The stability of a plane liquid sheet is studied experimentally and theoretically, with an emphasis on the effect of the surrounding gas. Co-blowing with a gas velocity of the same order of magnitude as the liquid velocity is studied, in order to quantify its effect on the stability of the sheet. Experimental results are obtained for a water sheet in air at Reynolds number Rel = 3000 and Weber number We = 300, based on the half-thickness of the sheet at the inlet, water mean velocity at the inlet, the surface tension between water and air and water density and viscosity. The sheet is excited with different frequencies at the inlet and the growth of the waves in the streamwise direction is measured. The growth rate curves of the disturbances for all air flow velocities under study are found to be within 20% of the values obtained from a local spatial stability analysis, where water and air viscosities are taken into account, while previous results from literature assuming inviscid air overpredict the most unstable wavelength with a factor 3 and the growth rate with a factor 2. The effect of the air flow on the stability of the sheet is scrutinized numerically and it is concluded that the predicted disturbance growth scales with (i) the absolute velocity difference between water and air (inviscid effect) and (ii) the square root of the shear from air on the water surface (viscous effect).

Two-dimensional (2D) and three-dimensional (3D) instabilities in the wake of a circular cylinder placed near to a moving wall are investigated using direct numerical simulation (DNS). The study covers a parameter space spanning a non-dimensional gap ratio ($G^{\ast }$) between 0.1 to 19.5 and Reynolds number ($Re$) up to 300. Variations in the flow characteristics with $Re$ and $G^{\ast }$ are studied, and their correlations with the hydrodynamic forces on the cylinder are investigated. It is also found that the monotonic increase of the critical $Re$ for 2D instability ($Re_{cr2D}$) with decreasing $G^{\ast }$ is influenced by variations in the mean flow rate around the cylinder, the confinement of the near-wake flow by the plane wall and the characteristics of the shear layer formed above the moving wall directly below the cylinder. The first factor destabilizes the wake flow at a moderate $G^{\ast }$ while the latter two factors stabilize the wake flow with decreasing $G^{\ast }$. In terms of 3D instability, the flow transition sequence of ‘2D steady $\rightarrow$ 3D steady $\rightarrow$ 3D unsteady’ for small gap ratios is analysed at $G^{\ast }=0.2$. It is found that the 3D steady and 3D unsteady flows are triggered by Mode C instability due to wall proximity. However, the Mode C structure is not sustained indefinitely, since interference with the shear layer leads to other 3D steady and unsteady flow structures.

Direct numerical simulations (DNS) of flow in a turbine cascade with heat transfer have been performed. The set-up of the simulations was chosen in close accordance with previous experiments. Three of the experimental situations were simulated: one without free-stream turbulence and two with periodically incoming wakes of different frequency and with different levels of background fluctuation. Hence, the calculations allow us to study the influence of impinging wakes and background fluctuations on the development of the boundary layers and the local Nusselt number along the surfaces of the heated blade. Along the suction side, the pressure gradient is first favourable and then turns adverse near the trailing edge and the boundary layer remains laminar for the case without free-stream turbulence with the Nusselt number showing the typical decay from the leading to the trailing edge. With periodic wakes and background turbulence, transition occurs when the pressure gradient turns adverse, but intermittency persists so that the boundary layer is not fully turbulent when the trailing edge is reached. In this region, the heat transfer is increased significantly by an amount comparable to that found in the experiments. In the pre-transitional region with favourable pressure gradient, the flow acceleration stretches the free-stream vortices, aligning their axis with the flow direction, thereby forming streamwise vortical structures. These increase the laminar heat transfer in this region by 20–30%, which is, however, much less than observed in the experiments. On the pressure side, the pressure gradient is favourable along the entire blade so that the boundary layer remains laminar. Here, the wakes, through their impingement, also generate streamwise vortical structures which, because of the low convection speed on this side, have a very long lifetime compared to the structures along the suction side. Also these structures increase the laminar heat transfer by about 30%, which for the case with the highest wake frequency is again much less than in the experiments. The simulated average level of fluctuations in the laminar parts of the boundary layers is comparable or even higher than that in the experiments so that it seems likely that a difference in the spectral contents causes the discrepancies. The wake turbulence entering the calculation domain corresponds to that in far wakes with relatively small-scale structures, whereas in the experiments the wakes most probably still carried some large-scale fluctuations of the size of the wake width, which have been found to be more effective in increasing laminar heat transfer.

