The Rayleigh number R, in a horizontal layer with temperature-dependent viscosity can be based on the viscosity at T0, the mean of the boundary temperatures. The critical Rayleigh number Roc for fluids with exponential and super-exponential viscosity variation is nearly constant at low values of the ratio of the viscosities at the top and bottom boundaries; increases at moderate values of the viscosity ratio, reaching a maximum at a ratio of about 3000, and then decreases. This behaviour is explained by a simple physical argument based on the idea that convection begins first in the sublayer with maximum Rayleigh number. The prediction of Palm (1960) that certain types of temperature-dependent viscosity always decrease Roc is confirmed by numerical results but is not relevant to the viscosity variations typical of real liquids. The infinitesimal-amplitude state assumed by linear theory in calculating Roc does not exist because the convection jumps immediately to a finite amplitude at R0c. We observe a heat-flux jump at R0c exceeding 10% when the viscosity ratio exceeds 150. However, experimental measurements of R0c for glycerol up to a viscosity ratio of 3400 are in good agreement with the numerical predictions when the effects of a temperature-dependent expansion coefficient and thermal diffusivity are included.

Three-dimensional numerical solutions have been obtained for steady, linear shear flow past a fixed, heated spherical particle over a wide range of Reynolds number (0.1 [les ] R [les ] 100) and dimensionless shear rates (0.005 [les ] α [les ] 0.4). The results indicate that at a fixed shear rate, the dimensionless lift coefficient is approximately constant over a wide range of intermediate Reynolds numbers, and the drag coefficient also remains constant when normalized by the known values of drag for a sphere in uniform flow. At lower values of the Reynolds number, the lift and drag coefficients increase sharply with decreasing R, with the lift coefficient being directly proportional to R−½. For the range of shear rates studied here, the rate of heat transfer to the particle surface was found to depend only on the Reynolds number, that is, it was insensitive to the shear rate. The dimensionless rate of heat transfer, the Nussel number Nu, was seen to increase monotonically with R.

A novel approach to the study of the kinematics and dynamics of turbulent flows is presented. The method involves tracking in time coherent structures, and provides all of the information required to characterize eddies from birth to death. Spatially and temporally well-resolved DNSs of channel data at $\mathit{Re}_{{\it\tau}}=930{-}4200$ are used to analyse the evolution of three-dimensional sweeps, ejections (Lozano-Durán et al., J. Fluid Mech., vol. 694, 2012, pp. 100–130) and clusters of vortices (del Álamo et al., J. Fluid Mech., vol. 561, 2006, pp. 329–358). The results show that most of the eddies remain small and do not last for long times, but that some become large, attach to the wall and extend across the logarithmic layer. The latter are geometrically and temporally self-similar, with lifetimes proportional to their size (or distance from the wall), and their dynamics is controlled by the mean shear near their centre of gravity. They are responsible for most of the total momentum transfer. Their origin, eventual disappearance, and history are investigated and characterized, including their advection velocity at different wall distances and the temporal evolution of their size. Reinforcing previous results, the symmetry found between sweeps and ejections supports the idea that they are not independent structures, but different manifestations of larger quasi-streamwise rollers in which they are embedded. Spatially localized direct and inverse cascades are respectively associated with the splitting and merging of individual structures, as in the models of Richardson (Proc. R. Soc. Lond. A, vol. 97(686), 1920, pp. 354–373) or Obukhov (Izv. Akad. Nauk USSR, Ser. Geogr. Geofiz., vol. 5(4), 1941, pp. 453–466). It is found that the direct cascade predominates, but that both directions are roughly comparable. Most of the merged or split fragments have sizes of the order of a few Kolmogorov viscous units, but a substantial fraction of the growth and decay of the larger eddies is due to a self-similar inertial process in which eddies merge and split in fragments spanning a wide range of scales.

The flow in the wakes behind two-dimensional perforated plates has been investigated in the Reynolds number range 2·5 × 104 to 9·0 × 104.

Measurements of drag and shedding frequency were made and a pulsed hotwire anemometer was used to measure the mean velocity and turbulent intensity variations in the highly turbulent regions immediately behind the plates.

The results indicate the existence of two distinct types of flows: one appropriate to high and the other to low values of plate porosity.

