A new experimental investigation of decaying turbulence generated by a low-blockage space-filling fractal square grid is presented. We find agreement with previous works by Seoud & Vassilicos (Phys. Fluids, vol. 19, 2007, 105108) and Mazellier & Vassilicos (Phys. Fluids, vol. 22, 2010, 075101) but also extend the length of the assessed decay region and consolidate the results by repeating the experiments with different probes of increased spatial resolution. It is confirmed that this moderately high Reynolds number turbulence (up to here) does not follow the classical high Reynolds number scaling of the dissipation rate and does not obey the equivalent proportionality between the Taylor-based Reynolds number and the ratio of integral scale to the Taylor microscale . Instead we observe an approximate proportionality between and during decay. This non-classical behaviour is investigated by studying how the energy spectra evolve during decay and examining how well they can be described by self-preserving single-length-scale forms. A detailed study of homogeneity and isotropy is also presented which reveals the presence of transverse energy transport and pressure transport in the part of the turbulence decay region where we take data (even though previous studies found mean flow and turbulence intensity profiles to be approximately homogeneous in much of the decay region). The exceptionally fast turbulence decay observed in the part of the decay region where we take data is consistent with the non-classical behaviour of the dissipation rate. Measurements with a regular square mesh grid as well as comparisons with active-grid experiments by Mydlarski & Warhaft (J. Fluid Mech., vol. 320, 1996, pp. 331–368) and Kang, Chester & Meveneau (J. Fluid Mech., vol. 480, 2003, pp. 129–160) are also presented to highlight the similarities and differences between these turbulent flows and the turbulence generated by our fractal square grid.

Forcing by steep regular water waves on a vertical circular cylinder at finite depth was investigated numerically by solving the two-phase incompressible Navier–Stokes equations. Consistently with potential flow theory, boundary layer effects were neglected at the sea bed and at the cylinder surface, but the strong nonlinear motion of the free surface was included. The numerical model was verified and validated by grid convergence and by comparison to relevant experimental measurements. First-order convergence towards an analytical solution was demonstrated and an excellent agreement with the experimental data was found. Time-domain computations of the normalized inline force history on the cylinder were analysed as a function of dimensionless wave height, water depth and wavelength. Here the dependence on depth was weak, while an increase in wavelength or wave height both lead to the formation of secondary load cycles. Special attention was paid to this secondary load cycle and the flow features that cause it. By visual observation and a simplified analytical model it was shown that the secondary load cycle was caused by the strong nonlinear motion of the free surface which drives a return flow at the back of the cylinder following the passage of the wave crest. The numerical computations were further analysed in the frequency domain. For a representative example, the secondary load cycle was found to be associated with frequencies above the fifth- and sixth-harmonic force component. For the third-harmonic force, a good agreement with the perturbation theories of Faltinsen, Newman & Vinje (J. Fluid Mech., vol. 289, 1995, pp. 179–198) and Malenica & Molin (J. Fluid Mech., vol. 302, 1995, pp. 203–229) was found. It was shown that the third-harmonic forces were estimated well by a Morison force formulation in deep water but start to deviate at decreasing depth.

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.

In this paper a theoretical investigation of convection currents in anisotropic porous media is performed. The porous layer is homogeneous and bounded by two infinite perfectly heat-conducting horizontal planes. The criterion for the onset of convection is derived. The supercritical steady two-dimensional motion, the heat transport and the stability of the motion are investigated. Some of the results may be applied in insulation techniques.

In order to elucidate the turbulent structure below a shear-free gas-liquid interface, turbulence measurements were made in a 50 cm square by 40 cm deep tank stirred by a vertically oscillating grid well below the surface, using a split-film anemometer probe rotating in a horizontal circle. This instrument is able to measure both vertical and horizontal velocity fluctuations to within 0.4 mm of the surface, from which spatial spectra and profiles of r.m.s. velocity fluctuations and integral lengthscales can be calculated. The turbulent structure is affected by the presence of the surface within a ‘surface-influenced layer’ roughly one integral scale, or ten per cent of the distance from the surface to the centre of the grid stroke, in thickness. The shapes of the spectra and profiles within the surface-influenced layer are predicted to a good first approximation by the source theory of Hunt & Graham (1978), which treats the turbulent structure as the superposition of homogeneous turbulence with an irrotational velocity field driven by a source distribution at the surface which cancels the vertical velocity fluctuations there. The magnitudes (as opposed to the shapes) of the profiles scale according to the values that would otherwise occur in the vicinity of the surface-influenced layer were the surface not present. These magnitudes are adequately predicted by the bulk relations determined by Hopfinger & Toly (1976) and Thompson & Turner (1975), with no apparent dependence on turbulent Reynolds number. There are some minor discrepancies between the measured profiles and those of Hunt & Graham. A thin layer of reduced velocity fluctuations below what would be expected from the theory was observed near the surface. Also, anisotropy in the velocity spectra at depths within the surface-influenced layer extended well into the inertial subrange, whereas the Hunt & Graham theory predicts no anisotropy at high wavenumbers.