Modal decomposition techniques are used to analyse the wake field past a marine propeller achieved by previous numerical simulations (Muscari et al. Comput. Fluids, vol. 73, 2013, pp. 65–79). In particular, proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) are used to identify the most energetic modes and those that play a dominant role in the inception of the destabilization mechanisms. Two different operating conditions, representative of light and high loading conditions, are considered. The analysis shows a strong dependence of temporal and spatial scales of the process on the propeller loading and correlates the spatial shape of the modes and the temporal scales with the evolution and destabilization mechanisms of the wake past the propeller. At light loading condition, due to the stable evolution of the wake, both POD and DMD describe the flow field by the non-interacting evolution of the tip and hub vortex. The flow is mainly associated with the ordered convection of the tip vortex and the corresponding dominant modes, identified by both decompositions, are characterized by spatial wavelengths and frequencies related to the blade passing frequency and its multiples, whereas the dynamic of the hub vortex has a negligible contribution. At high loading condition, POD and DMD identify a marked separation of the flow field close to the propeller and in the far field, as a consequence of wake breakdown. The tonal modes are prevalent only near to the propeller, where the flow is stable; on the contrary, in the transition region a number of spatial and temporal scales appear. In particular, the phenomenon of destabilization of the wake, originated by the coupling of consecutive tip vortices, and the mechanisms of hub–tip vortex interaction and wake meandering are identified by both POD and DMD.

Studies of the morphology of fragmentation of liquid drops in a liquid medium due to shock-induced flows are reported. For such high-density-ratio and high-interfacialtension systems, the fragmentation is found to be primarily due to the penetration of the drop by unstable waves. An envelope of conditions which characterize drop breakup in these systems has been deduced. Analytical models, which involve the time constants of both the drop-piercing Taylor instability and the wake-forming shear effects, reflect our experimental results well. In addition, it is found that the mercury/water system studied here should simulate the uranium dioxide/sodium pair, which is of interest in safety studies of fast reactors.

This work reports an experimental investigation of the dispersion of a low-diffusive dye within a homogeneous swarm of high-Reynolds-number rising bubbles at gas volume fractions ${\it\alpha}$ ranging from 1 % to 13 %. The capture and transport of dye within bubble wakes is found to be negligible and the mixing turns out to result from the bubble-induced turbulence. It is described well by a regular diffusion process. The diffusion coefficient corresponding to the vertical direction is larger than that corresponding to the horizontal direction, owing to the larger intensity of the liquid fluctuations in the vertical direction. Two regimes of diffusion have been identified. At low gas volume fraction, the diffusion time scale is given by the correlation time of the bubble-induced turbulence and the diffusion coefficients increase roughly as ${\it\alpha}^{0.4}$. At large gas volume fraction, the diffusion time scale is imposed by the time interval between two bubbles and the diffusion coefficients become almost independent of ${\it\alpha}$. The transition between the two regimes occurs sooner in the horizontal direction ($1\,\%\leqslant {\it\alpha}\leqslant 3\,\%$) than in the vertical direction ($3\,\%\leqslant {\it\alpha}\leqslant 6\,\%$). Physical models based on the hydrodynamic properties of the bubble swarm are introduced and guidelines for practical applications are suggested.

Slide coating is a means of rapidly depositing multilayered liquid films of precise thickness and uniformity, as in manufacture of photographic products. Liquid is metered through one or more slots onto the inclined surface of the coating die, flows down that face and across a gap onto a fast-moving smooth surface. In this paper the steady, two-dimensional slide coating flow of a Newtonian liquid is analysed by solving the full Navier–Stokes system with the Galerkin/finite-element technique, spine parametrization of free surfaces and full Newton iteration. The lower meniscus in the gap is assumed to remain pinned at the die edge and the wetting-line singularity on the surface being coated is relieved by introducing dynamic-slip and contact-angle parameters. Results include the effects of several design and operating parameters on free-surface profiles and details of the flow field; these are presented by means of contours of kinematic and dynamic variables and local force balances over subdomains. The profiles show standing waves on the slide, rapid film thinning just before the gap, and exponential approach to the final film thickness on the web. As Reynolds number is raised and/or web speed is lowered several recirculation regions are predicted, deleterious features that have also been detected in experiments.

