A fully three-dimensional model is proposed for the generation of tidal sand waves and sand banks from small bottom perturbations of a flat seabed subject to tidal currents. The model predicts the conditions leading to the appearance of both tidal sand waves and sand banks and determines their main geometrical characteristics. A finite wavelength of both sand waves and sand banks is found around the critical conditions, thus opening the possibility of performing a weakly nonlinear stability analysis able to predict the equilibrium amplitude of the bottom forms. As shown by previous works on the subject, the sand wave crests turn out to be orthogonal to the direction of the main tidal current. The present results also show that in the Northern Hemisphere sand bank crests are clockwise or counter-clockwise rotated with respect to the main tidal current depending on the counter-clockwise or clockwise rotation of the velocity vector induced by the tide. Only for unidirectional or quasi-unidirectional tidal currents are sand banks always counter-clockwise rotated. The predictions of the model are supported by comparisons with field data. Finally, the mechanisms leading to the appearance of sand waves and sand banks are discussed in the light of the model findings. In particular, it is shown that the growth of sand banks is not only induced by the depth-averaged residual circulation which is present around the bedforms and is parallel to the crests of the bottom forms: a steady drift of the sediment from the troughs towards the crests is also driven by the steady velocity component orthogonal to the crests which is present close to the bottom and can be quantified only by a three-dimensional model. While the former mechanism appears to trigger the formation of counter-clockwise sand banks only, the latter mechanism can give rise to both counter-clockwise and clockwise rotated sand banks.

Large-eddy simulations of the Richtmyer–Meshkov instability with reshock are pre- sented and the results are compared with experiments. Several configurations of shocks initially travelling from light (air) to heavy (sulfur hexafluoride, SF6) have been simulated to match previous experiments and good agreement is found in the growth rates of the turbulent mixing zone (TMZ). The stretched-vortex subgrid model used in this study allows for subgrid continuation modelling, where statistics of the unresolved scales of the flow are estimated. In particular, this multiscale modelling allows the anisotropy of the flow to be extended to the dissipation scale, $\eta$, and estimates to be formed for the subgrid probability density function of the mixture fraction of air/SF6 based on the subgrid variance, including the effect of Schmidt number.

We present experimental observations of hole formation in ejecta crowns, when a viscous drop impacts onto a thin film of low-viscosity liquid with significantly lower surface tension than the drop liquid. The holes are promoted by Marangoni-driven flows in the sheet, resulting from a spray of fine droplets ejected from the thin film hitting the inner side of the crown. The puncturing of the sheet takes place in three distinct steps. First a circular patch of the sheet thins by Marangoni-driven flows. Then this thinner film ruptures and a hole quickly opens up spanning the patch. Finally, the hole opens up further at an accelerated rate, driven by the unbalanced surface tension at its edge. The holes grow until they meet adjacent holes, thus leaving a foam-like network of liquid filaments, which then breaks up into a cloud of droplets.

