In this paper, two-dimensional problems of the diffraction and stability of shock waves are investigated using an approximate theory in which disturbances to the flow are treated as a wave propagation on the shocks. These waves carry changes in the slope and the Mach number of the shock. The equations governing the wave propagation are analogous in every way to the non-linear equations for plane waves in gas dynamics, and their solutions can be deduced by the same mathematical techniques. Since the propagation speed of the waves is found to be an increasing function of Mach number, waves carrying an increase in Mach number will eventually break and form what we may call a ‘shock’, corresponding to the breaking of a compression wave into a shock in the ordinary plane wave case. Such a ‘shock’ moving on the shock is called a shock-shock. The shock-shock is a discontinuity in Mach number and shock slope, and it must be fitted in to satisfy the appropriate relations between these are interpreted as the trace of cylindrical sound waves in the flow behind the shock. In particular a shock-shock is the trace of a genuine shock in the flow behind, and thus corresponds to Mach reflection.

The general theory of the wave propagation is set out in § 2. The subsequent sections contain applications of the theory to specific problems, including the motion of a shock along a curved wall, diffraction by a wedge, stability of plane shocks and the instability of a converging cylindrical shock.

We study the organization of turbulence in supersonic boundary layers through large-scale direct numerical simulations (DNS) at , and momentum-thickness Reynolds number up to (corresponding to ) which significantly extend the current envelope of DNS in the supersonic regime. The numerical strategy relies on high-order, non-dissipative discretization of the convective terms in the Navier–Stokes equations, and it implements a recycling/rescaling strategy to stimulate the inflow turbulence. Comparison of the velocity statistics up to fourth order shows nearly exact agreement with reference incompressible data, provided the momentum-thickness Reynolds number is matched, and provided the mean velocity and the velocity fluctuations are scaled to incorporate the effects of mean density variation, as postulated by Morkovin’s hypothesis. As also found in the incompressible regime, we observe quite a different behaviour of the second-order flow statistics at sufficiently large Reynolds number, most of which show the onset of a range with logarithmic variation, typical of ‘attached’ variables, whereas the wall-normal velocity exhibits a plateau away from the wall, which is typical of ‘detached’ variables. The modifications of the structure of the flow field that underlie this change of behaviour are highlighted through visualizations of the velocity and temperature fields, which substantiate the formation of large jet-like and wake-like motions in the outer part of the boundary layer. It is found that the typical size of the attached eddies roughly scales with the local mean velocity gradient, rather than being proportional to the wall distance, as happens for the wall-detached variables. The interactions of the large eddies in the outer layer with the near-wall region are quantified through a two-point amplitude modulation covariance, which characterizes the modulating action of energetic outer-layer eddies.

Direct numerical simulations of stably and strongly stratified turbulent flows with Reynolds number Re ≫ 1 and horizontal Froude number Fh ≪ 1 are presented. The results are interpreted on the basis of a scaling analysis of the governing equations. The analysis suggests that there are two different strongly stratified regimes according to the parameter . When , viscous forces are unimportant and lv scales as lv ∼ U/N (U is a characteristic horizontal velocity and N is the Brunt–Väisälä frequency) so that the dynamics of the flow is inherently three-dimensional but strongly anisotropic. When , vertical viscous shearing is important so that (lh is a characteristic horizontal length scale). The parameter is further shown to be related to the buoyancy Reynolds number and proportional to (lO/η)4/3, where lO is the Ozmidov length scale and η the Kolmogorov length scale. This implies that there are simultaneously two distinct ranges in strongly stratified turbulence when : the scales larger than lO are strongly influenced by the stratification while those between lO and η are weakly affected by stratification. The direct numerical simulations with forced large-scale horizontal two-dimensional motions and uniform stratification cover a wide Re and Fh range and support the main parameter controlling strongly stratified turbulence being . The numerical results are in good agreement with the scaling laws for the vertical length scale. Thin horizontal layers are observed independently of the value of but they tend to be smooth for < 1, while for > 1 small-scale three-dimensional turbulent disturbances are increasingly superimposed. The dissipation of kinetic energy is mostly due to vertical shearing for < 1 but tends to isotropy as increases above unity. When < 1, the horizontal and vertical energy spectra are very steep while, when > 1, the horizontal spectra of kinetic and potential energy exhibit an approximate k−5/3h-power-law range and a clear forward energy cascade is observed.

