It is well known that subgrid models such as Smagorinsky's cannot be used for the spatially growing simulation of the transition to turbulence of flat-plate boundary layers, unless large-amplitude perturbations are introduced at the upstream boundary: they are over-dissipative, and the flow simulated remains laminar. This is also the case for the structure-function model (SF) of Métais & Lesieur (1992). In the present paper we present a sequel to this model, the filtered-structure-function (FSF) model. It consists of removing the large-scale fluctuations of the field before computing its second-order structure function. Analytical arguments confirm the superiority of the FSF model over the SF model for large-eddy simulations of weakly unstable transitional flows. The FSF model is therefore used for the simulation of a quasi-incompressible (M∞ = 0.5) boundary layer developing spatially over an adiabatic flat plate, with a low level of upstream forcing. With the minimal resolution 650 × 32 × 20 grid points covering a range of streamwise Reynolds numbers Rex1 ε [3.4 × 105, 1.1 × 106], transition is obtained for 80 hours of time-processing on a CRAY 2 (whereas DNS of the whole transition takes about ten times longer). Statistics of the LES are found to be in acceptable agreement with experiments and empirical laws, in the laminar, transitional and turbulent parts of the domain. The dynamics of low-pressure and high-vorticity distributions is examined during transition, with particular emphasis on the neighbourhood of the critical layer (defined here as the height of the fluid travelling at a speed equal to the phase speed of the incoming Tollmien–Schlichting waves). Evidence is given that a subharmonic-type secondary instability grows, followed by a purely spanwise (i.e. time-independent) mode which yields peak-and-valley splitting and transition to turbulence. In the turbulent region, flow visualizations and local instantaneous profiles are provided. They confirm the presence of low- and high-speed streaks at the wall, weak hairpins stretched by the flow and bursting events. It is found that most of the vorticity is produced in the spanwise direction, at the wall, below the high-speed streaks. Isosurfaces of eddy viscosity confirm that the FSF model does not perturb transition much, and acts mostly in the vicinity of the hairpins.

Experiments have been performed, in capillary tubes, on the displacement of a viscous fluid (glycerine) by a less viscous one (a glycerine–water mixture) with which it is miscible in all proportions. A diagnostic measure of the amount of viscous fluid left behind on the tube wall has been found, for both vertical and horizontal tubes, as a function of the Péclet (Pe) and Atwood (At) numbers, as well as a parameter that is a measure of the relative importance of viscous and gravitational effects. The asymptotic value of this diagnostic quantity, for large Pe and an At of unity, has been found to agree with that found in immiscible displacements, while the agreement with the numerical results of Part 2 (Chen & Meiburg 1966), over the whole range of At, is very good. At values of the average Pe greater than 1000 a sharp interface existed so that it was possible to make direct comparisons between the present results and a prior experiment with immiscible fluids, in particular an effective surface tension at the diffusing interface could be evaluated. The effect of gravity on the amount of viscous fluid left on the tube wall has been investigated also, and compared with the results of Part 2. A subsidiary experiment has been performed to measure both the average value of the diffusion coefficient between pure glycerine and several glycerine–water mixtures, in order to be able to calculate a representative Péclet number for each experiment, and the local value as a function of the local concentration of glycerine, in the dilute limit.

Numerical simulations are presented which, in conjunction with the accompanying experimental investigation by Petitjeans & Maxworthy (1996), are intended to elucidate the miscible flow that is generated if a fluid of given viscosity and density displaces a second fluid of different such properties in a capillary tube or plane channel. The global features of the flow, such as the fraction of the displaced fluid left behind on the tube walls, are largely controlled by dimensionless quantities in the form of a Péclet number Pe, an Atwood number At, and a gravity parameter. However, further dimensionless parameters that arise from the dependence on the concentration of various physical properties, such as viscosity and the diffusion coefficient, result in significant effects as well.

