A numerical and experimental study of buoyancy-driven flow in the annulus between two horizontal coaxial cylinders at Rayleigh numbers approaching and exceeding the critical values is presented. The stability of the flow is investigated using linear theory and the energy method. Theoretical predictions of the critical Rayleigh number for onset of secondary flows are obtained for a wide range of radius ratio R and are verified by comparison with results of previous experimental studies. A subcritical Rayleigh number which provides a necessary condition for global flow stability is also determined. The three-dimensional transient equations of fluid flow and heat transfer are solved to study the manifestation of instabilities within annuli having impermeable endwalls, which are encountered in various applications. For the first time, a thorough examination of the development of spiral vortex secondary flow within a moderate gap annulus and its interaction with the primary flow is performed for air. Simulations are conducted to investigate factors influencing the size and number of post-transitional vortex cells. The evolution of stable three-dimensional flow and temperature fields with increasing Rayleigh number in a large gap annulus is also studied. The distinct flow structures which coexist in the large gap annulus at high Rayleigh numbers preceding transition to oscillatory flow, including transverse vortices at the end walls which have not been previously identified, are established numerically and experimentally. The solutions for the large-gap annulus are compared to those for the moderate-gap case to clarify fundamental differences in behaviour. Heat transfer results in the form of local Nusselt number distributions are presented for both the moderate- and large-gap cases. Results from a series of experiments performed with air to obtain data for validation of the numerical scheme and further information on the flow stability are presented. Additionally, the change from a crescent-shaped flow pattern to a unicellular pattern with centre of rotation at the top of the annulus is investigated numerically and experimentally for a Prandtl number of 100. Excellent agreement between the numerical and experimental results is shown for both Prandtl numbers studied. The present work provides, for the first time, quantitative three-dimensional descriptions of spiral convection within a moderate-gap annulus containing air, flow structures preceding oscillation in a large-gap annulus for air, and unicellular flow development in a large-gap annulus for large Prandtl number fluids.

Turbulent mass transfer to a wall at high Schmidt numbers is controlled by the velocity field within the viscous sublayer. Measurements have been obtained of the root-mean-square fluctuating mass transfer coefficient and the frequency spectrum of the fluctuating mass transfer coefficient for a Schmidt number of about 2300. From an order-of-magnitude analysis it is concluded that flow fluctuations in the direction of mean flow have little effect on the mass transfer fluctuations. A comparison of the mass transfer spectrum with the spectrum of the component of the velocity gradient in the transverse direction sz reveals that the high-frequency portion of the sz spectrum is not effective in transferring mass. Approximate relations between the mass transfer spectrum and the sz spectrum are developed for high frequencies and for low frequencies.

We derive a time-stepping method for unsteady fully nonlinear two-dimensional motion of a two-layer fluid. Essential parts of the method are: use of Taylor series expansions of the prognostic equations, application of spatial finite difference formulae of high order, and application of Cauchy's theorem to solve the Laplace equation, where the latter is found to be advantageous in avoiding instability. The method is computationally very efficient. The model is applied to investigate unsteady trans-critical two-layer flow over a bottom topography. We are able to simulate a set of laboratory experiments on this problem described by Melville & Helfrich (1987), finding a very good agreement between the fully nonlinear model and the experiments, where they reported bad agreement with weakly nonlinear Korteweg–de Vries theories for interfacial waves. The unsteady transcritical regime is identified. In this regime, we find that an upstream undular bore is generated when the speed of the body is less than a certain value, which somewhat exceeds the critical speed. In the remaining regime, a train of solitary waves is generated upstream. In both cases a corresponding constant level of the interface behind the body is developed. We also perform a detailed investigation of upstream generation of solitary waves by a moving body, finding that wave trains with amplitude comparable to the thickness of the thinner layer are generated. The results indicate that weakly nonlinear theories in many cases have quite limited applications in modelling unsteady transcritical two-layer flows, and that a fully nonlinear method in general is required.

