In this paper we propose a simple model that, by comparing different time scales, allows a prediction for the mean flow structure and its dynamics in confined thermal convection in a cylindrical cell of aspect ratio (diameter over cell height) $\Gamma\,{=}\,1/2$. It is shown that the break-up of the mean elongated recirculation into two counter-rotating unity-aspect-ratio rolls, sometimes referred to as flow bimodality, occurs only in a narrow range of Rayleigh numbers whose extrema depend on the Prandtl number. The predictions of the present model are consistent with the published literature, according to which the dual mean flow structure has been observed in numerical simulations at $\hbox{\it Pr}\,{=}\,0.7$ and experiments in gaseous helium ($\hbox{\it Pr}\,{\approx}\,0.7$) but never in water at ‘ambient’ temperature ($\hbox{\it Pr} \,{\approx}\,5$) and only once in water at $T\,{=}\,80\,^\circ$C ($\hbox{\it Pr}\,{=}\,2$). Another prediction of the model is that the thermal properties of the sidewall affect the mean flow unsteadiness and, sometimes, prevent transitions via a subtle anchoring mechanism that has been identified and verified by ad hoc numerical simulations.

Flow structure, momentum and heat transport in the wake of two tandem circular cylinders have been experimentally investigated. Measurements were conducted at $x/d\,{=}\, $10, 20 and 30 ($d$ is the cylinder diameter) at a Reynolds number of 7000 using a three-wire (one cross-wire plus a cold wire) probe, in conjunction with a cross-wire. The upstream cylinder was slightly heated. The flow behind two tandem cylinders is conventionally divided into three regimes based on whether the shear layers separated from the upstream cylinder overshoot or reattach on the downstream cylinder before forming a vortex street, or form vortices between the cylinders. The present investigation uncovers two remarkably different flow structures in the reattachment regime, depending on whether the shear layers from the upstream cylinder reattach on the downstream or upstream side of the downstream cylinder. As such, four cylinder centre-to-centre spacing ratios, i.e. $L/d\,{=}\,$1.3, 2.5, 4.0 and 6.0, were examined, each representing one distinct flow structure. The phase-averaged sectional streamlines and vorticity contours display a single vortex street, irrespective of different regimes. However, the detailed flow structure, in particular, the vortex strength, and its downstream development depend upon $L/d$. The cross-stream distributions of the Reynolds stresses and heat fluxes at a given $x/d$ vary from one to another. Such variation is also evident in the coherent contributions to the Reynolds stresses and heat fluxes. The results are connected to different initial conditions for the four flow structures. The momentum and heat transport characteristics are summarized for each flow structure.

This paper presents a generalization of the classical results of T. B. Benjamin (J. Fluid. Mech. vol. 31, 1968, p. 209) concerning the propagation of a steady gravity current into a homogeneous ambient, to the case of a stratified ambient. The current of thickness $h$ and density $\roh_c$ propagates, with speed $U$, at the bottom of a long horizontal channel of height $H$, into the unperturbed ambient whose density increases linearly from $\roh_o$ (at the top) to $\roh_b$ (at the bottom). The reduced gravity is $g^{\prime} \,{=}\, (\roh_c/\roh_o -1)g$ and the governing parameters are $a \,{=}\, h/H$ and $S \,{=}\, (\roh_b-\roh_o)/(\roh_c-\roh_o)$, with $0<a<1, 0 < S < 1 $; here $g$ is the acceleration due to gravity. For a Boussinesq high-Reynolds two-dimensional configuration, a flow-field solution of Long's model, combined with flow-force balance over the width of the channel, are used for obtaining the desired results, in particular: $\hbox{\it Fr} \,{=}\, U/(g^{\prime}h)^{1/2}$, head loss (dissipation), and criticality of $U$ with respect to the fastest internal wave mode. The classical results of Benjamin are fully recovered for $S \rightarrow 0$. For small $S$ and fixed $a$, the values of $\hbox{\it Fr}$ and head loss are shown to decrease with $S$ like $(1-2S/3)^{1/2}$ and $(1-2S/3)$, respectively, and the propagation is supercritical. For larger $S$ several solutions are possible (for a given geometry $a$), mostly in the subcritical regime. Considerations for the physical acceptability of the multiple results are presented, and the connection with observations from lock-release experiments are discussed. The conclusion is that the present results provide a reliable and versatile generalization of the classical unstratified problem.