The simulations identify two distinct Pe regimes, separated by a transitional region. For large values of Pe, typically above O(10), a quasi-steady finger forms, which persists for a time of O(Pe) before it starts to decay, and Poiseuille flow and Taylor dispersion are approached asymptotically. Depending on the strength of the gravitational forces, we observe a variety of topologically different streamline patterns, among them some that leak fluid from the finger tip and others with toroidal recirculation regions inside the finger. Simulations that account for the experimentally observed dependence of the diffusion coefficient on the concentration show the evolution of fingers that combine steep external concentration layers with smooth concentration fields on the inside. In the small-Pe regime, the flow decays from the start and asymptotically reaches Taylor dispersion after a time of O(Pe).

An attempt was made to evaluate the importance of the Korteweg stresses and the consequences of assuming a divergence-free velocity field. Scaling arguments indicate that these effects should be strongest when steep concentration fronts exist, i.e. at large values of Pe and At. However, when compared to the viscous stresses, Korteweg stresses may be relatively more important at lower values of these parameters, and we cannot exclude the possibility that minor discrepancies observed between simulations and experiments in these parameter regimes are partially due to these extra stresses.

The magnetohydrodynamic (MHD) flow through sharp 90° bends of rectangular cross-section, in which the flow turns from a direction almost perpendicular to the magnetic field to a direction almost aligned with the magnetic field, is investigated experimentally for high values of the Hartmann number M and of the interaction parameter N. The bend flow is characterized by strong three-dimensional effects causing a large pressure drop and large deformations in the velocity profile. Since such bends are basic elements of fusion reactors, the scaling laws of magnetohydrodynamic bends flows with the main flow parameters such as M and N as well as the sensitivity to small magnetic field inclinations are of major importance. The obtained experimental results are compared to those of an asymptotic theory.

In the case where one branch of the bend is perfectly aligned with the magnetic field good agreement between the results obtained by the asymptotic model and by the experiments was found at high M ≈ 8 × 10 and N ≈ 105 for pressure as well as for electric potentials on the duct surface. At lower values of N a significant influence of inertia has been detected. The pressure drop due to inertial effects was found to scale with N−1/3. The same – 1/3-power dependency on N has been found in the vicinity of the bend for the electric potentials at walls aligned with the magnetic field. At walls with a significant normal component of the field an influence neither of the Hartmann number nor of the interaction parameter has been found. This suggests that the inertial part of the pressure drop arises from inertial side layers, whereas the core flow remains inertialess and inviscid. A variation of the Hartmann number is of negligible influence compared to inertia effects with respect to pressure drop and surface potential distribution. The viscous part of the pressure drop scales with M−½.

Changes of the magnetic field orientation with respect to the bend lead in general to different flow patterns in the duct, because the electric current paths are changed. The inertia–electromagnetic interaction determines the magnitude of the inertial part of the pressure drop, which scales with N−1/3 for any magnetic field orientation. The dependence of the pressure drop on M remains proportional to M−½. With increasing M and N the measured data tend to those predicted by the asymptotic model. Local measurements within the liquid metal exhibit discrepancies with the model predictions for which no adequate explanation has been found. But they show that below a critical interaction parameter flow regions exist in which the flow is time dependent. These regions are highly localized, whereas the flow in the rest of the bend remains steady.