We investigate the basic mechanism whereby bars form in tidal channels or estuaries. The channel is assumed to be long enough to allow neglect of the effects of end conditions on the process of bar formation. In this respect, the object of the present analysis differs from that of Schuttelaars & de Swart (1999) who considered bars of length scaling with the finite length of the tidal channel. The channel bottom is assumed to be cohesionless and consisting of uniform sediments. Bars are shown to arise from a mechanism of instability of the erodible bed subject to the propagation of a tidal wave. Sediment is assumed to be transported both as bedload and as suspended load. A fully three-dimensional model is employed both for the hydrodynamics and for sediment transport. At the leading order of approximation considered, the effects of channel convergence, local inertia and Coriolis forces on bar instability are shown to be negligible. Unlike fluvial free bars, in the absence of mean currents tidal free bars are found to be non-migrating features (in the mean). Instability arises for large enough values of the mean width to depth ratio of the channel, for given mean values of the Shields parameter and of the relative channel roughness. The role of suspended load is such as to stabilize bars in the large-wavenumber range and destabilize them for small wavenumbers. Hence, for large values of the mean Shields stress, it turns out that the first critical mode (the alternate bar mode) is characterized by a very small value of the critical width to depth ratio. Furthermore, the order-m mode being characterized by a critical value of the width to depth ratio equal to m times the critical value for the first mode, it follows that for large values of the mean Shields stress several unstable modes are simultaneously excited for relatively low values of the aspect ratio. This suggests that the actual bar pattern observed in nature may arise from an interesting nonlinear competition among different unstable modes.

A structure capable of substantially amplifying water waves over a broad range of frequencies at selected locations is proposed. The structure consists of a small number of C-shaped cylinders in a line array, with the cylinder properties graded along the array. Using linear potential-flow theory, it is shown that the energy carried by a plane incident wave is amplified within specified cylinders for wavelengths comparable to the array length and for a range of incident directions. Transfer-matrix analysis is used to attribute the large amplifications to excitation of local Rayleigh–Bloch waves and gradual slowing down of their group velocity along the array.

An approximation to the profile of given area with smallest drag in laminar flow is obtained (for Reynolds numbers between 103 and 105). It was shown previously by Pironneau (1974) that the skin friction on such a profile has to satisfy certain optimality conditions; the method used is based on these results. It was found that the optimum profile is long and thin (thickness-to-chord ratio about 10%), the front end being shaped like a wedge of angle 90° and the rear end like a cusp. The drag is very close to the drag on a flat plate of equal length.

Experiments were performed in dump-stabilized axisymmetric flames to assess turbulent flame speeds (${S}_{T} $) and mean flamelets speeds (stretched laminar flame speeds, ${S}_{L, k} $). Fuels with significantly different thermodiffusive properties have been investigated, ranging from pure methane to syngas (${\mathrm{H} }_{2} \text{{\ndash}} \mathrm{CO} $ blends) and pure hydrogen, while the pressure was varied from 0.1 to 1.25 MPa. Flame front corrugation was measured with planar laser-induced fluorescence (PLIF) of the OH radical, while turbulence quantities were determined with particle image velocimetry (PIV). Two different analyses based on mass balance were performed on the acquired flame images. The first method assessed absolute values of turbulent flame speeds and the second method, by means of an improved fractal methodology, provided normalized turbulent flame speeds (${S}_{T} / {S}_{L, k} $). Deduced average Markstein numbers exhibited a strong dependence on pressure and hydrogen content of the reactive mixture. It was shown that preferential-diffusive-thermal (PDT) effects acted primarily on enhancing the stretched laminar flame speeds rather than on increasing the flame front corrugations. Interaction between flame front and turbulent eddies measured by the fractal dimension was shown to correlate with the eddy temporal activity.