The motion of a single Brownian probe particle subjected to a constant external body force and immersed in a dispersion of colloidal particles is studied with a view to providing a simple model for particle tracking microrheology experiments in the active and nonlinear regime. The non-equilibrium configuration of particles induced by the motion of the probe is calculated to first order in the volume fraction of colloidal particles over the entire range of Pe, accounting for hydrodynamic and excluded volume interactions between the probe and dispersion particles. Here, Pe is the dimensionless external force on the probe, or Péclet number, and is a characteristic measure of the degree to which the equilibrium microstructure of the dispersion is distorted. For small Pe, the microstructure (in a reference frame moving with the probe) is primarily dictated by Brownian diffusion and is approximately fore–aft symmetric about the direction of the external force. In the large Pe limit, advection is dominant except in a thin boundary layer in the compressive region of the flow where it is balanced by Brownian diffusion, leading to a highly non-equilibrium microstructure. The computed microstructure is employed to calculate the average translational velocity of the probe, from which the ‘microviscosity’ of the dispersion may be inferred via application of Stokes drag law. For small departures from equilibrium (Pe), the microviscosity ‘force-thins’ proportional to $\hbox{\it Pe}^2$ from its Newtonian low-force plateau. For particles with long-range excluded volume interactions, force-thinning persists until a terminal Newtonian plateau is reached in the limit $\hbox{\it Pe}\,{\rightarrow}\,\infty$. In the case of particles with very short-range excluded volume interactions, the force-thinning ceases at $\hbox{\it Pe}\,{\sim}\, O(1)$, at which point the microviscosity attains a minimum value. Beyond $\hbox{\it Pe}\,{\sim}\, O(1)$, the microstructural boundary layer coincides with the lubrication range of hydrodynamic interactions causing the microviscosity to enter a continuous ‘force-thickening’ regime. The qualitative picture of the microviscosity variation with Pe is in good agreement with theoretical and computational investigations on the ‘macroviscosity’ of sheared colloidal dispersions, and, after appropriate scaling, we are able to make a direct quantitative comparison. This suggests that active tracking microrheology is a valuable tool with which to explore the rich nonlinear rheology of complex fluids.

The dynamics of gases in permeable media is approached both experimentally and by numerical simulations. The experiments were performed in matrices made of packed beds of spheres in rarefied conditions and a model for the direct simulation of the molecular kinetics is proposed. Comparisons between experimental data and numerical results show the influence of the main parameters of the gas–solid interaction and the range of validity of the model. Moreover it is shown that there is a flow condition for the minimum permeability of the medium to the gas flow. Such a minimum depends upon the Knudsen number, and can be explained by the molecular dynamics as in the well-known Knudsen's experiment on capillaries.

The formation of columnar eddies in a rapidly rotating environment is often attributed to nonlinear processes, acting on the nonlinear time scale $l/|\bm{u}|$. We argue that this is not the whole story, and that linear wave propagation can play an important role, at least on the short time scale of $\Omega^{-1}$. In particular, we consider the initial value problem of a compact blob of vorticity (an eddy) sitting in a rapidly rotating environment. We show that, although the energy of the eddy disperses in all directions through inertial wave propagation, the axial components of its linear impulse and angular momentum disperse along the rotation axis only, remaining confined to the cylinder which circumscribes the initial vortex blob. This confinement has a crucial influence on the manner in which energy disperses from the eddy, with the energy density within the tangent cylinder remaining much higher than that outsid (i.e. decaying as $t^{-1}$ inside the cylinder and $t^{-3/2}$ outside). When the initial conditions consist of an array of vortex blobs the situation is more complicated, because the energy density within the tangent cylinder of any one blob is eventually swamped by the radiation released from all the other blobs. Nevertheless, we would expect that a turbulent flow which starts as a collection of blobs of vorticity will, for times of order $\Omega^{-1}$, exhibit columnar vortices, albeit immersed in a random field of inertial waves. Laboratory experiments are described which do indeed show the emergence of columnar eddies through linear mechanisms, though these experiments are restricted to the case of inhomogeneous turbulence. Since the Rossby number in the experiments is of the order of unity, this suggests that linear effects can still influence and shape turbulence when nonlinear processes are also operating.

This work aims at understanding the viscous effects of the outer potential flow on Prandtl's boundary layer. For a body moving with a constant velocity in an otherwise quiescent liquid, the non-zero viscous dissipation of the outer potential flow gives rise to an additional drag, increasing the drag calculated from the boundary layer alone. The drag is considered in three cases here, on a two-dimensional circular gas bubble in a streaming flow, at the edge of the boundary layer around a rapidly rotating cylinder in a uniform flow, and on an airfoil in a streaming flow. The drag may be computed using the dissipation method or the viscous pressure correction of the irrotational pressure. Such a pressure correction can be induced by the discrepancy between the irrotatinal shear stress and the zero shear stress at a fluid–gas interface, or by the discrepancy between the shear stress evaluated from the boundary-layer solution and that evaluated from the outer potential flow solution at the edge of the boundary layer.