Two time-periodic solutions with genuine three-dimensional structure are numerically discovered for the incompressible Navier–Stokes equation of a constrained plane Couette flow. One solution with strong variation in spatial and temporal structure exhibits a full regeneration cycle, which consists of the formation and breakdown of streamwise vortices and low-velocity streaks; the other one, of gentle variation, represents a spanwise standing-wave motion of low-velocity streaks. These two solutions are unstable and the corresponding periodic orbits in the phase space are connected with each other. A turbulent state wanders around the strong one for most of the time except for occasional escapes from it. As a result, the mean velocity profile and the root-mean-squares of velocity fluctuations of the plane Couette turbulence agree very well with the temporal averages of those of this periodic motion. After an occasional escape from the strong solution, the turbulent state reaches the gentle periodic solution and returns. On the way back, it experiences an overshoot accompanied by strong turbulence activity like an intermittent bursting phenomenon.

The velocity and shape of rising bubbles, with an equivalent radius of 0.33–1.00 mm, in ‘hyper clean’ water, have been experimentally determined. For the small bubbles there is perfect agreement with theory, proving that this water can be considered as pure (no surfactants). For the larger bubbles there is a small discrepancy due to an overestimation in the theory.

The drag on a cylindrical particle moving in a thin sheet of viscous fluid is calculated. It is supposed that the sheet is embedded in fluid of much lower viscosity. A finite steady drag is obtained, which depends logarithmically on the ratio of the viscosities. The Einstein relation is used to determine the diffusion coefficient for Brownian motion of the particle, with application to the movement of molecules in biological membranes. In addition, the Brownian motion is calculated using the Langevin equation, and a logarithmically time-dependent diffusivity is obtained for the case when the embedding fluid has zero viscosity.

Townsend's (1961) hypothesis that the turbulent motion in the inner region of a boundary layer consists of (i) an ‘active’ part which produces the shear stress τ and whose statistical properties are universal functions of τ and y, and (ii) an ‘inactive’ and effectively irrotational part determined by the turbulence in the outer layer, is supported in the present paper by measurements of frequency spectra in a strongly retarded boundary layer, in which the ‘inactive’ motion is particularly intense. The only noticeable effect of the inactive motion is an increased dissipation of kinetic energy into heat in the viscous sublayer, supplied by turbulent energy diffusion from the outer layer towards the surface. The required diffusion is of the right order of magnitude to explain the non-universal values of the triple products measured near the surface, which can therefore be reconciled with universality of the ‘active’ motion.

Dimensional analysis shows that the contribution of the ‘active’ inner layer motion to the one-dimensional wave-number spectrum of the surface pressure fluctuations varies as τ2w/k1 up to a wave-number inversely proportional to the thickness of the viscous sublayer. This result is strongly supported by the recent measurements of Hodgson (1967), made with a much smaller ratio of microphone diameter to boundary-layer thickness than has been achieved previously. The disagreement of the result with most other measurements is attributed to inadequate transducer resolution in the other experiments.

A series of experiments designed to reveal the properties of high Reynolds number vortex rings, using flow-visualization and laser-Doppler techniques, has uncovered several interesting and unexpected results. Starting at the beginning of the motion, at a nozzle, and proceeding downstream, these include the following.

