ALBERT

Description: The response of a Gaussian vortex to a weak time-dependent external strain field is studied numerically. The cases of an impulsive strain, an on-off step function, and a continuous random strain are considered. Transfers of enstrophy between mean and azimuthal components are observed, and the results are compared with an analogous passive scalar model and with Kida's elliptical vortex model. A 'rebound' phenomenon is seen: after enstrophy is transferred from mean to azimuthal component by the external straining field, there is a subsequent transfer of enstrophy back from the azimuthal component to the mean. Analytical support is given for this phenomenon using Lundgren's asymptotic formulation of the spiral wind-up of vorticity. Finally the decay of the vortex under a continuous random external strain is studied numerically and compared with the passive scalar model. The vorticity distribution decays more slowly than the scalar because of the rebound phenomenon.

Description: The properties of gravito-inertial waves propagating in a stably stratified rotating spherical shell or sphere are investigated using the Boussinesq approximation. In the perfect fluid limit, these modes obey a second-order partial differential equation of mixed type. Characteristics propagating in the hyperbolic domain are shown to follow three kinds of orbits: quasi-periodic orbits which cover the whole hyperbolic domain; periodic orbits which are strongly attractive; and finally, orbits ending in a wedge formed by one of the boundaries and a turning surface. To these three types of orbits, our calculations show that there correspond three kinds of modes and give support to the following conclusions. First, with quasi-periodic orbits are associated regular modes which exist at the zero-diffusion limit as smooth square-integrable velocity fields associated with a discrete set of eigenvalues, probably dense in some subintervals of [O, N], N being the Brunt-Väisälä frequency. Second, with periodic orbits are associated singular modes which feature a shear layer following the periodic orbit; as the zero-diffusion limit is taken, the eigenfunction becomes singular on a line tracing the periodic orbit and is no longer square-integrable; as a consequence the point spectrum is empty in some subintervals of [0,N], It is also shown that these internal shear layers contain the two scales E1/3 and E1/4 as pure inertial modes (E is the Ekman number). Finally, modes associated with characteristics trapped by a wedge also disappear at the zero-diffusion limit; eigenfunctions are not squareintegrable and the corresponding point spectrum is also empty.

Description: We consider the evolution under the action of surface tension of wedges and cones of viscous fluid whose initial semi-angles are close to π/2. A short time after the fluid is released from rest, there is an inner region, where surface tension and viscosity dominate, and an outer region, where inertia and viscosity dominate. We also find that the velocity of the tip of the wedge or cone is singular, of O(log(1/t)), as time, f, tends to zero. After a long time, the free surface asymptotes to a similarity form where deformations are of O(t2/3), and capillary waves propagate away from the tip. However, a distance of 0(t3/4) away from the tip, viscosity acts to damp out the capillary waves. We solve the linearized governing equations using double integral transforms, which we calculate numerically, and use asymptotic techniques to approximate the solutions for small and large times. We also compare the asymptotic solution for the inviscid fat wedge with a numerical solution of the nonlinear inviscid problem for wedges of arbitrary semi-angle.

Description: In this paper we investigate the stability of a bilayer exposed to air flow. The bilayer consists of a viscoelastic solid layer (mucus), which rests on a viscous fluid film (serous fluid). The motivation behind this work is to examine the coupled, fluid/elastic instabilities related to mucus clearance in the lung where breathing and cough apply shear forces from the air flow onto the bilayer. Previous research on mucus transport due to air flow has not addressed the effects of the underlying serous layer nor those of surface tension at the mucus-air interface, two new features incorporated into the model. Surface tension effects are governed by the new parameter K′ = (σ/dG′) where σ is the air-mucus surface tension, G′ is the elastic shear modulus of the mucus, and d is a characteristic thickness of the bilayer. The model predictions for the onset of unstable waves as a function of the parameters are compared to previous theories and experiments to provide physical interpretations and to compare results. The comparison with experiments show good qualitative and quantitative agreement. The results are compared, also, to flow over a single, viscoelastic layer, with no viscous fluid underneath, to demonstrate the appearance of new wave behaviour when the viscous fluid is added.

Description: The tendency of granular materials in rapid shear flow to form non-uniform structures is well documented in the literature. Through a linear stability analysis of the solution of continuum equations for rapid shear flow of a uniform granular material, performed by Savage (1992) and others subsequently, it has been shown that an infinite plane shearing motion may be unstable in the Lyapunov sense, provided the mean volume fraction of particles is above a critical value. This instability leads to the formation of alternating layers of high and low particle concentrations oriented parallel to the plane of shear. Computer simulations, on the other hand, reveal that non-uniform structures are possible even when the mean volume fraction of particles is small. In the present study, we have examined the structure of fully developed layered solutions, by making use of numerical continuation techniques and bifurcation theory. It is shown that the continuum equations do predict the existence of layered solutions of high amplitude even when the uniform state is linearly stable. An analysis of the effect of bounding walls on the bifurcation structure reveals that the nature of the wall boundary conditions plays a pivotal role in selecting that branch of non-uniform solutions which emerges as the primary branch. This demonstrates unequivocally that the results on the stability of bounded shear flow of granular materials presented previously by Wang et al. (1996) are, in general, based on erroneous base states.

Description: Model equations that govern the evolution of internal gravity waves at the interface of two immiscible inviscid fluids are derived. These models follow from the original Euler equations under the sole assumption that the waves are long compared to the undisturbed thickness of one of the fluid layers. No smallness assumption on the wave amplitude is made. Both shallow and deep water configurations are considered, depending on whether the waves are assumed to be long with respect to the total undisturbed thickness of the fluids or long with respect to just one of the two layers, respectively. The removal of the traditional weak nonlinearity assumption is aimed at improving the agreement with the dynamics of Euler equations for large-amplitude waves. This is obtained without compromising much of the simplicity of the previously known weakly nonlinear models. Compared to these, the fully nonlinear models' most prominent feature is the presence of additional nonlinear dispersive terms, which coexist with the typical linear dispersive terms of the weakly nonlinear models. The fully nonlinear models contain the Korteweg-de Vries (KdV) equation and the Intermediate Long Wave (ILW) equation, for shallow and deep water configurations respectively, as special cases in the limit of weak nonlinearity and unidirectional wave propagation. In particular, for a solitary wave of given amplitude, the new models show that the characteristic wavelength is larger and the wave speed is smaller than their counterparts for solitary wave solutions of the weakly nonlinear equations. These features are compared and found in overall good agreement with available experimental data for solitary waves of large amplitude in two-fluid systems.

Description: The intermittency of pressure and pressure gradient in stationary isotropic turbulence at low to moderate Reynolds numbers is studied by direct numerical simulation (DNS) and theoretically. The energy spectra scale in Kolmogorov units as required by the universal-equilibrium hypothesis, but the pressure spectra do not. It is found that the variances of the pressure and pressure gradient are larger than those computed using the Gaussian approximation for the fourth-order moments of velocity, and that the variance of the pressure gradient, normalized by Kolmogorov units, increases roughly as script R signλ1/2, where script R signλ is the Taylor microscale Reynolds number. A theoretical explanation of the Reynolds number dependence is presented which assumes that the small-scale pressure field is driven by coherent small-scale vorticity-strain domains. The variance of the pressure gradient given by the model is the product of the variance of Ui,jUj,i, the source term of the Poisson equation for pressure, and the square of an effective length of the small-scale coherent vorticity-strain structures. This length can be expressed in terms of the Taylor and Kolmogorov microscales, and the ratio between them gives the observed Reynolds number dependence. Formal asymptotic matching of the spectral scaling observed at small scales in the DNS with the classical scaling at large scales suggests that at high Reynolds numbers the pressure spectrum in these forced flows consists of three scaling ranges which are joined by two inertial ranges, the classical k-7/3 range and a k-5/3 range at smaller scale. It is not possible, within the classical Kolmogorov theory, to determine the length scale at which the inertial range transition occurs because information beyond the energy dissipation rate is required.

Description: Numerical studies of two-dimensional, transonic flows of dense gases of retrograde type, known as BZT gases, around thin airfoils are presented. The computations are guided by a recent asymptotic theory of Rusak & Wang (1997). It provides a uniformly valid solution of the flow around the entire airfoil surface which is composed of outer and inner solutions. A new transonic small-disturbance (TSD) equation solver is developed to compute the nonlinear BZT gas flow in the outer region around most of the airfoil. The flow in the inner region near the nose of the airfoil is computed by solving the problem of a sonic flow around a parabola. Numerical results of the composite solutions calculated from the asymptotic formula are compared with the solutions of the Euler equations. The comparison demonstrates that, in the leading order, the TSD solutions of BZT gas flows represent the essence of the flow character around the airfoil as computed from the Euler equations. Furthermore, guided by the asymptotic formula, the computational results demonstrate the similarity rules for transonic flows of BZT gases. There are differences between the self-similar cases that may be related to the error associated with the accuracy of the asymptotic solution. A discussion on the flow patterns around an airfoil at transonic speeds and at various upstream thermodynamic conditions is also presented. The paper provides important guidelines for future studies on this subject.