We study the streaming flow past a rapidly rotating circular cylinder. The starting point is the full continuity and momentum equations without any approximations. We assume that the solution is a boundary-layer flow near the cylinder surface with the potential flow outside the boundary layer. The order of magnitude of the terms in the continuity and momentum equations can be estimated inside the boundary layer. When terms of the order of $\delta/a$ and higher are dropped, where $\delta$ is the boundary-layer thickness and $a$ is the radius of the cylinder, the equations used by M. B. Glauert (Proc. R. Soc. Lond. A, vol. 242, 1957, p. 108) are recovered. Glauert's solution ignores the irrotational rotary component of the flow inside the boundary layer, which is consistent with dropping $\delta/a$ terms in the governing equations.

We propose a new solution to this problem, in which the velocity field is decomposed into two parts. Outside the boundary layer, the flow is irrotational and can be decomposed into a purely rotary flow and a potential flow past a fixed cylinder. Inside the boundary layer, the velocity is decomposed into an irrotational purely rotary flow and a boundary-layer flow. Inserting this decomposition of the velocity field inside the boundary layer into the governing equations, we obtain a new set of equations for the boundary-layer flow, in which we do not drop the terms of the order of $\delta/a$ or higher. The pressure can no longer be assumed to be a constant across the boundary layer, and the continuity of shear stress at the outer edge of the boundary layer is enforced. We solve this new set of equations using Glauert's method, i.e. to expand the solutions as a power series of $\alpha \,{=}\, 2 U_0 /Q$, where $U_0$ is the uniform stream velocity and $Q$ is the circulatory velocity at the outer edge of the boundary layer. The pressure from this boundary-layer solution has two parts, an inertial part and a viscous part. The inertial part comes from the inertia terms in the momentum equations and is in agreement with the irrotational pressure; the viscous part comes from the viscous stress terms in the momentum equations and may be viewed as a viscous pressure correction, which contributes to both drag and lift. Our boundary-layer solution is in reasonable to excellent agreement with the numerical simulation in the companion paper by Padrino & Joseph (2006).

Results from the numerical simulation of the two-dimensional incompressible unsteady Navier–Stokes equations for streaming flow past a rotating circular cylinder are presented in this study. The numerical solution of the equations of motion is conducted with a commercial computational fluid dynamics package which discretizes the equations applying the control volume method. The numerical set-up is validated by comparing results for a Reynolds number based on the free stream of $\hbox{\it Re}$ = 200 and dimensionless peripheral speed of $\tilde{q}$ = 3, 4 and 5 with results from the literature. After the validation stage, various pairs of $\hbox{\it Re}$ and $\tilde{q}$ are specified in order to carry out the numerical experiments. These values are $\hbox{\it Re}$ = 200 with $\tilde{q}$ = 4 and 5; $\hbox{\it Re}$ = 400 with $\tilde{q}$ = 4, 5 and 6, and $\hbox{\it Re}$ = 1000 with $\tilde{q}$ = 3. In all these cases, gentle convergence to fully developed steady state is reached. From the numerical vorticity distribution, the position of the outer edge of the vortical region is determined as a function of the angular coordinate. This position is found by means of a reasonable criterion set to define the outmost curve around the cylinder where the vorticity magnitude reaches a certain cut-off value. By considering the average value of this profile, a uniform vortical region thickness is specified for every pair of $\hbox{\it Re}$ and $\tilde{q}$.

