A coherent-vortex analysis is made of a computational solution for the free decay of homogeneous, Charney-isotropic geostrophic turbulence at large Reynolds number. The method of analysis is a vortex detection and measurement algorithm that we call a vortex census. The census demonstrates how, through non-conservative interactions among closely approaching vortices, the vortex population evolves towards fewer, larger, sparser, and more weakly deformed vortices. After emergence from random initial conditions and a further period of population adjustment, there is a period of approximately self-similar temporal evolution in the vortex statistics. This behaviour is consistent with a mean-vortex scaling theory based on the conservation of energy, vortex extremum, and vortex aspect ratio. This period terminates as the population approaches a late-time non-turbulent end-state vortex configuration. The end state develops out of merger and alignment interactions among like-sign vortices, and even during the scaling regime, local clusters of nearly aligned vortices are common.

This paper considers the role of long finite-amplitude Rossby waves in determining the evolution of flow along a rapidly rotating channel with an uneven floor. The Rossby waves travel on a potential vorticity interface in a channel with a cross-channel step change in depth, where step position varies slowly along the channel. A nonlinear wave equation is derived describing the evolution of the potential vorticity interface. To leading order this is the hydraulic equation derived by Haynes, Johnson & Hurst (1993). Dispersion appears at the next order. Various solution regimes are identified. As well as slowly varying hydraulic solutions, two further types of steady solutions appear: approach-controlled flows and twin supercritical leaps. Both these solutions are characterized by leaps between supercritical branches of the hydraulic function. It is shown how the position and size of these ‘supercritical leaps’ can be determined within the context of hydraulic theory. To fully resolve the internal structure of these leaps dispersive effects must be included and leaps are shown to correspond to kink soliton solutions of the steady unforced problem. It is also shown that increasing dispersion (decreasing topographic length scale) causes the loss of the subcritical solution branch in some subcritical flows. The only candidate for a steady solution in these regimes is then an approach-controlled flow. Integrations of initial value problems show that in general flows evolve towards the dispersive form of the solution predicted by hydraulic theory, at least near the topographic perturbation. However, in those subcritical flows where sufficiently large dispersion causes the subcritical branch to disappear, unsteady integrations evolve to approach-controlled flows even when the dispersion is sufficiently small that the subcritical branch still exists.

An experimental investigation on the initial instability of nonlinear deep-water wave trains including wind effects is reported. The experiment was conducted at the Ocean Engineering Laboratory wind-wave facility (50 m long, 4.2 m wide, 2.1 m deep), with a fully computer-controlled mechanical wave generator to explore the parameter space: steepness; sideband frequency; wind speed. The estimated growth rates of the Benjamin–Feir instability from seeded wind-free experiments agreed well with the theoretical prediction derived from Krasitskii's four-wave reduced equation as computed here. Wind was added to the same wave system; the growth rates of the sidebands were reduced for weak, and enhanced for strong wind forcing. Experiments with naturally selected sidebands, i.e. unseeded, were conducted as well; measurements showed that wind did not inhibit the growth of sidebands in the case of either two-dimensional or three-dimensional instabilities. A comparison of the results with earlier work suggests that there are two independent effects of wind: first, the alteration of the inviscid growth for a given modulational frequency as shown by comparison with the seeded experiments without wind; second, a change in the natural modulational frequency appearing in the presence of wind which is a function of the wave age, as observed in unseeded experiments. Both effects combined will determine whether the modulational instability is enhanced or suppressed; comparison of experimental results with theoretical predictions suggests that the effect of wind on the natural selection of the modulational frequency is the dominant effect. It was shown that for moderate to old waves, the net effect of wind on the modulational instability is small. For all the experiments except a few unseeded cases with weak breakers, the modulation was small and no breaking was observed within the tank.

