of low-speed experiments and comparisons are made between the LDV measurements and the predicted flow field. The computational technique reproduces all of the observed flow features, although it slightly under-predicts their magnitude and extent. By analysing the predicted flow fields the mechanism for the formation of the trailing vortex pair is established. Earlier flow visualization experiments, which exhibited vortex shedding, are also simulated by solving the time-dependent governing equations and it is found that the principal unsteady flow features are captured. Despite the challenge posed to the computational method by the diverse range of flow phenomena induced by discrete suction, the predictions provide good agreement with the measurements and observations. The computational tool is subsequently applied to predict the flow fields of single and multiple rows of actual-scale micro-perforations under low-speed and typical transonic flight conditions. A range of suction-induced flow features are predicted and a variety of distinct flow modes are identified. The low-speed critical suction limits are also measured and a design criterion, based on the sucked streamtube characteristics, is established. The basis of this critical suction criterion is also validated for transonic flight configurations.

Experiments were conducted in a wind tunnel in which a turbulent boundary layer was naturally grown over flat beds of three types of nearly mono-disperse spherical particles with different diameters, densities and coefficient of restitution (r) (snow, 0.48 mm, 910 kg m−3; mustard seeds, 1.82 mm, 1670 kg m−3, r = 0.7; ice particles, 2.80 mm, 910 kg m−3, r = 0.8–0.9). The surface wind speeds (defined by the friction velocity u∗) were varied between 1.0 and 1.9 times the threshold surface wind speed (defined by u∗t). The trajectories, and ejection and impact velocities of the particles were recorded and analysed, even those that were raised only about one diameter into the flow.

Measurements of the average horizontal flux of saltating particles per unit area, f(z), at each level z above the surface showed that, for u∗/u∗t [les ] 1.5, f(z) is approximately independent of the particle density and decreases exponentially over a vertical scale length lf, that is about 3 to 4 times the estimated mean height of the particle trajectories 〈h〉. Numerical simulations of saltating grains were computed using the measured probabilities of ejection velocities and the mean velocity profile of the air flow, but neglecting the direct effect of the turbulence. The calculated mean values of the impact velocities and the trajectory dimensions were found to agree with the measurements in the saltation range, where u∗/u∗t < 1.5. Similarly, in this range the simulations of the horizontal flux profile and integral are also consistent with the measurements and with Bagnold's u∗3 formula, respectively.

When u∗/u∗t [ges ] 1.5, and u∗/VT [ges ] 1/10, where VT is the settling velocity, a transition from saltation to suspension occurs. This is indicated by the change in the mean mass flux profile which effectively becomes uniform with height (z) up to the top of the boundary layer. An explanation is provided for this low value of turbulence at transition relative to the settling velocity in terms of the random motion of the particles under the action of the turbulence when they reach the tops of their parabolic trajectories. The experiments show that, as u∗/u∗t increases from 1.0 to 1.9 the normalized mean vertical impact velocity 〈V3I〉/u∗ decreases by nearly 60% to about 0.6, which is less than 50% of the value for fluid particles. There is also a decrease in the vertical and horizontal component of the ejection velocity to values of 0.8 and 2.3, which are much less than their values in the saltation regime. We hypothesize that at the transition from saltation to suspension the ejection process changes quite sharply from being determined by impact collisions to being the result of aerodynamic lift forces and upward eddy motions.

New three-dimensional finite-amplitude travelling-wave solutions are found in rotating Hagen–Poiseuille flow (RHPF[Ωa, Ωp]) where fluid is driven by a constant pressure gradient along a pipe rotating axially at rate Ωa and at Ωp about a perpendicular diameter. For purely axial rotation (RHPF[Ωa, 0]), the two-dimensional helical waves found by Toplosky & Akylas (1988) are found to become unstable to three-dimensional travelling waves in a supercritical Hopf bifurcation. The addition of a perpendicular rotation at low axial rotation rates is found only to stabilize the system. In the absence of axial rotation, the two-dimensional steady flow solution in RHPF[0, Ωp] which connects smoothly to Hagen–Poiseuille flow as Ωp → 0 is found to be stable at all Reynolds numbers below 104. At high axial rotation rates, the superposition of a perpendicular rotation produces a ‘precessional’ instability which here is found to be a supercritical Hopf bifurcation leading directly to three-dimensional travelling waves. Owing to the supercritical nature of this primary bifurcation and the secondary bifurcation found in RHPF[Ωa, 0], no opportunity therefore exists to continue these three-dimensional finite-amplitude solutions in RHPF back to Hagen–Poiseuille flow. This then contrasts with the situation in narrow-gap Taylor–Couette flow where just such a connection exists to plane Couette flow.