Next, the theoretical approach of Wang & Joseph (2006a; see the companion paper) and the numerical results are used to determine two different values of the effective vortical region thickness for every pair of $\hbox{\it Re}$ and $\tilde{q}$. One effective thickness $\delta_D/a$ is obtained from the match between the additional drag on the outer edge of the vortical region according to the viscous correction of viscous potential flow (VCVPF) and the corresponding numerical profile while the other thickness $\delta_L/a$ is determined from the match between the pressure lift on the cylinder obtained from Wang & Joseph (2006a)'s simple modification of the boundary-layer analysis due to Glauert (Proc. R. Soc. Lond. A, vol. 242, 1957, p. 108) and the numerical value of the pressure lift coefficient. The values of $\delta_D/a$ and $\delta_L/a$ are used in the computation of various parameters associated with the flow, namely, the torque on the rotating cylinder, the circulatory velocity at the edge of the vortical region, which links the cylinder's angular velocity with the circulation of the irrotational flow of the viscous fluid outside this region, and the viscous dissipation. Predictions from the approaches of Glauert (1957) and Wang & Joseph (2006a) are also included for comparison. The values of both effective thicknesses, $\delta_D/a$ and $\delta_L/a$, are found to have the same order of magnitude. Then, we show that choosing $\delta_D/a$ as a unique effective thickness, the modification of Glauert's boundary-layer analysis and the VCVPF approach as proposed by Wang & Joseph (2006a) produce results which are in better general agreement with the values from numerical simulation than those from Glauert's solution.

Viscous fingering in porous media at large Péclet numbers is subject to an unsolved selection problem, not unlike the Saffman–Taylor problem. The mixing zone predicted by the entropy solution is found to spread much faster than is observed experimentally or from fine-scale numerical simulations. In this paper we apply a recent approach by Menon & Otto (Commun. Math. Phys., vol. 257, 2005, p. 303), to develop bounds for the mixing zone. These give growth velocities smaller than the entropy solution result $(M-1/M)$. In particular, for an exponential viscosity-concentration mixing rule, the mixing zone velocity is bounded by $(M-1)^2/(M\ln M)$, which is smaller than $(M-1/M)$. An extension to a porous medium with an uncorrelated random heterogeneity is also given.

A theoretical study is made of fully localized solitary waves, commonly referred to as ‘lumps’, on the interface of a two-layer fluid system in the case that the upper layer is bounded by a rigid lid and lies on top of an infinitely deep fluid. The analysis is based on an extension, that allows for weak transverse variations, of the equation derived by Benjamin (J. Fluid Mech. vol. 245, 1992, p. 401) for the evolution in one spatial dimension of weakly nonlinear long waves in this flow configuration, assuming that interfacial tension is large and the two fluid densities are nearly equal. The phase speed of the Benjamin equation features a minimum at a finite wavenumber where plane solitary waves are known to bifurcate from infinitesimal sinusoidal wavetrains. Using small-amplitude expansions, it is shown that this minimum is also the bifurcation point of lumps akin to the free-surface gravity–capillary lumps recently found on water of finite depth. Numerical continuation of the two symmetric lump-solution branches that bifurcate there reveals that the elevation-wave branch is directly connected to the familiar lump solutions of the Kadomtsev–Petviashvili equation, while the depression-wave branch apparently features a sequence of limit points associated with multi-modal lumps. Plane solitary waves of elevation, although stable in one dimension, are unstable to transverse perturbations, and there is evidence from unsteady numerical simulations that this instability results in the formation of elevation lumps.

