We report on the results of a laboratory investigation using a rotating two-layer annulus experiment, which exhibits both large-scale vortical modes and short-scale divergent modes. A sophisticated visualization method allows us to observe the flow at very high spatial and temporal resolution. The balanced long-wavelength modes appear only when the Froude number is supercritical (i.e. $F\,{>}\,F_\mathrm{critical}\,{\equiv}\, \upi^2/2$), and are therefore consistent with generation by a baroclinic instability. The unbalanced short-wavelength modes appear locally in every single baroclinically unstable flow, providing perhaps the first direct experimental evidence that all evolving vortical flows will tend to emit freely propagating inertia–gravity waves. The short-wavelength modes also appear in certain baroclinically stable flows.

We infer the generation mechanisms of the short-scale waves, both for the baro-clinically unstable case in which they co-exist with a large-scale wave, and for the baroclinically stable case in which they exist alone. The two possible mechanisms considered are spontaneous adjustment of the large-scale flow, and Kelvin–Helmholtz shear instability. Short modes in the baroclinically stable regime are generated only when the Richardson number is subcritical (i.e. $\hbox{\it Ri}\,{<}\,\hbox{\it Ri}_\mathrm{critical}\,{\equiv}\, 1$), and are therefore consistent with generation by a Kelvin–Helmholtz instability. We calculate five indicators of short-wave generation in the baroclinically unstable regime, using data from a quasi-geostrophic numerical model of the annulus. There is excellent agreement between the spatial locations of short-wave emission observed in the laboratory, and regions in which the model Lighthill/Ford inertia–gravity wave source term is large. We infer that the short waves in the baroclinically unstable fluid are freely propagating inertia–gravity waves generated by spontaneous adjustment of the large-scale flow.

Forced stratified flows are shown to suffer two types of linear long-wave instability: a ‘viscous’ instability which is related to the classical instability of Kolmogorov flow, and a ‘conductive instability’, with the form of a large-scale, negative thermal diffusion. The nonlinear dynamics of both instabilities is explored with weakly nonlinear theory and numerical computations. The introduction of stratification suppresses the viscous instability, but also makes it subcritical. The second instability arises with stronger stratification and creates a prominent staircase in the buoyancy field; the steps of the staircase evolve over long timescales by approaching one another, colliding and merging (coarsening the staircase).

Transient growth properties are computed for a two-phase temporal mixing layer of immiscible fluids with interfacial tension. Large transient growth factors are found to occur at short times in parameter regimes characteristic of the primary breakup of a liquid. Optimal growth factors scale with the square of the Reynolds number, as for single-phase flow. The flow fields of optimal disturbances show liquid upflows and high-speed streamwise gas jets occurring together near the interface, suggesting transient growth as a possible mechanism for the formation of interfacial patterns. Optimal growth occurs for streamwise uniform disturbances with a spanwise wavelength proportional to the thickness of the gas boundary layer. For a coaxial jet, the predicted number of ligaments would be inversely proportional to the gas boundary layer thickness.

We investigate the unsteady motion of a long bubble advancing under either prescribed pressure $p_{\rm b}$ or prescribed volume flux $q_{\rm b}$ into a fluid-filled flexible-walled channel at zero Reynolds number, an idealized model for the reopening of a liquid-lined lung airway. The channel walls are held under longitudinal tension and are supported by external springs; the bubble moves with speed $U$. Provided $p_{\rm b}$ exceeds a critical pressure $p_{\rm crit}$, the system exhibits two types of steady motion. At low speeds, the bubble acts like a piston, slowly pushing a column of fluid ahead of itself, and $U$ decreases with increasing $p_{\rm b}$. At high speeds, the bubble rapidly peels the channel walls apart and $U$ increases with increasing $p_{\rm b}$. Using two independent time-dependent simulation techniques (a two-dimensional boundary-element method and a one-dimensional asymptotic approximation), we show that with prescribed $p_{\rm b}\,{>}\,p_{\rm crit}$, peeling motion is stable and the steady pushing solution is unstable; for $p_{\rm b}\,{<}\,p_{\rm crit}$ the system ultimately exhibits unsteady pushing behaviour for which $U$ continually diminishes with time. When $q_{\rm b}$ is prescribed, peeling motion (with large $q_{\rm b}$) is again stable, but pushing motion (with small $q_{\rm b}$) loses stability at long times to a novel instability leading to spontaneous relaxation oscillations of increasing amplitude and period, for which the bubble switches abruptly between slow unsteady pushing and rapid quasi-steady peeling. This stick–slip motion is characterized using a third-order lumped-parameter model which in turn is reduced to a nonlinear map. Implications for the inflation of occluded lung airways are discussed.