The improvement of heat transfer conditions in liquid-metal magnetohydrodynamic (MHD) flows is of prime importance for self-cooled fusion blanket design concepts. For many years the research was based on stationary inertialess assumptions since it was expected that time-dependent inertial flows would be suppressed by strong electromagnetic damping, especially in the extreme range of fusion relevant parameters. In the present analysis the stationary inertialess assumptions are abandoned. Nevertheless, the classical ideas usually used to obtain inertialess asymptotic solutions are drawn on. The basic inertial equations are reduced to a coupled two-dimensional problem by analytical integration along magnetic field lines. The magnetic field is responsible for a quasi-two-dimensional flow; the non-uniform distribution of the wall conductivity creates a wake-type profile, the MHD effect reducing to a particular forcing and friction. The solution for the two-dimensional variables, the field aligned component of vorticity, the stream function, and the electric potential are obtained by numerical methods. In a flat channel with non-uniform electrical wall conductivity, time-dependent solutions similar to the Kármán vortex street behind bluff bodies are possible. The onset of the vortex motion, i.e. the critical Reynolds number depends strongly on the strength of the magnetic field expressed by the Hartmann number. Stability analyses in viscous hydrodynamic wakes often use the approximation of a unidirectional flow which does not take into account the spatial evolution of the wake. The present problem exhibits a wake-type basic flow, which does not change along the flow path. It represents, therefore, an excellent example to which the simple linear analysis on the basis of Orr-Sommerfeld-type equations applies exactly. Once unstable, the flow first exhibits a regular time periodic vortex pattern which is rearranged further downstream. One can observe an elongation, pairing, or sometimes more complex merging of vortices. All these effects lead to larger flow structures with lower frequencies. The possibility for a creation and maintenance of time-dependent vortex-type flow pattern in MHD flows is demonstrated.

The interaction between a zero-pressure-gradient turbulent boundary layer and a pair of strong, common-flow-down, streamwise vortices with a sizeable velocity deficit is studied by large-eddy simulation. The subgrid-scale stresses are modelled by a localized dynamic eddy-viscosity model. The results agree well with experimental data. The vortices drastically distort the boundary layer, and produce large spanwise variations of the skin friction. The Reynolds stresses are highly three-dimensional. High levels of kinetic energy are found both in the upwash region and in the vortex core. The two secondary shear stresses are significant in the vortex region, with magnitudes comparable to the primary one. Turbulent transport from the immediate upwash region is partly responsible for the high levels of turbulent kinetic energy in the vortex core; its effect on the primary stress 〈u′v′〉 is less significant. The mean velocity gradients play an important role in the generation of 〈u′v′〉 in all regions, while they are negligible in the generation of turbulent kinetic energy in the vortex core. The pressure-strain correlations are generally of opposite sign to the production terms except in the vortex core, where they have the same sign as the production term in the budget of 〈u′v′〉. The results highlight the limitations of the eddy-viscosity assumption (in a Reynolds-averaged context) for flows of this type, as well as the excessive diffusion predicted by typical turbulence models.

Vorticity waves are wave-like motions occurring in various types of shear flows. We study the dynamics of these motions in alongshore shear currents in situations where it can be described within weakly nonlinear asymptotic theory. The principal mechanism of vorticity waves can be interpreted as potential vorticity conservation with the background vorticity gradient provided both by the mean current shear and the variation of depth. Under the assumption that the mean potential vorticity distibution is monotonic in the cross-shore direction, the nonlinear stage of the dynamics of weakly nonlinear vorticity waves, long in comparison with the current cross-shore scale, is found to be governed by an evolution equation of the generalized Benjamin–Ono type. The dispersive terms are given by an integro-differential operator with the kernel determined by the large-scale cross-shore depth and current dependence. The derived equations form a wide new class of nonlinear evolution equations. They all tend to the Benjamin–Ono equation in the short-wave limit, while in the long-wave limit their asymptotics depend on the specific form of the depth and current profiles. For a particular family of model bottom profiles the equations are ‘intermediate’ between Benjamin–Ono and Korteweg–de Vries equations, but are distinct from the Joseph intermediate equation. Solitary-wave solutions to the equations for these depth profiles are found to decay exponentially. Taking into account coastline inhomogeneity or/and alongshore depth variations adds a linear forcing term to the evolution equation, thus providing an effective generation mechanism for vorticity waves.