Description: In most real or numerically simulated turbulent flows, the energy dissipated at small scales is equal to that injected at very large scales, which are anisotropic. Despite this injection-scale anisotropy, one generally expects the inertial-range scales to be locally isotropic. For moderate Reynolds numbers, the isotropic relations between second-order and third-order moments for temperature (Yaglom's equation) or velocity increments (Kolmogorov's equation) are not respected, reflecting a non-negligible correlation between the scales responsible for the injection, the transfer and the dissipation of energy. In order to shed some light on the influence of the large scales on inertial-range properties, a generalization of Yaglom's equation is deduced and tested, in heated grid turbulence (Rλ = 66). In this case, the main phenomenon responsible for the non-universal inertial-range behaviour is the non-stationarity of the second-order moments, acting as a negative production term.

Description: A WKB method is used to extend RDT (rapid distortion theory) to initially inhomogeneous turbulence and unsteady mean flows. The WKB equations describe turbulence wavepackets which are transported by the mean velocity and have wavenumbers which evolve due to the mean strain. The turbulence also modifies the mean flow and generates large-scale vorticity via the averaged Reynolds stress tensor. The theory is applied to Taylor's four-roller flow in order to explain the experimentally observed reduction in the mean strain. The strain reduction occurs due to the formation of a large-scale vortex quadrupole structure from the turbulent spot confined by the four rollers. Both turbulence inhomogeneity and three-dimensionality are shown to be important for this effect. If the initially isotropic turbulence is either homogeneous in space or two-dimensional, it has no effect on the large-scale strain. Furthermore, the turbulent kinetic energy is conserved in the two-dimensional case, which has important consequences for the theory of two-dimensional turbulence. The analytical and numerical results presented here are in good qualitative agreement with experiment.

Description: Using a matched asymptotic expansion we analyse the two-dimensional, near-critical reflection of a weakly nonlinear internal gravity wave from a sloping boundary in a uniformly stratified fluid. Taking a distinguished limit in which the amplitude of the incident wave, the dissipation, and the departure from criticality are all small, we obtain a reduced description of the dynamics. This simplification shows how either dissipation or transience heals the singularity which is presented by the solution of Phillips (1966) in the precisely critical case. In the inviscid critical case, an explicit solution of the initial value problem shows that the buoyancy perturbation and the alongslope velocity both grow linearly with time, while the scale of the reflected disturbance is reduced as 1/t. During the course of this scale reduction, the stratification is Overturned' and the Miles-Howard condition for stratified shear flow stability is violated. However, for all slope angles, the Overturning' occurs before the Miles-Howard stability condition is violated and so we argue that the first instability is convective. Solutions of the simplified dynamics resemble certain experimental visualizations of the reflection process. In particular, the buoyancy field computed from the analytic solution is in good agreement with visualizations reported by Thorpe & Haines (1987). One curious aspect of the weakly nonlinear theory is that the final reduced description is a linear equation (at the solvability order in the expansion all of the apparently resonant nonlinear contributions cancel amongst themselves). However, the reconstructed fields do contain nonlinearly driven second harmonics which are responsible for an important symmetry breaking in which alternate vortices differ in strength and size from their immediate neighbours.

Description: The equations of magnetohydrodynamics (MHD) of an ideal fluid have two families of topological invariants: the magnetic helicity invariants and the cross-helicity invariants. It is first shown that these invariants define a natural foliation (described as isomagnetovortical, or imv for short) in the function space in which solutions {u(x,t), h(x,t)} of the MHD equations reside. A relaxation process is constructed whereby total energy (magnetic plus kinetic) decreases on an imv folium (all magnetic and cross-helicity invariants being thus conserved). The energy has a positive lower bound determined by the global cross-helicity, and it is thus shown that a steady state exists having the (arbitrarily) prescribed families of magnetic and cross-helicity invariants. The stability of such steady states is considered by an appropriate generalization of (Arnold) energy techniques. The first variation of energy on the imv folium is shown to vanish, and the second variation δ2E is constructed. It is shown that δ2E is a quadratic functional of the first-order variations δ1u, δ1h of u and h (from a steady state U(x), H(X)), and that δ2E is an invariant of the linearized MHD equations. Linear stability is then assured provided δ2E is either positive-definite or negative-definite for all imv perturbations. It is shown that the results may be equivalently obtained through consideration of the frozen-in 'modified' vorticity field introduced in Part 1 of this series. Finally, the general stability criterion is applied to a variety of classes of steady states {U(x),H(x)}, and new sufficient conditions for stability to three-dimensional imv perturbations are obtained.

Description: Axisymmetric pipeline transportation of oil and water is simulated numerically as an initial value problem. The simulations succeed in predicting the spatially periodic Stokes-like waves called bamboo waves, which have been documented in experiments of Bai, Chen & Joseph (1992) for up-flow. The numerical scheme is validated against linearized stability theory for perfect core-annular flow, and weakly nonlinear saturation to travelling waves. Far from onset conditions, the fully nonlinear saturation to steady bamboo waves is achieved. As the speed is increased, the bamboo waves shorten, and peaks become more pointed. A new time-dependent bamboo wave is discovered, in which the interfacial waveform is steady, but the accompanying velocity and pressure fields are time-dependent. The appearance of vortices and the locations of the extremal values of pressure are investigated for both up- and down-flows.

Description: Surface waves superimposed upon a larger-scale flow are blocked and reflected at the points where the group velocities balance the convection by the larger-scale flow. In this study, we first extended the theory of Shyu & Phillips (1990) to the situation when short deep-water gravity waves propagate obliquely upon a steady unidirectional irrotational current and are reflected by it. In this case, the uniformly valid solution and the WKBJ solution of the short waves were derived from the Laplace equation and the kinematical and dynamical boundary conditions. These solutions in terms of some parameters (the expressions for which have also been deduced in this case) take the same forms as those derived by Shyu & Phillips, which by referring to Smith's (1975) theory can even be proved to be valid for gravity waves in an intermediate-depth region and near a curved moving caustic induced by an unsteady multidirectional irrotational current. In this general case, the expressions for certain parameters in these solutions cannot be obtained so that their values must be estimated in a numerical calculation. The algorithm for estimates of some of these parameters that are responsible for the amplitude of the reflected wave not being equal to that of the incident wave in the vicinity of the caustic and therefore are crucial for the computer calculation of the ray solution to be continued after reflection, was illustrated through numerical tests. This algorithm can avoid the error magnification phenomenon that occurred in the previous estimates of the reflected wave in the vicinity of the caustic using the action conservation principle directly. The forms of the solutions have also been utilized to clarify the wave profiles near caustics in a general situation, which indicate that in storm conditions freak waves characterized by a steeper forward face preceded by a deep trough will probably occur in the caustic regions.

Description: The aerodynamic and acoustic properties of supersonic elliptic and circular jets are experimentally investigated. The jets are perfectly expanded with an exit Mach number of approximately 1.5 and are operated in the Reynolds number range of 25 000 to 50 000. The reduced Reynolds number facilitates the use of conventional hot-wire anemometry and a glow discharge excitation technique which preferentially excites the varicose or flapping modes in the jets. In order to simulate the high-velocity and low-density effects of heated jets, helium is mixed with the air jets. This allows the large-scale structures in the jet shear layer to achieve a high enough convective velocity to radiate noise through the Mach wave emission process. Experiments in the present work focus on comparisons between the cold and simulated heated jet conditions and on the beneficial aeroacoustic properties of the elliptic jet. When helium is added to the jet, the instability wave phase velocity is found to approach or exceed the ambient sound speed. The radiated noise is also louder and directed at a higher angle from the jet axis. In addition, near-field hotwire spectra are found to match the far-field acoustic spectra only for the helium/air mixture case. These results demonstrate that there are significant differences between unheated and heated asymmetric jets in the Mach 1.5 speed range, many of which have been found previously for circular jets. The elliptic jet was also found to radiate less noise than the round jet at comparable operating conditions.

