Large-eddy simulations of temporally evolving turbulent mixing layers have been carried out. The effect of the initial conditions and the size of the computational box on the turbulent statistics and structures is examined in detail. A series of calculations was initialized using two different realizations of a spatially developing turbulent boundary-layer with their free streams moving in opposite directions. Computations initialized with mean flow plus random perturbations with prescribed moments were also conducted. In all cases, the initial transitional stage, from boundary-layer turbulence or random noise to mixing-layer turbulence, was followed by a self-similar period. The self-similar periods, however, differed considerably: the growth rates and turbulence intensities showed differences, and were affected both by the initial condition and by the computational domain size. In all simulations the presence of quasi-two-dimensional spanwise rollers was clear, together with ‘braid’ regions with quasi-streamwise vortices. The development of these structures, however, was different: if strong rollers were formed early (as in the cases initialized by random noise), a well-organized pattern persisted throughout the self-similar period. The presence of boundary layer turbulence, on the other hand, inhibited the growth of the inviscid instability, and delayed the formation of the roller–braid patterns. Increasing the domain size tended to make the flow more three-dimensional.

We present the results of laboratory and field measurements on the stability of wind-driven water surfaces. The laboratory measurements show that when exposed to an increasing wind starting from rest, surface current and wave generation is accompanied by a variety of phenomena that occur over comparable space and time scales. Of particular interest is the generation of small-scale, streamwise vortices, or Langmuir circulations, the clear influence of the circulations on the structure of the growing wave field, and the subsequent transition to turbulence of the surface flow. Following recent work by Melville, Shear & Veron (1998) and Veron & Melville (1999b), we show that the waves that are initially generated by the wind are then strongly modulated by the Langmuir circulations that follow. Direct measurements of the modulated wave variables are qualitatively consistent with geometrical optics and wave action conservation, but quantitative comparison remains elusive. Within the range of parameters of the experiments, both the surface waves and the Langmuir circulations first appear at constant Reynolds numbers of 370 ± 10 and 530 ± 20, respectively, based on the surface velocity and the depth of the laminar shear layer. The onset of the Langmuir circulations leads to a significant increase in the heat transfer across the surface. The field measurements in a boat basin display the same phenomena that are observed in the laboratory. The implications of the measurements for air–sea fluxes, especially heat and gas transfer, and sea-surface temperature, are discussed.

The nonlinear stability of a two-fluid system consisting of a viscous film bounded above by a heavier and thicker layer, between two horizontal plates, with one of the plates oscillating horizontally about a fixed position, is investigated. An evolution equation governing the thickness of the viscous film is derived. Numerical simulations of this equation on extended spatial intervals demonstrate nonlinear small-amplitude saturation of the Rayleigh–Taylor instability in certain parametric regimes. In the low-frequency time-asymptotic regimes, the averaged properties of the extensive spatio-temporal chaos are not steady, but rather oscillate in time. A quasi-equilibrium theory is proposed in which the low-frequency results are interpreted by building upon the notions developed earlier for the simpler case of a non-oscillatory film governed by the classical, constant-coefficient Kuramoto–Sivashinsky equation. In contrast, the higher-frequency solutions exhibit piecewise linear profiles that have never been encountered in simulations of non-oscillatory films. The amplitude as a function of frequency has a single minimum point which is of order one. Also, preliminary results of numerical simulations of film evolution are given for the large-amplitude parametric regimes. At some parameter values, rupture is observed, similar to the case with no base flow; in other regimes the basic flows succeeds in preventing rupture. The complete characterization of the factors responsible for the particular asymptotic fate of the film, rupture or no rupture, remains an open question.

The flow induced by a sphere which undergoes unsteady motion in a Newtonian fluid at small Reynolds number is considered at distances large compared with sphere radius a. Previous solutions of the unsteady Oseen equations (Ockendon 1968; Lovalenti & Brady 1993b) for rectilinear motion are refined. Three-dimensional Fourier transforms of the disturbance field are integrated over Fourier space to derive new concise equations for the velocity field and history force in terms of single history integrals.

