High-pressure gases are used to study high-Rayleigh-number Rayleigh–Bénard convection in cylindrical horizontal enclosures. The Nusselt–Rayleigh heat transfer relationship is investigated for 1×109 < Ra < 1.7×1012. Schlieren video images of the flow field are recorded through optical viewports in the pressure vessel. The data set is well correlated by Nu = 0.071Ra0.328. The schlieren results confirm the existence of a large-scale flow that periodically interrupts the ascending and descending plumes. The intensity of both the plumes and the large-scale flow increases with Rayleigh number.

Motivated in part by the problem of large-scale lateral turbulent heat transport in the Earth's atmosphere and oceans, and in part by the problem of turbulent transport itself, we seek to better understand the transport of a passive tracer advected by various types of fully developed two-dimensional turbulence. The types of turbulence considered correspond to various relationships between the streamfunction and the advected field. Each type of turbulence considered possesses two quadratic invariants and each can develop an inverse cascade. These cascades can be modified or halted, for example, by friction, a background vorticity gradient or a mean temperature gradient. We focus on three physically realizable cases: classical two-dimensional turbulence, surface quasi-geostrophic turbulence, and shallow-water quasi-geostrophic turbulence at scales large compared to the radius of deformation. In each model we assume that tracer variance is maintained by a large-scale mean tracer gradient while turbulent energy is produced at small scales via random forcing, and dissipated by linear drag. We predict the spectral shapes, eddy scales and equilibrated energies resulting from the inverse cascades, and use the expected velocity and length scales to predict integrated tracer fluxes.

When linear drag halts the cascade, the resulting diffusivities are decreasing functions of the drag coefficient, but with different dependences for each case. When β is significant, we find a clear distinction between the tracer mixing scale, which depends on β but is nearly independent of drag, and the energy-containing (or jet) scale, set by a combination of the drag coefficient and β. Our predictions are tested via high- resolution spectral simulations. We find in all cases that the passive scalar is diffused down-gradient with a diffusion coefficient that is well-predicted from estimates of mixing length and velocity scale obtained from turbulence phenomenology.

Hot-wire measurements were made simultaneously in two homogeneous ‘horizontal’ planes in the far-wake region of a cylinder. A technique developed using hot-wire data to identify the spatial characteristics of the large-scale bulges at the interface between the internal turbulent motions and the external irrotational flow was used to unambiguously relate these outer intermittent bulges to the inner coherent structures. It was found that a turbulent bulge is made up of a combination of a horseshoe vortex (whose legs form one double-roller eddy) and the straining region present just upstream of this structure. The approach also allowed the evaluation of the two most prominent phenomenological models for the entrainment mechanism in the far-wake region: the Kelvin–Helmholtz instability and Townsend's growth–decay cycle. It was found that the decaying and re-forming of the bulges and entrainment structures is not likely to occur. Rather, the evidence is that the large-scale bulges remain coherent for long streamwise distances in equilibrium with the overall similarity of the flow.

The theoretical study and experimental investigation of the reflection of asymmetric shock waves in steady flows reported by Li et al. (1999) are complemented by a numerical simulation. All the findings reported in both the theoretical study and the experimental investigation were also evident in the numerical simulation. In addition to weak regular reflection and Mach reflection wave configurations, strong regular reflection and inverse-Mach reflection wave configurations were recorded numerically. The hysteresis phenomenon, which was hypothesized in the course of the theoretical study and then verified in the experimental investigation, was also observed in the numerical simulation.

We combine experimental, theoretical and numerical efforts to investigate the turbulent wake far behind a surface ship at model scales. Experimental measurements using digital particle image velocimetry (DPIV) are performed for the wakes of three towed hulls with beam-to-draught ratios b/d = 1, 2, 6. Based on model speed and beam, the Reynolds and Froude numbers are O(103) and O(10−2) respectively. Distinct surface features associated with persistent surface-normal vorticity have been identified, which are characterized by large-scale meandering structures. Both lateral and longitudinal scales of the meandering are quantified, with the former found to increase as b/d decreases and the latter independent of b/d. Based on measurements at multiple horizontal and vertical planes, profiles of the mean flow and fluctuation intensity for each velocity component are obtained. To understand the turbulence transition mechanism, an Orr–Sommerfeld stability analysis (OS) is formulated for the wake flow with free-surface boundary conditions, and solved by using a fourth-order finite-difference scheme. Unstable modes antisymmetric to the wake centre-plane are identified. Consistent with the experimental results, the growth rates of unstable modes increase substantially as b/d decreases, while the dependence of meandering wavelengths on b/d is found to be weak. Finally, we perform direct numerical simulation (DNS) of Navier–Stokes equations for the wake flow. The growth rates of unstable modes agree well with the predictions by OS analysis. Compared with experiments, DNS accurately captures the surface-normal vorticity signatures, the meandering features, as well as statistics of turbulence intensity. We also obtain from DNS a detailed description of enstrophy, turbulence length scales, and vortex structures for the wake flow.

