Results are presented from an experimental study into the fine-scale structure of generic, Sc≈1, dynamically passive, conserved scalar fields in turbulent shear flows. The investigation was based on highly resolved, two-dimensional imaging of laser Rayleigh scattering, with measurements obtained in the self-similar far field of an axisymmetric coflowing turbulent jet of propane issuing into air at local outer-scale Reynolds numbers Reδ≡uδ/v of 11000 and 14000. The resolution and signal quality of these measurements allowed direct differentiation of the scalar field data ζ(x, t) to determine the instantaneous scalar energy dissipation rate field (Re Sc)−1∇ζ·∇ζ(x, t). Results show that, as for large-Sc scalars (Buch & Dahm 1996), the scalar dissipation rate field consists entirely of strained, laminar, sheet-like diffusion layers, despite the fact that at Sc≈1 the scale on which these layers are folded by vorticity gradients is comparable to the layer thickness. Good agreement is found between the measured internal structure of these layers and the self-similar local solution of the scalar transport equation for a spatially uniform but time-varying strain field. The self-similar distribution of dissipation layer thicknesses shows that the ratio of maximum to minimum thicknesses is only 3 at these conditions. The local dissipation layer thickness is related to the local outer scale as λD/δ ≡ΛRe−3/4δSc−1/2, with the average thickness found to be 〈Λ〉=11.2, with both the largest and smallest layer thicknesses following Kolmogorov Re−3/4δ) scaling.

We present laboratory measurements of the generation and evolution of Langmuir circulations as an instability of a wind-driven surface shear layer. The shear layer, which is generated by an accelerating wind starting from rest above a quiescent water surface, both accelerates and deepens monotonically until the inception of the Langmuir circulations. The Langmuir circulations closely follow the initial growth of the wind waves and rapidly lead to vertical mixing of the horizontal momentum and a deceleration of the surface layer. Prior to the appearance of the Langmuir circulations, the depth of the shear layer scales with (vt)1/2 (v is the kinematic viscosity and t is time), in accordance with molecular rather than turbulent transport. For final wind speeds in the range 3 to 5 m s−1, the wavenumber of the most unstable Langmuir circulation normalized by the surface wavenumber, k*lc, is 0.68±0.24, at a reciprocal Langmuir number, La−1, of 52±21. The observations are compared with available theoretical results, although none are directly applicable to the conditions of the experiments. The implications of this work for the generation and evolution of Langmuir circulations in the ocean and other natural water bodies are discussed.

The stability of supercritical Couette flow has been studied extensively, but few measurements of the velocity field of flow have been made. Particle image velocimetry (PIV) was used to measure the axial and radial velocities in a meridional plane for non-wavy and wavy Taylor–Couette flow in the annulus between a rotating inner cylinder and a fixed outer cylinder with fixed end conditions. The experimental results for the Taylor vortex flow indicate that as the inner cylinder Reynolds number increases, the vortices become stronger and the outflow between pairs of vortices becomes increasingly jet-like. Wavy vortex flow is characterized by azimuthally wavy deformation of the vortices both axially and radially. The axial motion of the vortex centres decreases monotonically with increasing Reynolds number, but the radial motion of the vortex centres has a maximum at a moderate Reynolds number above that required for transition. Significant transfer of fluid between neighbouring vortices occurs in a cyclic fashion at certain points along an azimuthal wave, so that while one vortex grows in size, the two adjacent vortices become smaller, and vice versa. At other points in the azimuthal wave, there is an azimuthally local net axial flow in which fluid winds around the vortices with a sense corresponding to the axial deformation of the wavy vortex tube. These measurements also confirm that the shift-and-reflect symmetry used in computational studies of wavy vortex flow is a valid approach.