Various slip-velocity profiles are classified by the ratio A of the particle relative displacement, z′p(t′) − z′p(τ′), to the diffusion length, l′D = 2[v(t′ − τ′)]1/2, where v is the kinematic viscosity of the fluid. Most previous studies are concerned with large-displacement motions for which the ratio is large in the long-time limit. It is shown using asymptotic calculations that the flow at any point at large distance z past a sphere for arbitrary large-displacement and non-reversing motion is the same as the steady-state laminar wake if z is expressed in terms of the time elapsed since the particle was at that point in the laboratory frame. The point source solution for the remainder of the far flow is also valid for the unsteady case.

A start-up motion with slip velocity V′p = γ′(t′)−1/2, t′ > 0, is investigated for which A is finite. A self-similar solution for the flow field is obtained in terms of space coordinates scaled by the diffusion length, u′ = auss(η)/t′ where η = r′/2(vt′)1/2. The unsteady Oseen correction to the drag is inversely proportional to time.

When A is small in the long-time limit (a small-displacement motion) the flow field also depends on the space coordinates in terms of η. The distribution of the streamwise velocity uz is symmetrical in z.

The ideal solidification problem of the two-dimensional boundary layer flow of a superheated liquid-metal binary alloy over a conveying substrate is examined. Analytical results for the velocities, the solute concentration and the temperature field are found in the asymptotic limits of small Prandtl numbers and large Schmidt numbers. The growth of the solidifying front is shown to follow the square-root law.

The work, being the first in a series concerned with the evolution of small perturbations in shear flows, studies the linear initial-value problem for inviscid spatially harmonic perturbations of two-dimensional shear flows of boundary-layer type without inflection points. Of main interest are the perturbations of wavelengths 2π/k long compared to the boundary-layer thickness H, kH = ε [Lt ] 1. By means of an asymptotic expansion, based on the smallness of ε, we show that for a generic initial perturbation there is a long time interval of duration ∼ ε−3 ln(1/ε), where the perturbation representing an aggregate of continuous spectrum modes of the Rayleigh equation behaves as if it were a single discrete spectrum mode having no singularity to the leading order. Following Briggs et al. (1970), who introduced the concept of decaying wave-like perturbations due to the presence of the ‘Landau pole’ into hydrodynamics, we call this object a quasi-mode. We trace analytically how the quasi-mode contribution to the entire perturbation field evolves for different field characteristics. We find that over O(ε−3 ln(1/ε)) time interval, the quasi-mode dominates the velocity field. In particular, over this interval the share of the perturbation energy contained in the quasi-mode is very close to 1, with the discrepancy in the generic case being O(ε4) (O(ε4) for the Blasius flow). The mode is weakly decaying, as exp(−ε3t). At larger times the quasi-mode ceases to dominate in the perturbation field and the perturbation decay law switches to the classical t−2. By definition, the quasi-modes are singular in a critical layer; however, we show that in our context their singularity does not appear in the leading order. From the physical viewpoint, the presence of a small jump in the higher orders has little significance to the manner in which perturbations of the flow can participate in linear and nonlinear resonant interactions. Since we have established that the decay rate of the quasi-modes sharply increases with the increase of the wavenumber, one of the major conjectures of the analysis is that the long-wave components prevail in the large-time asymptotics of a wide class of initial perturbations, not necessarily the predominantly long-wave perturbations. Thus, the explicit expressions derived in the long-wave approximation describe the asymptotics of a much wider class of initial conditions than might have been anticipated. The concept of quasi-modes also enables us to shed new light on the foundations of the method of piecewise linear approximations widely used in hydrodynamics.

We analyse modulational (large-scale) perturbations of stationary solutions of the two-dimensional incompressible Navier–Stokes equations. The stationary solutions are cellular flows with stream function ϕ = sin y1 sin y2 + δ cos y1 cos y2, 0 [les ] δ [les ] 1. Using multiscale techniques we derive effective coefficients, including the eddy viscosity tensor, for the (averaged) modulation equations. For cellular flows with closed streamlines we give rigorous asymptotic bounds at high Reynolds number for the tensor of eddy viscosity by means of saddle-point variational principles. These results allow us to compare the linear and nonlinear modulational stability of cellular flows with no channels and of shear flows at high Reynolds number. We find that the geometry of the underlying cellular flows plays an important role in the stability of the modulational perturbations. The predictions of the multiscale analysis are compared with direct numerical simulations.