We present experimental results on high-Reynolds-number motion of partially non-wetting liquid drops on inclined plane surfaces using: (i) water on fluoro-alkyl silane (FAS)-coated glass; and (ii) mercury on glass. The former is a high-hysteresis ($3{5}^{\circ } $) surface while the latter is a low-hysteresis one (${6}^{\circ } $). The water drop experiments have been conducted for capillary numbers $0. 0003\lt Ca\lt 0. 0075$ and for Reynolds numbers based on drop diameter $137\lt Re\lt 3142$. The ranges for mercury on glass experiments are $0. 0002\lt Ca\lt 0. 0023$ and $3037\lt Re\lt 20\hspace{0.167em} 069$. It is shown that when $Re\gg 1{0}^{3} $ for water and $Re\gg 10$ for mercury, a boundary layer flow model accounts for the observed velocities. A general expression for the dimensionless velocity of the drop, covering the whole $Re$ range, is derived, which scales with the modified Bond number ($B{o}_{m} $). This expression shows that at low $Re$, $Ca\sim B{o}_{m} $ and at large $Re$, $Ca \sqrt{Re} \sim B{o}_{m} $. The dynamic contact angle (${\theta }_{d} $) variation scales, at least to first-order, with $Ca$; the contact angle variation in water, corrected for the hysteresis, collapses onto the low-$Re$ data of LeGrand, Daerr & Limat (J. Fluid Mech., vol. 541, 2005, pp. 293–315). The receding contact angle variation of mercury has a slope very different from that in water, but the variation is practically linear with $Ca$. We compare our dynamic contact angle data to several models available in the literature. Most models can describe the data of LeGrand et al. (2005) for high-viscosity silicon oil, but often need unexpected values of parameters to describe our water and mercury data. In particular, a purely hydrodynamic description requires unphysically small values of slip length, while the molecular-kinetic model shows asymmetry between the wetting and dewetting, which is quite strong for mercury. The model by Shikhmurzaev (Intl J. Multiphase Flow, vol. 19, 1993, pp. 589–610) is able to group the data for the three fluids around a single curve, thereby restoring a certain symmetry, by using two adjustable parameters that have reasonable values. At larger velocities, the mercury drops undergo a change at the rear from an oval to a corner shape when viewed from above; the corner transition occurs at a finite receding contact angle. Water drops do not show such a clear transition from oval to corner shape. Instead, a direct transition from an oval shape to a rivulet appears to occur.

We investigate viscous fingering instabilities of the double-front system which results when fluid is injected into a porous medium containing fluid of a different composition and temperature. We describe a linear stability analysis based on an eigenfunction expansion method which enables us to investigate the structure of the discrete eigenvalue spectrum. We investigate the extent to which the properties of each front contribute to the tendency of the system to become unstable: we find that instabilities on the compositional front dominate because of the high ratio of thermal to solute diffusion. It is difficult for a stable compositional front to stabilize an unstable thermal front; however, this situation can result in a new fingering phenomenon in which the perturbations undergo coupled oscillations of growing amplitude.

The behaviour of an inviscid gravity current which is released from behind a lock and then propagates over a horizontal boundary at the base of a stratified ambient fluid is considered. An extension of the shallow-water formulation for a homogeneous ambient to the stratified case is developed, without using any additional adjustable parameters. Attention is focused on the initial ‘slumping’ stage of a rectangular current which is typified by a constant speed of propagation. The analytical results are in good agreement with, and give a firm theoretical interpretation of, the corresponding experiments and numerical simulations of Maxworthy et al. (2002). Finite-difference solutions of the Navier–Stokes equations, using a different technique from that used by Maxworthy et al. (2002), are also presented and provide both good agreement with their results and further validation of the present shallow-water approach. The differences between currents in a homogeneous and stratified ambient, and possible implementation of the results to other configurations, are discussed.

The method of multiple scales is used to determine the temporal and spatial variation of the amplitudes and phases of capillary-gravity waves in a deep liquid at or near the third-harmonic resonant wave-number. This case corresponds to a wavelength of 2·99 cm in deep water. The temporal variation shows that the motion is always bounded, and the general motion is an aperiodic travelling wave. The analysis shows that pure amplitude-modulated waves are not possible in this case contrary to the second-harmonic resonant case. Moreover, pure phase-modulated waves are periodic even near resonance because the non-linearity adjusts the phases to yield perfect resonance. These periodic waves are found to be unstable, in the sense that any disturbance would change them into aperiodic waves.

