Receptivity is the process which describes how environmental disturbances (such as gusts, acoustic waves or wall roughness) are filtered by a boundary layer and turned into downstream-growing waves. It is closely related to the identification of initial conditions for the disturbances and requires knowledge of the characteristics of the specific external forcing field. Without such a knowledge, it makes sense to focus on worst case scenarios and search for those initial states which maximize the disturbance amplitude at a given downstream position, and hence to identify upper bounds on growth rates, which will be useful in predicting the transition to turbulence. This philosophical approach has been taken by Tempelmann, Hanifi & Henningson (J. Fluid Mech., 2010, vol. 646, pp. 5–37) in a remarkably complete parametric study of ‘optimal disturbances’ for a model of the flow over a swept wing; they pinpoint the crucial importance both of the spatial variation of the flow and of non-modal disturbances, even when the flow is ‘supercritical’ and hence subject to classical ‘normal mode’ instabilities.

A parabolized set of linear equations is derived, which, in combination with the proposed solution procedure, allows for the study of both non-modal and modal disturbance growth in three-dimensional boundary layers. The method is applicable to disturbance waves whose lines of constant phase are closely aligned with the external streamline. Moreover, strongly growing disturbances may fall outside the scope of application. These equations are used in conjunction with a variational approach to compute optimal disturbances in Falkner–Skan–Cooke boundary layers subject to adverse and favourable pressure gradients. The disturbances associated with maximum energy growth initially take the form of streamwise vortices which are tilted against the mean crossflow shear. While travelling downstream these vortical structures rise into an upright position and evolve into bent streaks. The physical mechanism responsible for non-modal growth in three-dimensional boundary layers is therefore identified as a combination of the lift-up effect and the Orr mechanism. Optimal disturbances smoothly evolve into crossflow modes when entering the supercritical domain of the flow. Non-modal growth is thus found to initiate modal instabilities in three-dimensional boundary layers. Optimal growth is first studied for stationary disturbances. Influences of parameters such as sweep angle, spanwise wavenumber and position of inception are studied, and the initial optimal amplification of stationary crossflow modes because of non-modal growth is investigated. Finally, general disturbances are considered, and envelopes yielding the maximum growth at each position are computed. In general, substantial growth is already found upstream of the first neutral point. The computations show that at supercritical conditions, maximum growth of optimal disturbances in accelerated boundary layers can exceed the growth predicted for modal instabilities by several orders of magnitude.

This paper describes the fluid mechanics within a box containing a vertical plane distributed source of buoyancy. A theoretical analysis is presented that models the development of plumes from such sources in an unconfined ambient of uniform density. Two extensions are considered. The first concerns a sealed box and the second involves the more general situation where the box is ventilated by openings at top and bottom. In the sealed box the stratification develops in much the same way as for a ‘filling box’ containing a single-point source of buoyancy on the floor. An initial front descends from the ceiling of the box and an asymptotic stratification eventually develops which is continuous in the vertical direction. In the case of the ventilated box it is found that a complex stratification develops where one or more horizontal intrusions are formed by detachment of the plume/boundary layer from the vertically distributed source where the buoyancy of the plume is less than, or equal to, that of the stratified ambient at a given height. Experimental results are presented to demonstrate the validity of the theory. The findings are relevant to both forced and naturally ventilated buildings containing non-adiabatic vertical surfaces.

The effect of fluid velocity fluctuations on the dynamics of the particles in a turbulent gas–solid suspension is analysed in the low-Reynolds-number and high Stokes number limits, where the particle relaxation time is long compared with the correlation time for the fluid velocity fluctuations, and the drag force on the particles due to the fluid can be expressed by the modified Stokes law. The direct numerical simulation procedure is used for solving the Navier–Stokes equations for the fluid, the particles are modelled as hard spheres which undergo elastic collisions and a one-way coupling algorithm is used where the force exerted by the fluid on the particles is incorporated, but not the reverse force exerted by the particles on the fluid. The particle mean and root-mean-square (RMS) fluctuating velocities, as well as the probability distribution function for the particle velocity fluctuations and the distribution of acceleration of the particles in the central region of the Couette (where the velocity profile is linear and the RMS velocities are nearly constant), are examined. It is found that the distribution of particle velocities is very different from a Gaussian, especially in the spanwise and wall-normal directions. However, the distribution of the acceleration fluctuation on the particles is found to be close to a Gaussian, though the distribution is highly anisotropic and there is a correlation between the fluctuations in the flow and gradient directions. The non-Gaussian nature of the particle velocity fluctuations is found to be due to inter-particle collisions induced by the large particle velocity fluctuations in the flow direction. It is also found that the acceleration distribution on the particles is in very good agreement with the distribution that is calculated from the velocity fluctuations in the fluid, using the Stokes drag law, indicating that there is very little correlation between the fluid velocity fluctuations and the particle velocity fluctuations in the presence of one-way coupling. All of these results indicate that the effect of the turbulent fluid velocity fluctuations can be accurately represented by an anisotropic Gaussian white noise.