1. A formation process that is strongly Reynolds number dependent.

2. The amount of vorticity that appears downstream is very close to that predicted by a simple ‘slug’ model. However flow-visualization studies show that such a model is an oversimplification and that an excess of ring vorticity is probably cancelled by the ingestion of vorticity of opposite sign at the nozzle lip.

3. (iii) A new, bimodal form of vortex-core instability has been observed at moderate but not high Reynolds numbers.

4. Azimuthal inhomogeneities in the breaking of these, and the normal instability waves, create an ‘axial’ flow along the vortex core in the turbulent ring. This axial flow takes the form of a propagating wave that has many characteristics of a solitary wave. It is hypothesized that this axial flow prevents further ring instability.

5. The long-term behaviour of the turbulent ring is marked by dramatic changes in its growth rate, which are probably related to changes in the ‘organization’ of the vortex core. The descriptive turbulent-ring model developed in Maxworthy (1974) is substantially confirmed by these experiments and by observation of ring propagation through a stratified ambient fluid.

The flow around two circular cylinders of equal diameter, arranged in a staggered configuration, was investigated using flow visualization and particle image velocimetry for centre-to-centre pitch ratio P/D = 1[ratio ]0 to 5.0 and angle of incidence. α = 0° to 90°. Experiments were conducted within the low subcritical Reynolds number regime, from Re = 850 to 1900. Nine flow patterns were identified, and processes of shear layer reattachment, induced separation, vortex pairing and synchronization, and vortex impingement, were observed. New insight was gained into previously published Strouhal number data, by considering the flow patterns involved. The study revealed that vortex shedding frequencies are more properly associated with individual shear layers than with individual cylinders; more specifically, the two shear layers from the downstream cylinder often shed vortices at different frequencies.

The initial-value problem for perturbations of a flat, constant-strength vortex sheet is linearly ill posed in the sense of Hadamard, owing to Kelvin-Helmholtz instability. Previous numerical studies of this problem have experienced difficulty in converging when the mesh was refined. The present work examines Rosenhead's point-vortex approximation and seeks to understand better the source of this difficulty. Using discrete Fourier analysis, it is shown that perturbations introduced spuriously by computer roundoff error are responsible for the irregular point-vortex motion that occurs at a smaller time as the number of points is increased. This source of computational error is controlled here by using either higher precision arithmetic or a new filtering technique. Computations are presented which use a linear-theory growing eigenfunction of small amplitude/wavelength ratio as the initial perturbation. The results indicate the formation of a singularity in the vortex sheet at a finite time as previously found for other initial data by Moore and Meiron, Baker & Orszag using different techniques of analysis. Numerical evidence suggests that the point-vortex approximation converges up to but not beyond the time of singularity formation in the vortex sheet. For large enough initial amplitude, two singularities appear along the sheet at the critical time.

Time-averaged LDA measurements and time-resolved numerical flow predictions were performed to investigate the laminar flow induced by the harmonic in-line oscillation of a circular cylinder in water at rest. The key parameters, Reynolds number Re and Keulegan–Carpenter number KC, were varied to study three parameter combinations in detail. Good agreement was observed for Re=100 and KC=5 between measurements and predictions comparing phase-averaged velocity vectors. For Re=200 and KC=10 weakly stable and non-periodic flow patterns occurred, which made repeatable time-averaged measurements impossible. Nevertheless, the experimentally visualized vortex dynamics was reproduced by the two-dimensional computations. For the third combination, Re=210 and KC=6, which refers to a totally different flow regime, the computations again resulted in the correct fluid behaviour. Applying the widely used model of Morison et al. (1950) to the computed in-line force history, the drag and the added-mass coefficients were calculated and compared for different grid levels and time steps. Using these to reproduce the force functions revealed deviations from those originally computed as already noted in previous studies. They were found to be much higher than the deviations for the coarsest computational grid or the largest time step. The comparison of several in-line force coefficients with results obtained experimentally by Kühtz (1996) for β=35 confirmed that force predictions could also be reliably obtained by the computations.