The wind-driven stably stratified mid-latitude oceanic surface turbulent boundary layer is computationally simulated in the presence of a specified surface gravity-wave field. The gravity waves have broad wavenumber and frequency spectra typical of measured conditions in near-equilibrium with the mean wind speed. The simulation model is based on (i) an asymptotic theory for the conservative dynamical effects of waves on the wave-averaged boundary-layer currents and (ii) a boundary-layer forcing by a stochastic representation of the impulses and energy fluxes in a field of breaking waves. The wave influences are shown to be profound on both the mean current profile and turbulent statistics compared to a simulation without these wave influences and forced by an equivalent mean surface stress. As expected from previous studies with partial combinations of these wave influences, Langmuir circulations due to the wave-averaged vortex force make vertical eddy fluxes of momentum and material concentration much more efficient and non-local (i.e. with negative eddy viscosity near the surface), and they combine with the breakers to increase the turbulent energy and dissipation rate. They also combine in an unexpected positive feedback in which breaker-generated vorticity seeds the creation of a new Langmuir circulation and instigates a deep strong intermittent downwelling jet that penetrates through the boundary layer and increases the material entrainment rate at the base of the layer. These wave effects on the boundary layer are greater for smaller wave ages and higher mean wind speeds.

The hydrodynamic stability of a pure liquid undergoing steady rapid evaporation at reduced pressure is examined using linear stability analysis. Results show that the rapidly evaporating liquid is unstable to local variations in evaporation rate, local surface depressions being produced by the force exerted on the surface by the rapidly departing vapour and sustained liquid flows being driven by the resultant shear exerted on the liquid surface by the vapour. The coupling of this ‘differential vapour recoil’ mechanism to the Marangoni effect is investigated and the importance of inertial heat transfer, fluid inertia and viscous dissipation at the interface to system stability is resolved.

The stability of short-wave displacement perturbations on a vortex filament of constant vorticity in a weak externally imposed strain field is considered. The circular cross-section of the vortex filament in this straining flow field becomes elliptical. It is found that instability of short waves on this strained vortex can occur only for wavelengths and frequencies at the intersection points of the dispersion curves for an isolated vortex. Numerical results show that the vortex is stable at some of these points and unstable at others. The vortex is unstable at wavelengths for which ω = 0, thus giving some support to the instability mechanism for the vortex ring proposed recently by Widnall, Bliss & Tsai (1974). The growth rate is calculated by linear stability theory. The previous work of Crow (1970) and Moore & Saffman (1971) dealing with long-wave instabilities is discussed as is the very recent work of Moore & Saffman (1975).

The present study is to investigate the structure of space-time correlations of bursting motions, such as ejections and sweeps in an open-channel flow, by a new conditional sampling analysis of the instantaneous velocity and Reynolds-stress signals measured simultaneously by two dual-sensor hot-film probes. One probe was fixed near the edge of the buffer layer, while the other probe was moved in the streamwise, vertical and spanwise directions. The sorted instantaneous Reynolds-stress signals obtained from the fixed probe were used as a detecting information of the occurrences of ejection or sweep events. The streamwise and vertical spatial characteristics of the ejection-sweep motions, and their convection process are investigated in detail. Also, the spanwise spatial properties of the high- and low-speed streaks in the bursting motions are examined experimentally by the present conditional sampling method.

Next, a qualitative model is proposed which attempts to explain the space-time structures of the bursting phenomenon, on the basis of the above anemometry information and other visual observations.

Cavitation microstreaming is a well-known phenomenon; however, few flow visualizations or measurements of the velocity fields have been conducted. In this paper micro-PIV (particle image velocimetry) measurements and streak photography were used to study the flow field around a single and two oscillating bubbles resting on a solid boundary. The mode of oscillation of the bubble was also measured in terms of the variation in the radius of the bubble and the movement of the bubble's centroid so that the streaming flow field could be accurately related to the bubble's oscillatory motion. The mode of oscillation was found to vary primarily with the applied acoustic frequency. Several modes of oscillation were investigated, including translating modes where the bubble's centroid moves along either a single axis, an elliptical orbit or a circular orbit. The flow field resulting from these oscillation modes contains closed streamlines representing vortical regions in the vicinity of the bubble. The translating modes were observed to occur in sequential order with the acoustic excitation frequency, changing from a translation along a single axis, to an elliptical orbit and finally to a circular orbit, or vice versa. Following this sequence, there is a corresponding transformation of the streaming pattern from a symmetrical flow structure containing four vortices to a circular vortex centred on the bubble. Despite some inconsistencies, there is general agreement between these streaming patterns and those found in existing theoretical models. Volume and shape mode oscillations of single bubbles as well as several different cases of multiple bubbles simultaneously oscillating with the same frequency and phase were also investigated and show a rich variety of streaming patterns.