A fluctuating-force model is developed for representing the effect of the turbulent fluid velocity fluctuations on the particle phase in a turbulent gas–solid suspension in the limit of high Stokes number, where the particle relaxation time is large compared with the correlation time for the fluid velocity fluctuations. In the model, a fluctuating force is incorporated in the equation of motion for the particles, and the force distribution is assumed to be an anisotropic Gaussian white noise. It is shown that this is equivalent to incorporating a diffusion term in the Boltzmann equation for the particle velocity distribution functions. The variance of the force distribution, or equivalently the diffusion coefficient in the Boltzmann equation, is related to the time correlation functions for the fluid velocity fluctuations. The fluctuating-force model is applied to the specific case of a Couette flow of a turbulent particle–gas suspension, for which both the fluid and particle velocity distributions were evaluated using direct numerical simulations by Goswami & Kumaran (2010). It is found that the fluctuating-force simulation is able to quantitatively predict the concentration, mean velocity profiles and the mean square velocities, both at relatively low volume fractions, where the viscous relaxation time is small compared with the time between collisions, and at higher volume fractions, where the time between collisions is small compared with the viscous relaxation time. The simulations are also able to predict the velocity distributions in the centre of the Couette, even in cases in which the velocity distribution is very different from a Gaussian distribution.

The onset of shear flow turbulence is characterized by turbulent patches bounded by regions of laminar flow. At low Reynolds numbers localized turbulence relaminarizes, raising the question of whether it is transient in nature or becomes sustained at a critical threshold. We present extensive numerical simulations and a detailed statistical analysis of the lifetime data, in order to shed light on the sources of the discrepancies present in the literature. The results are in excellent quantitative agreement with recent experiments and show that turbulent lifetimes increase super-exponentially with Reynolds number. In addition, we provide evidence for a lower bound below which there are no meta-stable characteristics of the transients, i.e. the relaminarization process is no longer memoryless.

The buoyancy-driven flushing of fluid from a rectangular box via connections in the base and top into quiescent surroundings of uniform density is examined. Our focus is on the transient flows that develop when the interior is either initially stably stratified in two homogeneous layers – a dense layer below a layer at ambient density, or is filled entirely with dense fluid. Experiments with saline stratifications show that four distinct patterns of flow are possible. We classify these patterns in terms of the direction of flow through the base opening and the propensity of replacement fluid through the top opening to induce interfacial mixing. Unidirectional or bidirectional flow through the base opening may occur and within these two flow types either weak or vigorous interfacial mixing. We identify the three controlling geometrical parameters that determine which flow pattern is established, namely the fractional initial layer depths, the relative areas of the top and base openings and the horizontal length scale of the top opening relative to the initial dense layer depth. We show that these parameters may be reduced to two Froude numbers – one based on the fluxes through the base opening and whose value sets the direction of flow, and a second based on conditions at the top opening whose value determines the vigour of interfacial mixing. Theoretical models are developed for predicting the conditions for transition between each flow pattern and expressed as critical values of the Froude numbers identified.

This work concerns oscillatory boundary layers over smooth beds. It comprises combined visual and quantitative techniques including bed shear stress measurements. The experiments were carried out in an oscillating water tunnel. The experiments reveal two significant coherent flow structures: (i) Vortex tubes, essentially two-dimensional vortices close to the bed extending across the width of the boundary-layer flow, caused by an inflectional-point shear layer instability. The imprint of these vortices in the bed shear stress is a series of small, insignificant kinks and dips. (ii) Turbulent spots, isolated arrowhead-shaped areas close to the bed in an otherwise laminar boundary layer where the flow ‘bursts’ with violent oscillations. The emergence of the turbulent spots marks the onset of turbulence. Turbulent spots cause single or multiple violent spikes in the bed shear stress signal, which has profound implications for sediment transport (in both the laboratory and the field). The experiments also show that similar coherent flow structures exist in the case of combined oscillatory flow and current.