We use silicon strip detectors (originally developed for the CLEO III high-energy particle physics experiment) to measure fluid particle trajectories in turbulence with temporal resolution of up to 70000 frames per second. This high frame rate allows the Kolmogorov time scale of a turbulent water flow to be fully resolved for 140 [ges ] Rλ [ges ] 970. Particle trajectories exhibiting accelerations up to 16000 m s −2 (40 times the r.m.s. value) are routinely observed. The probability density function of the acceleration is found to have Reynolds-number-dependent stretched exponential tails. The moments of the acceleration distribution are calculated. The scaling of the acceleration component variance with the energy dissipation is found to be consistent with the results for low-Reynolds-number direct numerical simulations, and with the K41-based Heisenberg–Yaglom prediction for Rλ [ges ] 500. The acceleration flatness is found to increase with Reynolds number, and to exceed 60 at Rλ = 970. The coupling of the acceleration to the large-scale anisotropy is found to be large at low Reynolds number and to decrease as the Reynolds number increases, but to persist at all Reynolds numbers measured. The dependence of the acceleration variance on the size and density of the tracer particles is measured. The autocorrelation function of an acceleration component is measured, and is found to scale with the Kolmogorov time τη.

An experimental laboratory study has been carried out to investigate the propagation of an internal solitary wave of depression and its distortion by a bottom ridge in a two-layer stratified fluid system. Wave profiles, density fields and velocity fields have been measured at three reference locations, namely upstream, downstream and over the ridge. Experiments have been performed with wave amplitudes in the range 0.2– 1.9 times the depth of the upper layer, and a ratio between the lower and the upper layer in the range 3.0–8.5. The ridge slope was varied from 0.1 to 0.33 and the maximum ridge height was two-thirds of the thicker fluid layer. Over the ridge, the flow has been classified into: (i) cases when the bottom ridge has little influence on the propagation and spatial structure of the internal solitary wave, (ii) cases where the internal solitary wave is significantly distorted by the blocking effect of the ridge (though no wave breaking occurs), and (iii) cases for which the internal solitary wave is broken as it encounters and passes over the bottom ridge. A detailed description of the processes leading to wave breaking is given. Breaking has been found to take place when the fluid velocity in the lower layer exceeds 0.7 of a local nonlinear wave speed, defined at the top of the ridge. The breaking condition is also expressed in terms of the amplitude of the incident wave, the layer thickness ratio and the relative height of the ridge. The wave breaking can be determined from the input parameters of the experiment. The transmitted waves have been found to always consist of a leading pulse (solitary wave) followed by a dispersive wavetrain. The (solitary) wave amplitude is significantly reduced only when breaking takes place at the ridge. Internal waves of mode two are generated in cases with strong breaking.

This paper presents two linear stability analyses for an electrically conducting liquid contained in a vertical cylinder with a thermally insulated vertical wall and with isothermal top and bottom walls. There is a steady uniform vertical magnetic field. The first linear stability analysis involves a hybrid approach which combines an analytical solution for the Hartmann layers adjacent to the top and bottom walls with a numerical solution for the rest of the liquid domain. The second linear stability analysis involves an asymptotic solution for large values of the Hartmann number. Numerically accurate predictions of the critical Rayleigh number can be obtained for Hartmann numbers from zero to infinity with the two solutions presented here and a previous numerical solution which gives accurate results for small values of the Hartmann number.

