We use a numerical approach to study the receptivity of the boundary layer flow over a slender body with a leading edge of finite radius of curvature to small streamwise velocity fluctuations of a given frequency. The body of interest is a parabola in order to exclude jumps in curvature, which are known sites of receptivity and which occur on elliptic leading edges matched to finite-thickness at plates. The infinitesimally thin flat plate is the limiting solution for the parabola as the nose radius of curvature goes to zero. The formulation of the problem allows the two-dimensional unsteady Navier–Stokes equations in stream function and vorticity form to be converted to two steady systems of equations describing the basic (nonlinear) flow and the perturbation (linear) flow. The results for the basic flow are in excellent agreement with those in the literature. As expected, the perturbation flow was found to be a combination of an unsteady Stokes flow and Orr–Sommerfeld modes. To separate these, the unsteady Stokes flow was solved separately and subtracted from the total perturbation flow. We found agreement with the streamwise wavelengths and locations of Branches I and II of the linear stability neutral growth curve for Tollmien–Schlichting waves. The results showed an increase in the leading-edge receptivity with decreasing nose radius, with the maximum occurring for an infinitely sharp flat plate. The receptivity coefficient was also found to increase with angle of attack. These results were in qualitative agreement with the asymptotic analysis of Hammerton & Kerschen (1992). Good quantitative agreement was also found with the recent numerical results of Fuciarelli (1997), and the experimental results of Saric, Wei & Rasmussen (1994).

Experiments on rotating channel flow, where both the primary (induced by a Coriolis instability) and the secondary instability are triggered independently, are described, focusing on the development of a secondary instability consisting of high-frequency travelling waves and their subsequent breakdown. Detailed hot-wire velocity measurements of the secondary disturbance are made and the phase speed and growth rate for various frequencies are determined. It is shown that the frequency of highest growth rate is close to that which is observed for naturally developing flow. Some information on the later stages in the transition process is obtained from frequency spectra, which show that interaction between various modes gives rise to stochastic low-frequency disturbances, which may play an important role in the transition process. A theoretical model of the disturbance structure is described which is used to explain some of the measured results and also allows the determination of the disturbance cross-stream flow field from only streamwise velocity measurements.

Low-Prandtl-number convection is investigated in vertical axisymmetric turbulent buoyant sodium jets discharging into a slowly moving ambient. Measurements of mean velocity, mean temperature and temperature fluctuations are performed simultaneously using a miniature permanent-magnet flowmeter probe. By varying the ratio of momentum to buoyancy flux, or the densimetric Froude number, different intensities of buoyancy are obtained giving a range of conditions encompassing forced-convection jets, buoyant jets and plumes. In line with the classical properties of jets the radial velocity and temperature profiles can be described by the Gaussian function, independent of the flow regime, at all axial measuring positions. The axial decay of the centreline mean velocity for sodium is the same as for fluids of higher Prandtl number, governed by power laws with indices of −1 for forced convection, −2/3 for the transitional buoyant region and −1/3 for plume flow. In contrast, the centreline mean temperatures for sodium plumes decrease with a power of −1 compared with the −5/3 decay for fluids of higher Prandtl number. The different behaviour in sodium is due to the dominance of molecular diffusion in heat transport, while momentum transport is dictated by turbulent diffusion, which gives a similarity solution for forced-convection jets but not for buoyant jets or plumes. The radial profiles of the temperature r.m.s. values can be described by an axisymmetric curve with two maxima, independent of the flow regime, at all axial measuring positions and the two maxima are more pronounced than in conventional fluids. The temperature fluctuations are analysed to give statistical parameters such as minimum and maximum values, skewness, flatness, probability density functions and spectral distribution. The spectral distributions display both a convective subrange and the conductive subrange predicted for fluids of low Prandtl number. Integral length scales of the temperature fluctuations are evaluated and found to be significantly smaller than turbulent velocity scales.