This study continues the investigation of wave boundary layers reported by Carstensen, Sumer & Fredsøe (J. Fluid Mech., 2010, part 1 of this paper). The present paper summarizes the results of an experimental investigation of turbulent solitary wave boundary layers, simulated by solitary motion in an oscillating water tunnel. Two kinds of measurements were made: bed shear stress measurements and velocity measurements. The experiments show that the solitary-motion boundary layer experiences three kinds of flow regimes as the Reynolds number is increased: (i) laminar regime; (ii) laminar regime where the boundary-layer flow experiences a regular array of vortex tubes near the bed over a short period of time during the deceleration stage; and (iii) transitional regime characterized with turbulent spots, revealed by single/multiple, or, sometimes, quite dense spikes in the bed shear stress traces. Supplementary synchronized flow visualization tests confirmed the presence of the previously mentioned flow features. Information related to flow resistance are also given in the paper.

The interactions and merging of two unequal co-rotating vortices in a viscous fluid are investigated. Two-dimensional numerical simulations of initially equal-sized vortices with differing relative strengths are performed. In the case of equal-strength vortices, i.e. symmetric vortex pairs (Brandt & Nomura, J. Fluid Mech., vol. 592, 2007, pp. 413–446), the mutually induced strain deforms and tilts the vortices, which leads to a core detrainment process. The weakened vortices are mutually entrained and rapidly move towards each other as they intertwine and destruct. The flow thereby develops into a single compound vortex. With unequal strengths, i.e. asymmetric pairs, the disparity of the vortices alters the interaction. Merger may result from reciprocal but unequal entrainment, which yields a compound vortex; however other outcomes are possible. The various interactions are classified based on the relative timing of core detrainment and core destruction of the vortices. Through scaling analysis and simulation results, a critical strain rate parameter which characterizes the establishment of core detrainment is identified and determined. The onset of merging is associated with the achievement of the critical strain rate by ‘both’ vortices, and a merging criterion is thereby developed. In the case of symmetric pairs, the critical strain rate parameter is shown to be related to the critical aspect ratio. In contrast with symmetric merger, which is in essence a flow transformation, asymmetric merger may result in the domination of the stronger vortex because of the unequal deformation rates. If the disparity of the vortex strengths is sufficiently large, the critical strain rate is not attained by the stronger vortex before destruction of the weaker vortex, and the vortices do not merge.

The behaviour of an isolated nearly spherical drop in an ambient linear flow is examined analytically at small but finite Reynolds numbers, and thereby the first effects of inertia on the bulk stress in a dilute emulsion of neutrally buoyant drops are calculated. The Reynolds numbers, Re = a2ρ/μ and , are the relevant dimensionless measures of inertia in the continuous and disperse(drop) phases, respectively. Here, a is the drop radius, is the shear rate, ρ is the common density and and μ are, respectively, the viscosities of the drop and the suspending fluid. The assumption of nearly spherical drops implies the dominance of surface tension, and the analysis therefore corresponds to the limit of the capillary number(Ca) based on the viscosity of the suspending fluid being small but finite; in other words, Ca ≪ 1, where Ca = μa/T, T being the coefficient of interfacial tension. The bulk stress is determined to O(φRe) via two approaches. The first one is the familiar direct approach based on determining the force density associated with the disturbance velocity field on the surface of the drop; the latter is determined to O(Re) from a regular perturbation analysis. The second approach is based on a novel reciprocal theorem formulation and allows the calculation, to O(Re), of the drop stresslet, and hence the emulsion bulk stress, with knowledge of only the leading-order Stokes fields. The first approach is used to determine the bulk stress for linear flows without vortex stretching, while the reciprocal theorem approach allows one to generalize this result to any linear flow. For the case of simple shear flow, the inertial contributions to the bulk stress lead to normal stress differences(N1, N2) at O(φRe), where φ(≪1) is the volume fraction of the disperse phase. Inertia leads to negative and positive contributions, respectively, to N1 and N2 at O(φRe). The signs of the inertial contributions to the normal stress differences may be related to the O(ReCa) tilting of the drop towards the velocity gradient direction. These signs are, however, opposite to that of the normal stress differences in the creeping flow limit. The latter are O(φCa) and result from an O(Ca2) deformation of the drop acting to tilt it towards the flow axis. As a result, even a modest amount of inertia has a significant effect on the rheology of a dilute emulsion. In particular, both normal stress differences reverse sign at critical Reynolds numbers(Rec) of O(Ca) in the limit Ca ≪ 1. This criterion for the reversal in the signs of N1 and N2 is more conveniently expressed in terms of a critical Ohnesorge number(Oh) based on the viscosity of the suspending fluid, where Oh = μ/(ρaT)1/2. The critical Ohnesorge number for a sign reversal in N1 is found to be lower than that for N2, and the precise numerical value is a function of λ. In uniaxial extensional flow, the Trouton viscosity remains unaltered at O(φRe), the first effects of inertia now being restricted to O(φRe3/2). The analytical results for simple shear flow compare favourably with the recent numerical simulations of Li & Sarkar (J. Rheol., vol. 49, 2005, p. 1377).