Theoretical predictions for the coagulation rate induced by turbulent shear have often been based on the hypothesis that the turbulent velocity gradient is persistent (Saffman & Turner 1956) and that hydrodynamic and interparticle interactions (van der Waals attraction and electrostatic double-layer repulsion) between colloidal particles can be neglected. In the present work we consider the effects of interparticle forces on the turbulent coagulation rate, and we explore the response of the coagulation rate to changes in the Lagrangian velocity gradient correlation time (i.e. the characteristic evolution time for the velocity gradient in a reference frame following the fluid motion). Stokes equations of motion apply to the relative motion of the particles whose radii are much smaller than the lengthscales of turbulence (i.e. small particle Reynolds numbers). We express the fluid motion in the vicinity of a pair of particles as a locally linear flow with a temporally varying velocity gradient. The fluctuating velocity gradient is assumed to be isotropic and Gaussian with statistics taken from published direct numerical simulations of turbulence (DNS). Numerical calculations of particle trajectories are used to determine the rate of turbulent coagulation in the presence and absence of particle interactions. Results from the numerical simulations correctly reproduce calculated coagulation rates for the asymptotic limits of small and large total strain where total strain is a term used to describe the product of the characteristic strain rate and its correlation time. Recent DNS indicate that the correlation times for the fluctuating strain and rotation rate are of the same order as the Kolmogorov time (Pope 1990), suggesting theories that assume either small or large total strain may poorly approximate the turbulent coagulation rate. Indeed, simulations for isotropic random flows with intermediate total strain indicate that the coagulation rate in turbulence is significantly different from the analytical limits for large and small total strain. The turbulent coagulation rate constant for non-interacting monodisperse particles scaled with the Kolmogorov time and the particle radius is 8.62±0.02, whereas the commonly used model of Saffman & Turner (1956) predicts a value of 10.35 for non-rotational flows in the limit of persistent turbulent velocity gradients. Additional simulations incorporating hydrodynamic interactions and van der Waals attraction were used to estimate the actual rate of particle coagulation. For typical values of these parameters, particle interactions reduced the coagulation rate constant by at least 50%. In general, the collision efficiency (the ratio of coagulation with particle interactions to that without) decreased with increasing particle size and Kolmogorov shear rate.

The detailed structure of the aqueous surface sublayer flow immediately adjacent to the wind-driven air–water interface is investigated in a laboratory wind-wave flume using particle image velocimetry (PIV) techniques. The goal is to investigate quantitatively the character of the flow in this crucial, very thin region which is often disrupted by microscale breaking events. In this study, we also examine critically the conclusions of Okuda, Kawai & Toba (1977), who argued that for very short, strongly forced wind-wave conditions, shear stress is the dominant mechanism for transmitting the atmospheric wind stress into the water motion – waves and surface drift currents. In strong contrast, other authors have more recently observed very substantial normal stress contributions on the air side. The availability of PIV and associated image technology now permits a timely re-examination of the results of Okuda et al., which have been influential in shaping present perceptions of the physics of this dynamically important region. The PIV technique used in the present study overcomes many of the inherent shortcomings of the hydrogen bubble measurements, and allows reliable determination of the fluid velocity and shear within 200 μm of the instantaneous wind-driven air–water interface.

The results obtained in this study are not in accord with the conclusions of Okuda et al. that the tangential stress component dominates the wind stress. It is found that prior to the formation of wind waves, the tangential stress contributes the entire wind stress, as expected. With increasing distance downwind, the mean tangential stress level decreases marginally, but as the wave field develops, the total wind stress increases significantly. Thus, the wave form drag, represented by the difference between the total wind stress and the mean tangential stress, also increases systematically with wave development and provides the major proportion of the wind stress once the waves have developed beyond their early growth stage. This scenario reconciles the question of relative importance of normal and tangential stresses at an air–water interface. Finally, consideration is given to the extrapolation of these detailed laboratory results to the field, where the present findings suggest that the sea surface is unlikely to become fully aerodynamically rough, at least for moderate to strong winds.

The simple shear-flow model of Stern & Adam (1973), in which a layer of uniform vorticity and depth overlies an infinitely deep fluid, is here extended by the addition of an upper fluid layer of uniform thickness and constant velocity. In this way many experimentally observed velocity profiles can be approximated. The normal mode instabilities of such a model can be found analytically, and their properties calculated through the solution of a quartic polynomial equation. The dispersion relation is here determined and illustrated in its dependence on the Froude number and on the ratio H1/H2, where H1 and H2 denote the mean depths of the surface layer and the base of the shear layer, respectively. It is found that two branches of instability which are distinct when H1/H2 is moderate or small can become merged when H1/H2[ges ]0.4924. Also calculated are the fastest-growing modes, and their wavelengths. The results are applied to some examples of surface flows generated by towed bodies, and to steady spilling breakers.