This paper presents theoretical results on the instability of a Blasius boundary layer perturbed by Klebanoff modes (i.e. the low-frequency streaks known to be induced by free-stream turbulence). Herein, the Klebanoff distortions are modelled as the signature of a three-dimensional convected gust that may be either isolated or periodic along the spanwise direction. Relatively weak Klebanoff fluctuations can produce $O(1)$ changes tothe near-wall curvature of the base flow profile and, hence, fundamentally alter the nature of its instability characteristics.The perturbed flow is shown to support instabilities that are predominantly inviscid and have significantly larger growth rates and characteristic frequencies than the Tollmien–Schlichting(T–S) modes of an unperturbed Blasius flow. The spanwise mode shape of instabilities in the perturbed flow is determined by theSchrödinger equation, with a potential function that corresponds to the skin friction perturbation due to the Klebanoff distortion. The growth ratesof these modes are determined by the near-wall torsion of the perturbed flow.The unsteadiness of the Klebanoff distortion is shown to be a crucial element in determining the overall instability characteristics.

A linear stability analysis of a shear flow in the presence of a continuous but steep variation of viscosity between two layers of nearly uniform viscosity is presented. This instability is investigated in relation to the known interfacial instability for the parallel flow of two superposed fluids of different viscosity. With respect to this configuration, the stability of our problem depends on two new parameters: the interface thickness $\delta$ and the Péclet number $\hbox{\it Pe}$, which accounts for diffusion effects when viscosity perturbations, coupled to the velocity perturbations, are allowed. We show that instability still exists for the continuous viscosity profile, provided the thickness of the interface is small enough and $\hbox{\it Pe}$ sufficiently large. Small and large wavenumbers are found to be stable, at variance with the discontinuous configuration. Of particular interest is also the possibility of obtaining higher growth rates than in the discontinuous case for suitable $\hbox{\it Pe}$ and $\delta$ ranges.

Experiments have been conducted to determine the effect of density stratification upon certain characteristic features of so-called Taylor columns. The interior structure of the homogeneous Taylor column is first of all described and compared with flow patterns obtained when the fluid is stratified. Qualitative features of the horizontal and vertical motion (in particular, the attenuation with height of the distortion created by the obstacle) are then described for values of the stratification parameter S (defined as S = N/2 Ω, whereN and Ω are the Brunt-Väisälä and rotation frequencies respectively) in the range 0 [les ] S [les ] 0·24. The effect of density stratification upon, specifically, the length of the column is then described. A working definition for the existence of a Taylor column in a given experimental situation is formulated, enabling the strength of the column to be quantified at a particular height above the obstacle. Using this method the column length is measured as a function of S in the range 0 [les ] S [les ] 0·24. It is shown that even very slight stratification is sufficient to produce noticeable modification of all aspects of the flow. In particular, the column length is considerably reduced by weak stratification.

Propagating surface evolution in isotropic turbulence is studied using velocity fields generated by direct numerical simulations. The statistics of tangential strain rate, fluid velocity, characteristic curvature and area-following propagating surface elements are investigated. The one-time statistics of strain rate and fluid velocity pass monotonically from Lagrangian value at low propagation speeds to Eulerian values at high speeds. The strain-rate statistics start deviating significantly from the Lagrangian values only for propagating velocities greater than the Kolmogorov velocity scale vη, whereas, with fluid-velocity statistics the deviation occurs only for velocities greater than the turbulence intensity u′. The average strain rate experienced by a propagating surface decreases from a positive value to near zero with increasing propagation velocity. The autocorrelation function and frequency spectrum of fluid velocity and strain rate scale as expected in the limits of small and large propagation velocities. It is also found that for the range of propagation velocities considered, an initially plane surface element in turbulence develops a cusp in finite time with probability nearly one. The evolution of curvature is studied using the concept of hitting time. Initially plane propagating surfaces end up being almost cylindrical in shape. Highly curved surface elements are associated with negative strain rates and small surface areas.

