Transition in a fully developed circular pipe flow was investigated experimentally by introducing periodic perturbations. The simultaneous excitation of helical modes having indices m = ±1, ±2 and ±3 was chosen. The experiments revealed that the late stage of transition is accompanied by the formation of streaky structures that are associated with peaks and valleys in the azimuthal distribution of the streamwise velocity disturbance. The breakdown to turbulence starts with the appearance of spikes in the temporal traces of the velocity. Spectral characteristics of these spikes and the direction of their propagation relative to the wall are similar to those in boundary layers. Analysis of the data suggests the existence of a high-shear layer in the instantaneous velocity profile.

Additional experiments in which a very weak, steady flow was added locally to the periodic axisymmetric perturbation were also carried out. These experiments resulted in the generation of a single peak in the azimuthal distribution of the disturbance amplitude. The characteristics of the transition process (spikes, vortical patterns etc.) within this peak were similar to ones observed in the helical excitation experiments. Based on these results one may conclude that late stages of transition in a pipe flow and in a boundary layer are similar. The present report is part of an ongoing investigation that was initiated by Eliahou, Tumin & Wygnanski (1998a).

A general theoretical account is proposed for the zigzag instability of a vertical columnar vortex pair recently discovered in a strongly stratified experiment.

The linear inviscid stability of the Lamb–Chaplygin vortex pair is analysed by a multiple-scale expansion analysis for small horizontal Froude number (Fh = U/LhN, where U is the magnitude of the horizontal velocity, Lh the horizontal lengthscale and N the Brunt–Väisälä frequency) and small vertical Froude number (Fv = U/LvN, where Lv is the vertical lengthscale) using the scaling of the equations of motion introduced by Riley, Metcalfe & Weissman (1981). In the limit Fv = 0, these equations reduce to two-dimensional Euler equations for the horizontal velocity with undetermined vertical dependence. Thus, at leading order, neutral modes of the flow are associated, among others, to translational and rotational invariances in each horizontal plane. To each broken invariance is related a phase variable that may vary freely along the vertical. Conservation of mass and potential vorticity impose at higher order the evolution equations governing the phase variables that we derive for Fh [Lt ] 1 and Fv [Lt ] 1 in the spirit of phase dynamics techniques established for periodic patterns. In agreement with the experimental observations, this asymptotic analysis shows the existence of an instability consisting of a vertically modulated rotation and a translation of the columnar vortex pair perpendicular to the travelling direction. The dispersion relation as well as the spatial eigenmode of the zigzag instability are determined. The analysis predicts that the most amplified vertical wavelength should scale as U/N and the maximum growth rate as U/Lh.

Our main finding is thus that the typical thickness of the ensuing layers will be such that Fv = O(1) and not Fv [Lt ] 1 as assumed by Riley et al. (1981) and Lilly (1983). This implies that such strongly stratified flows are not described by two- dimensional horizontal equations. These results may help to understand the layering commonly observed in stratified turbulence and the fundamental differences with strictly two-dimensional turbulence.

This paper investigates the three-dimensional stability of a Lamb–Chaplygin columnar vertical vortex pair as a function of the vertical wavenumber kz, horizontal Froude number Fh, Reynolds number Re and Schmidt number Sc. The horizontal Froude number Fh (Fh = U/NR, where U is the dipole travelling velocity, R the dipole radius and N the Brunt–Väisälä frequency) is varied in the range [0.033, ∞[ and three set of Reynolds-Schmidt numbers are investigated: {Re = 10 000, Sc = 1}, Re = 1000, Sc = 1}, {Re = 200, Sc = 637}. In the whole range of Fh and Re, the dominant mode is always antisymmetric with respect to the middle plane between the vortices but its physical nature and properties change when Fh is varied. An elliptic instability prevails for Fh > 0.25, independently of the Reynolds number. It manifests itself by the bending of each vortex core in the opposite direction to the vortex periphery. The growth rate of the elliptic instability is reduced by stratification effects but its spatial structure is almost unaffected. In the range 0.2 < Fh < 0.25, a continuous transition occurs from the elliptic instability to a different instability called zigzag instability. The transitional range Fhc = 0.2–0.25 is in good agreement with the value Fh = 0.22 at which the elliptic instability of an infinite uniform vortex is suppressed by the stratification. The zigzag instability dominates for Fh [les ] 0.2 and corresponds to a vertically modulated bending and twisting of the whole vortex pair. The experimental evidence for this zigzag instability in a strongly stratified fluid reported in the first part of this study (Billant & Chomaz 2000a) are therefore confirmed and extended. The numerically calculated wavelength and growth rate for low Reynolds number compare well with experimental measurements.

The present numerical stability analysis fully agrees with the inviscid asymptotic analysis carried out in the second part of this investigation (Billant & Chomaz 2000b) for small Froude number Fh and long wavelength. This confirms that the zigzag instability is related to the breaking of translational and rotational invariances. As predicted, the growth rate of the zigzag instability is observed to be self-similar with respect to the variable Fhkz, implying that the maximum growth rate is independent of Fh while the most amplified dimensional wavenumber varies with N/U. The numerically computed eigenmode and dispersion relation are in striking agreement with the analytical results.