Description: The Darcy model with the Boussinesq approximations is used to study double-diffusive instability in a horizontal rectangular porous enclosure subject to two sources of buoyancy. The two vertical walls of the cavity are impermeable and adiabatic while Dirichlet or Neumann boundary conditions on temperature and solute are imposed on the horizontal walls. The onset and development of convection are first investigated using the linear and nonlinear perturbation theories. Depending on the governing parameters of the problem, four different regimes are found to exist, namely the stable diffusive, the subcritical convective, the oscillatory and the augmenting direct regimes. The governing parameters are the thermal Rayleigh number, RT, buoyancy ratio, N, Lewis number, Le, normalized porosity of the porous medium, ε, aspect ratio of the enclosure, A, and the thermal and solutal boundary condition type, κ, applied on the horizontal walls. On the basis of the nonlinear perturbation theory and the parallel flow approximation (for slender or shallow enclosures), analytical solutions are derived to predict the flow behaviour. A finite element numerical method is introduced to solve the full governing equations. The results indicate that steady convection can arise at Rayleigh numbers below the supercritical value, indicating the development of subcritical flows. At the vicinity of the threshold of supercritical convection the nonlinear perturbation theory and the parallel flow approximation results are found to agree well with the numerical solution. In the overstable regime, the existence of multiple solutions, for a given set of the governing parameters, is demonstrated. Also, numerical results indicate the possible occurrence of travelling waves in an infinite horizontal enclosure.

Description: Flows with velocity profiles very different from the parabolic velocity profile can occur in the entrance region of a tube as well as in tubes with converging/diverging cross-sections. In this paper, asymptotic and numerical studies are undertaken to analyse the temporal stability of such 'non-parabolic' flows in a flexible tube in the limit of high Reynolds numbers. Two specific cases are considered : (i) developing flow in a flexible tube; (ii) flow in a slightly converging flexible tube. Though the mean velocity profile contains both axial and radial components, the flow is assumed to be locally parallel in the stability analysis. The fluid is Newtonian and incompressible, while the flexible wall is modelled as a viscoelastic solid. A high Reynolds number asymptotic analysis shows that the non-parabolic velocity profiles can become unstable in the inviscid limit. This inviscid instability is qualitatively different from that observed in previous studies on the stability of parabolic flow in a flexible tube, and from the instability of developing flow in a rigid tube. The results of the asymptotic analysis are extended numerically to the moderate Reynolds number regime. The numerical results reveal that the developing flow could be unstable at much lower Reynolds numbers than the parabolic flow, and hence this instability can be important in destabilizing the fluid flow through flexible tubes at moderate and high Reynolds number. For flow in a slightly converging tube, even small deviations from the parabolic profile are found to be sufficient for the present instability mechanism to be operative. The dominant non-parallel effects are incorporated using an asymptotic analysis, and this indicates that non-parallel effects do not significantly affect the neutral stability curves. The viscosity of the wall medium is found to have a stabilizing effect on this instability.

Description: An experimental configuration has been found which allows detailed observation of three-dimensional vortex-vortex and associated vortex-surface interactions which appear similar to those observed in fully turbulent flow. Hydrogen bubble visualization illustrates a complicated intertwining, or braiding, of two initially co-rotating vortices. It is observed that the three-dimensional interactions of the braided vortices induces a pair of local surface-fluid eruptions reminiscent of the 'bursting' behaviour characteristic of the near-wall regeneration process of fully turbulent boundary layers.

Description: Linear mode conversion is the partial transfer of wave energy from one wave type (a) to another (b) in a weakly non-uniform background state. For propagation in one dimension (x), the local wavenumber kj x of each wave (j = a,b) varies with x; if these are equal at some xR, the waves are locally in phase, and resonant energy transfer can occur. We model wave propagation in the Gulf of Guinea, where wave a is an equatorially trapped Rossby-gravity (Yanai) wave, and wave b is a coastal Kelvin wave along the (zonal) north coast of the Gulf, both propagating in zonal coordinate x. The coupling of the waves is due to the overlap of their eigenfunctions (normal modes in y, the meridional coordinate). We derive coupled mode equations from a variational principle, and obtain an analytic expression for the wave-energy conversion coefficient, in terms of the wave frequency and the scale length of the thermocline depth.

Description: This paper discusses numerical experiments in which an initially uniform columnar vortex is subject to several types of axisymmetric forcing that mimic the strain field of a turbulent flow. The mean value of the strain along the vortex axis is in all cases zero, and the vortex is alternately stretched and compressed. The emphasis is on identifying the parameter range in which the vortex survives indefinitely. This extends previous work in which the effect of steady single-scale non-uniform strains was studied. In a first series of experiments the effect of the unsteadiness of the forcing is analysed, and it is found that the vortex survives as a compact object if the ratio between the oscillation frequency and the strain itself is low enough. A theoretical explanation is given which agrees with the numerical results. The strain is then generalized to include several spatial scales and oscillation frequencies, with characteristics similar to those in turbulent flows. The largest velocities are carried by the large scales, while the highest gradients and faster time scales are associated with the shorter wavelengths. Also in these cases 'infinitely long' vortices are obtained which are more or less uniform and compact. Vorticity profiles averaged along their axes are approximately Gaussian. The radii obtained from these profiles are proportional to the Burgers' radius of the r.m.s. (small-scale) axial strain, while the azimuthal velocities are proportional to the maximum (large-scale) axial velocity differences. The study is motivated by previous observations of intense vortex filaments in turbulent flows, and the scalings found in the present experiments are consistent with those found in the turbulent simulations.

Description: Using a physical space (i.e. non-modal) approach, we investigate interactions between fast inertio-gravity (IG) waves and slow balanced flows in a shallow rotating fluid. Specifically, we consider a train of IG waves impinging on a steady, exactly balanced vortex. For simplicity, the one-dimensional problem is studied first; the limitations of one-dimensionality are offset by the ability to define balance in an exact way. An asymptotic analysis of the problem in the small-amplitude limit is performed to demonstrate the existence of interactions. It is shown that these interactions are not confined to the modification of the wave field by the vortex but, more importantly, that the waves are able to alter in a non-trivial way the potential vorticity associated with that vortex. Interestingly, in this one-dimensional problem, once the waves have traversed the vortex region and have propagated away, the vortex exactly recovers its initial shape and thus bears no signature of the interaction. Furthermore, we prove this last result in the case of arbitrary vortex and wave amplitudes. Numerical integrations of the full one-dimensional shallow-water equations in strongly nonlinear regimes are also performed: they confirm that time-dependent interactions exist and increase with wave amplitude, while at the final state the vortex bears no sign of the interaction. In addition, they reveal that cyclonic vortices interact more strongly with the wave field than anticyclonic ones.

Description: The dynamics of a passive tracer in the velocity field of four identical point vortices, moving under the influence of their self-induced advection, is investigated. Of interest is the change in mixing and transport properties of the tracer for the three different classes of vortex motion : periodic, quasi-periodic and chaotic. As a consequence of conservation laws, the vortex motion is confined to a finite region of phase space; therefore, the tracer phase space can be partitioned into an inner and an outer region. We find that in the case of quasi-periodic vortex motion the tracer phase space exhibits a well-defined barrier to transport between the central chaotic region and the outer region, where the trajectories are regular. In the case of chaotic vortex motion the barrier becomes permeable. The particle dynamics goes through an intermittent behaviour, where forays into the central region alternate with trapping in outer annular orbits. In the far field, an estimate of diffusion rates is made through a multipole expansion of the tracer velocity field. We make use of a specific stochastic model for the tracer velocity field which predicts no diffusion for the case of quasiperiodic vortex motion and, for the case of chaotic vortex motion, a diffusion rate that goes to zero at large distances from the vortex cluster.

Description: A premixed ducted flame, burning in the wake of a bluff-body flame-holder, is considered. For such a flame, interaction between acoustic waves and unsteady combustion can lead to self-excited oscillations. The concept of a time-invariant turbulent flame speed is used to develop a kinematic model of the response of the flame to flow disturbances. Variations in the oncoming flow velocity at the flame-holder drive perturbations in the flame initiation surface and hence in the instantaneous rate of heat release. For linear fluctuations, the transfer function between heat release and velocity can be determined analytically from the model and is in good agreement with experiment across a wide frequency range. For nonlinear fluctuations, the model reproduces the flame surface distortions seen in schlieren films. Coupling this kinematic flame model with an analysis of the acoustic waves generated in the duct by the unsteady combustion enables the time evolution of disturbances to be calculated. Self-excited oscillations occur above a critical fuel-air ratio. The frequency and amplitude of the resulting limit cycles are in satisfactory agreement with experiment. Flow reversal is predicted to occur during part of the limit-cycle oscillation and the flame then moves upstream of the flame-holder, just as in experimental visualizations. The main nonlinearity is identified in the rate of heat release, which essentially 'saturates' once the amplitude of the velocity fluctuation exceeds its mean. We show that, for this type of nonlinearity, describing function analysis can be used to give a good estimate of the limit-cycle frequency and amplitude from a quasi-nonlinear theory.