Two types of longitudinal vortices are found to arise from distorted, migrating wakes convecting through a low-pressure turbine stator passage. The primary vortices emerge within the free stream as the wake is subjected to irrotational strains. Their axes align approximately with the local mean wake velocity. They are dragged over the surface and induce secondary vortices near the wall, which have the opposite sense of rotation to the primary vortices. Although they form on the concave side, these secondary vortices are neither produced, nor sustained by Görtler instability; rather, they are a consequence of severely straining the passing wakes.

Evidence is drawn from a numerical simulation of the unsteady, incompressible flow through a turbine stator passage with and without upstream turbulent wakes. The computations were performed with 2.5 and 5.7 × 107 grid points on a parallel computer.

A computational study is reported of the close interaction of nominally anti-parallel vortex tubes with unequal strengths, Γ1 and −Γ2, where Γ2/Γ1 [les ] 1. The computations are performed using a spectral method, with periodic boundary conditions and vortex Reynolds number Re ≡ Γ1/v = 1500, and the vortices are perturbed by a wavelength for which the pair is unstable because of their mutual interaction. The numerical method is tested for the case of equal-strength vortices, which exhibits the classic vortex reconnection phenomenon typified by bridging between the vortex cores and formation of thin vorticity threads as the bridged sections advect away under their self-induced velocity. Computations for vortices of unequal strengths are reported for cases with small, moderate and large strength differences, for which Γ2/Γ1 = 0.82, 0.54 and 0.25 are chosen as representative values. The bridges between the vortex structures form loops that twist owing to the unequal vortex strengths. In the thread region, the vortex interaction is controlled by competition between the effects of stretching of the weak vortex as it wraps around the stronger vortex and the core distortion induced on each vortex owing to the straining imposed by the opposing vortex. For cases with large vortex strength difference, the strong vortex remains nearly straight as the weak vortex wraps around it, inducing an interlaced pattern of positive and negative vorticity spirals within the core of the strong vortex. Over long time, the bridge regions form loops that propagate away from the thread region for cases with small strength difference and wrap around the nearly columnar strong vortex for cases with large strength difference.

Experiments were conducted on the interactions of two different-sized deformable drops moving due to gravity in an immiscible viscous fluid at low Reynolds number. As the drops come close to each other, several interactions are possible: (i) separation of the drops, (ii) capture of the smaller drop behind the larger drop, (iii) breakup of the smaller drop into two or more drops, and (iv) pass-through of one drop through the other, with possible cycle interaction or leap-frogging. The interactions depend on several system parameters, including the drop-to-medium viscosity ratio, the radius ratio of the two drops, the initial horizontal offset of the two drops at large vertical separation, and the gravitational Bond number (which represents the ratio of buoyant forces to interfacial tension forces for the larger drop and describes how much the drops will deform). Experimental analysis was conducted by videotaping trajectories of glycerol–water drops of various compositions falling in castor oil. The results show good agreement with available theoretical results, both for interaction maps and individual trajectories. The results also provide data beyond the present limitations of theoretical algorithms and reveal the new pass-through phenomenon.

This paper is an extension of an experimental investigation by Alving & Fernholz (1996). In the present experiments the effects of free-stream turbulence were investigated on a boundary layer with an adverse pressure gradient and a closed reverse-flow region. By adding free-stream turbulence the mean reverse-flow region was shortened or completely eliminated and this was used to control the size of the separation bubble. The turbulence intensity was varied between 0.2% and 6% using upstream grids while the turbulence length scale was on the order of the boundary layer thickness. Mean and fluctuating velocities as well as spectra were measured by means of hot-wire and laser-Doppler anemometry and wall shear stress by wall pulsed-wire and wall hot-wire probes.