The effects of molecular diffusivities of heat and mass on the counter-gradient scalar and momentum transfer in strongly stable stratification are experimentally investigated in unsheared and sheared stratified water mixing-layer flows downstream of turbulence-generating grids. Experiments are carried out in two kinds of stably stratified water flows. In the case of thermal stratification, the difference between the turbulent fluxes of an active scalar (heat with the Prandtl number of Pr ≈ 6) and a passive scalar (mass with the Schmidt number of Sc ≈ 600) is investigated. In the case of salt stratification, the effects of the molecular diffusion of the active scalar (salt) with a very high Schmidt number of Sc ≈ 600 on the counter-gradient scalar transfer is studied. Comparisons of the effects of molecular diffusivities are also made between thermally stratified water and air (Pr ≈ 0.7) flows. Further, the effects of mean shear on the counter-gradient scalar and momentum transfer are investigated for both stratified cases. Instantaneous temperature, concentration and streamwise and vertical velocities are simultaneously measured using a combined technique with a resistance thermometer, a laser-induced fluorescence method, and a laser-Doppler velocimeter with high spatial resolution. Turbulent scalar fluxes, joint probability density functions, and cospectra are estimated.

The results of the first case show that both active heat and passive mass develop counter-gradient fluxes but that the counter-gradient flux of passive mass is about 10% larger than that of active heat, mostly due to molecular diffusion effects at small scales. The counter-gradient scalar transfer mechanism in stable stratification can be explained by considering the relative balance between the available potential energy and the turbulent kinetic energy as in Schumann (1987). In thermally and salt-stratified water mixing-layer flows with the active scalars of high Prandtl and Schmidt numbers, the buoyancy-induced motions with finger-like structures first contribute to the counter-gradient scalar fluxes at small scales, and then the large-scale motions, which bring fluid back to its original levels, generate the counter-gradient fluxes at large scales. The contribution of the small-scale motions to the counter-gradient fluxes in stratified water flows is quite different from that in stratified air flows. The higher Prandtl or Schmidt number of the active scalar generates both the stronger buoyancy effects and the longer time-oscillation period of the counter-gradient scalar fluxes. The time-oscillation occurs at large scales but the counter-gradient fluxes at small scales persist without oscillating. The mean shear acts to reduce the counter-gradient scalar and momentum transfer at large scales, and therefore the counter-gradient fluxes in sheared stratified flows can be seen only in very strong stratification. The behaviour of the counter-gradient momentum flux in strong stratification is quite similar to that of the counter-gradient scalar flux.

This paper reports an experimental study of the effects of an externally applied electric field on the dynamics of drop formation in the dripping mode from a vertical metal capillary. The fluid issuing out of the capillary is a viscous liquid, the surrounding ambient fluid is air, and the electric field is generated by establishing a potential difference between the capillary and a horizontal, circular electrode of large radius placed downstream of the capillary outlet. By means of an ultra-high-speed video system that is capable of recording up to 12000 frames per second, special attention is paid to the dynamics of the liquid thread that connects the primary drop that is about to detach and fall from the capillary to the rest of the conical liquid mass that is hanging from it. The experiments show that as the strength of the electric field increases, the volume of the primary drop decreases whereas the maximum length attained by the thread increases. The reduction in the volume of primary drops and the increase in the length of threads occur because the effective electromechanical surface tension of the fluid interface falls as the field strength rises. For the highly conducting drops of aqueous NaCl solutions studied in this work, the increase in thread length is due solely to the rising importance of normal electric stress relative to the falling importance of surface tension. However, as the conductivity of the drop liquid decreases, the thread length is further increased on account of the stabilizing influence exerted by the increasing electric shear stress that acts on the charged liquid–gas interface. Two new phenomena are also reported that have profound implications for electrohydrodynamics and practical applications. First, it is shown that whereas the liquid thread always ruptures at its downstream end in the absence of an applied electric field or when the field strength is low, it ruptures at its upstream end when the field strength is sufficiently high. Since satellite drops are produced directly from the thread once both of its ends have ruptured, the change in the mechanism of breakup with field strength influences the dynamics and fate of satellite drops. Second, it is demonstrated that the generation of satellites, which are often undesirable in applications, can be suppressed by the judicious application of an electric field. This is accomplished by using a field of moderate strength to induce charges of the opposite sign on the nearby surfaces of the satellite drop and the liquid that remains pendant from the tube following thread rupture. At high field strengths, induced charge effects are too weak to compete with net charge effects: the satellite is repelled by the pendant drop and falls under gravity as a distinct entity.