Description: Fourier and wavelet transformation techniques are utilized in a complementary manner in order to characterize temporal aspects of the transition of a planar jet shear layer. The subharmonic is found to exhibit an interesting temporal amplitude and phase variation that has not been previously reported. This takes the form of intermittent π-shifts in subharmonic phase between two fixed phase values. These phase jumps are highly correlated with local minima of the subharmonic amplitude. In contrast, the fundamental amplitude and phase show no such behaviour. The temporal phase behaviour of the subharmonic has the effect of intermittently disrupting the phase lock with the fundamental. A dynamical systems model is developed which is based on a classic vortex representation of the shear layer. The Hamiltonian formulation of the problem is shown to provide remarkable agreement with the experimental results. All the essential aspects of the temporal amplitude and phase behaviour of the subharmonic are reproduced by the model including amplitude-dependent effects. The model is also shown to provide a dynamical systems based explanation for time-averaged amplitude and phase behaviour observed in these as well as earlier experiments. The results of experiments involving both bimodal forcing at fundamental and subharmonic frequencies with prescribed initial effective phase angle as well as experiments involving only fundamental excitation over an amplitude range extending two orders of magnitude are presented. The temporal subharmonic amplitude and phase behaviour is observed in bimodal forcing experiments in those regions of the flow characterized by subharmonic mode suppression and vortex tearing events (even if the forcing amplitudes are quite large). In addition, temporal subharmonic amplitude and phase behaviour is the rule in experiments involving low-amplitude forcing of the fundamental and the natural development of the subharmonic.

Description: A finite-amplitude long-wave equation is derived to describe the effect of weak current shear on internal waves in a uniformly stratified fluid. This effect is manifested through the introduction of a nonlinear term into the amplitude evolution equation, representing a projection of the shear from physical space to amplitude space. For steadily propagating waves the evolution equation reduces to the steady version of the generalized Korteweg-de Vries equation. An analysis of this equation is presented for a wide range of possible shear profiles. The type of waves that occur is found to depend on the number and position of the inflection points of the representation of the shear profile in amplitude space. Up to three possible inflection points for this function are considered, resulting in solitary waves and kinks (dispersionless bores) which can have up to three characteristic lengthscales. The stability of these waves is generally found to decrease as the complexity of the waves increases. These solutions suggest that kinks and solitary waves with multiple lengthscales are only possible for shear profiles (in physical space) with a turning point, while instability is only possible if the shear profile has an inflection point. The unsteady evolution of a periodic initial condition is considered and again the solution is found to depend on the inflection points of the amplitude representation of the shear profile. Two characteristic types of solution occur, the first where the initial condition evolves into a train of rank-ordered solitary waves, analogous to those generated in the framework of the Korteweg-de Vries equation, and the second where two or more kinks connect regions of constant amplitude. The unsteady solutions demonstrate that finite-amplitude effects can act to halt the critical collapse of solitary waves which occurs in the context of the generalized Kortewegde Vries equation. The two types of solution are then used to quantitatively relate previously reported observations of shock formation on the internal tide propagating onto the Australian North West Shelf to the observed background current shear.

Description: We investigate the temporal evolution of the geometrical distribution of a passive scalar injected continuously into the far field of a turbulent water jet at a scale d smaller than the local integral scale of the turbulence. The concentration field is studied quantitatively by a laser-induced-fluorescence technique on a plane cut containing the jet axis. Global features such as the scalar dispersion from the source, as well as the fine structure of the scalar field, are analysed. In particular, we define the volume occupied by the regions whose concentration is larger than a given concentration threshold (support of the scalar field) and the surface in which this volume is enclosed (boundary of the support). The volume and surface extents, and their respective fractal dimensions are measured as a function of time t, and the concentration threshold is normalized by the initial concentration Cs/C0 for different injection sizes d. All of these quantities display a clear dependence on t, d and Cs, and their evolutions rescale with the variable ξ = (ut/d)(Cs/C0), the fractal dimension being, in addition, scale dependent. The surface-to-volume ratio and the fractal dimension of both the volume and the surface tend towards unity at large ξ, reflecting the sheet-like structure of the scalar at small scales. These findings suggest an original picture of the kinetics of turbulent mixing.

Description: History forces on a stationary cylinder in arbitrary unsteady rectilinear flow are calculated by means of a model based on the asymptotic properties of the steady-state wake. The results capture many features found in numerical solutions of the Navier-Stokes equation for the same flows, though quantitative agreement deteriorates as the Reynolds number increases over the range 2 to 40. The cases studied are the impulsive start, stop, and reverse, and oscillatory flow.

Description: We consider the reflection of oblique compression waves from a two-dimensional, steady, laminar boundary layer on a flat, adiabatic plate at free-stream pressures such that dense-gas effects are non-negligible. The full Navier-Stokes equations are solved through use of a dense-gas version of the Beam-Warming implicit scheme. The main fluids studied are Bethe-Zel'dovich-Thompson (BZT) fluids. These are ordinary gases which have specific heats large enough to cause the fundamental derivative of gasdynamics to be negative for a range of pressures and temperatures in the single-phase vapour regime. It is demonstrated that the unique dynamics of BZT fluids can result in a suppression of shock-induced separation. Numerical tests performed reveal that the physical mechanism leading to this suppression is directly related to the disintegration of any compression discontinuities originating in the flow. We also demonstrate numerically that the interaction of expansion shocks with the boundary layer produces no adverse effects.

Description: This paper describes the evolution of an incompressible turbulent boundary layer on the flat wall of an 'S'-shaped wind tunnel test section under the influence of changing streamwise and spanwise pressure gradients. The unit Reynolds number based on the mean velocity at the entrance of the test section was fixed to 106 m-1 resulting in Reynolds numbers Reδ2, based on the streamwise momentum thickness and the local freestream velocity, between 3.9 and 11 x 103. The particular feature of the experiment is the succession of two opposite changes of core flow direction which causes a sign change of the spanwise pressure gradient accompanied by a reversal of the spanwise velocity component near the wall, i.e. by the formation of so-called cross-over velocity profiles. The aim of the study is to provide new insight into the development of the mean and fluctuating flow field in three-dimensional pressure-driven boundary layers, in particular of the turbulence structure of the near-wall and the cross-over region. Mean velocities, Reynolds stresses and all triple correlations were measured with a newly developed miniature triple-hot-wire probe and a near-wall hot-wire probe which could be rotated and traversed through the test plate. Skin friction measurements were mostly performed with a wall hot-wire probe. The data from single normal wires extend over wall distances of y+ ≳ 3 (in wall units), while the triple-wire probe covers the range y+ ≳ 30. The data show the behaviour of the mean flow angle near the wall to vary all the way to the wall. Then, to interpret the response of the turbulence to the pressure field, the relevant terms in the Reynolds stress transport equations are evaluated. Finally, an attempt is made to assess the departure of the Reynolds stress profiles from local equilibrium near the wall.

Description: Three-dimensional large-amplitude oscillations of a mercury drop were obtained by electrical excitation in low gravity using a drop tower. Multi-lobed (from three to six lobes) and polyhedral (including tetrahedral, hexahedral, octahedral and dodecahedral) oscillations were obtained as well as axisymmetric oscillation patterns. The relationship between the oscillation patterns and their frequencies was obtained, and it was found that polyhedral oscillations are due to the nonlinear interaction of waves. A mathematical model of three-dimensional forced oscillations of a liquid drop is proposed and compared with experimental results. The equations of drop motion are derived by applying the variation principle to the Lagrangian of the drop motion, assuming moderate deformation. The model takes the form of a nonlinear Mathieu equation, which expresses the relationships between deformation amplitude and the driving force's magnitude and frequency.

Description: Natural convection in a rectangular cavity is examined, utilizing the second law of thermodynamics. Through an application of the second law the rate of entropy generation associated with the convective pattern changes is evaluated for the onset of natural convection in a cavity with free boundaries, for which an exact solution is sought, as well as with rigid boundaries which is studied numerically. Entropy to be generated from the perturbed temperature and velocity fields is shown to depend on AR (aspect ratio of the cavity), Rac (the critical Rayleigh number) and a non-dimensional parameter, Ω, which is related to the ratio of entropy generation by viscous friction to that by thermal transport. The convective pattern change is related to a change in the spatial distributions of the rate of entropy generation due to heat transfer and due to dissipation, demonstrating that an application of the second law helps examine convective pattern changes quantitatively by dealing with temperature and velocity fields in a unified manner.