The three-dimensional flow around a spherical bubble moving steadily in a viscous linear shear flow is studied numerically by solving the full Navier–Stokes equations. The bubble surface is assumed to be clean so that the outer flow obeys a zero-shear-stress condition and does not induce any rotation of the bubble. The main goal of the present study is to provide a complete description of the lift force experienced by the bubble and of the mechanisms responsible for this force over a wide range of Reynolds number (0.1[les ]Re[les ]500, Re being based on the bubble diameter) and shear rate (0[les ]Sr[les ]1, Sr being the ratio between the velocity difference across the bubble and the relative velocity). For that purpose the structure of the flow field, the influence of the Reynolds number on the streamwise vorticity field and the distribution of the tangential velocities at the surface of the bubble are first studied in detail. It is shown that the latter distribution which plays a central role in the production of the lift force is dramatically dependent on viscous effects. The numerical results concerning the lift coefficient reveal very different behaviours at low and high Reynolds numbers. These two asymptotic regimes shed light on the respective roles played by the vorticity produced at the bubble surface and by that contained in the undisturbed flow. At low Reynolds number it is found that the lift coefficient depends strongly on both the Reynolds number and the shear rate. In contrast, for moderate to high Reynolds numbers these dependences are found to be very weak. The numerical values obtained for the lift coefficient agree very well with available asymptotic results in the low- and high-Reynolds-number limits. The range of validity of these asymptotic solutions is specified by varying the characteristic parameters of the problem and examining the corresponding evolution of the lift coefficient. The numerical results are also used for obtaining empirical correlations useful for practical calculations at finite Reynolds number. The transient behaviour of the lift force is then examined. It is found that, starting from the undisturbed flow, the value of the lift force at short time differs from its steady value, even when the Reynolds number is high, because the vorticity field needs a finite time to reach its steady distribution. This finding is confirmed by an analytical derivation of the initial value of the lift coefficient in an inviscid shear flow. Finally, a specific investigation of the evolution of the lift and drag coefficients with the shear rate at high Reynolds number is carried out. It is found that when the shear rate becomes large, i.e. Sr=O(1), a small but consistent decrease of the lift coefficient occurs while a very significant increase of the drag coefficient, essentially produced by the modifications of the pressure distribution, is observed. Some of the foregoing results are used to show that the well-known equality between the added mass coefficient and the lift coefficient holds only in the limit of weak shears and nearly steady flows.

Horizontal exchange flows driven by spatial variation of buoyancy fluxes through the water surface are found in a variety of geophysical situations. In all examples of such flows the timescale characterizing the variability of the buoyancy fluxes is important and it can vary greatly in magnitude. In this laboratory study we focus on the effects of this unsteadiness of the buoyancy forcing and its influence on the resulting flushing and circulation processes in a cavity. The experiments described all start with destabilizing forcing of the flows, but the buoyancy fluxes are switched to stabilizing forcing at three different times spanning the major timescales characterizing the resulting cavity-scale flows. For destabilizing forcing, these timescales are the flushing time of the region of forcing, and the filling-box timescale, the time for the cavity-scale flow to reach steady state. When the forcing is stabilizing, the major timescale is the time for the fluid in the exchange flow to pass once through the forcing boundary layer. This too is a measure of the time to reach steady state, but it is generally distinct from the filling-box time. When a switch is made from destabilizing to stabilizing buoyancy flux, inertia is important and affects the approach to steady state of the subsequent flow. Velocities of the discharges from the end regions, whether forced in destabilizing or stabilizing ways, scaled as u∼(Bl)1/3 (where B is the forcing buoyancy flux and l is the length of the forcing region) in accordance with Phillips' (1966) results. Discharges with destabilizing and stabilizing forcing were, respectively, Q−∼(Bl)1/3H and Q+∼(Bl)1/3δ (where H is the depth below or above the forcing plate and δ is the boundary layer thickness). Thus Q−/Q+>O(1) provided H>O(δ), as was certainly the case in the experiments reported, demonstrating the overall importance of the flushing processes occurring during periods of cooling or destabilizing forcing.

Steady, gravity water waves on a constant-over-depth current, progressing over a slowly varying bed, are studied with the purpose of connecting the wave action flux concept with conventional energy flux considerations. The analysis is two-dimensional and dissipation is neglected. A new relation between integral properties containing the energy flux referred to a ‘global’ level, the so-called mean energy level, gives the surprising result that this flux is simply the product of absolute angular frequency and wave action flux. An alternative, less physical, proof of this result is also presented. A general equation for the action velocity is set out and for linear waves shown to equal a well-known expression. Also presented are new expressions for relative phase velocity in terms of kinetic energy and mean momentum for the wave, and the kinetic energy in terms of the characteristic velocities for the combined wave and current motion. In the Appendix a simple relation between energy and action fluxes for small-amplitude waves on a linear shear current is found which resembles the irrotational theory, finite-height result. A possible extension of this relation to finite-height waves on a general shear current is discussed.