In Jeffery–Hamel flow, the motion of a viscous incompressible fluid between rigid plane walls, unidirectional flow is impossible if the angle between the walls exceeds a critical value of 2α2 which depends on the Reynolds number. In this paper the nonlinear development of the flow near this critical value is studied through numerical solutions of the two-dimensional Navier–-Stokes equations for flow in divergent channels with piecewise straight walls. It is found that if the angle between the walls exceeds 2α2 then Jeffery–-Hamel flow does not occur, and the solution takes the form of a large-amplitude wave with eddies attached alternately to the upper and lower walls. When viewed in the appropriate coordinate system, far downstream the wave has constant wavelength and strength, although, physically, there is a linear increase in wavelength with distance downstream, i.e. the wavelength is proportional to the channel width. If the angle between the walls is less than 2α2, then the existence (or otherwise) of the wave depends on the conditions near the inlet, in particular the local geometry of the channel. Jeffery–-Hamel flow is obtained downstream of the inlet for angles well below 2α2, but close to but below the critical value, solutions have been obtained with the wave extending (infinitely) far downstream. The wavelengths obtained numerically were compared with those from linear theory with spatially developing steady modes. No agreement was found: the wavelengths from the steady Navier–-Stokes solutions are significantly larger than that predicted by the theory. However, in other important aspects the results of this study are consistent with those from previous studies of the development/existence of Jeffery–-Hamel flow, in particular as regards the importance of the upstream conditions and the subcritical nature of the spatial development of the flow near the critical boundary in the Reynolds number–wall angle parameter space.

A high-speed video system was used to study the interaction between sediment particles and turbulence in the wall region of an open channel flow with both smooth and transitionally rough beds. In smooth flows, particles immersed within the viscous sublayer were seen to accumulate along low-speed wall streaks; apparently due to the presence of quasi-streamwise vortices in the wall region. Larger particles did not tend to group along streaks, however their velocity was observed to respond to the streaky structure of the flow velocity in the wall region. In transitionally rough flows particle sorting was not observed. Coherent flow structures in the form of shear layers typically observed in the near-wall region interacted with sediment particles lying on the channel bottom, resulting in the particles being entrained into suspension. Although there has been some speculation that this process would not be effective in entraining particles totally immersed in the viscous sublayer, the results obtained demonstrate the opposite. The entrainment mechanism appears to be the same independent of the roughness condition of the bottom wall, smooth or transitionally rough. In the latter case, however, hiding effects tend to preclude the entrainment of particles with sizes finer than that of the roughness elements. The analysis of particle velocity during entrainment shows that the streamwise component tends to be much smaller than the local mean flow velocity, while the vertical component tends to be much larger than the local standard deviation of the vertical flow velocity fluctuations, which would indicate that such particles are responding to rather extreme flow ejection events.