Description: Large-scale structures in a plane turbulent mixing layer are studied through the use of the proper orthogonal decomposition (POD). Extensive experimental measurements are obtained in a turbulent plane mixing layer by means of two cross-wire rakes aligned normal to the direction of the mean shear and perpendicular to the mean flow direction. The measurements are acquired well into the asymptotic region. From the measured velocities the two-point spectral tensor is calculated as a function of separation in the cross-stream direction and spanwise and streamwise wavenumbers. The continuity equation is then used for the calculation of the non-measured components of the tensor. The POD is applied using the cross-spectral tensor as its kernel. This decomposition yields an optimal basis set in the mean square sense. The energy contained in the POD modes converges rapidly with the first mode being dominant (49% of the turbulent kinetic energy). Examination of these modes shows that the first mode contains evidence of both known flow organizations in the mixing layer, i.e. quasi-two-dimensional spanwise structures and streamwise aligned vortices. Using the shot-noise theory the dominant mode of the POD is transformed back into physical space. This structure is also indicative of the known flow organizations.

Description: A theoretical framework is developed to predict the rate of geometric collision and the collision velocity of small size inertialess particles in general turbulent flows. The present approach evaluates the collision rate for small size, inertialess particles in a given instantaneous flow field based on the local eigenvalues of the rate-of-strain tensor. An ensemble average is then applied to the instantaneous collision rate to obtain the average collision rate. The collision rates predicted by Smoluchowski (1917) for laminar shear flow and by Saffman & Turner (1956) for isotropic turbulence are recovered. The collision velocities presently predicted in both laminar shear flow and isotropic turbulence agree well with the results from numerical simulations for particle collision in both flows. The present theory for evaluating the collision rate and the collision velocity is also applied to a rapidly sheared homogeneous turbulence to assess the effect of strong anisotropy on the collision rate. Using (ε/ν)1/2, in which ε is the average turbulence energy dissipation rate and ν is the fluid kinematic viscosity, as the characteristic turbulence shear rate to normalize the collision rate, the effect of the turbulence structure on the collision rate and collision velocity can be reliably described. The combined effects of the mean flow shear and the turbulence shear on the collision rate and collision velocity are elucidated.

Description: We show that the class of constitutive relations for turbulence models put forward by Wang (1997) in this journal conflicts with dimensional analysis, unless the turbulent Reynolds stresses were to be tied to the molecular viscous stresses everywhere in the flow. We then reiterate, using counter-examples, that the controversial postulate of material frame-indifference is unfounded for turbulence, and is counter-productive in the quest for accuracy. We add a comment on the role of acceleration.

Description: An experimental study of entrainment of sand-sized particles in turbulent boundary layers has been performed in a high-speed wind tunnel at square-pulse flow speeds of 27 to 101ms-1 and for soil bed lengths varying from 2.1 to 5.8m. Because of high particle drag-to-weight ratios (D/W = 100-1000) and friction velocities (uf) well above soil threshold friction velocities (uft;10 ≤ uf/uft ≤ 40), the present results correspond to the suspension regime of dust lofting, in contrast to low-speed saltation flows (1 ≤ uf/uft ≤ 5; D/W ≤ 15). Results are obtained characterizing particle entrainment for both a natural soil (White Sands Missile Range (WSMR) sand; 50% finer-by-weight diameter, D̄50 = 180μm) and a monosized sand sample (Ottawa sand, D̄50 = 250 μm). Measurements of local boundary layer velocities and dust densities were performed with traversing state-of-the-art diagnostics. Scouring rate data (0.015 ≤ m̄s ≤ 0.30gcm-2 s-1) and streamwise soil flux (10 ≤ Q ≤ 150gcm-1 s-1) as a function of bed length and velocity were determined. Scouring rates were found to increase as the 3/2-power of velocity, but decay as the inverse square root of dust bed length. Corresponding streamwise soil fluxes (also known as soil loss rates) increased to the 3/2-power of velocity in contrast to the cube power dependence for low-speed results (ufree-stream ≤ 15m s-1; Q ≤ 1.5gcm-1 s-1). Comparison of scouring rate data (from pre/post-test soil loss measurements) with derived data based on the rate of change of streamwise flux with distance was favourable. WSMR rates were always lower than Ottawa sand rates, a result consistent with the lower repose angle for the Ottawa sand sample. Both sets of soil data demonstrate that dust edges extend vertically to higher elevations than corresponding velocity edges. This result implies that the turbulent Schmidt number for the present flows is less than unity and of the order of 0.7. Favourable collapsing of the scouring rate data base was achieved when measured rates were normalized by the friction velocity mass flux, square root of edge Mach number and sand repose angle ratio. A universal rate of 0.3±0.1 correlated well with the bulk of the data.

Description: Stably stratified flows past three-dimensional orography have been investigated using a stratified towing tank. Flows past idealized axisymmetric orography in which the Froude number, Fh = U/Nh (where U is the towing speed, N is the buoyancy frequency and h is the height of the obstacle) is less than unity have been studied. The orography considered consists of two sizes of hemisphere and two cones of different slope. For all the obstacles measurements show that as Fh decreases, the drag coefficient increases, reaching between 2.8 and 5.4 times the value in neutral flow (depending on obstacle shape) for Fh ≲ 0.25. Local maxima and minima in the drag also occur. These are due to the finite depth of the tank and can be explained by linear gravity-wave theory. Flow visualization reveals a lee wave train downstream in which the wave amplitude is O(Fhh), the smallest wave amplitude occurring for the steepest cone. Measurements show that for all the obstacles, the dividing-streamline height, zs, is described reasonably well by the formula zs/h = 1 - Fh. Flow visualization and acoustic Doppler velocimeter measurements in the wake of the obstacles show that vortex shedding occurs when Fh ≲ 0.4 and that the period of the vortex shedding is independent of height. Based on velocity measurements in the wake of both sizes of hemisphere (plus two additional smaller hemispheres), it is shown that a blockage-corrected Strouhal number, S2c. = fL2/Uc, collapses onto a single curve when plotted against the effective Froude number, Fhc = Uc/Nh. Here, Uc is the blockage-corrected free-stream speed based on mass-flux considerations, f is the vortex shedding frequency and L2 is the obstacle width at a height zs/2. Collapse of the data is also obtained for the two different shapes of cone and for additional measurements made in the wake of triangular and rectangular flat plates. Indeed, the values of S2c for all these obstacles are similar and this suggests that despite the fact that the obstacle widths vary with height, a single length scale determines the vortex-street dynamics. Experiments conducted using a splitter plate indicate that the shedding mechanism provides a major contribution to the total drag (∼ 25%). The addition of an upstream pointing 'verge region' to a hemisphere is also shown to increase the drag significantly in strongly stratified flow. Possible mechanisms for this are discussed.

Description: The reflection of asymmetric shock waves in steady flows is studied both theoretically and experimentally. While the analytical model was two-dimensional, three-dimensional edge effects influenced the experiments. In addition to regular and Mach reflection wave configurations, an inverse-Mach reflection wave configuration, which has been observed so far only in unsteady flows (e.g. shock wave reflection over concave surfaces or over double wedges) has been recorded. A hysteresis phenomenon similar to the one that exists in the reflection of symmetric shock waves has been found to also exist in the reflection of asymmetric shock waves. The domains and transition boundaries of the various types of overall reflection wave configurations are analytically predicted.

Description: Atmospheric and oceanic convection often occurs over areas occupied by many localized circulation elements known as plumes. The convective transports therefore may depend not only on the individual elements, but also on the interactions between plumes and the turbulent environment created by other plumes. However, many attempts to understand these plumes focus on individual isolated elements, and the behaviour of an ensemble is not understood. Geophysical convection may be influenced by rotation when the transit time of a convecting element is long compared to an inertial period (for example in deep oceanic convection). Much recent attention has been given to the effect of rotation on individual plumes, but the role of rotation in modifying the behaviour of an ensemble is not fully understood. Here we examine the behaviour of plumes within an ensemble, both with and without rotation, to identify the influence of rotation on ensemble plume dynamics. We identify the coherent structures (plumes) present in numerical solutions of turbulent Rayleigh-Bénard convection, a canonical example of a turbulent plume ensemble. We use a conditional sampling compositing technique to extract the typical structure in both non-rotating and rotating solutions. The dynamical balances of these composite plumes are evaluated and compared with entraining plume models. We find many differences between non-rotating and rotating plumes in their transports of mass, buoyancy and momentum. As shown in previous studies, the expansion of the turbulent plume by entrainment of exterior fluid is suppressed by strong rotation. Our most significant new result is quantification of the continuous mixing between the plume and ambient fluid which occurs at high rotation without any net changes in plume volume. This mixing is generated by the plume-plume interactions and acts to reduce the buoyancy anomaly of the plume. By contrast, in the non-rotating case, no such loss of buoyancy by mixing occurs. As a result, the total buoyancy transport by upwardly moving plumes diminishes across the layer in the rotating case, while remaining approximately constant in the non-rotating case. At high values of rotation, the net vertical acceleration is considerably reduced compared to the non-rotating case due to loss of momentum through entrainment and mixing and a decelerating pressure gradient which partially balances the buoyancy-driven acceleration of plumes. As a result of the dilution of buoyancy, the pressure-gradient deceleration and the loss of momentum due to mixina with the environment in the rotating solutions, the conversion of potential energy to kinetic energy is significantly less than that of non-rotating plumes. The combination of efficient lateral mixing and slow vertical movement by the plumes accounts for the unstable mean temperature gradient that occurs in rotating Rayleigh-Bénard convection, while the less penetrative convection found at low Rossby number is a consequence of the reduced kinetic energy transport. Within the ensemble of plumes identified by the conditional sampling algorithm, distributions of vertical velocity, buoyancy and vorticity mimic those of the volume as a whole. Plumes cover a small fraction of the total area, yet account for most of the vertical heat flux.