Receptivity, disturbance growth and transition to turbulence of the three-dimensional boundary layer developing on a swept flat plate are studied by means of numerical simulations. The flow is subject to a favourable pressure gradient and represents a model for swept-wing flow downstream of the leading edge and upstream of the pressure minimum of the wing. The boundary layer is perturbed by free-stream turbulence and localized surface roughness with random distribution in the spanwise direction. The intensity of the turbulent free-stream fluctuations ranges from conditions typical for free flight to higher levels usually encountered in turbo-machinery applications. The free-stream turbulence initially excites non-modal streak-like disturbances as in two-dimensional boundary layers, soon evolving into modal instabilities in the form of unsteady crossflow modes. The crossflow modes grow faster than the streaks and dominate the downstream disturbance environment in the layer. The results show that the receptivity mechanism is linear for the disturbance amplitudes under consideration, while the subsequent growth of the primary disturbances rapidly becomes affected by nonlinear saturation in particular for free-stream fluctuations with high intensity. Transition to turbulence occurs in the form of localized turbulent spots randomly appearing in the flow. The main features of the breakdown are presented for the case of travelling crossflow vortices induced by free-stream turbulence. The flow is also receptive to localized roughness strips, exciting stationary crossflow modes. The mode with most efficient receptivity dominates the downstream disturbance environment. When both free-stream fluctuations and wall roughness act on the boundary layer at the same time, transition is dominated by steady crossflow waves unless the incoming turbulence intensity is larger than about 0.5% for roughness amplitudes of about one tenth of the boundary-layer displacement thickness. The results show that a correct prediction of the disturbance behaviour can be obtained considering the receptivity and evolution of individual modes. In addition, we provide an estimate for the amplitudes of the external disturbance sources above which a fully nonlinear receptivity analysis is necessary.

Computations are presented of the upward force on a two-dimensional wedge descending towards a plane surface due to the Stokes flow of an intervening viscous fluid. The predictions are compared with those of lubrication theory and an approximate analytical solution; all three predict a logarithmic divergence of the force with the minimum separation. An object falling vertically under gravity will therefore make contact with an underlying plane surface in finite time if roughened by asperities with sharp corners (with smooth surfaces, contact is made only after infinite time). Contact is still made in finite time if the roughened object also moves horizontally or rotates as it falls.

A lubrication theory is presented for the effect of fluid compressibility and solid elasticity on the descent of a two-dimensional smooth object falling under gravity towards a plane wall through a viscous fluid. The final approach to contact, which takes infinite time in the absence of both effects, is determined by numerical and asymptotic methods. Compressibility can lead to contact in finite time either during inertially generated oscillations or if the viscosity decreases sufficiently quickly with increasing pressure. The approach to contact is invariably slowed by allowing the solids to deform elastically; specific results are presented for an underlying elastic wall modelled as a foundation, half-space, membrane or beam.

The origin and the resultant properties of the strong pulses or shocks emitted by collapsing gas bubbles into a surrounding liquid are investigated numerically. The compressible flow in both phases is resolved. Results are presented for micron- and millimetre-sized bubbles and for bubble collapse triggered either by an acoustic driving or by an initially imposed spherical shock in the liquid. The origin of the diverging shocks is investigated, and the results of a parametric study for the acoustically driven collapse reveal a predominant linear dependence of the shock strength and width on the maximum bubble radius. The results compare favourably with experimental data and agree well with acoustic theory in the limit of weak forcing.

