The spatial evolution of the disturbances that lead to boundary-layer transition on a swept wedge is computed by large-eddy simulations (LES). Stationary and travelling crossflow-vortex disturbances are generated using steady and random-amplitude suction and blowing on the wedge. For a fixed initial amplitude of the stationary vortex and low-amplitude unsteady disturbances, the LES show the evolution of stationary-dominated crossflow disturbances similar to previous simulations and experiments: linear amplification is followed by vortex roll-over and doubly inflectional velocity profiles just prior to transition. A high-frequency secondary instability is associated with the double inflection points in the velocity profiles. The harmonic modes of the primary disturbance were found to be amplified, while no energy was found in any subharmonic mode. The physical phenomena were significantly different when the stationary and travelling vortices have comparable initial amplitudes: in this case, the vortex roll-over does not occur and transition is dominated by the travelling-wave component.

We study spatial and spatio-temporal pattern formation emergent from reaction–diffusion–advection systems formed by considering reaction–diffusion systems coupled to prescribed fluid flows. While there have been a number of studies on the planar dynamics of such systems and the resulting instabilities and spatio-temporal patterning in the plane, less has been done on complicated flows in complex domains. We consider a general approach for the study of bounded domains in order to model two- and three-dimensional geometries which are more likely to be of relevance for modelling dynamics within fluid vessels used in experiments. Considering a variety of problem geometries with finite cross-sections, such as two-dimensional channels, three-dimensional ducts and three-dimensional pipes, we demonstrate the role cross-section geometry plays in pattern formation under such systems. We find that the generic instability is that of an oscillatory or wave Turing instability, resulting in patterns which change in time, often being advected with the fluid flow. As in previous works, we observe a change in patterns formed when progressing from zero to weak to strong advection for uniform advection across the domain, with particularly strong advection destroying patterns. One novel finding is that heterogeneous fluid flow can induce qualitatively different patterns across the domain. For instance, Poiseuille flow with maximal advection in the centre of a vessel and zero advection at the boundary of a vessel is shown to exhibit patterns in the centre of the vessel which are different from patterns near the boundary, with differences attributed to the differential local advection within each region of the vessel. Additionally, we observe sheared patterns, which appear due to gradients in the fluid velocity, and cannot be obtained via any kind of uniform flow. Finally we also explore flow in more complex domains, including wavy-walled channels, continuous stirred-tank reactors, U-shaped pipes and a toroidal domain, in order to demonstrate behaviours when the flow is both heterogeneous and bidirectional, as well as to demonstrate that our results still apply for complex finite domains. Our analysis suggests that such non-trivial advection results in moving patterns which are more complex than observed in simpler reaction–diffusion–advection, and may be more characteristic of realistic flow regimes in biological media.

A nonlinear non-stationary problem of the development of two-dimensional convective motions in a plane horizontal fluid layer bounded by free surfaces and heated from below is studied. Horizontal dependences of the velocity and temperature are not assumed to be periodic or almost periodic; they have continuous wavenumber spectra and are represented by Fourier integrals. Vertical dependence of each variable is represented by several Fourier harmonics. Spectrum evolution is studied by means of the numerical integration of an initial-value problem. Initial disturbances of two qualitatively different classes are considered; viz those localized in horizontal extent within a narrow part of the layer as well as having the form of a roll set throughout a rather wide region. In both cases convective flows tend to evolve towards the arrays of well-established rolls with the same horizontal wavenumber ap, which apparently seems to be the physically optimal (preferred) one for two-dimensional convection at given Rayleigh number R and Prandtl number P. We see no indications that the deviation of the initial roll-disturbance wavenumber from ap must exceed some threshold value for giving rise to the flow readjustment in wavenumber. At sufficiently small P a decrease in ap with increasing R is observed which agrees with experiments. A comparison is made of various theoretical models and various experimental circumstances, whence it is seen that the less stable the flow (i.e. the easier it can readjust), the better the preferred wavenumber manifests itself. In particular, roll flows periodic in a horizontal direction all over the infinite layer are highly stable, and when only such flows are considered, as has most often been the case, the preferred wavenumber is not revealed.