Some general results are derived for the efficiency of energy absorption of a system of uniform oscillatory surface pressure distributions. The results, which are based on classical linear water-wave theory, show the close analogies which exist with theories for systems of absorbing oscillatory rigid bodies and a number of new reciprocal relations for pressure distributions are suggested and proved. Some simple examples illustrating the general results are given and compared with the corresponding results for rigid bodies.

This paper presents experimental measurements of the approach and rebound of a particle colliding with a wall in a viscous fluid. The particle's trajectory was controlled by setting the initial inclination angle of a pendulum immersed in a fluid. The resulting collisions were monitored using a high-speed video camera. The diameters of the particles ranged from 3 to 12 mm, and the ratio of the particle density to fluid density varied from 1.2 to 7.8. The experiments were performed using a thick glass or Lucite wall with different mixtures of glycerol and water. With these parameters, the Reynolds number defined using the velocity just prior to impact ranged from 10 to approximately 3000. A coefficient of restitution was defined from the ratio of the velocity just prior to and after impact.

The experiments clearly demonstrate that the rebound velocity depends on the impact Stokes number (defined from the Reynolds number and the density ratio) and weakly on the elastic properties of the material. Below a Stokes number of approximately 10, no rebound of the particle occurred. For impact Stokes number above 500 the coefficient of restitution appears to asymptote to the values for dry collisions. The coefficients of restitution were also compared with previous experimental studies. In addition, the approach of the particle to the wall indicated that the particle slowed prior to impacting the surface. The distance at which the particle's trajectory varied due to the presence of the wall was dependent on the impact Stokes number. The particle surface roughness was found to affect the repeatability of some measurements, especially for low impact velocities.

A method for the numerical solution of steadily progressing periodic waves on irrotational flow over a horizontal bed is presented. No analytical approximations are made. A finite Fourier series, similar to Dean's stream function series, is used to give a set of nonlinear equations which can be solved using Newton's method. Application to laboratory and field situations is emphasized throughout. When compared with known results for wave speed, results from the method agree closely. Results for fluid velocities are compared with experiment and agreement found to be good, unlike results from analytical theories for high waves.

The problem of shoaling waves can conveniently be studied using the present method because of its validity for all wavelengths except the solitary wave limit, using the conventional first-order approximation that on a sloping bottom the waves at any depth act as if the bed were horizontal. Wave period, energy flux and mass flux are conserved. Comparisons with experimental results show good agreement.

Consistency and uniqueness questions raised by both the 1941 and 1962 Kolmogorov inertial-range theories are examined. The 1941 theory, although unlikely from the viewpoint of vortex-stretching physics, is not ruled out just because the dissipation fluctuates; but self-consistency requires that dissipation fluctuations be confined to dissipation-range scales by a spacewise mixing mechanism. The basic idea of the 1962 theory is a self-similar cascade mechanism which produces systematically increasing intermittency with a decrease of scale size. This concept in itself requires neither the third Kolmogorov hypothesis (log-normality of locally averaged dissipation) nor the first hypothesis (universality of small-scale statistics as functions of scale-size ratios and locally averaged dissipation). It does not even imply that the inertial range exhibits power laws. A central role for dissipation seems arbitrary since conservation alone yields no simple relation between the local dissipation rate and the corresponding proper inertial-range quantity: the local rate of energy transfer. A model rate equation for the evolution of probability densities is used to illustrate that even scalar nonlinear cascade processes need not yield asymptotic log-normality. The approximate experimental support for Kolmogorov's hypothesis takes on added significance in view of the wide variety of a priori admissible alternatives.

If the Kolmogorov law $E(k) \propto k^{-\frac{5}{3}-\mu}$ is asymptotically valid, it is argued that the value of μ depends on the details of the nonlinear interaction embodied in the Navier–Stokes equation and cannot be deduced from overall symmetries, invariances and dimensionality. A dynamical equation is exhibited which has the same essential invariances, symmetries, dimensionality and equilibrium statistical ensembles as the Navier–Stokes equation but which has radically different inertial-range behaviour.