We study drop impact on a deep pool of the same fluid, with an emphasis on the air layer trapped under the droplets from its formation to its rupture. The penetration velocity of the air layer at a very short time scale prior to its rupture is shown, using an energy argument and experimental verification, to be one-half of the impact velocity. We then deduce the dependence of the rupture position on the liquid viscosity and the impact velocity. We show that the volume of the resulting air bubbles can be related to both those resulting from droplets impacting on solid surfaces and those resulting from rigid spheres impacting on liquid surfaces.

We present the results of the inviscid spatial stability of a parallel compressible mixing layer. The parameters of this study are the Mach number of the moving stream, the ratio of the temperature of the stationary stream to that of the moving stream, the frequency, and the direction of propagation of the disturbance wave. Stability characteristics of the flow as a function of these parameters are given. It is shown that if the Mach number exceeds a critical value there are always two groups of unstable waves. One of these groups is fast with phase speeds greater than ½ and is supersonic with respect to the stationary stream. The other is slow with phase speeds less than ½ and supersonic with respect to the moving stream. Phase speeds for the neutral and unstable modes are given, as well as growth rates for the unstable modes. Finally, we show that three-dimensional modes have the same general behaviour as the two-dimensional modes but with higher growth rates over some range of propagation direction.

Caflisch & Luke (1985) showed that, owing to the long range of the hydrodynamie interactions, the variance of the sedimentation velocity in a random suspension with uniform probability for the positions of the particles is divergent in the sense that it grows without bound as the macroscopic linear dimension of the settling vessel is increased. It is shown here, however, that a Debye-like screening of a particle's velocity disturbance, leading to a finite variance, will occur if the pair probability reflects a net deficit of one particle in the vicinity of each particle. The three-particle interactions, which determine the structure of a dilute, monodisperse suspension of spheres, lead to a deficit of neighbouring particles. The magnitude and range of this deficit are shown to be sufficient to lead to a Debye-like screening of the velocity disturbance at a radial distance of order aϕ−1, where a is the particle radius and ϕ their volume fraction. A self-consistent approximation to the screened conditional average velocity field and pair distribution is presented. The screening leads to a variance of the particle velocity and a particle tracer diffusion coefficient that are finite and of order Us2 and Usaϕ−1, respectively, where Vs is the Stokes settling velocity of the particles in unbounded fluid.

The fine-scale structure of turbulence in a fully developed turbulent duct flow is examined by considering the three-dimensional velocity derivative field obtained from direct numerical simulations at two relatively small Reynolds numbers. The magnitudes of all mean-square derivatives (normalized by wall variables) increase with the Reynolds number, the increase being largest at the wall. These magnitudes are not consistent with the assumption of local isotropy except perhaps near the duct centre-line. When the assumption of local isotropy is relaxed to one of local axisymmetry, or invariance with respect to rotation about a coordinate axis (here chosen in the streamwise direction), satisfactory agreement is indicated by the data outside the wall region. Support for axisymmetry is demonstrated by anisotropy invariant maps of the dissipation and vorticity tensors. The departure from axisymmetry does not appear to be affected by the Reynolds number. Expressions are proposed for approximations to the average energy dissipation and components of the mean-square vorticity. These proposals should allow these quantities to be measured accurately, at least in the present flow.

A large-eddy-based methodology for the simulation of turbulent sprays is discussed. The transport equations for the spatially filtered gas phase variables, in which source terms accounting for the droplet effects are added, are solved together with a probabilistic description of the liquid phase. The probabilistic approach for the liquid phase is based on the transport equation for the spatially filtered joint probability density function of the variables required in order to describe the state of the liquid phase. In this equation, unclosed terms representing the filtered Lagrangian rates of change of the variables describing the spray are present. General modelling ideas for subgrid-scale (SGS) effects are proposed. The capabilities of the approach and the validity of the closure models, with particular with respect to the SGS dispersion, are investigated through application to a dilute particle-laden turbulent mixing layer. It is demonstrated that the formulation is able to reproduce very closely the measured properties of both the continuous and dispersed phases. The large-eddy simulation (LES) results are also found to be entirely consistent with the experimentally observed characteristics of droplet–gas turbulence interactions. Consistent with direct numerical simulation (DNS) studies of isotropic turbulence laden with particles where the entire turbulence spectrum is found to be modulated by the presence of particles, the present investigation, which comprises the effects of particle transport upon the large-scale vortical structures of a turbulent shear flow, highlights what appears to be a selective behaviour; few large-scale frequencies gain energy whereas the remaining modes are damped.