We consider the parallel flow of two immiscible fluids in a Hele-Shaw cell. The evolution of disturbances on the fluid interfaces is studied both theoretically and experimentally in the large-capillary-number limit. It is shown that such interfaces support wave motion, the amplitude of which for long waves is governed by a set of KdV and Airy equations. The waves are dispersive provided that the fluids have unequal viscosities and that the space occupied by the inner fluid does not pertain to the Saffman-Taylor conditions (symmetric interfaces with half-width spacing). Experiments conducted in a long and narrow Hele-Shaw cell appear to validate the theory in both the symmetric and the non-symmetric cases.

We examine weak second-sound waves in He II at temperatures and pressures near one of the zeros of the Khalatnikov steepening parameter Γ. An extension of the reductive perturbation scheme of Taniuti & Wei is employed to derive the cubic Burgers’ equation governing these waves. It is shown that mixed nonlinearity may occur in disturbances in which the local value of Γ remains strictly positive or strictly negative. Further new results include expressions for the shock speed, shock structure and the conditions under which the shock thickness increases, rather than decreases, with strength. The fundamental existence conditions for temperature shocks are also delineated and related to the shock disintegration process observed in experimental studies.

Moore's (1978) equation for following the evolution of a thin layer of uniform vorticity in two dimensions is extended to the case of a non-uniform, instantaneously known, vorticity distribution, using the method of matched asymptotic expansions. In general, the vorticity distribution satisfies a boundary-layer equation. This has a similarity solution in the case of a vortex layer of small thickness in a viscous fluid. Using this solution, an equation of motion of a diffusing vortex sheet is obtained. The equation retains the simplicity of Birkhoff's integro-differential equation for a vortex sheet, while incorporating the effect of viscous diffusion approximately. The equation is used to study the growth of long waves on a Rayleigh layer.

In the present paper we obtain a theoretical expression for the temperature fluctuation spectrum for a Prandtl number of approximately one and for the region where both viscosity and molecular heat conductivity are important. An asymptotic theory for very large wavenumbers of the temperature spectrum in the turbulent flow is developed. The assumption of smallness of the correlation coefficient between the product of small-scale components of velocities at two points and the corresponding product of small-scale components of temperatures is used. The results of simultaneous measurements of streamwise velocity fluctuations and temperature fluctuations carried out in the plane of symmetry of a two-dimensional wake behind a slightly heated cylinder (Rλ = 270) in a wind tunnel is consistent with this assumption.

The main result of the theory developed is the appearance of a bump in the temperature spectrum for a Prandtl number of approximately one.

Slightly heated swirling jets have been investigated experimentally in order to determine the effect of swirl degree on the mean and turbulent flow pattern. In particular, it has been shown that the swirl action strongly intensifies the turbulent momentum and heat transport processes in the close vicinity of the jet outlet and tends to quench them further downstream.

In this work we consider how nonlinear hydrodynamic effects can lead to a mean force of interaction between two spheres of equal radius a undergoing translational fluctuations parallel or perpendicular to their line of centres. Motivated by amplitudes and Reynolds numbers characteristic of Brownian motion in colloidal systems, nonlinearities due to motion of the boundaries and to inertia throughout the fluid are treated as regular perturbations of the time-dependent Stokes equations. This formulation ultimately leads to a prescription for computing, at leading order, the time-average nonlinear force for the case of pure oscillatory modes – which represents the Fourier decomposition of more general motions. The associated hydrodynamic problems are solved numerically using a least-squares boundary singularity method. Frequency-dependent results over the whole spectrum are presented for a sphere-sphere gap equal to one radius; illustrative calculations are also carried out at other separations. Subsequently we extend the analysis of nonlinear drift to a Langevin equation formulation of the more complex problem of stochastic motion due to thermal fluctuations in the suspending fluid, i.e. Brownian motion. By integrating (numerically) over the spectrum of frequencies, we quantify how the mutual interactions of all translational disturbance modes give rise, on ensemble average, to a stochastic nonlinear force of interaction between the particles. It is particularly interesting that this net interaction – arising from a zero-mean random force – is of O(1) on the Brownian scale kT/a, even though it represents a small O(Re) correction at each frequency of pure oscillations. Finally, we discuss how the presence of stochastic nonlinear drift would lead to non-uniform equilibrium distributions of dilute colloidal suspensions, unless one adds to the random force in the Langevin equation a cancelling non-zero mean component.