Laboratory experiments were conducted in a water tank to investigate the rise through the atmosphere of thermals generated by the detonation of surplus military munitions. The fall of a dense volume of fluid through the water in the tank follows the same governing equations as the rise of a buoyant volume of gas in the atmosphere. By filling the tank with a layer of water and a layer of salt water, an elevated step change in density was obtained, simulating a temperature inversion or jump in the atmosphere. The growth of the linear dimensions of the thermal and its volume were determined and used in the development of a criterion for predicting when a thermal will fully penetrate the inversion. Replacing the second layer of fluid with water of gradually increasing salinity, an elevated constant density gradient was obtained. In these cases, the maximum penetration distance of the thermal was observed and the equilibrium position and vertical spread were determined experimentally.

These observations and the empirical relationships determined from them should prove useful in the development and evaluation of air pollution dispersion models for predicting the atmospheric transport and diffusion of material released during such detonations.

The initial stage of the water flow caused by an impact on a floating body is considered. The vertical velocity of the body is prescribed and kept constant after a short acceleration stage. The present study demonstrates that impact on a floating and non-flared body gives acoustic effects that are localized in time behind the front of the compression wave generated at the moment of impact and are of major significance for explaining the energy distribution throughout the water, but their contribution to the flow pattern near the body decays with time. We analyse the dependence on the body acceleration of both the water flow and the energy distribution – temporal and spatial. Calculations are performed for a half-submerged sphere within the framework of the acoustic approximation. It is shown that the pressure impulse and the total impulse of the flow are independent of the history of the body motion and are readily found from pressure-impulse theory. On the other hand, the work done to oppose the pressure force, the internal energy of the water and its kinetic energy are essentially dependent on details of the body motion during the acceleration stage. The main parameter is the ratio of the time scale for the acoustic effects and the duration of the acceleration stage. When this parameter is small the work done to accelerate the body is minimal and is spent mostly on the kinetic energy of the flow. When the sphere is impulsively started to a constant velocity (the parameter is infinitely large), the work takes its maximum value: Longhorn (1952) discovered that half of this work goes to the kinetic energy of the flow near the body and the other half is taken away with the compression wave. However, the work required to accelerate the body decreases rapidly as the duration of the acceleration stage increases. The optimal acceleration of the sphere, which minimizes the acoustic energy, is determined for a given duration of the acceleration stage. Roughly speaking, the optimal acceleration is a combination of both sudden changes of the sphere velocity and uniform acceleration.

If only the initial velocity of the body is prescribed and it then moves freely under the influence of the pressure, the fraction of the energy lost in acoustic waves depends only on the ratio of the body's mass to the mass of water displaced by the hemisphere.

The goals of this study were to develop a set of Reynolds-averaged governing equations for turbulent free-surface flow, and to use the resulting equations to determine the origin of the surface current in high-Froude-number jet flows. To develop the Reynolds-averaged equations, free-surface turbulent flow is treated as a two-fluid flow separated by an interface. It is shown that the general Navier–Stokes equations written for variable property flow embody the field equations applicable to each fluid, as well as the boundary conditions for the interface and, therefore, can be applied across the entire fluid domain, including the interface. With this as a starting point, a formulation of the Reynolds-averaged governing equations for turbulent free-surface flows can be developed rigorously. The resulting Reynolds-averaged equations are written in terms of density-weighted averages, their derivatives, and the probability density function for the free-surface position. These equations are similar to the conventional Reynolds-averaged equations, but include additional terms which represent the average effect of the forces acting instantaneously on the free surface, forces normally associated with the boundary conditions. These averaged equations are applied to the interaction of a turbulent jet with the free surface in order to establish, for arbitrary-Froude-number flows, the origin of the surface current, the large outward velocity which occurs in a thin layer adjacent to the surface. It is shown via an order-of-magnitude analysis that the outward acceleration associated with the surface current results from a combination of the Reynolds-stress anisotropy and the free-surface fluctuations. For low Froude number, the surface current is mainly driven by the Reynolds stress anisotropy, consistent with the results of Walker (1997); when the Froude number is large, the Reynolds-stress anisotropy is smaller and the free-surface fluctuations make a significant contribution.