A methodology termed the ‘filtered mass density function’ (FMDF) is developed and implemented for large-eddy simulation (LES) of variable-density chemically reacting turbulent flows at low Mach numbers. This methodology is based on the extension of the ‘filtered density function’ (FDF) scheme recently proposed by Colucci et al. (1998) for LES of constant-density reacting flows. The FMDF represents the joint probability density function of the subgrid-scale (SGS) scalar quantities and is obtained by solution of its modelled transport equation. In this equation, the effect of chemical reactions appears in a closed form and the influences of SGS mixing and convection are modelled. The stochastic differential equations (SDEs) which yield statistically equivalent results to those of the FMDF transport equation are derived and are solved via a Lagrangian Monte Carlo scheme. The consistency, convergence, and accuracy of the FMDF and the Monte Carlo solution of its equivalent SDEs are assessed. In non-reacting flows, it is shown that the filtered results via the FMDF agree well with those obtained by the ‘conventional’ LES in which the finite difference solution of the transport equations of these filtered quantities is obtained. The advantage of the FMDF is demonstrated in LES of reacting shear flows with non-premixed reactants. The FMDF results are appraised by comparisons with data generated by direct numerical simulation (DNS) and with experimental measurements. In the absence of a closure for the SGS scalar correlations, the results based on the conventional LES are significantly different from those obtained by DNS. The FMDF results show a closer agreement with DNS. These results also agree favourably with laboratory data of exothermic reacting turbulent shear flows, and portray several of the features observed experimentally.

The objective of this study is to develop a method of controlling vortex shedding behind a bluff body using control theory. A suboptimal feedback control procedure for local sensing and local actuation is developed and applied to the flow behind a circular cylinder. The location of sensors for feedback is limited to the cylinder surface and the control input from actuators is the blowing and suction on the cylinder surface. Three different cost functionals to be minimized (J1 and J2) or maximized (J3) are investigated: J1 is proportional to the pressure drag of the cylinder, J2 is the square of the difference between the target pressure (inviscid flow pressure) and real flow pressure on the cylinder surface, and J3 is the square of the pressure gradient on the cylinder surface, respectively. Given the cost functionals, the flow variable to be measured by the sensors and the control input from the actuators are determined from the suboptimal feedback control procedure. Several cases for each cost functional have been numerically simulated at Re = 100 and 160 to investigate the performance of the control algorithm. For all actuations, vortex shedding becomes weak or disappears, and the mean drag and drag/lift fluctuations significantly decrease. For a given magnitude of the blowing/suction, reducing J2 provides the largest drag reduction among the three cost functionals.

The transient evolution of the bubble-size probability density functions resulting from the breakup of an air bubble injected into a fully developed turbulent water ow has been measured experimentally using phase Doppler particle sizing (PDPA) and image processing techniques. These measurements were used to determine the breakup frequency of the bubbles as a function of their size and of the critical diameter Dc defined as Dc = 1.26 (σ/ρ)3/5ε−2/5, where ε is the rate of dissipation per unit mass and per unit time of the underlying turbulence. A phenomenological model is proposed showing the existence of two distinct bubble size regimes. For bubbles of sizes comparable to Dc, the breakup frequency is shown to increase as (σ/ρ)−2/5ε−3/5 √D/Dc−1, while for large bubbles whose sizes are greater than 1.63Dc, it decreases with the bubble size as ε1/3D−2/3. The model is shown to be in good agreement with measurements performed over a wide range of bubble sizes and turbulence intensities.

Based on energy principles, we propose a statistical model to describe the bubble size probability density function of the daughter bubbles resulting from the shattering of a mother bubble of size D0 immersed in a fully developed turbulent water flow. The model shows that the bubble size p.d.f. depends not only on D0, but also on the value of the dissipation rate of turbulent kinetic energy of the underlying turbulence of the water, ε. The phenomenological model is simple, yet it predicts detailed experimental measurements of the transient bubble size p.d.f.s performed over a range of bubble sizes and dissipation rates ε in a very consistent manner. The agreement between the model and the experiments is particularly good for low and moderate bubble turbulent Weber numbers, Wet = ρΔu2(D0)D0/σ where the assumption of the binary breakup is shown to be consistent with the experimental observations. At larger values of Wet, it was found that the most probable number of daughter bubbles increases and the assumption of tertiary breakup is shown to lead to a better fit of the experimental measurements.