The flow field and mixing in an expansion-ramp geometry is studied using large-eddy simulation (LES) with subgrid scale (SGS) modelling. The expansion-ramp geometry was developed to investigate enhanced mixing and flameholding characteristics while maintaining low total-pressure losses. Passive mixing was considered without taking into account the effects of chemical reactions and heat release, an approximation that is adequate for experiments conducted in parallel. The primary objective of the current work is to validate the LES–SGS closure in the case of passive turbulent mixing in a complex configuration and, if successful, to rely on numerical simulation results for flow details unavailable via experiment. Total (resolved-scale plus subgrid contribution) probability density functions (p.d.f.s) of the mixture fraction are estimated using a presumed beta-distribution model for the subgrid field. Flow and mixing statistics are in good agreement with the experimental measurements, indicating that the mixing on a molecular scale is correctly predicted by the LES–SGS model. Finally, statistics are shown to be resolution-independent by computing the flow for three resolutions, at twice and four times the resolution of the coarsest simulation.

A vortex placed at a density interface winds it into an ever-tighter spiral. We show that this results in a combination of a centrifugal Rayleigh–Taylor (CRT) instability and a spiral Kelvin–Helmholtz (SKH) type of instability. The SKH instability arises because the density interface is not exactly circular, and dominates at large times. Our analytical study of an inviscid idealized problem illustrates the origin and nature of the instabilities. In particular, the SKH is shown to grow slightly faster than exponentially. The predicted form lends itself for checking by a large computation. From a viscous stability analysis using a finite-cored vortex, it is found that the dominant azimuthal wavenumber is smaller for lower Reynolds number. At higher Reynolds numbers, disturbances subject to the combined CRT and SKH instabilities grow rapidly, on the inertial time scale, while the flow stabilizes at low Reynolds numbers. Our direct numerical simulations are in good agreement with these studies in the initial stages, after which nonlinearities take over. At Atwood numbers of 0.1 or more, and a Reynolds number of 6000 or greater, both stability analysis and simulations show a rapid destabilization. The result is an erosion of the core, and breakdown into a turbulence-like state. In studies at low Atwood numbers, the effect of density on the inertial terms is often ignored, and the density field behaves like a passive scalar in the absence of gravity. The present study shows that such treatment is unjustified in the vicinity of a vortex, even for small changes in density when the density stratification is across a thin layer. The study would have relevance to any high-Péclet-number flow where a vortex is in the vicinity of a density-stratified interface.

We study the turbulence transition of plane Couette flow in large domains where localized perturbations are observed to generate growing turbulent spots. Extending previous studies on the boundary between laminar and turbulent dynamics we determine invariant structures intermediate between laminar and turbulent flow. In wide but short domains we find states that are localized in spanwise direction, and in wide and long domains the states are also localized in downstream direction. These localized states act as critical nuclei for the transition to turbulence in spatially extended domains.

The stochastic coherent adaptive large eddy simulation (SCALES) methodology is a novel approach to the numerical simulation of turbulence, where a dynamic grid adaptation strategy based on wavelet threshold filtering is utilized to solve for the most ‘energetic’ eddies. The effect of the less energetic unresolved motions is simulated by a model. Previous studies have demonstrated excellent predictive properties of the SCALES approach for decaying homogeneous turbulence. In this paper the applicability of the method is further explored for statistically steady turbulent flows by considering linearly forced homogeneous turbulence at moderate Reynolds number. A local dynamic subgrid-scale eddy viscosity model based on the definition of the kinetic energy associated with the unresolved motions is used as closure model. The governing equations for the wavelet filtered velocity field, along with the additional evolution equation for the subgrid-scale kinetic energy, are numerically solved by means of a dynamically adaptive wavelet collocation method. It is demonstrated that adaptive simulations closely match results from a reference pseudo-spectral fully de-aliased direct numerical simulation, by using only about 1% of the corresponding computational nodes. In contrast to classical non-adaptive large eddy simulation, the agreement with direct solution holds for the mean flow statistics as well as in terms of energy and enstrophy spectra up to the dissipative wavenumbers range.

In this paper, we analyse by numerical simulations the nonlinear dynamics of the elliptic instability in the configurations of a single strained vortex and a system of two counter-rotating vortices. We show that although a weakly nonlinear regime associated with a limit cycle is possible, the nonlinear evolution far from the instability threshold is, in general, much more catastrophic for the vortex. In both configurations, we put forward some evidence of a universal nonlinear transition involving shear layer formation and vortex loop ejection, leading to a strong alteration and attenuation of the vortex, and a rapid growth of the vortex core size.