We investigate theoretically the first effects of inertia on the orientation dynamics of a torque-free spheroidal particle in simple shear flow when the deviation from sphericity is small. The inertialess motion of any axisymmetric particle in simple shear represents a degenerate limit, the spheroidal geometry being a special case; as originally found by Jeffery (Proc. R. Soc. Lond. A, vol. 102, 1922, p. 161), the orientation vector moves indefinitely along any one of a single-parameter family of closed orbits centred around the vorticity axis, the distribution across orbits being determined by initial conditions. We consider both the inertia of the particle and that of the suspending fluid, characterized by the Stokes ($\hbox{\it St}$) and Reynolds numbers ($\hbox{\it Re}\,{=}\, \rho_f/\rho_p\hbox{\it St}$, $\rho_p$ and $\rho_f$ being the particle and fluid densities), respectively, as mechanisms for breaking the aforementioned degeneracy. The former is defined as $\hbox{\it St} \,{=}\, a^2\dot{\gamma}\rho_p/\mu$, where $\dot{\gamma}$ is the shear rate, $a$ is the radius of the unperturbed sphere and $\mu$ is the fluid viscosity. When the particles are much denser than the suspending fluid, as is the case for aerosols, $\hbox{\it St} \,{\gg}\, \hbox{\it Re}$ (both parameters being much less than unity), inertial forces in the fluid may be neglected. It is then found, in the absence of gravity, that a slightly prolate spheroid drifts toward the shearing plane, while the axis of a slightly oblate spheroid tends toward the vorticity axis, both on a time scale of $O(|\epsilon | \hbox{\it St} \dot{\gamma})^{-1}$, where $\epsilon ({\ll}\,1)$ is the deviation from sphericity. For the case of neutrally buoyant particles ($\hbox{\it St} \,{=}\,\hbox{\it Re}$), inertia of both the particle and fluid come into play. In contrast to the small but finite $\hbox{\it St}$ zero $\hbox{\it Re}$ case, the orientation vector of a neutrally buoyant prolate spheroid now migrates toward the direction of vorticity, while that of an oblate spheroid drifts towards the shearing plane. The time scale of drift towards the asymptotic state is $O(| \epsilon | \hbox{\it Re} \dot{\gamma})^{-1}$ in both cases. Thereafter, we also examine the rotations of prolate and oblate spheroids in the presence of both gravity and shear, the analysis again being restricted to weak inertial effects. A wide range of interesting orientational behaviour arises, and the long-time orientation dynamics of the spheroids are determined as a function of both the density ratio $\rho_p/\rho_f$ and a shear parameter $N$, defined as $N\,{=}\, 2a \rho_f g/(9\mu \dot{\gamma})$.

We use Brownian dynamics to investigate the effect of chain flexibility on the cross-streamline migration of short polymers in a pressure-driven flow between two infinite flat plates. A simulation method is described that models a polymer molecule at the Kuhn step level as a chain of $N$ freely jointed Brownian rods, and includes multibody hydrodynamic interactions between the chain segments and the surrounding channel walls. Our study confirms the existence of shear-induced migration away from the solid boundaries toward the channel centreline as a result of wall hydrodynamic interactions. At a fixed ratio $H/R_{g}$ of the channel width to the bulk radius of gyration, and at a fixed value of the Weissenberg number $Wi$, simulations show that migration is not significantly influenced by flexibility for chains of length $N\,{\ge}\, 2$. Much weaker migration is observed however for fully rigid chains ($N\,{=}\,1$), and a mechanism is discussed to explain migration in that case.

Direct numerical simulations of thermal convection over grooved plates are presented and discussed, in comparison with the standard flat-plate case, in order to gain a better understanding of the altered near-wall dynamics and of the enhancement of the heat transfer. The simulations are performed in a cylindrical cell of aspect-ratio (diameter over cell height) Γ = 1/2 at fixed Prandtl number Pr = 0.7 with the Rayleigh number Ra ranging from 2 × 106 to 2 × 1011. The results show an increase of heat transfer, or in non-dimensional form the Nusselt number Nu when the mean thermal boundary-layer thickness becomes smaller than the groove height, in agreement with earlier experimental investigations available from the literature. The present increase, however, results in a steeper power law of the Nu vs. Ra law rather than a simple upward shift of the Nu law of the flat plate. This finding agrees with some studies, but it is at variance with others. Possible causes for this difference are discussed with the help of an electrical analogy.

Results are presented of experimental investigations into the motion of a heavy sphere in a rotating cylinder which is completely filled with highly viscous fluid. For a given cylinder rotation rate, the sphere adopts a fixed position and rotates adjacent to the cylinder wall. For the case of a smooth sphere the motion is consistent with that predicted by a Stokes flow model. Artificially roughened spheres exhibit particle–boundary contact caused by impacts of surface asperities with the boundary for low cylinder surface speeds. For higher cylinder surface speeds the behaviour of the roughened spheres crosses smoothly from the particle–boundary contact regime to motion with hydrodynamically lubricated flow.