The breakup process of a drop or a bubble immersed in a straining flow at high Reynolds numbers, is studied numerically with the aim at comparing the breakup frequencies obtained with those measured in real flows. We assume that both the inner and the outer velocity fields are axisymmetric and irrotational. Under these assumptions the time evolution of the drop's interface is computed with a boundary integral method for a wide range of the inner-to-outer density ratios, $\Lambda$. Despite the simplicity of the model, it qualitatively displays some of the features of the turbulent breakup of drops and bubbles observed experimentally. Furthermore, when $\Lambda \sim O(1)$, the slender geometry of the droplets observed in the numerical simulations suggests the use of a simplified theoretical analysis that reproduces accurately the time evolution of the drop radius obtained numerically.

We present a linear stability analysis of density-driven miscible flow in porous media in the context of carbon dioxide sequestration in saline aquifers. Carbon dioxide dissolution into the underlying brine leads to a local density increase that results in a gravitational instability. The physical phenomenon is analogous to the thermal convective instability in a semi-infinite domain, owing to a step change in temperature at the boundary. The critical time for the onset of convection in such problems has not been determined accurately by previous studies. We present a solution, based on the dominant mode of the self-similar diffusion operator, which can accurately predict the critical time and the associated unstable wavenumber. This approach is used to explain the instability mechanisms of the critical time and the long-wave cutoff in a semi-infinite domain. The dominant mode solution, however, is valid only for a small parameter range. We extend the analysis by employing the quasi-steady-state approximation (QSSA) which provides accurate solutions in the self-similar coordinate system. For large times, both the maximum growth rate and the most dangerous mode decay as $t^{1/4}$. The long-wave and the short-wave cutoff modes scale as $t^{1/5}$ and $t^{4/5}$, respectively. The instability problem is also analysed in the nonlinear regime by high-accuracy direct numerical simulations. The nonlinear simulations at short times show good agreement with the linear stability predictions. At later times, macroscopic fingers display intense nonlinear interactions that significantly influence both the front propagation speed and the overall mixing rate. A dimensional analysis for typical aquifers shows that for a permeability variation of 1—3000 mD, the critical time can vary from 2000 yrs to about 10 days while the critical wavelength can be between 200 m and 0.3 m.

Consecutive images of the fragmentation of capillary waves in an ultrasonic field were obtained using a high-speed video camera through a microscope at a frame rate of 500000 frames per second. The images showed that micro bubbles of uniform diameter from 4 to 15$\,\mu$m were generated at a constant periodic rate when a small amount of gas was introduced (via a needle) into a highly viscous liquid whose kinematic viscosity was between 5 and 100 mm$^{2}\,{\rm s}^{-1}$. Conditions for stable generation of micro bubbles of uniform diameter were also studied by changing the inner diameter of the needle between 0.08 and 0.34 mm, excitation frequency of around 18.77 and 42.15 kHz, kinematic viscosity of liquid between 5 and 100 mm$^{2}\,{\rm s}^{-1}$, surface tension between 20 and 34 mN m$^{-1}$, and viscosity of gas between 9.0 and 31.7$\,\mu$Pa s. Results revealed that (i) a projection is formed on the oscillatory gas–liquid interface and micro bubbles are released from the tip of the projection; (ii) gas viscosity critically affects the formation of the projection and should be around 20.0$\,\mu $Pa s for stable mother bubble oscillation; (iii) conditions for stable generation of micro bubbles are also affected by excitation frequency, surface tension and viscosity of the liquid, and dimensions of the needle; (iv) two controlling parameters for stable generation are the Weber number (${\it We}\,{=}\,\rho {f}^{2}d_{\hbox{\scriptsize{\it in}}}^{3}/\sigma $, where $\rho $ is the density of the liquid, $f$ is the excitation frequency, $d_{\hbox{\scriptsize{\it in}}}$ is the inner diameter of the needle, and $\sigma $ is the surface tension) and the Womersley number (${\it Wo}\,{=}\,d_{\hbox{\scriptsize{\it in}}}(f/{\nu })^{1 / 2}$, where $\nu $ is the kinematic viscosity of liquid); and (v) uniform-diameter micro bubbles are generated stably when ${\it We}<300$ and $2<$${\it Wo}<5$. Under the conditions where micro bubbles of uniform diameter were stably generated, the bubble diameter increased almost linearly with increasing gas pressure inside the needle. The gradient of this linear function can be expressed as a function of WoWe, and the normalized outer diameter of the needle, and decreases either with decreasing inner diameter of the needle or with increasing excitation frequency, surface tension and viscosity of the liquid, and outer diameter of the needle.