In this article we present new experimental and theoretical results which were obtained for the flow between two concentric cylinders, with the inner one rotating and in the presence of an axial, stable density stratification. This system is characterized by two control parameters: one destabilizing, the rotation rate of the inner cylinder; and the other stabilizing, the stratification.

Two oscillatory linear stability analyses assuming axisymmetric flow conditions are presented. First an eigenmode linear stability analysis is performed, using the small-gap approximation. The solutions obtained give insight into the instability mechanisms and indicate the existence of a confined internal gravity wave mode at the onset of instability. In the second stability analysis, only diffusion is neglected, predicting accurately the instability threshold as well as the critical pulsation for all the stratifications used in the experiments.

Experiments show that the basic, purely azimuthal flow (circular Couette flow) is destabilized through a supercritical Hopf bifurcation to an oscillatory flow of confined internal gravity waves, in excellent agreement with the linear stability analysis. The secondary bifurcation, which takes the system to a pattern of drifting non-axisymmetric vortices, is a saddle-node bifurcation. The proposed bifurcation diagram shows a global bifurcation, and explains the discrepancies between previous experimental and numerical results. For slightly larger values of the rotation rate, weakly turbulent spectra are obtained, indicating an early appearance of weak turbulence: stationary structures and defects coexist. Moreover, in this regime, there is a large distribution of structure sizes. Visualizations of the next regime exhibit constant-wavelength structures and fluid exchange between neighbouring cells, similar to wavy vortices. Their existence is explained by a simple energy argument.

The generalization of the bifurcation diagram to hydrodynamic systems with one destabilizing and one stabilizing control parameter is discussed. A qualitative argument is derived to discriminate between oscillatory and stationary onset of instability in the general case.

The instabilities of Saffman–Taylor viscous fingers are revisited experimentally in the standard linear channel as well as in wedges of angle θ0. The local destabilization of a finger occurs by a splitting of its tip and results in the formation of two branches separated by a fjord. It is shown that, in a first approximation, the central line of a fjord follows a curve normal to the successive profiles of stable fingers. These normal curves are computed analytically for the Saffman–Taylor finger in a linear cell and numerically for the wedges. The length of a fjord is critically sensitive to the position of the initial destabilization of the finger. The nearer to the tip it occurs, the longer the fjord will be. Assuming a uniform spatial distribution of the disturbances in the central part of the finger front it is possible to predict the size distribution of the lateral branches. In the linear channel the probability of branches larger than the channel width is negligible. For wedges of increasing angle the probability of large secondary branches increases. Finally, for wedges with θ0 larger than approximately 90° infinite fjords separating two long-lived structures are observed. Our experimental results also suggest a generalization of the definition of virtual cells. With this new definition it is possible to show that the increasing complexity of the patterns corresponds to a hierarchy of virtual cells of various sizes.

A theoretical model of an aeroengine intake–fan system is developed in order to show the existence of acoustic resonance in the intake. In general this phenomenon can be linked to instabilities in aircraft engine inlets.

The model incorporates a slowly varying duct intake and accounts for the swirling flow downstream of the fan. The slow axial variation in cross-section gives rise to turning points where upstream-propagating acoustic modes are totally reflected into downstream-propagating modes. The effect of the swirling flow downstream can be to cut off a mode which is cut on upstream of the fan. It is shown that these two aspects of the flow, coupled with the effects of the fan (represented by an actuator disc), can lead to acoustic modes becoming trapped in the intake, thus giving rise to pure acoustic resonance.

A whole range of system parameters, such as axial, fan and swirl Mach numbers, which satisfy the conditions for resonance are identified. The effects of a stationary blade row behind the fan are also considered leading to a second family of resonant states. In addition we find resonance due to reflection of acoustic modes at the open (inlet) end of the duct.

The sources of sound during the interactions of two identical two-dimensional inviscid vortex pairs are investigated numerically by using the vortex sound theory and the method of contour dynamics. The sound sources are identified and then separated into two independent components, which represent the contributions from the vortex centroid dynamics and the microscopic vortex core dynamics. Results show that the sound generation mechanism of the latter is independent of the type of vortex pair interaction, while that of the former depends on the jerks, accelerations and vortex forces on the vortex pairs. The power developed by the vortex forces is found to be important in the generation of sound when the vortex cores are severely deformed and their centroids are close to each other. The isolated source terms also explain the appearance of wavy oscillations on the time variations of the sound source strengths in the vortex ring and the two-dimensional vortex interaction systems.