A turbulent flame–wall interaction (FWI) configuration is studied using three-dimensional direct numerical simulation (DNS) and detailed chemical kinetics. The simulations are used to investigate the effects of the wall turbulent boundary layer (i) on the structure of a hydrogen–air premixed flame, (ii) on its near-wall propagation characteristics and (iii) on the spatial and temporal patterns of the convective wall heat flux. Results show that the local flame thickness and propagation speed vary between the core flow and the boundary layer, resulting in a regime change from flamelet near the channel centreline to a thickened flame at the wall. This finding has strong implications for the modelling of turbulent combustion using Reynolds-averaged Navier–Stokes or large-eddy simulation techniques. Moreover, the DNS results suggest that the near-wall coherent turbulent structures play an important role on the convective wall heat transfer by pushing the hot reactive zone towards the cold solid surface. At the wall, exothermic radical recombination reactions become important, and are responsible for approximately 70% of the overall heat release rate at the wall. Spectral analysis of the convective wall heat flux provides an unambiguous picture of its spatial and temporal patterns, previously unobserved, that is directly related to the spatial and temporal characteristic scalings of the coherent near-wall turbulent structures.

Turbulent Taylor–Couette flow with arbitrary rotation frequencies ω1, ω2 of the two coaxial cylinders with radii r1 < r2 is analysed theoretically. The current Jω of the angular velocity ω(x,t) = uϕ(r,ϕ,z,t)/r across the cylinder gap and and the excess energy dissipation rate ϵw due to the turbulent, convective fluctuations (the ‘wind’) are derived and their dependence on the control parameters analysed. The very close correspondence of Taylor–Couette flow with thermal Rayleigh–Bénard convection is elaborated, using these basic quantities and the exact relations among them to calculate the torque as a function of the rotation frequencies and the radius ratio η = r1/r2 or the gap width d = r2 − r1 between the cylinders. A quantity σ corresponding to the Prandtl number in Rayleigh–Bénard flow can be introduced, . In Taylor–Couette flow it characterizes the geometry, instead of material properties of the liquid as in Rayleigh–Bénard flow. The analogue of the Rayleigh number is the Taylor number, defined as Ta ∝ (ω1 − ω2)2 times a specific geometrical factor. The experimental data show no pure power law, but the exponent α of the torque versus the rotation frequency ω1 depends on the driving frequency ω1. An explanation for the physical origin of the ω1-dependence of the measured local power-law exponents α(ω1) is put forward. Also, the dependence of the torque on the gap width η is discussed and, in particular its strong increase for η → 1.

The Rayleigh number at which steady convective flow changes to time-dependent flow is determined experimentally for several fluids with Prandtl numbers from 1 to 104. The time dependence is of two forms: (i) a slow tilting of the cell boundary, with time scale of the vertical thermal diffusion time, (ii) an oscillation with a faster time scale determined by the orbit time of the fluid around the cell. The nature of this oscillation is one of hot (or cold) spots advected with the original cellular motion. At a fixed point in the fluid this produces a time periodic oscillation of the temperature. A discrete change of slope of the heat flux curve accompanies this transition. As the Rayleigh number is increased, transition to disorder is seen to result from an increase in the frequency and number of these oscillations.

The influence of an external flow velocity on the instability of a circular jet has been investigated by means of the inviscid linearized stability theory. The instability properties of spatially growing axisymmetric and first-order azimuthal disturbances show that the external flow inhibits the instability of the circular jet, but increases the unstable frequency range. Similarity considerations lead to the result that, in a first approximation, the disturbed flow field is independent of the external flow velocity, if the axial co-ordinate is contracted by a suitably chosen stretching factor and if the disturbance frequency is reduced by the same factor. It is concluded that the large-scale structure of jet turbulence is modified in the same manner by the external flow.