The full set of velocity derivatives, $\partial u_{i}/\partial x_{j}$, is measured experimentally in a Lagrangian way in quasi-homogeneous isotropic turbulence. This is achieved by applying the three-dimensional particle tracking velocimetry (3D-PTV) technique to an electromagnetically forced flow with $\hbox{\it Re}_{\lambda}\,{\thickapprox}\,50$. Checks based on precise kinematic relations show that the technique presented measures the velocity derivatives with good accuracy. In a study on vorticity, characteristic properties of turbulent flows known from direct numerical simulations are reproduced. These are the positive skewness of the intermediate eigenvalue of the rate of strain tensor, $s_{ij}$, $\langle \Lambda_{2}\rangle \,{>}\,0$, the predominance of vortex stretching over vortex compression, $\langle \omega_{i}\omega_{j}s_{ij}\rangle \,{>}\,0$ and the predominant alignment of vorticity, ${\bm \omega}$, with the intermediate principal axis of strain, ${\bm \lambda}_{2}$. Results on the evolution in time of material lines, ${\bm l}$, compared to vortex lines, ${\bm \omega}$, are presented. They show that the nonlinear interaction of vorticity with the surrounding flow assists viscosity in maintaining this predominant ${\bm \lambda}_{2}$-alignment of vorticity. Lagrangian measurements of enstrophy budget terms suggest that there is no pointwise balancing of production and viscous reduction of enstrophy and that the role played by viscosity is of great importance.

The onset of convection in a rotating cylindrical annulus with sloping conical boundaries is studied in the case where this slope increases with the radius. The critical modes assume the form of drifting spiralling columns attached to the inner cylindrical wall at moderate and large Prandtl numbers, but they become attached to the outer wall at low Prandtl numbers. These latter ‘equatorially attached’ modes are multicellular at intermediate rotation rates. Through a perturbation analysis which is validated by a numerical code, we show that all equatorially attached modes are quasi-inertial modes and analyse the physical mechanisms leading to multicells. This is done for both stress-free and no-slip boundary conditions. At finite amplitudes the convection generates a Reynolds stress which leads to the development of a mean zonal flow, and a geometrical analysis of the mechanisms leading to this zonal flow is presented. The influence of Ekman friction on the zonal flow is also studied.

The mixing of a scalar (temperature) emitted from a concentrated line source in fully developed high-aspect-ratio turbulent channel flow is studied. The motivation for the work is to study the effect of the inhomogeneity on the scalar dispersion. It is most readily carried out in a flow that is inhomogeneous in only one direction, i.e. channel flow. Experiments were performed at two Reynolds numbers ($\hbox{\it Reacute;\,{\equiv}\,\langle U(y=h)\rangle h/\nu\,{=}\,10\,400$ and 22800), three wall-normal source locations ($y_s/h\,{=}\,0.067$, 0.17 and 1.0) and six downstream distances ($4.0 \,{\le}\, x/h \,{\le}\,22.0$). Both the mean and r.m.s. temperature profiles were found to be described well by truncated Gaussian distributions. In contrast to homogeneous flows, (i) the growth rates of the mean profile widths did not exhibit power law behaviours, (ii) the centres of the r.m.s. profiles were found to drift towards the centre of the channel for plumes emanating from off-centreline source locations and (iii) the r.m.s. profiles showed no tendency towards double peaks far downstream, as are observed in homogeneous flows. For near-wall source locations, the probability density function (PDF) of the scalar fluctuations evolved from a quasi-Gaussian distribution near the wall to a strongly positively skewed PDF (with a large spike at the cold-fluid temperature) for transverse locations away from the wall. Increasing the Reynolds number was found to improve the mixing, even though this decreases the amount of time for which the scalar can mix (owing to the more rapid advection). For the centreline source location, the PDF shape was, in general, more spiked, indicating the importance of the flapping of the plume in this case. The effect of the meandering of the plume was less significant when the plume was bounded by the wall. Second- and third-order velocity–temperature correlations were presented. The differences in their profiles for the near-wall and centreline source locations were distinct.