The velocity distribution function for a two-dimensional vibro-fluidized bed of particles of radius r is calculated using asymptotic analysis in the limit where (i) the dissipation of energy during a collision due to inelasticity or between successive collisions due to viscous drag is small compared to the energy of a particle and (ii) the length scale for the variation of density is large compared to the particle size. In this limit, it is shown that the parameters εG=rg/T0 and ε=U20/T0[Lt ]1, and ε and εG are used as small parameters in the expansion. Here, g is the acceleration due to gravity, U0 is the amplitude of the velocity of the vibrating surface and T0 is the leading-order temperature (divided by the particle mass). In the leading approximation, the dissipation of energy and the separation of the centres of particles undergoing a binary collision are neglected, and the system is identical to a gas of rigid point particles in a gravitational field. The leading-order particle number density is given by the Boltzmann distribution ρ0∝exp(−gz/T0, and the velocity distribution function is given by the Maxwell–Boltzmann distribution f(u)=(2πT0)−1exp [−u2/(2T0)], where u is the particle velocity. The temperature cannot be determined from the leading approximation, however, and is calculated by a balance between the rate of input of energy at the vibrating surface due to particle collisions with this surface, and the rate of dissipation of energy due to viscous drag or inelastic collisions. The first correction to the distribution function due to dissipative effects is calculated using the moment expansion method, and all non-trivial first, second and third moments of the velocity distribution are included in the expansion. The correction to the density, temperature and moments of the velocity distribution are obtained analytically. The results show several systematic trends that are in qualitative agreement with previous experimental results. The correction to the density is negative at the bottom of the bed, increases and becomes positive at intermediate heights and decreases exponentially to zero as the height is increased. The correction to the temperature is positive at the bottom of the bed, and decreases and assumes a constant negative value as the height is increased. The mean-square velocity in the vertical direction is greater than that in the horizontal direction, thereby facilitating the transport of energy up the bed. The difference in the mean-square velocities decreases monotonically with height for a system where the dissipation is due to inelastic collisions, but it first decreases and then increases for a system where the dissipation is due to viscous drag.

The character and stability of two- and three-dimensional thermocapillary driven convection are investigated by numerical simulations. In two dimensions, Hopf bifurcation neutral curves are delineated for fluids with Prandtl numbers (Pr) 10.0, 6.78, 4.4 and 1.0 in the Reynolds number (Re)–cavity aspect ratio (Ax) plane corresponding to Re[les ]1.3×104 and Ax[les ]7.0. It is found that time-dependent motion is only possible if Ax exceeds a critical value, Axcr, which increases with decreasing Pr. There are two coexisting neutral curves for Pr[ges ]4.4. Streamline and isotherm patterns are presented at different Re and Ax corresponding to stationary and oscillatory states. Energy analyses of oscillatory flows are performed in the neighbourhood of critical points to determine the mechanisms leading to instability. Results are provided for flows near both critical points of the first unstable region with Ax=3.0 and Pr=10.

In three dimensions, attention is focused on the influence of sidewalls, located at y=0 and y=Ay, and spanwise motion on the transition. In general, sidewalls have a damping effect on oscillations and hence increase the magnitude of the first critical Re. However, the existence of spanwise waves can reduce this critical Re. At large aspect ratios Ax=Ay=15, our results with Pr=13.9 at the lower critical Reynolds number of the first unstable region are in good agreement with those from infinite layer linear stability analysis.

Two identical circular cylinders are submerged to the same depth in tandem in a stream. There are separation distances between the cylinder centres such that the combination makes no downstream waves, and hence is subject to zero net wave drag. In general there is then a non-zero equal and opposite horizontal force on each cylinder. However, there are special depths of submergence such that this interaction force between the cylinders also vanishes, and hence each cylinder is separately free of horizontal force. The parameter range for this phenomenon is explored both by linear theory for cylinders of small radius, and by a fully nonlinear computer program. For example, a configuration with a separation distance of approximately one half-wavelength gives zero force on each cylinder when the depth of submergence is approximately three-quarters of the separation distance.

Numerical simulations of benthic gravity-driven currents along continental shelves suggest they exhibit considerable time and spatial variability and tend to organize themselves into large-scale bottom-intensified cold domes or eddies. Attempts to derive simple relations governing the evolution of the spatial moments of the mass equation for baroclinic eddies have failed because it is not clear how to express the form or wave drag stresses associated with the excited (topographic) Rossby wave field in the surrounding fluid in terms of the eddy moments. We develop a simple model for the leading-order time evolution of a cold dome configuration which initially nearly satisfies the Mory–Stern isolation constraint. As the topographic Rossby wave field in the surrounding fluid interacts with the cold dome, higher azimuthal modes are excited within the cold dome which develop into spiral-like filamentary structures on the eddy boundary. The trajectory followed by the position of the maximum height of the cold dome corresponds to sub-inertial along- and cross-slope oscillations superimposed on a mean along-slope drift (well described by the Nof velocity). Nevertheless, the theory suggests that there are no oscillations (at least to second order) in the horizontal spatial moments of the eddy height, that is, the centre of mass of the eddy moves steadily in the along- and down-slope directions (i.e. ‘southwestward’ relative to the topographic β-plane). The theoretical analysis is in good agreement with a nonlinear numerical simulation which we present.