We report experimental results for the influence of a tilt angle $\beta$ relative to gravity on turbulent Rayleigh–Bénard convection of cylindrical samples. The measurements were made at Rayleigh numbers $R$ up to $10^{11}$ with two samples of height $L$ equal to the diameter $D$ (aspect ratio $\Gamma \,{\equiv}\, D/L \,{\simeq}\, 1$), one with $L \,{\simeq}\, 0.5$ m (the ‘large’ sample) and the other with $L \,{\simeq}\, 0.25$m (the ‘medium’ sample). The fluid was water with a Prandtl number $\sigma \,{=}\, 4.38$.

In contrast to the experiences reported by Chillà Eur. Phys. J. B, vol. 40, 2004 p. 223) for a similar sample but with $\Gamma \,{\simeq}\, 0.5$ ($D\,{=}\,0.5$ and $L\,{=}\,1.0$m), we found no long relaxation times. For $R\,{=}\,9.4\,{\times}\,10^{10}$ we measured the Nusselt number $ \cal N$ as a function of $\beta$ and obtained a small $\beta$ dependence given by ${\cal N}(\beta)\,{=}\,{\cal N}_0 [1-(3.1\,{\pm}\,0.1)\,{\times}\, 10^{-2}|\beta|]$ when $\beta$ is in radians. This reduction of $\cal N$ is about a factor of 50 smaller than the result found by Chillà et al. (2004) for their $\Gamma\,{\simeq}\,0.5$ sample.

We measured sidewall temperatures at eight equally spaced azimuthal locations on the horizontal mid-plane of the sample and used them to obtain cross-correlation functions between opposite azimuthal locations. The correlation functions had Gaussian peaks centred about $t_1^{cc} \,{>}\, 0$ that corresponded to half a turnover time of the large-scale circulation (LSC) and yielded Reynolds numbers $\hbox{\it Re}^{cc}$ of the LSC. For the large sample and $R \,{=}\, 9.4\,{\times}\, 10^{10}$ we found $\hbox{\it Re}^{cc}(\beta) \,{=}\, \hbox{\it Re}^{cc}(0)\,{\times}\, [1 + (1.85\,{\pm}\, 0.21) |\beta| - (5.9\,{\pm}\, 1.7) \beta^2]$. Similar results were obtained from the auto-correlation functions of individual thermometers. These results are consistent with measurements of the amplitude $\delta$ of the azimuthal sidewall temperature variation at the mid-plane that gave $\delta(\beta) \,{=}\, \delta(0)\,{\times}\, [1 + (1.84 \,{\pm}\, 0.45) |\beta| - (3.1 \,{\pm}\, 3.9) \beta^2]$ for the same $R$. An important conclusion is that the increase of the speed (i.e. of $\hbox{\it Re}$) of the LSC with $\beta$ does not significantly influence the heat transport. Thus the heat transport must be determined primarily by the instability mechanism operative in the boundary layers, rather than by the rate at which ‘plumes’ are carried away by the LSC. This mechanism is apparently independent of $\beta$.

Over the range $10^9 \,{\lesssim}\, R \,{\lesssim}\, 10^{11}$ the enhancement of $\hbox{\it Re}^{cc}$ at constant $\beta$ due to the tilt could be described by a power law of $R$ with an exponent of $-1/6$, consistent with a simple model that balances the additional buoyancy due to the tilt angle by the shear stress across the boundary layers. Even a small tilt angle dramatically suppressed the azimuthal meandering and the sudden reorientations characteristic of the LSC in a sample with $\beta \,{=}\, 0$. For large $R$ the azimuthal mean of the temperature at the horizontal mid-plane differed significantly from the average of the top- and bottom-plate temperatures due to non-Boussinesq effects, but within our resolution was independent of $\beta$.