The rise speed of supercritical carbon dioxide spheres in aqueous surfactant solution was investigated experimentally. Decanonic acid was added to water as a surfactant, and then the rise speed of the CO$_{2}$ spheres was measured at pressures 5 MPa to 10 MPa and at temperatures 28 $^{\circ}$C to 40 $^{\circ}$C generating different phases of spheres, namely gas, liquid, and supercritical. The results revealed that gas bubbles and liquid droplets showed the same behaviour as a rising rigid sphere due to the adsorption of surfactant onto the surface of the spheres. In contrast, due to the absorption of surfactant into the spheres, the drag coefficient of spheres of supercritical fluid was lower than that of a rigid sphere and was similar to that of a fluid sphere with a moving boundary. This behaviour mainly occurred in the rising process of the spheres composed of supercritical CO$_{2}$.

When a paintbrush is dipped into a pot of paint and pulled out, surface tension forces cause the individual hairs in the brush to coalesce even as the brush becomes impregnated with paint. We study a simple model of this elastocapillary interaction in the context of the surface-tension-driven vertical rise of a liquid between two long flexible hydrophilic sheets that are held a small distance apart at one end. We provide an analytic theory for the static shapes of the sheets as well as the liquid rise height which is different from that of the classical law of Jurin, and show that our experiments are quantitatively consistent with the theory.

For free-surface water flows with a vorticity that is monotone with depth, we show that any critical point of a functional representing the total energy of the flow adjusted with a measure of the vorticity, subject to the constraints of fixed mass and horizontal momentum, is a steady water wave.

We present a detailed investigation of the parametric subharmonic resonance mechanism that leads a plane, monochromatic, small-amplitude internal gravity wave, also referred to as the primary wave, to instability. Resonant wave interaction theory is used to derive a simple kinematic model for the parametrically forced perturbation, and direct numerical simulations of the Boussinesq equations in a vertical plane permit the nonlinear simulation of the internal gravity wave field. The processes that eventually drive the wave field to breaking are also addressed.

We show that parametric instability may be viewed as an optimized scenario for drawing energy from the primary wave, that is, from a periodic flow with both oscillating shear and density gradient. Optimal energy exchange maximizing perturbation growth is realized when the perturbation has a definite spatio-temporal structure: its energy is phase-locked with the vorticity of the primary wave. This organization allows the perturbation energy to alternate between kinetic form when locally the primary wave shear is negative, then maximizing kinetic energy extraction from the primary wave, and potential form when the primary wave shear is positive, then minimizing the reverse transfer to that wave. The perturbation potential energy increases through the primary wave density gradient whether the latter is positive, that is when the medium is of reduced static stability, or negative (increased static stability). When the primary wave amplitude is small, all energy transfer terms are predicted well by the kinematic model. One important result is that the rate of potential energy transfer from the primary wave to the perturbation is always larger than the rate of kinetic energy transfer, whatever the primary wave.