Description: The present paper is devoted to an analysis of tip vortex cavitation under confined situations. The tip vortex is generated by a three-dimensional foil of elliptical planform, and the confinement is achieved by flat plates set perpendicular to the span, at an adjustable distance from the tip. In the range of variation of the boundary-layer thickness investigated, no significant interaction was observed between the tip vortex and the boundary layer which develops on the confinement plate. In particular, the cavitation inception index for tip vortex cavitation does not depend significantly upon the length of the plate upstream of the foil. On the contrary, tip clearance has a strong influence on the non-cavitating structure of the tip vortex and consequently on the inception of cavitation in its core. The tangential velocity profiles measured by a laser-Doppler velocimetry (LDV) technique through the vortex, between the suction and the pressure sides of the foil, are strongly asymmetric near the tip. They become more and more symmetric downstream and the confinement speeds up the symmetrization process. When the tip clearance is reduced to a few millimetres, the two extrema of the velocity profiles increase. This increase results in a decrease of the minimum pressure in the vortex centre and accounts for the smaller resistance to cavitation observed when tip clearance is reduced. For smaller values of tip clearance, a reduction of tip clearance induces on the contrary a significant reduction in the maxima of the tangential velocity together with a significant increase in the size of the vortex core estimated along the confinement plate. Hence, the resistance to cavitation is much higher for such small values of tip clearance and in practice, no tip vortex cavitation is observed for tip clearances below 1.5mm. The cavitation number for the inception of tip vortex cavitation does not correlate satisfactorily with the lift coefficient, contrary to classical results obtained without any confinement. Owing to the specificity introduced by the confinement, the usual procedure developed in an infinite medium to estimate the vortex strength from LDV measurements is not applicable here. Hence, a new quantity homogeneous to a circulation had to be defined on the basis of the maximum tangential velocity and the core size, which proved to be better correlated to the cavitation inception data.

Description: A linear stability analysis is made of a family of natural convection flows in an arbitrarily inclined rectangular enclosure. The flow is driven by prescribed heat or mass fluxes along two opposing walls. The analysis allows for perturbations in arbitrary directions; however, the purely longitudinal or transverse modes are numerically found to be the most unstable. For the numerical treatment, a finite difference method with automatically calculated differencing molecules, variable order of accuracy, and accurate boundary treatment is developed. In cases with boundary layers, a special scaling is applied. For base solutions with natural (bottom heavy) stratification, critical conditions are solved for as a function of the Rayleigh number, Ra, and the angle of inclination to the bottom-heated case, a, for different Prandtl numbers (Pr), with complete results for Pr = 0.025,0.1,0.7,7,1000, and Pr → ∞. The uniform flux case is found to be much more stable than that of Hart (1971) with fixed wall temperatures, a fact which is attributed to the much larger stratification which occurs in the base solution. As could be expected, instabilities tend to be favoured by a decrease in Pr, an increase in Ra, and a decrease in a; however, exceptions to all these rules could be found. Cases in which the wavenumber is zero, or approaches zero in different ways, are studied analytically. Integral conditions, derived from the unresolved end regions, are applied in the analysis. The results show that all the base solutions with unnatural (top heavy) stratification are unstable to large-wavelength stationary rolls whose axes are parallel with the base flow. Real-valued perturbations are constructed and visualized for some of the modes considered.

Description: The propagation of a two-dimensional weakly nonlinear wavefront into a polytropic gas in a uniform state and at rest has been studied. Successive positions of the wavefront and the distribution of amplitude on it are obtained by solving a system of conservation forms of the equations of weakly nonlinear ray theory (WNLRT) using a TVB scheme based on the Lax-Friedrichs flux. The predictions of the WNLRT are found to be qualitatively quite different from the predictions of the linear theory. The linear wavefronts leading to the formation of caustics are replaced by nonlinear wavefronts with kinks. By varying the initial shape of the wavefront and the amplitude distribution on it, the formation and separation of kinks on the wavefront has been studied.

Description: Extended mild-slope equations for the propagation of small-amplitude water waves over variable bathymetry regions, recently proposed by Massel (1993) and Porter & Staziker (1995), are shown to exhibit an inconsistency concerning the sloping-bottom boundary condition, which renders them non-conservative with respect to wave energy. In the present work, a consistent coupled-mode theory is derived from a variational formulation of the complete linear problem, by representing the vertical distribution of the wave potential as a uniformly convergent series of local vertical modes at each horizontal position. This series consists of the vertical eigenfunctions associated with the propagating and all evanescent modes and, when the slope of the bottom is different from zero, an additional mode, carrying information about the bottom slope. The coupled-mode system obtained in this way contains an additional equation, as well as additional interaction terms in all other equations, and reduces to the previous extended mild-slope equations when the additional mode is neglected. Extensive numerical results demonstrate that the present model leads to the exact satisfaction of the bottom boundary condition and, thus, it is energy conservative. Moreover, it is numerically shown that the rate of decay of the modal-amplitude functions is improved from O(n-2), where n is the mode number, to O(n-4), when the additional sloping-bottom mode is included in the representation. This fact substantially accelerates the convergence of the modal series and ensures the uniform convergence of the velocity field up to and including the boundaries.

Description: The phenomenon of a succession of upstream-advancing solitary waves generated by underwater disturbances moving steadily with a transcritical velocity in two-dimensional shallow water channels is investigated. The two-dimensional Navier-Stokes (NS) equations with the complete set of viscous boundary conditions are solved numerically by the finite-difference method to simulate the phenomenon. The overall features of the phenomenon illustrated by the present numerical results are unanimous with observations in nature as well as in laboratories. The relations between amplitude and celerity, and between amplitude and period of generation of solitary waves can be accurately simulated by the present numerical method, and are in good agreement with predictions of theoretical formulae. The dependence of solitary wave radiation on the blockage and on the body shape is investigated. It furnishes collateral evidence of the experimental findings that the blockage plays a key role in the generation of solitary waves. The amplitude increases while the period of generation decreases as the blockage coefficient increases. It is found that in a viscous flow the shape of an underwater object has a significant effect on the generation of solitary waves owing to the viscous effect in the boundary layer. If a change in body shape results in increasing the region of the viscous boundary layer, it enhances the viscous effect and so does the disturbance forcing; therefore the amplitudes of solitary waves increase. In addition, detailed information of the flow, such as the pressure distribution, velocity and vorticity fields, are given by the present NS solutions.

Description: This paper presents the first experimental results on Marangoni-Bénard instability in a symmetrical three-layer system. A pure thermocapillary phenomenon has been observed by performing the experiment in a microgravity environment where buoyancy forces can be neglected. This configuration enables the hydrodynamic stability of two identical liquid-liquid interfaces subjected to a normal gradient of temperature to be studied. The flow is driven by one interface only and obeys the criterion based on the heat diffusivity ratio proposed by Scriven & Sternling (1959) and Smith (1966). The measured critical temperature difference for the onset of convection is compared to the value obtained from two-dimensional numerical simulations. The results of the simulations are in reasonable agreement with the velocimetry and the thermal experimental data for moderate supercriticality. Numerically and experimentally, the convective pattern exhibits a transition between different convective regimes for similar temperature gradients. Their common detailed features are discussed.

Description: Numerical experiments on modified turbulent channels at moderate Reynolds numbers are used to differentiate between several possible regeneration cycles for the turbulent fluctuations in wall-bounded flows. It is shown that a cycle exists which is local to the near-wall region and does not depend on the outer flow. It involves the formation of velocity streaks from the advection of the mean profile by streamwise vortices, and the generation of the vortices from the instability of the streaks. Interrupting any of those processes leads to laminarization. The presence of the wall seems to be only necessary to maintain the mean shear. The generation of secondary vorticity at the wall is shown to be of little importance in turbulence generation under natural circumstances. Inhibiting its production increases turbulence intensity and drag.