A new third-order solution for bichromatic bi-directional water waves in finite depth is presented. Earlier derivations of steady bichromatic wave theories have been restricted to second-order in finite depth and third-order in infinite depth, while third-order theories in finite depth have been limited to the case of monochromatic short-crested waves. This work generalizes these earlier works. The solution includes explicit expressions for the surface elevation, the amplitude dispersion and the vertical variation of the velocity potential, and it incorporates the effect of an ambient current with the option of specifying zero net volume flux. The nonlinear dispersion relation is generalized to account for many interacting wave components with different frequencies and amplitudes, and it is verified against classical expressions from the literature. Limitations and problems with these classical expressions are identified. Next, third-order resonance curves for finite-amplitude carrier waves and their three-dimensional perturbations are calculated. The influence of nonlinearity on these curves is demonstrated and a comparison is made with the location of dominant class I and class II wave instabilities determined by classical stability analyses. Finally, third-order resonance curves for the interaction of nonlinear waves and an undular sea bottom are calculated. On the basis of these curves, the previously observed downshift/upshift of reflected/transmitted class III Bragg scatter is, for the first time, explained.

The concept of impulse is employed with conformal mapping to yield relatively simple relations for the force exerted on a two-dimensional stationary object by an incompressible irrotational and unsteady flow with moving vortices. An explicit relation for symmetric vortex flows is found, involving the vortex strength and the first and second derivatives of the mapping function evaluated at the vortex position. Furthermore an expression for not-necessarily symmetric vortex flows is derived, containing vortex strength, the first derivative of the mapping function evaluated at the vortex position, and the vortex velocity.

We report measurements of the Nusselt number, $\hbox{\it Nu}$, in turbulent thermal convection in a cylindrical container of aspect ratio 4. The highest Rayleigh number achieved was $\hbox{\it Ra} \,{=}\, 2 \,{\times}\, 10^{13}$. Except for the last half a decade or so of $\hbox{\it Ra}$, experimental conditions obey the Boussinesq approximation accurately. For these conditions, the data show that the $\log\hbox{\it Nu}$–$\log\hbox{\it Ra}$ slope saturates at a value close to 1/3, as observed previously by us in experiments with smaller aspect ratios. The increasing slope over the last half a decade of $\hbox{\it Ra}$ is inconclusive because the corresponding conditions are non-Boussinesq. Finally, we report a modified scaling relation between the plume advection frequency and ${\hbox{\it Ra}}$ that collapses data for different aspect ratios.

The effects of localized periodic blowing on a turbulent boundary layer were investigated by direct numerical simulation. Time-periodic blowing was applied through a spanwise slot by varying the wall-normal velocity in a cyclic manner from 0 to $2A^ + $. Time-periodic blowing was applied at frequencies in the range $0\,{ \le}\, f^ +\,{ \le}\, 0.08$ at a fixed blowing amplitude of $A^ + $ = 0.5. Simulations of a spatially evolving turbulent boundary layer were carried out for two Reynolds numbers, $\hbox{\it Re}_{\theta, in} $ = 300 and 670. Before investigating the effects of periodic blowing, the effects of steady blowing were examined. A new parameter, $\sigma ^ + $, was proposed for describing local blowing; the usefulness of this parameter was that the responses of the flow variables at the two Reynolds numbers were the same for the same $\sigma ^ + $. The effects of varying the blowing frequency were scrutinized by examining the phase- or time-averaged turbulent statistics. For both Reynolds numbers, application of blowing at a frequency of $f^ + $ = 0.035 was found to give the maximum increases in Reynolds shear stress, streamwise vorticity fluctuations and energy redistribution. Analysis of the Reynolds stress budget showed that this effective blowing frequency induced the greatest enhancement of the pressure–strain term, which is closely related to the energy redistribution. Analysis of the phase-averaged stretching and tilting terms revealed that the stretching term is significantly enhanced in the ‘downward’ motion that is induced by the spanwise vortical motion. The correlation between the streamwise vorticity and the stretching term changed in magnitude and length scale as the blowing phase was varied, whereas the correlation between the streamwise vorticity and the tilting term did not.