As the perturbation amplifies, overturned isopycnals first appear in reduced static stability regions, implying that the total field should become unstable through a buoyancy induced (or Rayleigh–Taylor) instability. Hence, a two-dimensional model is no longer valid for studying the subsequent flow development.

The hypothesis that turbulence evolution can be described by kinetic energy, dissipation, and the Reynolds stresses alone is the basis of current second moment closures. This hypothesis is shown to be inconsistent with the Navier–Stokes equations at the two-component limit, because it implies that initially two-component turbulence remains two-component for all times. The inconsistency can be traced to the Markovian stress evolution imposed by the hypothesis; even Markovian spectral closures cannot predict evolution from a two-component initial state. Some implications of these findings for imposing realizability constraints on turbulence models are discussed.

The effect of an insoluble surfactant on the stability of the core–annular flow of two immiscible fluids is investigated by a normal-mode linear analysis and by numerical simulations based on the immersed-interface method for axisymmetric perturbations. The results reveal that, although the Marangoni stress due to surfactant concentration variations is unable to initiate a new type of instability as in the case of two-dimensional two-layer channel flow, it does destabilize the interface by broadening the range of growing wavenumbers and by raising the growth rate of unstable perturbations. Numerical simulations for large-amplitude disturbances reveal that the surfactant plays an important role in determining the morphology of the interfacial structures developing in the nonlinear stages of the motion.

An algebraically exact solution to the problem of linear water-wave scattering by a periodic array of scatterers is presented in which the scatterers may be of arbitrary shape. The method of solution is based on an interaction theory in which the incident wave on each body from all the other bodies in the array is expressed in the respective local cylindrical eigenfunction expansion. We show how to calculate the slowly convergent terms efficiently which arise in the formulation and how to calculate the scattered field far from the array. The application to the problem of linear acoustic scattering by cylinders with arbitrary cross-section is also discussed. Numerical calculations are presented to show that our results agree with previous calculations. We present some computations for the case of fixed, rigid and elastic floating bodies of negligible draft concentrating on presenting the amplitudes of the scattered waves as functions of the incident angle.

We present a mesoscopic model of the fluid–wall interactions for flows in microchannel geometries. We define a suitable implementation of the boundary conditions for a discrete version of the Boltzmann equations describing a wall-bounded single-phase fluid. We distinguish different slippage properties on the surface by introducing a slip function, defining the local degree of slip for hydrodynamical fields at the boundaries. The slip function plays the role of a renormalizing factor which incorporates, with some degree of arbitrariness, the microscopic effects on the mesoscopic description. We discuss the mesoscopic slip properties in terms of slip length, slip velocity, pressure drop reduction (drag reduction), and mass flow rate in microchannels as a function of the degree of slippage and of its spatial distribution and localization, the latter parameter mimicking the degree of roughness of the ultra-hydrophobic material in real experiments. We also discuss the increment of the slip length in the transition regime, i.e. at ${O}(1)$ Knudsen numbers.

Finally, we compare our results with molecular dynamics investigations of the dependence of the slip length on the mean channel pressure and local slip properties and with the experimental dependence of the pressure drop reduction on the percentage of hydrophobic material deposited on the surface.