A stochastic one-dimensional model for thermal convection is formulated and applied to high-Rayleigh-number convection. Comparisons with experimental data for heat transfer in Rayleigh–Bénard cells are used to estimate two model parameters. Reasonable agreement with experimental results is obtained over a wide range of physical parameter values (six orders of magnitude in Rayleigh number, five orders of magnitude in Prandtl number). Using the model, the statistics of fluctuations in the core of the convection cell are studied. Good agreement with available experimental data is obtained. Two distinct p.d.f. shapes are seen; one at low Prandtl number which matches experimental observations, and another at high Prandtl number for which no experimental data exists. The model results are interpreted in terms of two distinct mechanisms for the production of core fluctuations.

For rapidly flowing granular mixtures, existing kinetic-theory descriptions based on an assumed form of the velocity distribution function typically contain one of two simplifying assumptions: a Maxwellian velocity distribution or an equipartition of energy. In the current work, the influence of non-equipartition effects is explored in the context of two flow types: flow in which species segregation does not occur (namely, simple shear flow) and a segregating flow. For the former case, a comparison between existing kinetic theories and molecular-dynamics simulations of a binary system indicates that the incorporation of a non-Maxwellian velocity distribution is critical for reliable stress predictions, as is consistent with previous findings. However, the predictions are fairly insensitive to the equipartition versus non-equipartition treatment, despite the presence of a significant non-equipartition of energy. Nevertheless, an analysis of the diffusion equation for a segregating flow indicates that the presence of a non-equipartition of energy gives rise to additional components of the driving forces associated with size segregation. These additional components involve gradients of the species temperature, whereas theories based on an equipartition assumption only involve gradients in the mixture temperature. Molecular-dynamics simulations of the segregating flow, in conjunction with kinetic theory of binary systems, show that the non-equipartition effects are non-negligible for systems characterized by moderate values of mass differences and restitution coefficients. These simulations also reveal that the more massive particle may exhibit a lower species temperature than its lighter counterpart, contrary to previous observations in non-segregating systems. A physical explanation for this behaviour is provided.

We report on lattice-Boltzmann simulations of slow fluid flow past mono- and bidisperse random arrays of spheres. We have measured the drag force on the spheres for a range of diameter ratios, mass fractions and packing fractions; in total, we studied 58 different parameter sets. Our simulation data for the permeability agrees well with previous simulation results and the experimental findings. On the basis of our data for the individual drag force, we have formulated new drag force relations for both monodisperse and polydisperse systems, based on the Carman–Kozeny equations; the average deviation of our binary simulation data with the new relation is less than 5%. We expect that these new relations will significantly improve the numerical modelling of gas–solid systems with polydisperse particles, in particular with respect to mixing and segregation phenomena. For binary systems with large diameter ratios (1:4), the individual drag force on a particle, as calculated from our relations, can differ by up to a factor of five compared with predictions presently favoured in chemical engineering.

Collisionless suspensions of inertial particles (finite-size impurities) are studied in two- and three-dimensional spatially smooth flows. Tools borrowed from the study of random dynamical systems are used to identify and to characterize in full generality the mechanisms leading to the formation of strong inhomogeneities in the particle concentration.

Phenomenological arguments are used to show that in two dimensions, the positions of heavy particles form dynamical fractal clusters when their Stokes number (non-dimensional viscous friction time) is below some critical value. Numerical simulations provide strong evidence for the presence of this threshold in both two and three dimensions and for particles not only heavier but also lighter than the carrier fluid. In two dimensions, light particles are found to cluster at discrete (time-dependent) positions and velocities in some range of the dynamical parameters (the Stokes number and the mass density ratio between fluid and particles). This regime is absent in the three-dimensional case for which evidence is that the (Hausdorff) fractal dimension of clusters in phase space (position–velocity) always remains above two.

After relaxation of transients, the phase-space density of particles becomes a singular random measure with non-trivial multiscaling properties, whose exponents cannot be predicted by dimensional analysis. Theoretical results about the projection of fractal sets are used to relate the distribution in phase space to the distribution of the particle positions. Multifractality in phase space implies also multiscaling of the spatial distribution of the mass of particles. Two-dimensional simulations, using simple random flows and heavy particles, allow the accurate determination of the scaling exponents: anomalous deviations from self-similar scaling are already observed for Stokes numbers as small as $10^{-4}$.