Parallel inviscid O(1) shear interacting with O(ε) spanwise-independent neutral rotational Rayleigh waves are used to model turbulent boundary layer flow over small-amplitude rigid wavy terrain. Of specific interest is the instability of the flow to spanwise-periodic initially exponentially growing longitudinal vortex modes via the Craik–Leibovich CL2-O(1) instability mechanism and whether it is this instability mechanism that gives rise to longitudinal vortices evident in the recent experiments of Gong et al. (1996). In modelling the flow, wave and turbulence length scales are assumed sufficiently disparate to cause minimal interaction. This allows the primary mean velocity profile to be specified. Two profiles were chosen: a power law and the logarithmic law of the wall. Important in wave–mean interactions of this class are the effect of wave-induced fluctuations upon the mean state and the influence of the developing mean flow on the fluctuating part of the motion. The former is described by a generalized Lagrangian-mean formulation; the latter by a modified Rayleigh equation. Together they comprise an eigenvalue problem for the growth rate appropriate to the initial stages of the instability. Both primary mean flows are unstable to longitudinal vortex form in the presence of Rayleigh waves whose amplitudes diminish with altitude. Moreover the interaction is most unstable for streamwise wavenumbers α = O(1), the growth rate increasing with increased spanwise wavenumber. In comparing the results with experiment, it is first shown that spanwise-independent waves excited in Gong et al.'s experiment depict velocity fluctuations whose amplitudes diminish with altitude in accord with those for appropriate Rayleigh waves. Concordantly, the longitudinal vortices depict transverse velocity components that are weaker by a factor of ε than the axial perturbation and are observed to grow at a rate consistent with exponential growth. All are key features of CL2-O(1), although the observed growth rate is not in accord with the maximal suggested by inviscid instability theory. Rather it appears that the spanwise wavenumber takes a value at which energy is extracted from the mean motion in an optimal volume-averaged sense while minimizing energy loss to both viscous dissipation and small-scale turbulence. It is concluded that the CL2-O(1) instability mechanism is physically realizable and that the data of Gong et al. represent the first documented observations thereof.

We show that Kolmogorov's (1941b) inertial-range law for the third-order structure function can be derived from a dynamical equation including pressure terms and mean flow gradient terms. A new inertial-range law, relating the two-point pressure–velocity correlation to the single-point pressure–strain tensor, is also derived. This law shows that the two-point pressure–velocity correlation, just like the third-order structure function, grows linearly with the separation distance in the inertial range. The physical meaning of both this law and Kolmogorov's law is illustrated by a Fourier analysis. An inertial-range law is also derived for the third-order velocity–enstrophy structure function of two-dimensional turbulence. It is suggested that the second-order vorticity structure function of two-dimensional turbulence is constant and scales with $\epsilon ^{2/3}_\omega$ in the enstrophy inertial range, εω being the enstrophy dissipation. Owing to the constancy of this law, it does not imply a Fourier-space inertial-range law, and therefore it is not equivalent to the k−1 law for the enstrophy spectrum, suggested by Kraichnan (1967) and Batchelor (1969).

Simulations of decaying two-dimensional turbulence suggest that the one-point vorticity density has the self-similar form $P_\omega \sim t\;\;f(\omega t)$

implied by Batchelor's (1969) similarity hypothesis, except in the tails. Specifically, similarity holds for |ω| < ωm, while pω falls off rapidly above. The upper bound of the similarity range, ωm, is also nearly conserved in high-Reynolds-number hyperviscosity simulations and appears to be related to the average amplitude of the most intense vortices (McWilliams 1990), which was an important ingredient in the vortex scaling theory of Carnevale et al. (1991).

The universal function f also appears to be hyperbolic, i.e. $f(x) \sim c/2\vert x \vert^{1+q_c}$ for |x| > x*, where qc = 0.4 and x* = 70, which along with the truncated similarity form implies a phase transition in the vorticity moments $\langle \vert \omega\vert ^q\rangle \sim \left\{\begin{array}{ll} c_q t^{-q}, & -1 < q < q_c\cr c(q - q_c)^{-1} \omega _m^{q-q_c} t^{-q_c} & q > q_c, \end{array}\right.$ between the self-similar 'background sea' and the coherent vortices. Here Cq and c are universal. Low-order moments are therefore consistent with Batchelor's similarity hypothesis whereas high-order moments are similar to those predicted by Carnevale et al. (1991). A self-similar but less well-founded expression for the energy spectrum is also proposed.

It is also argued that ωc = x*/t represents 'mean sea-level', i.e. the (average) threshold separating the vortices and the sea, and that there is a spectrum of vortices with amplitudes in the range (ωs,ωm). The total area occupied by vortices is also found to remain constant in time, with losses due to mergers of large-amplitude vortices being balanced by gains due to production of weak vortices. By contrast, the area occupied by vortices above afixed threshold decays in time as observed by McWilliams (1990).