Numerical studies are described of the flows generated by a sphere moving vertically in a uniformly stratified fluid. It is found that the axisymmetric standing vortex usually found in homogeneous fluids at moderate Reynolds numbers (25 [les ] Re [les ] 200) is completely collapsed by stable stratification, generating a strong vertical jet. This is consistent with our experimental visualizations. For Re = 200 the complete collapse of the vortex occurs at Froude number F ≃ 19, and the critical Froude number decreases slowly as Re increases. The Froude number and the Reynolds number are here defined by F = W/Na and Re = 2Wa/v, with W being the descent velocity of the sphere, N the Brunt–Väisälä frequency, a the radius of the sphere and v the kinematic viscosity coefficient. The inviscid processes, including the generation of the vertical jet, have been investigated by Eames & Hunt (1997) in the context of weak stratification without buoyancy effects. They showed the existence of a singularity of vorticity and density gradient on the rear axis of the flow and also the impossibility of realizing a steady state. When there is no density diffusion, all the isopycnal surfaces which existed initially in front of the sphere accumulate very near the front surface because of density conservation and the fluid in those thin layers generates a rear jet when returning to its original position. In the present study, however, the fluid has diffusivity and the buoyancy effects also exist. The density diffusion prevents the extreme piling up of the isopycnal surfaces and allows the existence of a steady solution, preventing the generation of a singularity or a jet. On the other hand, the buoyancy effect works to increase the vertical velocity to the rear of the sphere by converting the potential energy to vertical kinetic energy, leading to the formation of a strong jet. We found that the collapse of the vortex and the generation of the jet occurs at much weaker stratifications than those necessary for the generation of strong lee waves, showing that jet formation is independent of the internal waves. At low Froude numbers (F [les ] 2) the lee wave patterns showed good agreement with the linear wave theory and the previous experiments by Mowbray & Rarity (1967). At very low Froude numbers (F [les ] 1) the drag on a sphere increases rapidly, partly due to the lee wave drag but mainly due to the large velocity of the jet. The jet causes a reduction of the pressure on the rear surface of the sphere, which leads to the increase of pressure drag. High velocity is induced also just outside the boundary layer of the sphere so that the frictional drag increases even more significantly than the pressure drag.

The theoretical linear stability of a shock wave moving in an unlimited homogeneous environment has been widely studied during the last fifty years. Important results have been obtained by Dýakov (1954), Landau & Lifchitz (1959) and then by Swan & Fowles (1975) where the fluctuating quantities are written as normal modes. More recently, numerical studies on upwind finite difference schemes have shown some instabilities in the case of the motion of an inviscid perfect gas in a rectangular channel. The purpose of this paper is first to specify a mathematical formulation for the eigenmodes and to exhibit a new mode which was not found by the previous stability analysis of shock waves. Then, this mode is confirmed by numerical simulations which may lead to a new understanding of the so-called carbuncle phenomenon.

Even though the first theoretical example of chaotic advection was a three-dimensional flow (Hénon 1966), the number of theoretical studies addressing chaos and mixing in three-dimensional flows is small. One problem is that an experimentally tractable three-dimensional system that allows detailed experimental and computational investigation had not been available. A prototypical, bounded, three-dimensional, moderate-Reynolds-number flow is presented; this system lends itself to detailed experimental observation and allows high-precision computational inspection of geometrical and dynamical effects. The flow structure, captured by means of cuts with a laser sheet (experimental Poincaré section), is visualized via continuously injected fluorescent dye streams, and reveals detailed chaotic structures and chains of high-period islands. Numerical experiments are performed and compared with particle image velocimetry (PIV) and flow visualization results. Predictions of existing theories for chaotic advection in three-dimensional volume-preserving flows are tested. The ratio of two frequencies of particle motion – the frequency of motion around the vertical axis and the frequency of recirculation in the plane containing the axis – is identified as the crucial parameter. Using this parameter, the number of islands in the chain can be predicted. The same parameter – using as a base-case the integrable motion – allows the identification of operating conditions where small perturbations lead to nearly complete mixing.

A linear stability analysis is carried out to examine the initial stage of sand-wave growth under tidal flows and the occurrence of a preferred length scale. The fact that these bedforms typically have length scales small compared to the tidal excursion is exploited by adopting an asymptotic approach to solve the hydrodynamic part of the problem, i.e. to find the hydrodynamic response to an initially small bed perturbation. This method is shown to have important advantages over previously used methods, since it allows an exploration of the complete sand-wave regime (whereas other methods fail for short sand waves), and in general it is also more accurate. It is found that the selection of a preferred length scale depends mainly on only two parameters (the bed-slope coefficient, and the ratio of friction velocity to eddy viscosity), whereas there appears to be almost no dependence on the water depth.

The one-dimensional Burgers equation in the inviscid limit with white noise initial condition is revisited. The one- and two-point distributions of the Burgers field as well as the related distributions of shocks are obtained in closed analytical forms. In particular, the large distance behaviour of spatial correlations of the field is determined. Since higher-order distributions factorize in terms of the one- and two- point functions, our analysis provides an explicit and complete statistical description of this problem.