An experimental study on the interaction of the internal wave field generated by oscillating cylinders in a stratified fluid with a pancake dipole is presented. The experiments are carried out in a salt-stratified water tank with constant Brunt–Väisälä frequency ($N$). Experimental observations of the deformation of the wave beams owing to the interaction with the dipole are presented. When the wave and the dipole propagate horizontally in opposite directions (counterpropagating case), the phase line of the gravity wave beam steepens towards the vertical as it enters the dipolar field and it may even reach a turning point where the wave is reflected. When the dipole and the wave propagate in the same direction (copropagating case), the wave beam is bent towards the horizontal and may be absorbed by the dipole. These observations are in good agreement with a two-dimensional ray-theoretic model, even if the flow is fully three-dimensional and, the vertical shear induced by the dipole being too strong, the hypothesis of slow variation assumed in the WKB approximation is not verified. When the waves encounter a critical layer, we show by rigorous measurement that momentum is transferred to the dipole. New three-dimensional effects of the dipolar velocity field on the propagating internal waves are also discussed. In particular, focusing and refraction of a wave beam occurring because of the horizontal structure of the background dipolar flow allow us to explain some of the observed features that cannot be accounted for through the two-dimensional ray theory.

The bottom friction and the limited vertical extent of the water depth play a significant role in the dynamics of shallow wakes. These effects along with the effect of the strength of the shear layer define the wake parameter $S$. A nonlinear model, based on a second-order explicit finite volume solution of the depth-averaged shallow water equation in which the fluxes are obtained from the solution of the Bhatnagar–Gross–Krook (BGK) Boltzmann equation, is developed and applied to shallow wake flows for which laboratory data are available. The velocity profiles, size of the recirculating wake, oscillation frequency, and wake centreline velocity are studied. The computed and measured results are in reasonable agreement for the vortex street (VS) and unsteady bubble (UB) regimes, but not for the steady bubble (SB). The computed length of the recirculation region is about 60% shorter than the measured value when $S$ belongs to the SB regime. As a result, the stability investigation performed in this paper is restricted to $S$ values away from the transition between SB and UB. Linear analysis of the VS time-averaged velocity profiles reveals a region of absolute instability in the vicinity of the cylinder associated with large velocity deficit, followed by a region of convective instability, which is in turn followed by a stable region. The frequency obtained from Koch's criterion is in good agreement with the shedding frequency of the fully developed VS. However, this analysis does not reveal the mechanism that sets the global shedding frequency of the VS regime because the basic state is obtained from the VS regime itself. The mechanism responsible for VS shedding is sought by investigating the stability behaviour of velocity profiles in the UB regime as $S$ is decreased towards the critical value which defines the transition from the UB to the VS. The results show that the near wake consists of a region of absolute instability sandwiched between two convectively unstable regions. The frequency of the VS appears to be predicted well by the selection criteria given in Pier & Huerre (2001) and Pier (2002), suggesting that the ‘wave-maker’ mechanism proposed in Pier & Huerre (2001) in the context of deep wakes remains valid for shallow wakes. The amplitude spectra produced by the nonlinear model are characterized by a narrow band of large-amplitude frequencies and a wide band of small-amplitude frequencies. Weakly nonlinear analysis indicates that the small amplitude frequencies are due to secondary instabilities. Both the UB and VS regimes are found to be insensitive to random forcing at the inflow boundary. The insensitivity to random noise is consistent with the linear results which show that the UB and VS flows contain regions of absolute instabilities in the near wake where the velocity deficit is large.

While non-Boussinesq hexagonal convection patterns are known to be stable close to threshold (i.e. for Rayleigh numbers $R \,{\approx}\, R_c$), it has often been assumed that they are always unstable to rolls for slightly higher Rayleigh numbers. Using the incompressible Navier–Stokes equations for parameters corresponding to water as the working fluid, we perform full numerical stability analyses of hexagons in the strongly nonlinear regime ($\epsilon\,{\equiv}\, (R-R_c)/R_c\,{=}\,{O}(1)$). We find ‘re-entrant’ behaviour of the hexagons, i.e. as $\epsilon$ is increased they can lose and regain stability. This can occur for values of $\epsilon$ as low as $\epsilon\,{=}\,0.2$. We identify two factors contributing to the re-entrance: (i) far above threshold there exists a hexagon attractor even in Boussinesq convection as has been shown recently and (ii) the non-Boussinesq effects increase with $\epsilon$. Using direct simulations for circular containers we show that the re-entrant hexagons can prevail even for sidewall conditions that favour convection in the form of competing stable rolls. For sufficiently strong non-Boussinesq effects hexagons even become stable over the whole $\epsilon$-range considered, $0 \,{\le}\, \epsilon \,{\le}\, 1.5$.