Building on our recent work on induced-charge electro-osmosis (ICEO) and electrophoresis (ICEP), as well as the Russian literature on spherical metal colloids, we examine the rich consequences of broken geometric and field symmetries upon the ICEO flow around conducting bodies. Through a variety of paradigmatic examples involving ideally polarizable (e.g. metal) bodies with thin double layers in weak fields, we demonstrate that spatial asymmetry generally leads to a net pumping of fluid past the body by ICEO, or, in the case of a freely suspended colloidal particle, translation and/or rotation by ICEP. We have chosen model systems that are simple enough to admit analysis, yet which contain the most important broken symmetries. Specifically, we consider (i) symmetrically shaped bodies with inhomogeneous surface properties, (ii) ‘nearly symmetric’ shapes (using a boundary perturbation scheme), (iii) highly asymmetric bodies composed of two symmetric bodies tethered together, (iv) symmetric conductors in electric-field gradients, and (v) arbitrarily shaped conductors in general non-uniform fields in two dimensions (using complex analysis). In non-uniform fields, ICEO flow and ICEP motion exist in addition to the more familiar dielectrophoretic forces and torques on the bodies (which also vary with the square of the electric field). We treat all of these problems in two and three dimensions, so our study has relevence for both colloids and microfluidics. In the colloidal context, we describe principles to ‘design’ polarizable particles which rotate to orient themselves and translate steadily in a desired direction in a DC or AC electric field. We also describe ‘ICEO spinners’ that rotate continuously in AC fields of arbitrary direction, although we show that ‘near spheres’ with small helical perturbations do not rotate, to leading order in the shape perturbation. In the microfluidic context, strong and steady flows can be driven by small AC potentials applied to systems containing asymmetric structures, which holds promise for portable or implantable self-powered devices. These results build upon and generalize recent studies in AC electro-osmosis (ACEO). Unlike ACEO, however, the inducing surfaces in ICEO can be physically distinct from the driving electrodes, increasing the frequency range and geometries available.

The effect of initial conditions on the decay exponent and coefficient and virtual origin in the decay power-law form for the variation of the variance of the turbulent velocity downstream of biplane grids constructed of rods of both round and square cross-section is determined. This effect is determined for data obtained as part of the present study as well as from previous studies. These studies cover a Reynolds number range from 6000 to 68000, mesh sizes of 2.54 and 5.08 cm, and solidities of 0.34 and 0.44.

It is shown that the choice of the virtual origin and the use of data in the non-homogeneous portion of the flow can have a significant influence on the value of the parameters in the decay power-law. Criteria are developed to identify the nearly homogeneous and isotropic portion of the flow. These criteria include low values of the velocity skewness, constancy of the skewness of the velocity derivative and balance of the turbulent kinetic energy equation.

Results based on data selected by means of these criteria show that the decay exponent and virtual origin are independent of initial conditions such as Reynolds number, mesh size, solidity, and rod shape and surface roughness with values of respectively 1.30 and 0. In contrast and as expected, the decay coefficient is found to be a function of these initial conditions. Thus, the downstream variation of the variance of the turbulent velocity is universally self-similar.