A derivation is given of the amplitude equations governing pattern formation in surface tension gradient-driven Bénard–Marangoni convection. The amplitude equations are obtained from the continuity, the Navier–Stokes and the Fourier equations in the Boussinesq approximation neglecting surface deformation and buoyancy. The system is a shallow liquid layer heated from below, confined below by a rigid plane and above with a free surface whose surface tension linearly depends on temperature. The amplitude equations of the convective modes are equations of the Ginzburg–Landau type with resonant advective non-variational terms. Generally, and in agreement with experiment, above threshold solutions of the equations correspond to an hexagonal convective structure in which the fluid rises in the centre of the cells. We also analytically study the dynamics of pattern formation leading not only to hexagons but also to squares or rolls depending on the various dimensionless parameters like Prandtl number, and the Marangoni and Biot numbers at the boundaries. We show that a transition from an hexagonal structure to a square pattern is possible. We also determine conditions for alternating, oscillatory transition between hexagons and rolls. Moreover, we also show that as the system of these amplitude equations is non-variational the asymptotic behaviour (t→∞) may not correspond to a steady convective pattern. Finally, we have determined the Eckhaus band for hexagonal patterns and we show that the non-variational terms in the amplitude equations enlarge this band of allowable modes. The analytical results have been checked by numerical integration of the amplitude equations in a square container. Like in experiments, numerics shows the emergence of different hexagons, squares and rolls according to values given to the parameters of the system.

We find a quantitative approximation which explains the appearance and amplification of surface waves in a highly viscous fluid when it is subjected to vertical accelerations (Faraday's instability). Although stationary surface waves with frequency equal to half of the frequency of the excitation are observed in fluids of different kinematical viscosities we show here that the mechanism which produces the instability is very different for a highly viscous fluid as compared with a weakly viscous fluid. This is achieved by deriving an exact equation for the linear evolution of the surface which is non-local in time. We show that for a highly viscous fluid this equation becomes local and of second order and is then a Mathieu equation which is different from the one found for weak viscosity. Analysing the new equation we find an intimate relation with the Rayleigh–Taylor instability.

A detailed investigation of the hydrodynamic stability to transverse linear disturbances of a steady, one-dimensional detonation in an ideal gas undergoing an irreversible, unimolecular reaction with an Arrhenius rate constant is conducted via a normal-mode analysis. The method of solution is an iterative shooting technique which integrates between the detonation shock and the reaction equilibrium point. Variations in the disturbance growth rates and frequencies with transverse wavenumber, together with two-dimensional neutral stability curves and boundaries for all unstable low- and higher frequency modes, are obtained for varying detonation bifurcation parameters. These include the detonation overdrive, chemical heat release and reaction activation energy. Spatial perturbation eigenfunction behaviour and phase and group velocities are also obtained for selected sets of unstable modes. Results are presented for both Chapman–Jouguet and overdriven detonation velocities. Comparisons between the earlier pointwise determination of stability and interpolated neutral stability boundaries obtained by Erpenbeck are made. Possible physical mechanisms which govern the wavenumber selection underlying the initial onset of either regular or irregular cell patterns are also discussed.

The Galerkin and the finite element methods are used to study the onset of the double-diffusive convective regime in a rectangular porous cavity. The two vertical walls of the cavity are subject to constant fluxes of heat and solute while the two horizontal ones are impermeable and adiabatic. The analysis deals with the particular situation where the buoyancy forces induced by the thermal and solutal effects are opposing each other and of equal intensity. For this situation, a steady rest state solution corresponding to a purely diffusive regime is possible. To demonstrate whether the solution is stable or unstable, a linear stability analysis is carried out to describe the oscillatory and the stationary instability in terms of the Lewis number, Le, normalized porosity, ε, and the enclosure aspect ratio, A. Using the Galerkin finite element method, it is shown that there exists a supercritical Rayleigh number, RsupTC, for the onset of the supercritical convection and an overstable Rayleigh number, RoverTC, at which overstability may arise. Furthermore, the overstable regime is shown to exist up to a critical Rayleigh number, RoscTC, at which the transition from the oscillatory to direct mode convection occurs. By using an analytical method based on the parallel flow approximation, the convective heat and mass transfer is studied. It is found that, below the supercritical Rayleigh number, RsupTC, there exists a subcritical Rayleigh number, RsubTC, at which a stable convective solution bifurcates from the rest state through finite-amplitude convection. In the range of the governing parameters considered in this study, a good agreement is observed between the analytical predictions and the finite element solution of the full governing equations. In addition, it is found that, for a given value of the governing parameters, the converged solution can be permanent or oscillatory, depending on the porous-medium porosity value, ε.