We have carried out three-dimensional Navier–Stokes simulations, from quiescent conditions to the limit cycle, of a theoretical travelling-wave thermoacoustic heat engine (TAE) composed of a long variable-area resonator shrouding a smaller annular tube, which encloses the hot (HHX) and ambient (AHX) heat exchangers, and the regenerator (REG). Simulations are wall-resolved, with no-slip and adiabatic conditions enforced at all boundaries, while the heat transfer and drag due to the REG and HXs are modelled. HHX temperatures have been investigated in the range 440–500 K with the AHX temperature fixed at 300 K. The initial exponential growth of acoustic energy is due to a network of travelling waves thermoacoustically amplified by looping around the REG/HX unit in the direction of the imposed temperature gradient. A simple analytical model demonstrates that such instability is a localized Lagrangian thermodynamic process resembling a Stirling cycle. An inviscid system-wide linear stability model based on Rott’s theory is able to accurately predict the operating frequency and the growth rate, exhibiting properties consistent with a supercritical Hopf bifurcation. The limit cycle is governed by acoustic streaming – a rectified steady flow resulting from high-amplitude nonlinear acoustics. Its key features are explained with an axially symmetric incompressible model driven by the wave-induced stresses extracted from the compressible calculations. These features include Gedeon streaming, Rayleigh streaming in the resonator, and mean recirculations due to flow separation. The first drives the mean advection of hot fluid from the HHX to a secondary heat exchanger (AHX2), in the thermal buffer tube (TBT), necessary to achieve saturation of the acoustic energy growth. The direct evaluation of the nonlinear energy fluxes reveals that the efficiency of the device deteriorates with the drive ratio and that the acoustic power in the TBT is balanced primarily by the mean advection and thermoacoustic heat transport.

We present numerical simulations of randomly forced internal gravity waves in a uniformly stratified Boussinesq fluid, and compare the resulting vertical wavenumber energy spectra with the saturation spectrum $E_z(k_z)\,{=}\,c\,N^2k_z^{-3}$ ($N$ is the Brunt–Väisälä frequency) observed in the atmosphere and ocean. Overall, we have been unsuccessful at reproducing the observed spectrum in our simulations. Our spectra are shallower than $k_z^{-3}$, although they steepen towards it with increasing stratification as long as wave breaking (in the form of static instability) is resolved. The spectral amplitude increases like $N^{1.1}$ rather than $N^2$. For a single stratification, our spectrum agrees well with the saturation spectrum with $c\,{=}\,0.1$, but only because it is spuriously steepened by insufficient resolution. We show that overturning occurs when the length scale $l_c\,{=}\,u_{rms}/N$ is larger than the dissipation scale, where $u_{rms}$ is the root mean square velocity. This scale must be at least three times larger than the dissipation scale for the energy spectrum to be independent of Reynolds number in our simulations. When this condition is not satisfied, the computed energy spectrum must be interpreted with caution. Finally, we show that for strong stratifications, the presence of vortical energy can have a dramatic effect on the spectrum of wave energy due to the efficiency of interactions between two waves and a vortical mode. Any explanation of the energy spectrum involving resonant interactions must take into account the effect of vortical motion.

The weakly nonlinear, two-dimensional problem for the disturbance due to a slender obstacle in a uniformly stratified, Boussinesq fluid moving past the obstacle with constant basic horizontal velocity U, is considered up to second order in the amplitude ε of the disturbance. Analogous rotating problems are also treated. Particular attention is given to calculating explicitly the columnar-disturbance strengths upstream and downstream of the obstacle, both in the stratified and in the rotating problems, with a view to discussing the truth or otherwise of Long's hypothesis (LH).

Whether or not columnar disturbances are found far upstream, violating LH, depends, interalia, on whether or not the flow is externally bounded by rigid horizontal planes (or by a tube or annulus, in the rotating problem), and on whether the problem is made determinate by means of an ‘inviscid transient’ formulation, or by means of a ‘viscous’ one.

The inviscid, transient, bounded problem, for time-development of lee waves from a state of no initial disturbance, always exhibits columnar disturbances oforder ε2 somewhere in the fluid. They are generated, not near the obstacle, but in the ‘tails’ or transient terminal zones of the lee-wave trains. The columnar-disturbance strengths are largely independent of how the flow is set up from an initially undisturbed state. I n all but one instance the effect is non-zero far up-stream. The exception is the singly-subcritical stratified (or narrow-gap rotating) case, in which the excitation has modal structure sin(2z), the fluid region being 0 [les ] z [les ] π in this case the only columnar disturbance that can penetrate up-stream has structure sinz and so is not excited.