An erodible surface exposed to supercritical flow often devolves into a series of steps that migrate slowly upstream. Each step delineates a headcut with an associated hydraulic jump. These steps can form in a bed of cohesive material which, once eroded, is carried downstream as washload without redeposition. Here the case of purely erosional, one-dimensional periodic, or cyclic steps in cohesive material is considered. The St. Venant shallow-water equations combined with a formulation for sediment erosion are used to construct a complete theory of the erosional case. The solution allows wavelength, wave height, migration speed and bed and water surface profiles to be determined as functions of imposed parameters. The analysis also admits a solution for a solitary step, or single headcut of self-preserving form.

A linear stability analysis of incipient channellization on hillslopes is performed using the shallow-water equations and a description of the erosion of a cohesive bed. The base state consists of a laterally uniform Froude-subcritical sheet flow down a smooth, downward-concave hillslope profile. The downstream boundary condition consists of the imposition of a Froude number of unity. The process of channellization is thus driven from the downstream end. The flow and bed profiles describe a base state that migrates at constant, slow speed in the upstream direction due to bed erosion. Transverse perturbations corresponding to a succession of parallel incipient channels are introduced. It is found that these perturbations grow in time, so describing incipient channellization, only when the characteristic spacing between incipient channels is on the order of 6–100 times the Froude-critical depth divided by the resistance coefficient. The characteristic wavelength associated with maximum perturbation growth rate is found to scale as 10 times the Froude-critical depth divided by the resistance coefficient. Evaluating the friction coefficient as on the order of 0.01, an estimate of incipient channel spacing on the order of 1000 times the Froude-critical depth is obtained. The analysis reveals that downstream-driven channellization becomes more difficult as (a) the critical shear stress required to erode the bed becomes so large that it approaches the Froude-critical shear stress reached at the downstream boundary and (b) the Froude number of the subcritical equilibrium flow attained far upstream approaches unity. Alternative mechanisms must be invoked to explain channellization on slopes high enough to maintain Froude-supercritical sheet flow.

The desorption of bubbles during solidification of a melt occurs in processes as diverse as the making of ice cubes, the formation of igneous rocks and the casting of metals. In both the metal casting and rock formation processes, careful observation of the final solid suggests that the desorbed bubbles often form regular spatial patterns. Understanding and quantifying the mechanisms by which such patterns arise is important. In the geological context, comparison between field measurements and the predictions of a model will allow geologists to estimate in-situ magma properties. In the metal casting context, engineers would like to be able to specify mould geometries and cooling conditions to ensure that the distribution of bubbles will not compromise the strength of critical sections of the casting.

In the present study, we develop a detailed mathematical model to predict the distribution of desorbed bubbles in a solidified melt. Our new model builds upon previous knowledge on this phenomenon in the geological context (Toramaru et al. 1996, 1997). We describe desorption of a dissolved gas in a semi-infinite melt, solidified by a one-dimensional heat flux. In the absence of convection, the transfer of heat and solute occurs mainly by a diffusive mechanism and the crystallization proceeds most rapidly near the cooled boundary. The crystals formed contain almost no dissolved gas and hence the concentration of gas dissolved in the melt increases progressively towards the cooled boundary. Diffusion of dissolved gas from the crystallizing zone is slow and, as a result, the local melt becomes supersaturated and gas bubbles desorb. The full equations for this coupled solidification and desorption processes are solved numerically.

We find that bubbles desorb forming a sequence of layers parallel to the cooled boundary. The spacing between these bubble layers increases geometrically from the cooled boundary. We give a physical interpretation for this geometric pattern and analyse the effect of physical parameters on the layer spacing. We show that our theoretical model captures the important physical mechanisms involved in the solidification and desorption processes by comparing its predictions with available measurements from a geological formation.

The interaction between magnetic fields and convection is interesting both because of its astrophysical importance and because the nonlinear Lorentz force leads to an especially rich variety of behaviour. We present several sets of computational results for magnetoconvection in a square box, with periodic lateral boundary conditions, that show transitions from steady convection with an ordered planform through a regime with intermittent bursts to complicated spatiotemporal behaviour. The constraints imposed by the square lattice are relaxed as the aspect ratio is increased. In wide boxes we find a new regime, in which regions with strong fields are separated from regions with vigorous convection. We show also how considerations of symmetry and associated group theory can be used to explain the nature of these transitions and the sequence in which the relevant bifurcations occur.

The dependence of the Nusselt number Nu on the Rayleigh Ra and Prandtl Pr number is determined for 104 < Ra < 107 and 0.07 < Pr < 7 using DNS with no-slip upper and lower boundaries and free-slip sidewalls in a 8 × 8 × 2 box. Nusselt numbers, velocity scales and boundary layer thicknesses are calculated. For Nu there are good comparisons with experimental data and scaling laws for all the cases, including Ra2/7 laws at Pr = 0.7 and Pr = 7 and at low Pr, a Ra1/4 regime. Calculations at Pr = 0.3 predict a new Nu ∼ Ra2/7 regime at slightly higher Ra than the Pr = 0.07 calculations reported here and the mercury Pr = 0.025 experiments.