A class of explicit solutions of the two-dimensional Euler equations consisting of a finite-area patch of uniform vorticity surrounded by a finite distribution of co- rotating satellite line vortices is constructed. The results generalize the classic study of co-rotating vortex arrays by J. J. Thomson. For N satellite line vortices (N [ges ] 3) a continuous one-parameter family of rotating vortical equilibria is derived in which different values of the continuous parameter correspond to different shapes and areas of the central patch. In an appropriate limit, vortex patch equilibria with cusped boundaries are found. A study of the linear stability is performed and a wide range of the solutions found to be linearly stable. Contour dynamics methods are used to calculate the typical nonlinear evolution of the configurations. The results are believed to provide the only known exact solutions involving rotating vortex patches besides the classical Kirchhoff ellipse.

We study the viscous fingering or Saffman–Taylor instability in two different dilute or semi-dilute polymer solutions. The different solutions exhibit only one non-Newtonian property, in the sense that other non-Newtonian effects can be neglected. The viscosity of solutions of stiff polymers has a strong shear rate dependence. Relative to Newtonian fluids, narrower fingers are found for rigid polymers. For solutions of flexible polymers, elastic effects such as normal stresses are dominant, whereas the shear viscosity is almost constant. Wider fingers are found in this case. We characterize the non-Newtonian flow properties of these polymer solutions completely, allowing for separate and quantitative investigation of the influence of the two most common non-Newtonian properties on the Saffman–Taylor instability. The effects of the non-Newtonian flow properties on the instability can in all cases be understood quantitatively by redefining the control parameter of the instability.

The dynamics of a non-neutrally buoyant particle moving in a rotating cylinder filled with a Newtonian fluid is examined analytically. The particle is set in motion from the centre of the cylinder due to the acceleration caused by the presence of a gravitational field. The problem is formulated in Cartesian coordinates and a relevant fractional Lagrangian equation is proposed. This equation is solved exactly by recognizing that the eigenfunctions of the problem are Mittag–Leffler functions. Virtual mass, gravity, pressure, and steady and history drag effects at low particle Reynolds numbers are considered and the balance of forces acting on the particle is studied for realistic cases. The presence of lift forces, both steady and unsteady, is taken into account. Results are compared to the exact solution of the Maxey–Riley equation for the same conditions. Substantial differences are found by including lift in the formulation when departing from the infinitesimal particle Reynolds number limit. For particles lighter than the fluid, an asymptotically stable equilibrium position is found to be at a distance from the origin characterized by X ≈ −Vτ/Ω and Y/X ≈ (CS/3π√2) Res1/2, where Vτ is the terminal velocity of the particle, Ω is the positive angular velocity of the cylinder, Res is the shear Reynolds number a2Ω/v, and CS is a constant lift coefficient. To the knowledge of the authors this work is the first to solve the particle Lagrangian equation of motion in its complete form, with or without lift, for a non-uniform flow using an exact method.

We examine the critical merging distance between two equal-volume, equal-potential- vorticity quasi-geostrophic vortices. We focus on how this distance depends on the vertical offset between the two vortices, each having a unit mean height-to-width aspect ratio. The vertical direction is special in the quasi-geostrophic model (used to capture the leading-order dynamical features of stably stratified and rapidly rotating geophysical flows) since vertical advection is absent. Nevertheless vortex merger may still occur by horizontal advection.

In this paper, we first investigate the equilibrium states for the two vortices as a function of their vertical and horizontal separation. We examine their basic properties together with their linear stability. These findings are next compared to numerical simulations of the nonlinear evolution of two spheres of potential vorticity. Three different regimes of interaction are identified, depending on the vertical offset. For a small offset, the interaction differs little from the case when the two vortices are horizontally aligned. On the other hand, when the vertical offset is comparable to the mean vortex radius, strong interaction occurs for greater horizontal gaps than in the horizontally aligned case, and therefore at significantly greater full separation distances. This perhaps surprising result is consistent with the linear stability analysis and appears to be a consequence of the anisotropy of the quasi-geostrophic equations. Finally, for large vertical offsets, vortex merger results in the formation of a metastable tilted dumbbell vortex.