A completely different result holds for ‘viscous’ formulations for unseparated, bounded régimes (with steady lee waves spatially attenuated by effects of small molecular diffusion). The strengths of all columnar disturbances, upstream and downstream, vanish in the limit of small diffusivity.

In the inviscid, transient, unbounded problem, the upstream influence is, likewise, evanescent, being O(ε2t−2) as time t → ∞.

The basic expansion in powers of ε will be invalid for times ∝ ε−1 or greater, because of resonant-interactive instability of the lee waves.

An experimental investigation of the explosions of 2 in. diameter glass spheres under high internal pressure has been made. The spheres were initially filled with air or helium at 400 and 326 p.s.i., respectively, and were exploded in air at atmospheric pressure. Experiments on the simulation of high-altitude explosions are also described. Schlieren and spark shadowgraph records of explosion phenomena, and pressure records of the reflexion of the spherical shock wave at various radii, are presented.

An account of some initial experiments on the implosion of 5 in. diameter glass spheres is given. The results were not very satisfactory because of the failure of the spheres to shatter in a desirable manner while under an external pressure of 65 p.s.i.

Numerical solutions to the air and helium sphere explosions are described and the experimental wave phenomena are shown to be in good quantitative agreement with the theoretical predictions, in that they exhibit all the main features that were predicted and are modified only by the physical limitations of the glass diaphragm. A formation process is associated with the spherical shock waves in practice, resulting in initial shock velocities which are lower than the theoretical values.

The influence of turbulence on the orientation state of a dilute density matched suspension of stiff fibres at high Reynolds number in a planar contraction is investigated. High-speed imaging and laser-Doppler velocimetry techniques are used to quantify fibre orientation distribution and turbulent characteristics. A nearly homogeneous isotropic grid-generated turbulent flow is introduced at the contraction inlet. Flow Reynolds number and inlet turbulent characteristics are varied in order to determine their effects on orientation distribution. The orientation anisotropy is shown to be accurately modelled by a Fokker-Planck type equation. Results show that rotational diffusion is highly influenced by inlet turbulent characteristics and decays exponentially with convergence ratio. Furthermore, the effect of turbulent energy production in the contraction is shown to be negligible. Also, the results show that the flow Reynolds number has negligible effect on the development of orientation anisotropy, and the influence of turbulence on fibre rotation is negligible for rotational Péclet number $\gt$10.

Solidification of a double-diffusive liquid from a vertical wall in a rectangular cavity is investigated using both mathematical and physical modelling. A two-dimensional mathematical model is developed to simulate the process of solidification in a two-component liquid containing a denser solute. Observations on the solidification behaviour of aqueous sodium carbonate solutions are used to verify the results of the mathematical analysis. The theory and experiments provide a clear picture of the role of double-diffusion in producing vertical compositional and density stratification in an initially homogeneous liquid during solidification. The development of horizontally oriented convection cells in the stratified liquid is correlated with the magnitude of the destabilizing lateral temperature difference across the liquid region.

According to the celebrated Bolgiano–Obukhov (Bolgiano, J. Geophys. Res., vol. 64 (12), 1959, pp. 2226–2229; Obukhov, Dokl. Akad. Nauk SSSR, vol. 125, 1959, p. 1246) phenomenology for moderately stably stratified turbulence, the energy spectrum in the inertial range shows a dual scaling: the kinetic energy follows (i) ${\sim}k^{-11/5}$ for $k<k_{B}$, and (ii) ${\sim}k^{-5/3}$ for $k>k_{B}$, where $k_{B}$ is the Bolgiano wavenumber. The $k^{-5/3}$ scaling, akin to passive scalar turbulence, is a direct consequence of the assumption that buoyancy is insignificant for $k>k_{B}$. We revisit this assumption, and using the constancy of kinetic and potential energy fluxes and simple theoretical analysis, we find that the $k^{-5/3}$ spectrum is absent. This is because the velocity field at small scales is too weak to establish a constant kinetic energy flux as in passive scalar turbulence. A quantitative condition for the existence of the second regime is also derived in the paper.