An analytical and computational study is presented on solution properties of strongly nonlinear vortex/wave interactions involving Tollmien–Schlichting waves, in boundary-layer transition. The longitudinal vortex part, i.e. the total mean flow, is governed by a three-dimensional vortex system but coupled, through an effective spanwise slip condition at the surface, with the accompanying wave part, so that both the vortex and the wave parts are unknowns. Terminal forms of the space-marching or time-marching problem are proposed first, yielding either a lift-off separation singularity or a strong-attachment singularity. Second, a similarity version of the complete system is addressed numerically and analytically. This leads to a number of interesting solution features as the typical wave pressure is increased into the strongly nonlinear regime. In particular, lift-off separation and attachment forms seem to emerge which are analogous with those proposed above. The flow developments beyond the terminal forms are discussed, together with the links of the work with recent computational results and, tentatively, with experimental observations including the creation of lambda vortices (as a form of lift-off separation).

The effect of wall cooling on hypersonic boundary-layer separation near a compression ramp is considered. Two cases are identified corresponding to the value of the average Mach number $\overline{M}$ across the upstream boundary layer approaching the compression ramp. The flow is referred to as supercritical for $\overline > 1$ and subcritical for $\overline{M} < 1$. The interaction is described by triple-deck theory, and numerical results are given for both cases for various ramp angles and levels of wall cooling. The effect of wall cooling on the absolute instability described recently by Cassel, Ruban & Walker (1995) for an uncooled wall is of particular interest; a stabilizing effect is observed for supercritical boundary layers, but a strong destabilizing influence occurs in the subcritical case. Wall cooling also influences the location and size of the separated region. For supercritical flow, progressive wall cooling reduces the size of the recirculating-flow region, the separation point moves downstream, and upstream influence is diminished. In contrast for the subcritical case downstream influence is reduced with increased cooling. In either situation, a sufficient level of wall cooling eliminates separation altogether for the ramp angles considered. The present numerical results closely confirm the strong wall cooling theory of Kerimbekov, Ruban & Walker (1994).

Results of an experimental and numerical study of parametrically excited nonlinear cross-waves in the vicinity of the cut-off frequency, are reported. Experiments are performed at three cross-wave modes and in the whole range of existence of cross-waves. Numerical studies are based on the solution of the nonlinear Schrödinger equation with a boundary condition at the wavemaker which corresponds to parametric excitation. The validity of the scaling procedure adopted in the model is verified experimentally. Dissipation is incorporated in the model equation and in the wavemaker boundary condition. The influence of the wave breaking on the range of existence of cross-waves is discussed and the relation between the maximum possible steepness of cross-waves and the limits of their existence is obtained.

The temporal evolution of baroclinic basin-scale waves in a rotating circular basin following an initial forcing event is investigated using a laboratory study. Experiments conducted in the circular domain containing a two-layer fluid with a flat bottom and vertical sidewalls demonstrate that the response is essentially linear with frictional effects at the boundaries steadily dissipating wave energy. Experiments conducted in the same configuration but with the addition of simple topographic features, either a radially protruding cape or a bathymetric ridge, exhibit wave/topography interactions that result in the formation of an eddy field and an offshore flow, respectively. The rate of wave decay, as well as the amount of horizontal mixing occurring within the basin, is significantly enhanced by such interactions. The results of this study are then considered in terms of their implications for the baroclinic basin-scale wave energy pathways in large stratified lakes influenced by the Earth's rotation.