Direct numerical simulations of turbulence resulting from Kelvin–Helmholtz instability in stratified shear flow are used to examine the geometry of the dissipation range in a variety of flow regimes. As the buoyancy and shear Reynolds numbers that quantify the degree of isotropy in the dissipation range increase, alignment statistics evolve from those characteristic of parallel shear flow to those found previously in studies of stationary, isotropic, homogeneous turbulence (e.g. Ashurst et al. 1987; She et al. 1991; Tsinober et al. 1992). The analysis yields a limiting value for the mean compression rate of scalar gradients that is expected to be characteristic of all turbulent flows at sufficiently high Reynolds number.

My main focus is the value of the constant q that appears in both the Batchelor (1959) and Kraichnan (1968) theoretical forms for the passive scalar spectrum. Taking account of the effects of time-dependent strain, I propose a revised estimate of q, denoted qe, which appears to agree with spectral shapes derived from simulations and observations better than do previous theoretical estimates. The revised estimate is qe = 7.3±4, and is expected to be valid whenever the buoyancy Reynolds number exceeds O(102). The Kraichnan (1968) spectral form, in which effects of intermittency are accounted for, provides a better fit to the DNS results than does the Batchelor (1959) form.

The behaviour of the long-time self-diffusion tensor in concentrated colloidal dispersions is studied using dynamic simulation. The simulations are of a suspension of monodisperse Brownian hard spheres in simple shear flow as a function of the Péclet number, Pe, which measures the relative importance of shear and Brownian forces, and the volume fraction, ϕ. Here, Pe = &γdot;a2/D0, where &γdot; is the shear rate, a the particle size and D0 = kT/6πηa is the Stokes–Einstein diffusivity of an isolated particle of size a with thermal energy kT in a solvent of viscosity η. Two simulations algorithms are used: Stokesian Dynamics for inclusion of the many-body hydrodynamic interactions, and Brownian Dynamics for suspensions without hydrodynamic interactions. A new procedure for obtaining high-quality diffusion data based on averaging the results of many short simulations is presented and utilized. At low shear rates, low Pe, Brownian diffusion due to a random walk process dominates and the characteristic scale for diffusion is the Stokes–Einstein diffusivity, D0. At zero Pe the diffusivity is found to be a decreasing function of ϕ. As Pe is slowly increased, O(Pe) and O(Pe3/2) corrections to the diffusivity due to the flow are clearly seen in the Brownian Dynamics system in agreement with the theoretical results of Morris & Brady (1996). At large shear rates, large Pe, both systems exhibit diffusivities that grow linearly with the shear rate by the non-Brownian mechanism of shear-induced diffusion. In contrast to the behaviour at low Pe, this shear-induced diffusion mode is an increasing function of ϕ. Long-time rotational self-diffusivities are of interest in the Stokesian Dynamics system and show similar behaviour to their translational analogues. An off-diagonal long-time self-diffusivity, Dxy, is reported for both systems. Results for both the translational and rotational Dxy show a sign change from low Pe to high Pe due to different mechanisms in the two regimes. A physical explanation for the off-diagonal diffusivities is proposed.

Using a combination of bifurcation theory for two-dimensional dynamical systems and numerical simulations, we systematically determine the possible flow topologies of the steady vortex breakdown in axisymmetric flow in a cylindrical container with rotating end-covers. For fixed values of the ratio of the angular velocities of the covers in the range from −0.02 to 0.05, bifurcations of recirculating bubbles under variation of the aspect ratio of the cylinder and the Reynolds number are found. Bifurcation curves are determined by a simple fitting procedure of the data from the simulations. For the much studied case of zero rotation ratio (one fixed cover) a complete bifurcation diagram is constructed. Very good agreement with experimental results is obtained, and hitherto unresolved details are determined in the parameter region where up to three bubbles exist. For non-zero rotation ratios the bifurcation diagrams are found to change dramatically and give rise to other types of bifurcations.