We compare the results of two-dimensional, biased random walk models of individual swimming micro-organisms with advection–diffusion models for the whole population. In particular, we consider the influence of the local flow environment (gyrotaxis) on the resulting motion. In unidirectional flows, the results of the individual and population models are generally in good agreement, even in flows in which the cells can experience a range of shear environments, and both models successfully predict the phenomena of gravitactic focusing. Numerical results are also compared with asymptotic expressions for weak and strong shear. Discrepancies between the models arise in two cases: (i) when reflective boundary conditions change the orientation distribution in the random walk model from that predicted by the long-term asymptotics used to derive the advection–diffusion model; (ii) when the spatial and temporal scales are not large enough for the advection–diffusion model to apply. We also use a simple two-dimensional flow containing a variety of flow regimes to explore what happens when there are localized regions in which the generalized Taylor dispersion theory used in the derivation of the population model does not apply. For spherical cells, we find good agreement between the models outside the ‘break-down’ regions, but comparison of the results within these regions is complicated by the presence of nearby boundaries and their influence on the random walk model. In contrast, for rod-shaped cells which are reorientated by both vorticity and strain, we see qualitatively different spatial patterns between individual and advection–diffusion models even in the absence of gyrotaxis, because cells are advected between regions of differing rates of strain.

A theory is developed for the hydrodynamic interactions of surfactant-covered spherical drops in creeping flows. The surfactant is insoluble, and flow-induced changes of surfactant concentration are small, i.e. the film of adsorbed surfactant is incompressible.

For a single surfactant-covered drop in an arbitrary incident flow, the Stokes equations are solved using a decomposition of the flow into surface-solenoidal and surface-irrotational components on concentric spherical surfaces. The surface-solenoidal component is unaffected by surfactant; the surface-irrotational component satisfies a slip-stick boundary condition with slip proportional to the surfactant diffusivity. Pair hydrodynamic interactions of surfactant-covered bubbles are computed from the one-particle solution using a multiple-scattering expansion. Two terms in a lubrication expansion are derived for axisymmetric near-contact motion.

The pair mobility functions are used to compute collision efficiencies for equal-size surfactant-covered bubbles in linear flows and in Brownian motion. An asymptotic analysis is presented for weak surfactant diffusion and weak van der Waals attraction. In the absence of surfactant diffusion, collision efficiencies for surfactant-covered bubbles are higher than for rigid spheres in straining flow and lower in shear flow. In shear flow, the collision efficiency vanishes for surfactant diffusivities below a critical value if van der Waals attraction is absent.

Penetrative convection in a two-layer system in which a layer of fluid overlies and saturates a porous medium is simulated via internal heating. The motion in the porous medium is described via Darcy's law and in the fluid layer by the Navier–Stokes equations with a Boussinesq approximation. The lower porous surface is held fixed at a temperature $T_L$, while the upper fluid surface is stress free and held at $T_U\,{>}\,T_L$. Internal heating takes place in both layers and allows the model to describe penetrative convection. The strength of heating has a dramatic effect on both the onset of convection and the nature of the ensuing convection cells. It is found that a heat source/sink $Q$ in the fluid layer has a destabilizing effect on the porous layer whereas one in the porous medium $Q_m$ has a stabilizing influence on the fluid. The effect of $Q$ and $Q_m$ on their respective layers, however, depends strongly upon the temperature difference $T_U\,{-}\,T_L$, and the strength and type of heating in the opposite layer. When $Q$ and $Q_m$ are varied, a range of streamlines are presented that exhibit novel behaviour. The model is compared with an alternative in which the density is assumed to have quadratic temperature dependence and there is no internal heating. When the two models are mathematically adjoint they are shown to yield the same critical instability threshold but different eigenfunctions. It is also shown that the initiating cell is not necessarily the strongest one. This curious behaviour is explained and illustrated with a range of streamlines for variable permeability.

Optical microscopy and multi-particle tracking are used to study hydrodynamic interactions of monodisperse polymethylmethacrylate (PMMA) spheres at a decalin–water interface. The short-time self-diffusion coefficient measured at low surface coverage has the form DSS(n) = αD0(1 − βn), where n is the area fraction occupied by the particles, and D0 is the Stokes–Einstein diffusion coefficient in the bulk suspension of PMMA spheres in decalin. The measured values of α are found to be in good agreement with the numerical calculation for the drag coefficient of interfacial particles. The measured values of β differ from that obtained for bulk suspensions, indicating that hydrodynamic interactions between the particles have interesting new features at the interface.

A hot film flowmeter, a thin film resistance thermometer and several thermistors were mounted on a submarine and used to study turbulence and temperature microstructure in the open sea. The surface layer is found to be continuously turbulent and patches of turbulence have been found in and below the main ocean thermocline.