Since Bödewadt's (1940) seminal work on the boundary layer flow produced by a fluid in solid-body rotation over a stationary disk of infinite radius there has been much interest in determining the stability of such flows. To date, it appears that there is no theoretical study of the stability of Bödewadt's self-similar solution to perturbations that are not self-similar. Experimental studies have been compromised due to the difficulty in establishing these steady flows in the laboratory. Savaç (1983, 1987) has studied the endwall boundary layers of flow in a circular cylinder following impulsive spin-down. During the first few radians of rotation, the endwall boundary layers have a structure very similar to Bödewadt layers. For certain conditions, SavaÇ has observed a series of axisymmetric waves travelling radially inwards in the endwall boundary layers. The conjecture is that these waves represent a mode of instability of the Bödewadt layer. Within a few radians of rotation however, the centrifugal instability of the sidewall layer dominates the spin-down process and the endwall waves are difficult to examine further.

Here, the impulsive spin-down problem is examined numerically for Savaç’ (1983, 1987) conditions and good agreement with his experiments is achieved. New experimental results are also presented, which include quantitative space-time information regarding the axisymmetric waves. These agree well with both the numerics and the earlier experimental work. Further, a related problem is considered numerically. This flow is also initially in solid-body rotation, but only the endwalls are impulsively stopped, keeping the sidewall rotating. This results in a flow virtually identical to the usual spin-down flow for the first few radians of rotation, except in the immediate vicinity of the sidewall. The sidewall layer is no longer centrifugally unstable and the circular waves on the endwalls are observed without the influence of the sidewall instability.

The stability of steady axisymmetric convection in cylinders heated from below and insulated laterally is investigated numerically using a mixed finite-difference/Chebyshev collocation method to solve the base flow and the linear stability equations. Linear stability boundaries are given for radius to height ratios γ from 0.9 to 1.56 and for Prandtl numbers Pr = 0.02 and Pr = 1. Depending on γ and Pr, the azimuthal wavenumber of the critical mode may be m = 1, 2, 3, or 4. The dependence of the critical Rayleigh number on the aspect ratio and the instability mechanisms are explained by analysing the energy transfer to the critical modes for selected cases. In addition to these results the onset of buoyant convection in liquid bridges with stress-free conditions on the cylindrical surface is considered. For insulating thermal boundary conditions, the onset of convection is never axisymmetric and the critical azimuthal wavenumber increases monotonically with γ. The critical Rayleigh number is less then 1708 for most aspect ratios.

We investigate the Eulerian and Lagrangian spectral scaling properties of vortex tubes, and the consistency of these properties with Tennekes’ (1975) statistical advection analysis and universal equilibrium arguments. We consider three different vortex tubes with power-law wavenumber spectra: a Burgers vortex tube, an inviscid Lundgren single spiral vortex sheet, and a vortex tube solution of the Euler equation. While the Burgers vortex is a steady solution of the Navier–Stokes equation, the other two are unsteady solutions of, respectively, the Navier–Stokes and the Euler equations. In our numerical experiments we study the vortex tubes by subjecting each of them to external ‘large-scale’ sinusoidal advection of characteristic frequency f and length scale ρ.

Not only do we find that the Eulerian frequency spectrum ϕE(ω) can be derived from the wavenumber spectrum E(k) using the simple Tennekes advection relation ω ∼ k for all finite advection frequencies f when the vortex is steady, but also when the vortex is unsteady, and in the Lundgren case even when f = 0 owing to the self-advection of the Lundgren vortex by its own differential rotation.

An analytical calculation using the method of stationary phases for f = 0 shows that for large enough Reynolds numbers the combination of strain with differential rotation implies that ϕL(ω) ∼ ω−2+Const for large values of ω. We verify numerically that ϕL(ω) does not change when f ≠ 0. With the Burgers vortex tube we are in a position to investigate the spectral broadening of the Eulerian frequency spectrum with respect to the Lagrangian frequency spectrum. A spectral broadening does exist but is different from the spectral broadening predicted by Tennekes (1975).