Description: An experimental and theoretical investigation of the flow and density distribution arising from the upward turbulent injection of a dense fluid into a stratified environment of finite extent is presented. Initially, the rising fluid reaches a maximum height before the flow reverses direction and intrudes either along the base of the tank or at an intermediate height in the environment. As more dense fluid is added through either a point or line source, both the fountain and the environment evolve with time. We determine expressions for the motion of the ascending and descending 'fronts' that mark the vertical extent of the spreading layer. We also consider the changes to the environmental density profile and determine an expression for the rate at which the top of the fountain rises due to these changes. Finally, we apply our results quantitatively to two physical problems: the replenishment of magma chambers and the heating or cooling of a room.

Description: A parametric study of multiple steady states, their stability, onset of oscillatory instability, and some supercritical unsteady regimes of convective flow of a Boussinesq fluid in laterally heated rectangular cavities is presented. Cavities with four no-slip boundaries, isothermal vertical and perfectly insulated horizontal boundaries are considered. Four distinct branches of steady-state flows are found for this configuration. A complete study of stability of each branch is performed for the aspect ratio A (length/height) of the cavity varying continuously from 1 to 11 and for two fixed values of the Prandtl number: Pr = 0 and Pr = 0.015. The results are represented as stability diagrams showing the critical parameters (critical Grashof number and the frequency at the onset of the oscillatory instability) corresponding to transitions from steady to oscillatory states, appearance of multi-roll states, merging of multiple states and backwards transitions from multi-roll to single-roll states. For better comparison with the existing experimental data, an additional stability study for varying Prandtl number (0.015 ≤ Pr ≤ 0.03) and fixed value of the aspect ratio A = 4 was carried out. It was shown that the dependence of the critical Grashof number on the aspect ratio and the Prandtl number is very complicated and a very detailed parametric study is required to reproduce it correctly. Comparison with the available experimental data for A = 4 shows that the results of a two-dimensional stability analysis are in good agreement with the experimental results if the width ratio (width/height) of the experimental container is sufficiently large. The study is carried out numerically with the use of two independent numerical approaches based on the global Galerkin and finite-volume methods.

Description: Linear Rossby waves in a two-layer ocean with a corrugated bottom relief (the isobaths are straight parallel lines) are investigated. The case of a rough bottom relief (the wave scale L is much greater than the bottom relief scale Lb) is studied analytically by the method of multiple scales. A special numerical technique is developed to investigate the waves over a periodic bottom relief for arbitrary relationships between L and Lb. There are three types of modes in the two-layer case: barotropic, topographic, and baroclinic. The structure and frequencies of the modes depend substantially on the ratio Δ= (Δh/h2)/(L/a) measuring the relative strength of the topography and β-effect. Here Δh/h2 is the typical relative height of topographic inhomogeneity and a is the Earth's radius. The topographic and barotropic mode frequencies depend weakly on the stratification for small and large Δ and increase monotonically with increasing Δ. Both these modes become close to pure topographic modes for Δ ≫ 1. The dependence of the baroclinic mode on Δ is more non-trivial. The frequency of this mode is of the order of f0L2 i/aL (Li is the internal Rossby scale) irrespective of the magnitude of Δ. At the same time the spatial structure of the mode depends strongly on Δ. With increasing Δ the relative magnitude of motion in the lower layer decreases. For Δ ≫ 1 the motion in the mode is confined mainly to the upper layer and is very weak in the lower one. A similar concentration of mesoscale motion in an upper layer over an abrupt bottom topography has been observed in the real ocean many times. Another important physical effect is the so-called 'screening'. It implies that for Lb, 〈 Li the small-scale component of the wave with scale Lb is confined to the lower layer, whereas in the upper layer the scale of the motion L is always greater than or of the order of, Li. In other words, the stratification prevents the ingress of motion with scale smaller than the internal Rossby scale into the main thermocline.

Description: This work brings new insight to the question of heat transfer in near-critical fluids under Earth gravity conditions. The interplay between buoyant convection and thermoacoustic heat transfer (piston effect) is investigated in a two-dimensional non-insulated cavity containing a local heat source, to reproduce the conditions used in recent experiments. The results were obtained by means of a finite-volume numerical code solving the Navier-Stokes equations written for a low-heat-diffusing near-critical van der Waals fluid. They show that hydrodynamics greatly affects thermoacoustics in the vicinity of the upper thermostated wall, leading to a rather singular heat transfer mechanism. Heat losses through this wall govern a cooling piston effect. Thus, the thermal plume rising from the heat source triggers a strong enhancement of the cooling piston effect when it strikes the middle of the top boundary. During the spreading of the thermal plume, the cooling piston effect drives a rapid thermal quasi-equilibrium in the bulk fluid since it is further enhanced so as to balance the heating piston effect generated by the heat source. Then, homogeneous fluid heating is cancelled and the bulk temperature stops increasing. Moreover, diffusive and convective heat transfers into the bulk are very weak in such a low-heat-diffusing fluid. Thus, even though a steady state is not obtained owing to the strong and seemingly continuous instabilities present in the flow, the bulk temperature is expected to remain quasi-constant. Comparisons performed with a supercritical fluid at initial conditions further from the critical point show that this thermalization process is peculiar to near-critical fluids. Even enhanced by the thermal plume, the cooling piston effect does not balance the heating piston effect. Thus, overall piston-effect heating lasts much longer, while convection and diffusion progressively affect the thermal field much more significantly. Ultimately, a classical two-roll convective-diffusive structure is obtained in a perfect gas, without thermoacoustic heat transfer playing any role.

Description: Analytical expressions are derived for the velocity vector, the stress components and the viscosity function in fully developed channel and pipe flow of Phan-Thien-Tanner (PTT) fluids; both the linearized and the exponential forms of the PTT equation are considered. The solution shows that the wall shear stress of a PTT fluid is substantially smaller than the corresponding value for a Newtonian or upper-convected Maxwell fluid, with implications for comparing predicted and measured values in a non-dimensional form.

Description: Convection in a vertical magnetic field occurs in narrow cells in the physically relevant limit where the Chandrasekhar number Q becomes large, corresponding to a strong field or small diffusion. This allows asymptotic solutions to be developed for fully nonlinear convection, requiring only the solution of a nonlinear boundary value problem. Solutions for steady and oscillatory magnetoconvection are obtained with different scalings. In the steady case, the heat flux and the fluid velocity are found at leading order in the asymptotic expansion and the vertical velocity scales as Q 1/6 . In the oscillatory case, where it is necessary to continue to second order, the vertical velocity is of order Q 1/3 and the frequency of the oscillations is always greater than that predicted by linear theory. The heat flux does not depend on either the wavenumber or the planform.

Description: In this paper we study a dynamical system consisting of a rigid body and an inviscid incompressible fluid. Two general configurions of the system are considered: (a) a rigid body with a cavity completely filled with a fluid and (b) a rigid body surrounded by a fluid. In the first case the fluid is confined to an interior (for the body) domain and in the second case it occupies an exterior domain, which may, in turn, be bounded by some fixed rigid boundary or may extend Io infinity. The aim of the paper is twofold: (i) to develop Arnold's technique for the system 'body + fluid' and (ii) to obtain sufficient conditions for the stability of steady states of the system. We first establish an energy-type variational principle for an arbitrary steady state of the system. Then we generalize this principle for states that are steady either in translationally moving in some fixed direction or rotating around some fixed axis coordinate system. The second variations of the corresponding functionals are calculated. The general results are applied to a number of particular stability problems. The first is the stability of a steady translational motion of a two-dimensional body in an irrotational flow. Here we have found that (for a quite wide class of bodies) the presence of non-zero circulation about the body does not affect its stability - a result that seems to be new. The second problem concerns the stability of a steady rotation of a force-free rigid body with a cavity containing an ideal fluid. Here we rediscover the stability criterion of Rumyantsev (see Moiseev & Rumyantsev 1965). The complementary problem - when a body is surrounded by a fluid and both body and fluid rotate with constant angular velocity around a fixed axis passing through the centre of mass of the body - is also considered and the corresponding sufficient conditions for stability are obtained.