A cylinder containing liquid with a free surface is subjected to a vertical oscillation of amplitude εg/ω2 and frequency 2ω, where ω is within O(εω) of the natural frequency of a particular (primary) mode in the surface-wave spectrum and 0 < ε 1. A Lagrangian formulation, which includes terms of second and fourth order in the primary mode and second order in the secondary modes (which are excited by the primary mode), together with the method of averaging, leads to a Hamiltonian system for the slowly varying amplitudes of the primary mode. The fixed points (which correspond to harmonic motions) and phase-plane trajectories and their perturbations due to linear damping are determined. It is shown that ε > δ, where δ is the damping ratio (actual/critical) of the primary mode, is a necessary condition for subharmonic response of that mode. Explicit results are given for the dominant axisymmetric and antisymmetric modes in a circular cylinder. Internal resonance, in which a pair of modes have frequencies that approximate ω and 2ω, is discussed separately, and the fixed points and their stability for the special case ω2 = 2ω1 are determined. Internal resonance for ω2 = ω1 is discussed in an appendix.

Coherent features of a turbulent Mach 0.9, Reynolds number $10^{6}$ jet are educed from a high-fidelity large eddy simulation. Besides the well-known Kelvin–Helmholtz instabilities of the shear layer, a new class of trapped acoustic waves is identified in the potential core. A global linear stability analysis based on the turbulent mean flow is conducted. The trapped acoustic waves form branches of discrete eigenvalues in the global spectrum, and the corresponding global modes accurately match the educed structures. Discrete trapped acoustic modes occur in a hierarchy determined by their radial and axial order. A local dispersion relation is constructed from the global modes and found to agree favourably with an empirical dispersion relation educed from the simulation data. The product between direct and adjoint modes is then used to isolate the trapped waves. Under certain conditions, resonance in the form of a beating occurs between trapped acoustic waves of positive and negative group velocities. This resonance explains why the trapped modes are prominently observed in the simulation and as tones in previous experimental studies. In the past, these tones were attributed to external factors. Here, we show that they are an intrinsic feature of high-subsonic jets that can be unambiguously identified by a global linear stability analysis.

The mechanism of turbulence development in periodic Klebanoff transition in a boundary layer has been studied experimentally and in a direct numerical simulation (DNS) with controlled disturbance excitation. In order to compare the results quantitatively, the flow parameters were matched in both methods, thus providing complementary data with which the origin of turbulence in the transition process could be explained. Good agreement was found for the development of the amplitude and shape of typical disturbance structures, the Λ-vortices, including the development of ring-like vortices and spikes in the time traces. The origin and the spatial development of random velocity perturbations were measured in the experiment, and are shown together with the evolution of local high-shear layers. Since the DNS is capable of providing the complete velocity and vorticity fields, further conclusions are drawn based on the numerical data. The mechanisms involved in the flow randomization process are presented in detail. It is shown how the random perturbations which initially develop at the spike-positions in the outer part of the boundary layer influence the flow randomization process close to the wall. As an additional effect, the interaction of vortical structures and high-shear layers of different disturbance periods was found to be responsible for accelerating the transition to a fully developed turbulent flow. These interactions lead to a rapid intensification of a high-shear layer very close to the wall that quickly breaks down because of the modulation it experiences through interactions with vortex structures from the outer part of the boundary layer. The final breakdown process will be shown to be dominated by locally appearing vortical structures and shear layers.

Amplitude equations (including nonlinear damping terms) are derived which describe the evolution of patterns in large-aspect-ratio driven capillary wave experiments. For drive strength just above threshold, a reduction of the number of marginal modes (from travelling capillary waves to standing waves) leads to simpler amplitude equations, which have a Lyapunov functional. This functional determines the wavenumber and symmetry (square) of the most stable uniform state. The original amplitude equations, however, have a secondary instability to transverse amplitude modulation (TAM), which is not present in the standing-wave equations. The TAM instability announces the restoration of the full set of marginal modes.

On a vertically vibrating fluid interface, a droplet can remain bouncing indefinitely. When approaching the Faraday instability onset, the droplet couples to the wave it generates and starts propagating horizontally. The resulting wave–particle association, called a walker, was shown previously to have remarkable dynamical properties, reminiscent of quantum behaviours. In the present article, the nature of a walker's wave field is investigated experimentally, numerically and theoretically. It is shown to result from the superposition of waves emitted by the droplet collisions with the interface. A single impact is studied experimentally and in a fluid mechanics theoretical approach. It is shown that each shock emits a radial travelling wave, leaving behind a localized mode of slowly decaying Faraday standing waves. As it moves, the walker keeps generating waves and the global structure of the wave field results from the linear superposition of the waves generated along the recent trajectory. For rectilinear trajectories, this results in a Fresnel interference pattern of the global wave field. Since the droplet moves due to its interaction with the distorted interface, this means that it is guided by a pilot wave that contains a path memory. Through this wave-mediated memory, the past as well as the environment determines the walker's present motion.