This paper pertains to the steady condensation on to, or evaporation from, a liquid droplet suspended in a mixture of its vapour and an inert gas. The treatment is from a kinetic theory viewpoint. The Maxwell moment method is used with Lees’ two stream Maxwellian representations for the distribution functions, to obtain a closed form result valid through the whole range from free molecule to continuum. In the diffusion control limit the formula reduces to approximate results obtained by Maxwell, Fuchs and others. In the limit where no inert gas is present the formula reduces to a result obtained earlier by the author. The formulae presented here for the mass and energy flux can now be used to calculate the growth rate of very small droplets, under a wide range of conditions.

A Mach-Zehnder interferometer was used to stud two-dimensional Bénard convection cells. The experiments were performed with distilled water and sea water in the region where density is a linear function of temperature. Two-dimensional convection rolls were formed with Rayleigh numbers as great as 23400. Reversal in the temperature profile was obtained for R/Rc ≥ 3·8, and an overshoot of about 6% was observed at R/Rc = 9·2 and 13·8. This agrees with the values predicted theoretically by Veronis (1966) for stress-free boundaries and Royal (1969) for rigid boundaries. This disagrees with the experimental results of Gille (1967), who reports an overshoot of only 1 ½% at R/Rc = 16. Many of the other results agree with those of other experimenters, such as the relation between the cell height-to-width ratio and Rayleigh number, the relation between the Nusselt number and Rayleigh number, and the value of the critical Rayleigh number.

We consider viscoelastic flows over topography in the presence of inertia. Such flows are modelled by an integral-boundary-layer approximation of the equations of motion and wall/free-surface boundary conditions. Steady states for flows over a step-down in topography are characterized by a capillary ridge immediately before the entrance to the step. A similar capillary ridge has also been observed for non-inertial Newtonian flows over topography. The height of the ridge is found to be a monotonically decreasing function of the Deborah number. Further, we examine the interaction between capillary ridges and excited non-equilibrium inertia/viscoelasticity-driven solitary pulses. We demonstrate that ridges have a profound influence on the drainage dynamics of such pulses: they accelerate the drainage process so that once the pulses pass the topographical feature they become equilibrium ones and are no longer excited.

Instabilities and flow characteristics in the far wake of a circular cylinder are examined through direct numerical simulations. The transitions to the two-layered and secondary vortex streets are quantified by a new method based on the time-averaged transverse velocity field. Two processes for the transition to the secondary vortex street are observed: (i) the merging of two same-sign vortices over a range of low Reynolds numbers ($Re$) between 200 and 300, and (ii) the pairing of two opposite-sign vortices, followed by the merging of the paired vortices into subsequent vortices, over a range of $Re$ between 400 and 1000. Single vortices may be generated between the merging cycles due to mismatch of the vortices. The irregular merging process results in flow irregularity and an additional frequency signal $f_{2}$ (in addition to the primary vortex shedding frequency $f_{1}$) in the two-layered and secondary vortex streets. In particular, a gradual energy transfer from $f_{1}$ to $f_{2}$ with distance downstream is observed in the two-layered vortex street prior to the merging. The frequency spectra of $f_{2}$ are broad-band for $Re=200$–300 but become increasingly sharp-peaked with increasing $Re$ because the vortex merging process becomes increasingly regular. The ratio of the sharp-peaked frequencies $f_{2}$ and $f_{1}$ is equal to the ratio of the numbers of vortices observed after and before the merging. The general conclusions drawn from a circular cylinder are expected to be applicable to other bluff bodies.

The equations for planar two-dimensional steady flow of an ideal dissociating gas are linearized, assuming small disturbances to a free stream in chemical equilibrium.

As an example of their solution, the flow past a sharp corner in a supersonic stream is evaluated and the variations of flow properties in the relaxation zone are found. Numerical illustrations are provided using an ‘oxygen-like’ ideal gas and comparisons made with a characteristics solution. The flow past a sharp corner can be studied in a conventional shock tube and it may be possible to verify the present theory experimentally. In particular it may prove feasible to use the results to obtain a measure of the reaction rates in the gas mixture.