Direct numerical simulations (DNS) have been performed of turbulent flow in a plane channel with a solid top wall and a permeable bottom wall. The permeable wall is a packed bed, which is characterized by the mean particle diameter and the porosity. The main objective is to study the influence of wall permeability on the structure and dynamics of turbulence. The flow inside the permeable wall is described by means of volume-averaged Navier–Stokes equations. Results from four simulations are shown, for which only the wall porosity ($\epsilon_c$) is changed. The Reynolds number based on the thickness of the boundary layer over the permeable wall and the friction velocity varies from $\hbox{\it Re}_{\tau}^p\,{=}\,176$ for $\epsilon_c\,{=}\,0$ to $\hbox{\it Re}_{\tau}^p\,{=}\,498$ for $\epsilon_c\,{=}\,0.95$. The influence of wall permeability can be characterized by the permeability Reynolds number, $\hbox{\it Re}_K$, which represents the ratio of the effective pore diameter to the typical thickness of the viscous sublayers over the individual wall elements. For small $\hbox{\it Re}_K$, the wall behaves like a solid wall. For large $\hbox{\it Re}_K$, the wall is classified as a highly permeable wall near which viscous effects are of minor importance. It is observed that streaks and the associated quasi-streamwise vortices are absent near a highly permeable wall. This is attributed to turbulent transport across the wall interface and the reduction in mean shear due to a weakening of, respectively, the wall-blocking and the wall-induced viscous effect. The absence of streaks is consistent with a decrease in the peak value of the streamwise root mean square (r.m.s.) velocity normalized by the friction velocity at the permeable wall. Despite the increase in the peak values of the spanwise and wall-normal r.m.s. velocities, the peak value of the turbulent kinetic energy is therefore smaller. Turbulence near a highly permeable wall is dominated by relatively large vortical structures, which originate from a Kelvin–Helmholtz type of instability. These structures are responsible for an exchange of momentum between the channel and the permeable wall. This process contributes strongly to the Reynolds-shear stress and thus to a large increase in the skin friction.

An experimental study of Rayleigh–Bénard convection in the strongly turbulent regime is presented. We report results obtained at low Prandtl number (in mercury, Pr = 0.025), covering a range of Rayleigh numbers 5 × 106 < Ra < 5 × 109, and compare them with results at Pr∼1. The convective chamber consists of a cylindrical cell of aspect ratio 1.

Heat flux measurements indicate a regime with Nusselt number increasing as Ra0.26, close to the 2/7 power observed at Pr∼1, but with a smaller prefactor, which contradicts recent theoretical predictions. A transition to a new turbulent regime is suggested for Ra ≃ 2 × 109, with significant increase of the Nusselt number. The formation of a large convective cell in the bulk is revealed by its thermal signature on the bottom and top plates. One frequency of the temperature oscillation is related to the velocity of this convective cell. We then obtain the typical temperature and velocity in the bulk versus the Rayleigh number, and compare them with similar results known for Pr∼1.

We review two recent theoretical models, namely the mixing zone model of Castaing et al. (1989), and a model of the turbulent boundary layer by Shraiman & Siggia (1990). We discuss how these models fail at low Prandtl number, and propose modifications for this case. Specific scaling laws for fluids at low Prandtl number are then obtained, providing an interpretation of our experimental results in mercury, as well as extrapolations for other liquid metals.

The interaction between a conventional rectangular (primary) air jet and a co-flowing synthetic jet is investigated experimentally. The nozzles of both jets have the same long dimension but the aspect ratio of the synthetic jet orifice is 25 times larger. Detailed particle image velocimetry (PIV) measurements of the flow in the midspan plane show that primary jet fluid is directed into the synthetic jet orifice and the interaction between the jets leads to the formation of a closed recirculating flow domain. The concomitant formation of a low-pressure region between the jets results in deflection of the primary jet toward the actuator jet despite the absence of an extended control surface (e.g. a diffuser or collar) and is balanced by a force on the primary jet conduit. For a given synthetic jet strength and primary jet speed, the vectoring force depends mainly on the volume flow rate of primary jet fluid that is diverted into the synthetic jet actuator. This flow rate is regulated by restricting the flow of entrained ambient fluid using a small streamwise extension of the synthetic jet orifice that scales with the orifice width. The response of the primary jet to the imposed vectoring is investigated using stepped modulation of the driving signal. The characteristic vectoring time and vectoring angle decrease monotonically with primary jet speed.