Free-stream turbulence had a small effect on the boundary layer in the mild adverse-pressure-gradient region but in the vicinity of separation and along the reverse-flow region mean velocity profiles, skin friction and turbulence structure were strongly affected. Downstream of the mean or instantaneous reverse-flow regions highly disturbed boundary layers developed in a nominally zero pressure gradient and converged to a similar turbulence structure in all three cases at the end of the test section. This state was, however, still very different from that in a canonical boundary layer.

An experimental investigation was conducted to study the wall boundary condition for large-eddy simulation (LES) of a turbulent boundary layer at Rθ = 3500. Most boundary condition formulations for LES require the specification of the instantaneous filtered wall shear stress field based upon the filtered velocity field at the closest grid point above the wall. Three conventional boundary conditions are tested using simultaneously obtained filtered wall shear stress and streamwise and wall-normal velocities, at locations nominally within the log region of the flow. This was done using arrays of hot-film sensors and X-wire probes. The results indicate that models based on streamwise velocity perform better than those using the wall-normal velocity, but overall significant discrepancies were found for all three models. A new model is proposed which gives better agreement with the shear stress measured at the wall. The new model is also based on the streamwise velocity but is formulated so as to be consistent with ‘outer-flow’ scaling similarity of the streamwise velocity spectra. It is therefore expected to be more generally applicable over a larger range of Reynolds numbers at any first-grid position within the log region of the boundary layer.

A theory is presented for the transport in open-channel flow of a chemical species under the influence of kinetic sorptive exchange between phases that are dissolved in water and sorbed onto suspended sediments. The asymptotic method of homogenization is followed to deduce effective transport equations for both phases. The transport coefficients for the solute are shown to be functions of the local sediment concentration and therefore vary with space and time. The three important controlling parameters are the suspension number, the bulk solid–water distribution ratio and the sorption kinetics parameter. It is illustrated with a numerical example that when values of these parameters are sufficiently high, the advection and dispersion of the solute cloud can be dominated by the sorption effects. The concentration distribution can exhibit an appreciable deviation from Gaussianity soon after discharge, which develops into a long tailing as the solute cloud gradually moves ahead of the sediment cloud.

Among the important physical phenomena associated with the jet in crossflow is the formation and evolution of vortical structures in the flow field, in particular the counter-rotating vortex pair (CVP) associated with the jet cross-section. The present computational study focuses on the mechanisms for the dynamical generation and evolution of these vortical structures. Transient numerical simulations of the flow field are performed using three-dimensional vortex elements. Vortex ring rollup, interactions, tilting, and folding are observed in the near field, consistent with the ideas described in the experimental work of Kelso, Lim & Perry (1996), for example. The time-averaged effect of these jet shear layer vortices, even over a single period of their evolution, is seen to result in initiation of the CVP. Further insight into the topology of the flow field, the formation of wake vortices, the entrainment of crossflow, and the effect of upstream boundary layer thickness is also provided in this study.

The parabolized stability equations (PSE) are used to investigate issues of nonlinear flow development and mixing in compressible reacting shear layers, which are modelled with an infinitely fast-chemistry assumption. Particular emphasis is placed on investigating the change in flow structure that occurs when compressibility and heat release are added to the flow. These conditions allow the ‘outer’ instability modes – one associated with each of the fast and slow streams – to dominate over the ‘central’, Kelvin–Helmholtz mode that exists unaccompanied in incompressible non-reacting mixing layers. Analysis of scalar probability density functions in flows with dominant outer modes demonstrates the ineffective, one-sided nature of mixing that accompanies these flow structures. Colayer conditions, where two modes have equal growth rate and the mixing layer is formed by two sets of vortices, offer some opportunity for mixing enhancement. Their extent, however, is found to be limited in the mixing layer's parameter space. Extensive validation of the PSE technique also provides a unique perspective on central-mode vortex pairing, further supporting the view that pairing is primarily governed by instability growth rates; mutual induction appears to be a secondary process. This perspective sheds light on how linear stability theory is able to provide such an accurate prediction of experimentally observed, fully nonlinear flow phenomenon.