The instability to longitudinal vortices of two-dimensional density-stratified temporally evolving wavy shear flow is considered. The problem is posited in the context of Langmuir circulations, LCs, beneath wind-driven surface waves and the instability mechanism is generalized Craik–Leibovich, either CLg or CL2. Of interest is the influence of non-stationary base flows on the instability according to linear theory. It is found that the instability is described by a family of similarity solutions and that the growth rate of the instability, in non-stationary base flows, is doubly exponential in time, although the growth rate reduces to exponential when the base flow is stationary. An example is given for weakly sheared wind-driven flow evolving in the presence of growing irrotational surface waves. Waves aligned both with the wind and counter to it are considered, as is the role of stratification. Antecedent to the example is an initial value problem posed by Leibovich & Paolucci (1981) for neutral waves in slowly evolving shear. Here, however, the waves and shear may grow (or decay) at rates comparable with the LCs. Furthermore the current here has two components: a wind-driven portion due to the wind stress applied at the free surface and a second due to the diffusion of momentum due to the wave-amplitude-squared free-surface stress condition. Using the case for neutral waves in non-stratified uniform shear for reference, it is found, in general, that growing waves are stabilizing while decaying waves are destabilizing to the formation of LCs, although the latter applies only for sufficiently large spanwise spacings and is subject to a globally stable lower bound. Decaying waves in the absence of wind can also be destabilizing to LCs. When the wind is counter to the waves, however, only decaying waves are unstable to LCs. Furthermore, while growing waves are stable to the formation of LCs in the presence of stable stratification, decaying waves are unstable in both aligned and opposed wind-wave conditions. Unstable stratification on the other hand, is destabilizing to LCs for all temporal waves in both aligned and opposed wind-wave conditions.

A shallow fluid-filled cavity with a longitudinal applied temperature gradient is subjected to spanwise accelerations (g-jitter) representing the space-based microgravity environment. A simplified slot model is introduced to describe the buoyancy-driven flow and advected temperature fields produced in the cavity. Numerical solutions indicate that boundary layer behaviour can manifest itself in the limit of strong g-jitter (large Rayleigh number Ra). However, boundary layer thicknesses do not obey the conventional Ra−1/4 scaling that typically arises in free thermal convection problems. This anomalous scaling results from the three-dimensional complexity of the flow and advected temperature fields, which are not themselves produced by a single fixed applied temperature change. Three different regimes are identified at large Rayleigh number characterized by the shapes of the advected temperature profiles. These regimes are selected according to the values of the Biot number Bi and an aspect ratio parameter. Simple models are presented of the boundary layer behaviour which reproduce, in each regime, the numerically predicted scalings for boundary layer thickness and advected temperature. These models give a succinct overall picture of the slot behaviour in the buoyancy-dominated limit.

We have experimentally examined the effects of a common soluble surfactant on gas bubbles in liquid flows in inclined tubes. Air bubbles of known size (λ = 0.8, 1.0, 1.5) are held stationary under minimum flow conditions in tubes inclined at fixed angles (ω = 25°, 45°, 65°, 90°). Sodium dodecyl sulphate (SDS) is infused into the bulk flow at two bulk concentrations (C = 10% or 100% critical micelle concentration (CMC)). In addition to recording pressure and flow waveforms, we capture video images of bubbles before and during exposure to the surfactant. Modification of the interfacial properties by the surfactant results in extremely dynamic bubble behaviour including interfacial deformation, deformation plus axial translation, and bubble detachment from the wall plus translation. We measure the corresponding time-dependent pressure gradient within the tube. The surfactant mediated responses observed are dependent upon the interrelated effects of C, λ and ω. A high bulk concentration of surfactant may produce more rapid modification of bubble shape and influence wetting, thus increasing the potential for bubble detachment. The likelihood that detachment will occur increases further as bubble volume in increased. In both vertical tubes in which contact forces are absent and in non-vertical tubes, the infusion of surfactant may result in axial translation either in the direction of, or opposite to, the direction of the bulk flow. Critical to the translation and/or detachment of the bubble is the surfactant-mediated modification of contact line mechanics. Contact line velocities corresponding to rates of shrinkage of dewetted surface area are extracted from experimental data. We also explore the potential effects of surfactants on interfacial remobilization. This investigation demonstrates the potential use of surfactants to be used for dislodging dewetted gas bubbles by the intentional manipulation of interfacial properties.

In Robinet et al. (2000), an instability for shock waves in gas dynamics is exhibited. The fluid was assumed to obey the polytropic perfect gas pressure law. We show in this note that such an instability does not occur, as proved in the extensive work of Majda (1983). Two arguments are developed: we give first a mathematical result that is violated by the conclusions of Robinet et al. (2000). Then we detail the results of Robinet et al. (2000) and show why they do not yield any conclusions. Finally, we develop a general calculation and show that an instability cannot occur.