When two drops of radius R touch, surface tension drives an initially singular motion which joins them into a bigger drop with smaller surface area. This motion is always viscously dominated at early times. We focus on the early-time behaviour of the radius rm of the small bridge between the two drops. The flow is driven by a highly curved meniscus of length 2πrm and width Δ[Lt ]rm around the bridge, from which we conclude that the leading-order problem is asymptotically equivalent to its two-dimensional counterpart. For the case of inviscid surroundings, an exact two-dimensional solution (Hopper 1990) shows that Δ∝r3m and rm∼(tγ/πη) ln [tγ(ηR)]; and thus the same is true in three dimensions. We also study the case of coalescence with an external viscous fluid analytically and, for the case of equal viscosities, in detail numerically. A significantly different structure is found in which the outer-fluid forms a toroidal bubble of radius Δ∝r3/2m at the meniscus and rm∼(tγ/4πη) ln [tγ/(ηR)]. This basic difference is due to the presence of the outer-fluid viscosity, however small. With lengths scaled by R a full description of the asymptotic flow for rm(t)[Lt ]1 involves matching of lengthscales of order r2m, r3/2m, rm, 1 and probably r7/4m.

In this paper we investigate the linear stability of detonations in which the underlying steady one-dimensional solutions are of the pathological type. Such detonations travel at a minimum speed, which is greater than the Chapman–Jouguet (CJ) speed, have an internal frozen sonic point at which the thermicity vanishes, and the unsupported wave is supersonic (i.e. weak) after the sonic point. Pathological detonations are possible when there are endothermic or dissipative effects present in the system. We consider a system with two consecutive irreversible reactions A→B→C, with an Arrhenius form of the reaction rates and the second reaction endothermic. We determine analytical asymptotic solutions valid near the sonic pathological point for both the one-dimensional steady equations and the equations for linearized perturbations. These are used as initial conditions for integrating the equations. We show that, apart from the existence of stable modes, the linear stability of the pathological detonation is qualitatively the same as for CJ detonations for both one- and two-dimensional disturbances. We also consider the stability of overdriven detonations for the system. We show that the frequency of oscillation for one-dimensional disturbances, and the cell size based on the wavenumber with the highest group velocity for two-dimensional disturbances, are both very sensitive to the detonation speed for overdriven detonations near the pathological speed. This dependence on the degree of overdrive is quite different from that obtained when the unsupported detonation is of the CJ type.

The high Reynolds number flow through a circular pipe divided along a diameter by a semi-infinite splitter plate is considered. Matched asymptotic expansions are used to analyse the developing flow, which is decomposed into four regions: a boundary layer of Blasius type growing along the plate, an inviscid core, a viscous layer close to the curved wall and a nonlinear corner region. The core solution is found numerically, initially in the long-distance down-pipe limit and thereafter the full problem is solved using down-pipe Fourier transforms. The accuracy in the corners of the semicircular cross-section is improved by subtracting out the singularity in the velocity perturbation. The linear viscous wall layer is solved analytically in terms of a displacement function determined from the core. A plausible structure for the corner region and equations governing the motion therein are presented although no solution is attempted. The presence of the plate has little effect ahead of the bifurcation, but wall shear on the curved wall is found to increase from its undisturbed value downstream.

The fully developed flow in a vertical cavity or duct subject to horizontal heating is considered. Solutions of the Boussinesq equations are obtained for rectangular and elliptic sections, in terms of Fourier series and polynomials, respectively. Both generalize the familiar odd-symmetric cubic profile of the plane cavity. Uniqueness is demonstrated under the restriction that the flow is independent of height. For cavities with rectangular sections, it is predicted and verified that the flow in the plane of spanwise symmetry is practically independent of the span if this exceeds 1.7 times the breadth.