An experimental study is presented on contact angle dynamics during spreading/recoiling of mm-sized water droplets impacting orthogonally on various surfaces with $\hbox{\it We}\,{=}\,O(0.1)-O(10)$, $Ca\,{=}\,O(0.001)-O(0.01)$, $\hbox{\it Re}\,{=}\,O(100)-O(1000)$, $Oh\,{=}\,O(0.001)$ and $Bo\,{=}\,O(0.1)$. In this impact regime, inertial, viscous and capillary phenomena act in unison to influence contact angle dynamics. The wetting properties of the target surfaces range from wettable to non-wettable. The experiments feature accelerating and decelerating wetting lines, capillary surface waves in the early impact stages, contact angle hysteresis, and droplet rebound under non-wetting conditions. The objective of the work is to provide insight into the dynamic behaviour of the apparent (macroscopic) contact angle $\theta$ and its dependence on contact line velocity $V_{\hbox{\scriptsize{\it CL}}}$ at various degrees of surface wetting. By correlating the temporal behaviours of $\theta$ and $V_{\hbox{\scriptsize{\it CL}}}$, the angle vs. speed relationship is established for each case examined. The results reveal that surface wettability has a critical influence on dynamic contact angle behaviour. The hydrodynamic wetting theory of Cox (J. Fluid Mech. vol. 357, 1998, p. 249) and the molecular-kinetic theory of wetting by Blake & Haynes (J. Colloid Interface Sci.) vol. 30, 1969, p. 421) are implemented to extract values of the corresponding microscopic wetting parameters required to match the experimentally observed $\theta$vs. $V_{\hbox{\scriptsize{\it CL}}}$ data. Application of hydrodynamic theory indicates that in the slow stage of forced spreading the slip length and the microscopic contact angle should be contact line velocity dependent. The hydrodynamic theory performs well during kinematic (fast) spreading, in which solid/liquid interactions are weak. Application of the molecular kinetic theory yields physically reasonable molecular wetting parameters, which, however, vary with impact conditions. The results indicate that even for a single liquid there is no universal expression to relate contact angle with contact line speed. Finally, analysis of the spreading dynamics on the non-wettable surfaces shows that it conforms to the Cassie-Baxter regime (only partial liquid/solid contact is maintained). The present results offer guidance for numerical or analytical studies, which require careful attention to the implementation of boundary conditions at the moving contact line, including the need to specify the dependence of contact angle on contact line speed.

The laminar separation bubble on an SD7003 aerofoil at a Reynolds number Re = 66000 was investigated to determine the dominant frequencies of the transition process and the flapping of the bubble. The measurements were performed with a high-resolution time-resolved particle image velocimetry (TR-PIV) system. Contrary to typical measurements performed through conventional PIV, the different modes can be identified by applying TR-PIV. The interaction between the shed vortices is analysed, and their significance for the production of turbulence is presented. In the shear layer above the bubble the generation and amplification of vortices due to Kelvin–Helmholtz instabilities is observed. It is found that these instabilities have a weak coherence in the spanwise direction. In a later stage of transition these vortices lead to a three-dimensional breakdown to turbulence.

A new stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence is proposed. The model is three-dimensional and its formulation takes account of recent improvements in the understanding of one-particle models. In particular the model is designed so that if the particle pairs are initially well mixed in the fluid, they will remain so. In contrast to previous models, the new model leads to a prediction for the particle separation probability density function which is in qualitative agreement with inertial subrange theory. The values of concentration variance from the model show encouraging agreement with experimental data. The model results suggest that, at large times, the intensity of concentration fluctuations (i.e. standard deviation of concentration divided by mean concentration) tends to zero in stationary conditions and to a constant in decaying turbulence.