The injection of liquid from a central well into a partially saturated aquifer of finite thickness is described using similarity solutions. The solutions illustrate that injection leads to a growing zone around the source in which the fluid fills the whole depth of the aquifer. Beyond this zone, the current adjusts to the depth in the far field as the fluid slumps under gravity. The radial extent of the zone in which the aquifer is fully flooded depends on the ratio of the buoyancy-driven flow speed to the pressure-driven flow speed associated with the injection. New laboratory experiments, using a model porous medium, support the model predictions in the case of an initially unsaturated layer. The analysis is then developed to allow for a fully saturated aquifer, containing fluid of lower density than the injectate, and a further class of similarity solutions is developed. Again, these are shown to be consistent with new laboratory experiments. In concluding, we briefly consider how the results may be combined, to explore the self-similar dynamics of a relatively dense fluid injected into an aquifer which is partially saturated with fluid of smaller density.

The method of characteristics is used to calculate the unsteady one-dimensional flow produced by the convection of a temperature disturbance, contained between two contact surfaces, in high subsonic flow through a nozzle. The results show the development of the pressure disturbances which are associated with the force perturbation needed to accelerate the temperature disturbance at the same rate as the surrounding gas and propagate in both the upstream and downstream directions. Using a simple flow model, expressions for the mean pressure disturbances are derived and shown to be in good agreement with numerical solutions. Other numerical solutions which show the effect of changing the various flow parameters and the nozzle shape are presented. In all of the cases considered, the pressure disturbances caused by a 10 % change in the inlet temperature are significantly large and may be an important part of ‘excess’ noise.

In this work, an attempt is made to gain a better understanding of the breakage of low-viscosity drops in turbulent flows by determining the dynamics of deformation of an inviscid drop in response to a pressure variation acting on the drop surface. Known scaling relationships between wavenumbers and frequencies, and between pressure fluctuations and velocity fluctuations in the inertial subrange are used in characterizing the pressure fluctuation. The existence of a maximum stable drop diameter dmax follows once scaling laws of turbulent flow are used to correlate the magnitude of the disruptive forces with the duration for which they act.

Two undetermined dimensionless quantities, both of order unity, appear in the equations of continuity, motion, and the boundary conditions in terms of pressure fluctuations applied on the surface. One is a constant of proportionality relating root-mean-square values of pressure and velocity differences between two points separated by a distance l. The other is a Weber number based on turbulent stresses acting on the drop and the resisting stresses in the drop due to interfacial tension. The former is set equal to 1, and the latter is determined by studying the interaction of a drop of diameter equal to dmax with a pressure fluctuation of length scale equal to the drop diameter. The model is then used to study the breakage of drops of diameter greater than dmax and those with densities different from that of the suspending fluid.

It is found that, at least during breakage of a drop of diameter greater than dmax by interaction with a fluctuation of equal length scale, a satellite drop is always formed between two larger drops. When very large drops are broken by smaller-length-scale fluctuations, highly deformed shapes are produced suggesting the possibility of further fragmentation due to instabilities. The model predicts that as the dispersed-phase density increases, dmax decreases.

A linear, stationary internal-wave model is used to determine the vertical propagation of gravity waves produced by submerged moving bodies in a stratified fluid. The turbulent wake generated by the moving object is taken into account parameterically. If the stability of the stratification above the body increases (i.e. the body is under a thermocline) resonance effects occur. Large-amplitude internal waves are generated and they give rise to strong divergence fields at the water surface. These will persist far downstream of the object and are potentially detectable. Limitations of the linear model and comparisons with experimental results available in the literature are discussed.

The instability of an initially homogeneous suspension of spheroids, settling due to gravity, is reconsidered. For non-spherical particles, previous studies in the literature report that normal-mode density perturbations of maximum growth rate are those of arbitrarily large, horizontal wavelength. Using the ‘mixture theory’ for two-phase flow we show that the maximum growth rate for horizontal density perturbations is obtained for a finite wavelength if the inertia of the bulk motion associated with the normal-mode density perturbation is accounted for. We find that for long wavelengths, , the growth rate approaches zero as . The maximum growth rate is obtained for , where is the axis perpendicular to the axis of rotational symmetry of the spheroid, is the undisturbed volume fraction of particles and , typically , is a Reynolds number of the bulk motion on a typical length scale and a velocity scale on the order of the undisturbed settling speed. The theoretical results for the wavelength selection agree qualitatively well with previous experimental results in the literature of measured correlation lengths of vertical streamers in settling fibre suspensions.