The non-hydrostatic description of three-dimensional waves incident over a plane beach at a long straight coastline is considered in terms of the inverse Kontorovich–Lebedev integral transform. This is seen as a natural extension to earlier work by the author where the two-dimensional (normal incidence) flow is expressed as an inverse Mellin transform, and similar simplifications in the description here are encountered. In particular computations are undertaken for a variety of beach slopes of the form α=π/2m where m is an integer and for a range of incidence angles. These computations have previously only been practical for beaches whose slope α is regarded as asymptotically small thereby allowing versions of the mild-slope equation to be used. For the chosen slope angles, the solution is established with rigour and methods of estimating near- and far-field asymptotics arise naturally in this discussion. For the case of perfect reflection, a previously known solution is recovered in closed form as a finite sum of exponential terms, and a shoreline ‘amplification factor’ aγ is considered for these waves and is computed for a range of beach slopes through the entire spectrum of incidence angles. It is shown analytically that, in the limit of normal incidence, the value of aγ approaches the well-known classical result a0=m1/2 and, for glancing incidence, Whitham's (1979) result is confirmed where the value approaches either 1 or 0 depending on whether the beach angle is or is not an angle at which a new Ursell edge wave mode appears (m odd).

As applications of the new development, comprehensive near-field expansions for arbitrary reflection are written and verified by computation. These permit the construction of refracted wavefronts and wave rays for arbitrary beach slope without the usual phase velocity assumptions. Instability is indicated at very oblique incidence where nonlinear modelling (Peregrine & Ryrie 1983) predicts ‘anomalous refraction’. Results are presented graphically and computation of derivatives of the potential enables estimation of the (second order) set-down seaward of the breaker zone. This is found to decrease as wave attack becomes increasingly oblique.

Numerous laboratory and field experiments on nonlinear surface wave trains propagating in deep water (Lake & Yuen 1978; Ramamonjiarisoa & Mollo-Christensen 1979; Mollo-Christensen & Ramamonjiarisoa 1982; Melville 1983) have showed a specific wave modulation that so far has not been explained by nonlinear theories. Typical effects were the so-called wave phase reversals, negative frequencies, and crest pairing, experimentally observed in some portions of the modulated wave train. In the present paper, in order to explain these modulation manifestations, the equations for wavenumber, frequency, and velocity potential amplitude are derived consistently in the third-order approximation related to the wave steepness. The resulting model generalizes, for instance, the well-known nonlinear Schrödinger equation theory, to which it transforms at certain values of the governing parameters.

The stationary solutions to the derived set of equations are found in quadrature and then analysed. Within well-defined ranges of the model parameters, these solutions explicitly manifest the above-mentioned wave modulation effects. In particular, they show the wave phase kinks to arise on areas of relatively small free-surface displacement in complete accordance with the experiments.

The model with deeply modulated wavenumber and frequency permits one also to analyse the appropriately short surface wavepackets and modulation periods. In this case, a variety of new interesting wave solutions arises revealing complicated alteration of smooth and rough portions of the free surface. Of special importance are solitary waves, naturally generalizing envelope solitons of the nonlinear Schrödinger equation, but having a varying frequency (as a principle of the proposed theory) and a non-zero wave ‘pedestal’ at infinity. These new types of modulated surface waves should be also observable in laboratory tanks and under field conditions, because the relevant free parameters of theory are not extreme.

Linear eigenvalue calculations and spatial direct numerical simulations (DNS) of disturbance growth in Falkner–Skan–Cooke (FSC) boundary layers have been performed. The growth rates of the small-amplitude disturbances obtained from the DNS calculations show differences compared to linear local theory, i.e. non-parallel effects are present. With higher amplitude initial disturbances in the DNS calculations, saturated cross-flow vortices are obtained. In these vortices strong shear layers appear. When a small random disturbance is added to a saturated cross-flow vortex, a low-frequency mode is found located at the bottom shear layer of the cross-flow vortex and a high-frequency secondary instability is found at the upper shear layer of the cross-flow vortex. The growth rates of the secondary instabilities are found from detailed analysis of simulations of single-frequency disturbances. The low-frequency disturbance is amplified throughout the domain, but with a lower growth rate than the high-frequency disturbance, which is amplified only once the cross-flow vortices have started to saturate. The high-frequency disturbance has a growth rate that is considerably higher than the growth rates for the primary instabilities, and it is conjectured that the onset of the high-frequency instability is well correlated with the start of transition.

The motion of two drops in a uniform electric field is considered using the leaky dielectric model. The drops are assumed to have no native charge and a dielectrophoretic effect favours translation of the drops toward one another. However, circulatory flows that stem from electrohydrodynamic stresses may either act with or against this dielectrophoretic effect. Consequently, both prolate and oblate drop deformations may be generated and significant deformation occurs near drop contact owing to enhancement of the local electric field. For sufficiently widely spaced drops, electrohydrodynamic flows dominate direct electrical interactions so drops may be pushed apart, though closely spaced drops almost always move together as a result of the electrical interaction or deformation.