The main objective of this study is to clarify the effect of thermal wall boundary conditions on turbulence statistics and structures in a compressible turbulent flow. This work is an extension of Morinishi et al. (J. Fluid Mech. vol. 502, 2004, p. 273), who performed DNS of compressible turbulent channel flow between adiabatic and isothermal walls at Mach number $M\,{=}\,1.5$ (Case 2). We address the question of whether the modification of turbulence statistics is attributable to the effect of the adiabatic wall boundary condition or the effect of the increase of wall temperature caused by the adiabatic wall boundary condition. New DNS of the compressible turbulent channel flow between isothermal walls with the wall temperature difference at the Mach number $M=1.5$ (Case 1) and DNS of the corresponding incompressible turbulent flow with passive scalar transport (Case I) are performed. The present study shows that the mean temperature profile near the high-temperature wall for Case 1 has an additional maximum due to the friction work, while such an additional maximum does not appear for Cases 2 and I. The additional maximum leads to a corresponding near-wall maximum of temperature fluctuations. We find the direction of energy transfer due to pressure work near the adiabatic wall for Case 2 being opposite to that near the isothermal wall to be due to the effect of the high-temperature wall, not to the effect of the adiabatic wall. These findings are explained by using the budgets of internal energy and temperature variance transport equations.

Data from the near-wall-turbulent region of the high-Reynolds-number atmospheric surface layer are used to analyse the attached-eddy model of wall turbulence. All data were acquired during near-neutral conditions at the Surface Layer Turbulence and Environmental Science Test (SLTEST) facility located in the western Utah Great Salt Lake Desert. Instantaneous streamwise and wall-normal components of velocity were collected with a wall-normal array of two-component hot wires within the first 2 m above the surface of the salt flats. Streamwise and wall-normal turbulence intensities and spectra are directly compared to corresponding laboratory data and similarity formulations hypothesized from the attached-eddy model of wall turbulence. This affords the opportunity to compare results with Reynolds numbers varying over three orders of magnitude. The wall-normal turbulence-intensity similarity formulation is extended. The results show good support for the similarity arguments forwarded by the attached-eddy model as well as Townsend's (1956) Reynolds-number similarity hypothesis and lack of the ‘inactive’ motion influence on the wall-normal velocity component. The effects of wall roughness and the spread in the convection velocity due to this roughness are also discussed.

A physical model for compressible flows with phase transition is described, in which all the processes of phase transition, i.e. nucleation, droplet growth, droplet evaporation and de-nucleation, are incorporated. The model is focused on dilute mixtures of vapour and droplets in a carrier gas with typical maximum liquid mass fraction smaller than 0.02. The new model is based on a reinterpretation of Hill's method of moments of the droplet size distribution function. Starting from the general dynamic equation, it is emphasized that nucleation or de-nucleation correspond to the rates at which droplets enter or leave droplet size space, respectively. Nucleation and de-nucleation have to be treated differently in agreement with their differences in physical nature. Attention is given to the droplet growth model that takes into account Knudsen effects and temperature differences between droplets and gas. The new phase transition model is then combined with the Euler equations and results in a new numerical method: ASCE2D. The numerical method is first applied to the problem of shock/expansion wave formation in a closed shock tube with humid nitrogen as a driver gas. Nucleation and droplet growth are induced by the expansion wave, and in turn affect the structure of the expansion wave. When the main shock, reflected from the end wall of the low-pressure section, passes the condensation zone, evaporation and de-nucleation occur. As a second example, the problem of the flow of humid nitrogen in a pulse-expansion wave tube, designed to study nucleation and droplet growth in monodisperse clouds, is investigated experimentally and numerically.