Two-dimensional global eigenmodes are used as a projection basis both for analysing the dynamics and building a reduced model for control in a prototype separated boundary-layer flow. In the present configuration, a high-aspect-ratio smooth cavity-like geometry confines the separation bubble. Optimal growth analysis using the reduced basis shows that the sum of the highly non-normal global eigenmodes is able to describe a localized disturbance. Subject to this worst-case initial condition, a large transient growth associated with the development of a wavepacket along the shear layer followed by a global cycle related to the two unstable global eigenmodes is found. The flow simulation procedure is coupled to a measurement feedback controller, which senses the wall shear stress at the downstream lip of the cavity and actuates at the upstream lip. A reduced model for the control optimization is obtained by a projection on the least stable global eigenmodes, and the resulting linear-quadratic-Gaussian controller is applied to the Navier–Stokes time integration. It is shown that the controller is able to damp out the global oscillations.

The impact of fluidic actuation on the wake and drag of a three-dimensional blunt body is investigated experimentally. Jets blowing tangentially to the main flow force the wake with variable frequency and amplitude. Depending on the forcing conditions, two flow regimes can be distinguished. First, in the case of broadband actuation with frequencies comprising the natural wake time scale, the convection of the jet structures enhances wake entrainment, shortens the length of the recirculating flow and increases drag. Secondly, at higher actuation frequencies, shear-layer deviation leads to fluidic boat tailing of the wake. It additionally lowers its turbulent kinetic energy thus reducing the entrainment of momentum towards the recirculating flow. The combination of both mechanisms produces a rise in the base pressure and reduces the drag of the model. Both actuation regimes are characterized by complementary velocity, pressure and drag measurements at several upstream conditions and control parameters. By adding curved surfaces to deviate the jets by the Coanda effect, periodic actuation is reinforced and drag reductions of approximately 20 % are achieved. The unsteady Coanda blowing not only intensifies the flow deviation and the base pressure recovery but also preserves the unsteady high-frequency forcing effect on the turbulent field. The present results encourage further development of fluidic control to improve the aerodynamics of road vehicles and provide a complementary insight into the relation between wake dynamics and drag.

A WKB-perturbation technique is applied to study the slow modulation of a Stokes wave train on the surface of water. It is found that new terms directly representing modulation rates must be included to extend the scope of validity of Whitham's theory based on an averaged Lagrangian. Two examples are discussed. In the first, a monochromatic wave normally incident on a mild beach is studied and the local rate of depth variation is found to affect the wave phase. In the second, the ‘side-band instability’ problem of Benjamin & Feir is discussed from both linear and non-linear points of view.

Experiments were performed studying the formation of a dip on the surface of an initially stationary liquid draining from a cylindrical tank through an axisym-metrically placed circular orifice. Based upon the information obtained from the experiments, a simple analytical expression was derived predicting the height of the liquid surface in the tank at which this dip forms. A comparison was made between the experimental data and the results of the analysis and good agreement was found between theory and data.

Results are given for experiments on water-lubricated pipelining of 6.01 P cylinder oil in a vertical apparatus in up- and downflow in regimes of modest flow rates, less than 3 ft/s. Measured values of the flow rates, holdup ratios, pressure gradients and flow types are presented and compared with theoretical predictions based on ideal laminar flow and on the predictions of the linear theory of stability. New flow types, not achieved in horizontal flows, are observed: bamboo waves in upflow and corkscrew waves in downflow. Nearly perfect core–annular flows are observed in downflows and these are nearly optimally efficient with values close to the ideal. The holdup ratio in upflow and fast downflow is a constant independent of the value and the ratio of values of the flow rates of oil and water. A vanishing holdup ratio can be achieved by fluidizing a long lubricated column of oil in the downflow of water. The results of experiments are compared with computations from ideal theory for perfect core–annular flow and from the linear theory of stability. Satisfactory agreements are achieved for the celerity and diagnosis of flow type. The wave is shown to be nearly stationary, convected with the oil core in this oil and all oils of relatively high viscosity. These results are robust with respect to moderate changes in the values of viscosity and surface tension. The computed wavelengths are somewhat smaller than the average length of the bamboo waves which are observed. This is explained by stretching effects of buoyancy and lubrication forces induced by the wave. Other points of agreement and disagreement are reviewed.