Large-amplitude oscillations of drops and bubbles immersed in an immiscible liquid host have been investigated using ultrasonic radiation pressure techniques. Single levitated or trapped drops and bubbles with effective radius between 0.2 and 0.8 cm have been driven into resonant shape oscillations of the first few orders. The direct coupling of driven drop shape oscillations between the axisymmetric l=6 and l=3 modes has been documented as well as the interaction between axisymmetric and non-axisymmetric l=3 and l=2 modes. Effective resonant energy transfer from higher- to lower-order modes has been observed together with a much less efficient energy transfer in the reverse direction. The first three resonant modes for bubbles trapped in water have also been excited, and mode coupling during driven and free-decaying oscillations has been measured. The evidence gathered thus far indicates that efficient drop resonant coupling between a higher- and a lower-order mode occurs when the characteristic frequency of the latter mode roughly coincides with a harmonic resonance.

A linear stability analysis of vapour–liquid counter flow in porous media is carried out. For the vapour-dominated basic state the development in time of both pressure and saturation disturbances is studied. The pressure field is shown to be asymptotically stable for all choices of thermal boundary conditions, excluding the insulating–insulating boundary condition for which it is neutrally stable. The saturation field is proven to be Lyapunov stable: the saturation disturbance remains bounded by an infinitesimal number at all times. For both vapour- and liquid-dominated basic states the direction of propagation of small saturation disturbances is determined. These results explain the formation of two-layer geothermal structures and why alternative structures cannot develop within homogeneous reservoirs.

A class of exact solutions of the Navier–Stokes equations is derived. Each of them represents the velocity field v=U+u of a thin vortical layer (a planar jet) under a uniform strain velocity field U in three-dimensional infinite space, and provides a simple flow model in which nonlinear coupling between small eddies plays a key role in small-scale vortex dynamics. The small-scale structure of the velocity field is studied by numerically analysing the Fourier spectrum of u. It is shown that the Fourier spectrum of u falls off exponentially with wavenumber k for large k. The Taylor expansion in powers of the coordinate (say y) in the direction perpendicular to the vortical layer suggests that the solution may be well approximated by a function with certain poles in the complex y-plane. The Fourier spectrum based on the singularities is in good agreement with that obtained numerically, where the exponential decay rate is given by the distance of the poles from the real axis of y.

Martel et al. (1998) have shown that interior damping may be comparable with boundary-layer damping for surface waves in small cylinders and that its incorporation yields predictions in agreement with the experimental results of Henderson & Miles (1994) for non-axisymmetric waves on a clean surface with a fixed contact line. In the present note, Henderson & Miles's boundary-layer calculation is supplemented by a calculation of interior damping based on Lamb's dissipation integral for an irrotational flow. The analysis, which omits second-order boundary-layer effects, is simpler than that of Martel et al. (which includes these effects and is based on an expansion in an inverse Reynolds number), but yields results of comparable accuracy within the parametric domain of the experiments. The corresponding calculations for a fully contaminated (inextensible) surface reduce the discrepancy between calculation and experiment but, in contrast to the results for a clean surface, leave a significant residual discrepancy. An unexplained discrepancy also remains for axisymmetric waves on either a clean or a contaminated surface.

The unsteady fully nonlinear free-surface flow due to an impulsively started submerged point sink is studied in the context of incompressible potential flow. For a fixed (initial) submergence h of the point sink in otherwise unbounded fluid, the problem is governed by a single non-dimensional physical parameter, the Froude number, [Fscr ]≡Q/4π(gh5)1/2, where Q is the (constant) volume flux rate and g the gravitational acceleration. We assume axisymmetry and perform a numerical study using a mixed-Eulerian–Lagrangian boundary-integral-equation scheme. We conduct systematic simulations varying the parameter [Fscr ] to obtain a complete quantification of the solution of the problem. Depending on [Fscr ], there are three distinct flow regimes: (i) [Fscr ]<[Fscr ]1≈0.1924 – a ‘sub-critical’ regime marked by a damped wave-like behaviour of the free surface which reaches an asymptotic steady state; (ii) [Fscr ]1<[Fscr ]<[Fscr ]2≈0.1930 – the ‘trans-critical’ regime characterized by a reversal of the downward motion of the free surface above the sink, eventually developing into a sharp upward jet; (iii) [Fscr ]>[Fscr ]2 – a ‘super-critical’ regime marked by the cusp-like collapse of the free surface towards the sink. Mechanisms behind such flow behaviour are discussed and hydrodynamic quantities such as pressure, power and force are obtained in each case. This investigation resolves the question of validity of a steady-state assumption for this problem and also shows that a small-time expansion may be inadequate for predicting the eventual behaviour of the flow.