We develop a mathematical model for the spreading of a thin volatile liquid droplet on a uniformly heated surface. The model accounts for the effects of surface tension, evaporation, thermocapillarity, gravity and disjoining pressure for both perfectly wetting and partially wetting liquids. Previous studies of non-isothermal spreading did not include the effects of disjoining pressure and therefore had to address the difficult issue of imposing proper boundary conditions at the contact line where the droplet surface touches the heated substrate. We avoid this difficulty by taking advantage of the fact that dry areas on the heated solid surface are typically covered by a microscopic adsorbed film where the disjoining pressure suppresses evaporation. We use a lubrication-type approach to derive a single partial differential equation capable of describing both the time-dependent macroscopic shape of the droplet and the microscopic adsorbed film; the contact line is then defined as the transition region between the two. In the framework of this model we find that both evaporation and thermocapillary stresses act to prevent surface-tension-driven spreading. Apparent contact angle, defined by the maximum interfacial slope in the contact-line region, decays in time as a droplet evaporates, but the rate of decay is different from that predicted in earlier studies of evaporating droplets. We attribute the difference to nonlinear coupling between different physical effects contributing to the value of the contact angle; previous studies used a linear superposition of these effects. We also discuss comparison of our results with experimental data available in the literature.

A forward-facing cavity is examined as a means of reducing the severe heating and delaying ablation onset at the nose-tip of a hypersonic vehicle. Whereas previous studies have concentrated on the nature of flow-induced Hartmann-whistle oscillations or on heating rates alone, the present study addresses the effect of the cavity on ablation onset times through experiments and coupled flow field/heat conduction simulation. Using our previously developed experimental technique, a parametric study is undertaken to optimize the forward-facing cavity geometry for the most delayed ablation onset. The geometric parameters of cavity length, lip radius and diameter are independently optimized for a given nose-tip diameter. Then using benchmarked linked flow field/heat conduction simulations, numerical simulations are conducted for each parametrically optimized configuration in order to investigate the flow physics. The impact of the forward-facing cavity on aerodynamic drag is also considered.

Results from experiments on wave interaction with a rigid plate are reported. The plate is projected from one of the sidewalls of the basin. The sidewall acts as a plane of symmetry, thereby doubling the widths of the plate and of the basin. The tests are carried out in regular waves of varying periods and steepnesses. At wavelengths comparable with the width of the plate, strong run-ups are observed at the plate–wall intersection, increasing with the wave steepness. These run-ups take many wave cycles to develop, with no steady state being reached in some cases. It is advocated that these phenomena result from third-order interactions between the incident and reflected wave fields, over a wide area on the weather side of the plate. A theoretical model is proposed, based on tertiary wave interaction. A parabolic equation is derived that describes the transformation of the incoming waves through their interaction with the reflected wave field. A steady-state solution is obtained through iterations. Results from the theoretical model are compared with the experimental data, with good agreement.

Turbulence quantities have been measured for a low-Reynolds-number fully developed two-dimensional channel flow subjected to system rotation. Turbulence intensities, Reynolds shear stress, correlation coefficient, skewness and flatness factors, four-quadrant analysis, autocorrelation coefficient and power spectra are investigated. According to the dimensional analysis, the relevant parameters of this flow are the Reynolds number $\hbox{\it Re}^{\ast}\,{=}\,u^{*}D/\nu$ and the Coriolis parameter $Rc\,{=}\,\Omega \nu/u^{*2}$ for the wall region, and $Re^*$ and $\Omega D/u^*$ for the turbulent core-region. The existence of a Coriolis region where turbulence intensities are defined by a new variable $y^{*}_{c}\,{=}\,y/\delta _{c}$ has been clarified on the pressure side in the rotating channel flow. The amount of turbulent kinetic energy transported by the Coriolis term is extremely small compared to the production term in the transport equation of Reynolds normal stress. However, the Coriolis term makes a large contribution to Reynolds shear stress transport on the pressure side of the channel. It is caused by the strong ejection which occurs periodically on the pressure side even though the ejection frequency is low. The strong ejection is conjectured to be caused by a large-scale longitudinal structure like a roll cell on the pressure side of the channel.