A stabilizing gradient of solute inhibits the onset of convection in a fluid which is subjected to an adverse temperature gradient. Furthermore, the onset of instability may occur as an oscillatory motion because of the stabilizing effect of the solute. These results are obtained from linear stability theory which is reviewed briefly in the following paper before finite-amplitude results for two-dimensional flows are considered. It is found that a finite-amplitude instability may occur first for fluids with a Prandtl number somewhat smaller than unity. When the Prandtl number is equal to unity or greater, instability first sets in as an oscillatory motion which subsequently becomes unstable to disturbances which lead to steady, convecting cellular motions with larger heat flux. A solute Rayleigh number, Rs, is defined with the stabilizing solute gradient replacing the destabilizing temperature gradient in the thermal Rayleigh number. When Rs is large compared with the critical Rayleigh number of ordinary Bénard convection, the value of the Rayleigh number at which instability to finite-amplitude steady modes can set in approaches the value of Rs. Hence, asymptotically this type of instability is established when the fluid is marginally stratified. Also, as Rs → ∞ an effective diffusion coefficient, Kρ, is defined as the ratio of the vertical density flux to the density gradient evaluated at the boundary and it is found that κρ = √(κκs) where κ, κs are the diffusion coefficients for temperature and solute respectively. A study is made of the oscillatory behaviour of the fluid when the oscillations have finite amplitudes; the periods of the oscillations are found to increase with amplitude. The horizontally averaged density gradients change sign with height in the oscillating flows. Stably stratified fluid exists near the boundaries and unstably stratified fluid occupies the mid-regions for most of the oscillatory cycle. Thus the step-like behaviour of the density field which has been observed experimentally for time-dependent flows is encountered here numerically.

The averaged equations of slow flow in random arrays of fixed spheres are developed as a hierarchy of integro-differential equations, and an iteration procedure is described for obtaining the mean drag in the case of small volume concentration c. The leading approximation is that given by Brinkman's model of flow past a single fixed sphere, in which the effects of all other spheres are treated as a Darcy resistance. The higher approximations take account of the modification to the mean flow, particularly in the near field, due to the localized nature of the actual resistance. Thus the second approximation finds the change due to a second sphere, and averages over all its possible positions. The result for the mean drag confirms Childress’ terms in clogc and c (apart from an arithmetical correction to the latter), but indicates that for practical values of c numerical evaluation of integrals is needed, rather than expansion in powers of c and log c. The last section of the paper develops the corresponding results for flow through random arrays of fixed parallel circular cylinders.

Large-eddy simulations of a subsonic three-dimensional cavity flow with self-sustaining oscillations are carried out for a Reynolds number based on the length of the cavity equal to $7\,{\times}\,10^6$. Meticulous comparisons with available experimental data corresponding to the same configuration demonstrate a high level of accuracy. Special attention is paid to the mixing layer that develops over the cavity and two different zones are identified. The first one is dominated by Kelvin–Helmholtz instability, and the linear as well as quadratic energy transfers leading to the filling of velocity spectra are described. The Kelvin–Helmholtz instability also appears to be forced near the origin of the layer, and it is postulated that the small recirculation bubble located in this area is responsible for the forcing. Downstream of the first zone and up to the vicinity of the aft wall, the layer behaves very similarly to a free mixing layer by exhibiting a linear spreading. An influence of the recirculating flow inside the cavity upon the growth of the layer is nevertheless observed at downstream stations. Analysis of the pressure on the floor of the cavity reveals that the self-sustaining oscillation-related pressure modes (Rossiter modes) are independent of their spanwise location inside the cavity. On the contrary, Rossiter modes exhibit streamwise modulations and it is demonstrated that a very simple two-wave model is able to reproduce the spatial shape of the modes. Nonlinear interactions between Rossiter modes are encountered, as well as nonlinear interactions with low-frequency components. A joint time–frequency analysis shows a temporal modulation of the Rossiter mode levels at similar low frequencies, resulting in a special form of intermittency with competitive energy exchanges between modes.

Recently a new system of field equations for the accurate description of flows in rarefied gases, called regularized 13-moment equations, was obtained by means of a hybrid gas kinetic approach. The first part of this paper discusses the relationship of the new system to classical high-order theories like the Burnett and super-Burnett equations as well as to modified models like the augmented and regularized Burnett equations. In the second part, shock structure calculations with the new theory are presented and compared to direct-simulation Monte Carlo (DSMC) solutions and to solutions of the Burnett models. Owing to additional higher-order dissipation in the system, the profiles are smooth for any Mach number, in contrast to the results of Grad's 13-moment case. The results show reliable quantitative agreement with DSMC simulations for Mach numbers up to $M_{0}\approx 3.0$. The agreement is better for Maxwell molecules than for hard spheres. The results of the augmented Burnett equations are comparable, but these equations are shown to be spatially unstable. Additionally, a validiation procedure for the new equations is presented by investigating the positivity of Grad's distribution function.