Description: The evolution of a single hairpin vortex-like structure in the mean turbulent field of a low-Reynolds-number channel flow is studied by direct numerical simulation. The structure of the initial three-dimensional vortex is extracted from the two-point spatial correlation of the velocity field by linear stochastic estimation given a second-quadrant ejection event vector. Initial vortices having vorticity that is weak relative to the mean vorticity evolve gradually into omega-shaped vortices that persist for long times and decay slowly. As reported in Zhou, Adrian & Balachandar (1996), initial vortices that exceed a threshold strength relative to the mean flow generate new hairpin vortices upstream of the primary vortex. The detailed mechanisms for this upstream process are determined, and they are generally similar to the mechanisms proposed by Smith et al. (1991), with some notable differences in the details. It has also been found that new hairpins generate downstream of the primary hairpin, thereby forming, together with the upstream hairpins, a coherent packet of hairpins that propagate coherently. This is consistent with the experimental observations of Meinhart & Adrian (1995). The possibility of autogeneration above a critical threshold implies that hairpin vortices in fully turbulent fields may occur singly, but they more often occur in packets. The hairpins also generate quasi-streamwise vortices to the side of the primary hairpin legs. This mechanism bears many similarities to the mechanisms found by Brooke & Hanratty (1993) and Bernard, Thomas & Handler (1993). It provides a means by which new quasi-streamwise vortices, and, subsequently, new hairpin vortices can populate the near-wall layer.

Description: We analyse and improve a recently-proposed two-phase flow model for the statistical evolution of two-fluid mixing. A hyperbolic equation for the volume fraction, whose characteristic speed is the average interface velocity υ*, plays a central role. We propose a new model for υ* in terms of the volume fraction and fluid velocities, which can be interpreted as a constitutive law for two-fluid mixing. In the incompressible limit, the two-phase equations admit a self-similar solution for an arbitrary scaling of lengths. We show that the constitutive law for υ* can be expressed directly in terms of the volume fraction, and thus it is an experimentally measurable quantity. For incompressible Rayleigh-Taylor mixing, we examine the self-similar solution based on a simple zero-parameter model for υ*. It is shown that the present approach gives improved agreement with experimental data for the growth rate of a Rayleigh-Taylor mixing layer. Closure of the two-phase flow model requires boundary conditions for the surfaces that separate the two-phase and single-phase regions, i.e. the edges of the mixing layer. We propose boundary conditions for Rayleigh-Taylbr mixing based on the inertial, drag, and buoyant forces on the furthest penetrating structures which define these edges. Our analysis indicates that the compatibility of the boundary conditions with the two-phase flow model is an important consideration. The closure assumptions introduced here and their consequences in relation to experimental data are compared to the work of others.

Description: We present direct numerical simulations of the spatial development of normal mode perturbations to boundary layers with Falkner-Skan velocity profiles. Values of the pressure gradient parameter considered range from very small, i.e. nearly flat-plate conditions, to relatively large values corresponding to incipient separation. In almost all cases, we find that the most effective perturbation is one composed of a plane wave and a pair of oblique waves inclined at equal and opposite angles to the primary flow direction. The frequency of the oblique waves is half that of the fundamental plane wave and because the conditions for resonance are satisfied exactly, all modes share a common critical layer, thus facilitating a strong interaction. The oblique waves initially undergo a parametric type of subharmonic resonance, but in accordance with recent analyses of non-equilibrium critical layers, the system subsequently becomes fully coupled. From that point on, the amplification of all modes, including the plane wave, substantially exceeds the predictions of linear stability theory. Good agreement is obtained with the experimental small pressure gradient results of Corke & Gruber (1996). Our growth rates are slightly larger owing to slight differences in initial conditions (e.g. the angle of inclination of the oblique waves). The spectral element method was used to discretize the Navier-Stokes equations and the preconditioned conjugate gradient method was used to solve the resulting system of algebraic equations. At the inflow boundary, Orr-Sommerfeld modes were employed to provide the initial forcing, whereas the buffer domain technique was used at the outflow boundary to prevent convective wave reflection or upstream propagation of spurious information.

Description: Trapped modes around a row of bottom-mourned vertical circular cylinders in a channel are examined. The cylinders are identical, and their axes equally spaced in a plane perpendicular to the channel walk. The analysis has been made by employing the multipole expansion method under the assumption of linear water wave theory. Al least the same number of trapped modes is shown to exist as the number of cylinders for both Neumann and Dirichlet trapped modes, with the exception that for cylinders having large radius the mode corresponding to the Dirichlet trapped mode for one cylinder will disappear. Close similarities between the Dirichlet trapped modes around a row of cylinders in a channel and the near-resonant phenomenon in the wave diffraction around a long array of cylinders in the open sea are discussed. An analogy with a mass-spring oscillating system is also presented.

Description: We divide the interaction between wind and ocean surface waves into three parameter regimes, namely slow, intermediate and fast waves, that are distinguished by the ratio c/u. (c is the wave phase speed and u. is the friction velocity in the wind). We develop here an analytical model for linear changes to the turbulent air flow caused by waves of small slope that is applicable to slow and to fast waves. The wave-induced turbulent shear stress is parameterized here with a damped mixing-length model, which tends to the mixing-length model in an inner region that lies dose to the surface, and is then damped exponentially to zero in an outer region that lies above the inner region. An adjustable parameter in the damped mixing-length model controls the rate of decay of the wave-induced stress above the inner region, and shows bow the results van from a model with no damping, which corresponds to using the mixing-length model throughout the flow, to a model with full damping, which, following previous suggestions, correctly represents rapid distortion of the wave-induced turbulence in the outer region. Solutions for air flow over fast waves are obtained by analysing the displacement of streamlines over the wave: they show that fast waves are damped, thereby giving their energy up to the wind. There is a contribution to this damping from a counterpart of the non-separated sheltering mechanism that gives rise to growth of slow waves (Belcher & Hunt 1993). This sheltering contribution is smaller than a contribution from the wave-induced surface stress working against the orbital motions in the water. Solutions from the analysis for both slow and fast waves are in excellent agreement with values computed by Mastenbroek (1996) from the nonlinear equations of motion with a full second-order closure model for the turbulent stresses. Comparisons with data for slow and intermediate waves show that the results agree well with laboratory measurements over wind-ruffled paddle-generated waves, but give results that are a factor of about two smaller than measurements of purely wind-generated waves. We know of no data for fast waves with which to compare the model. The damping rates we find for fast waves lead to e-folding times for the decay of the waves that are a day or longer. Although this wind-induced damping of fast waves is small, we suggest that it might control low-frequency waves in a fully-developed sea.

Description: The surface current generated by internal waves in the ocean affects surface gravity waves. The propagation of short surface waves is studied using both simple ray theory for linear waves and a fully nonlinear numerical potential solver. Attention is directed to the case of short waves with initially uniform wavenumber. as may be generated by a strong gust of wind. In general, some of the waves are focused by the surface current and in these regions the waves steepen and may break. Comparisons are made between ray theory and the more accurate solutions. For ray theory, the occurrence of focusing is examined in some detail and exact analytic solutions are found for rays on currents with linear and quadratic spatial variation - only the latter giving focusing for our initial conditions. With regard to interpretation of remote sensing of the sea surface, we find that enhanced wave steepness is not necessarily associated with a particular phase of the internal wave, and simplistic interpretations may sometimes be misleading.

Description: A three-dimensional analysis is performed to investigate the effects of an electric field on the steady deformation and small-amplitude oscillation of a bubble in dielectric liquid. To deal with a general class of electric fields, an electric field near the bubble is approximately represented by the sum of a uniform field and a linear field. Analytical results have been obtained for steady deformation and modification of oscillation frequency by using the domain perturbation method with the angular momentum operator approach. It has been found that, to the first order, the steady shape of a bubble in an arbitrary electric field can be represented by a linear combination of a finite number of spherical harmonics Y1 m. where 0 ≤ 1 ≤ 4 and |m| ≤ 1. For the oscillation about the deformed steady shape, the overall frequency modification from the value of free oscillation about a spherical shape is obtained by considering two contributions separately: (i) that due to the deformed steady shape (indirect effect), and (ii) that due to the direct effect of an electric field. Both the direct and indirect effects of an electric field split the (2l+1)-fold degenerate frequency of Y1 m modes, in the case of free oscillation about a spherical shape, into different frequencies that depend on m. However, when the average is taken over the (2l+1) values of m, the frequency splitting due to the indirect effect via the deformed steady shape preserves the average value, while the splitting due to the direct effect of an electric field does not. The oscillation characteristics of a bubble in a uniform electric field under the negligible compressibility assumption are compared with those of a conducting drop in a uniform electric field. For axisymmetric oscillation modes, deforming the steady shape into a prolate spheroid has the same effect of decreasing the oscillation frequency in both the drop and the bubble. However, the electric field has different effects on the oscillation about a spherical shape. The oscillation frequency increases with the increase of electric field in the case of a bubble, while it decreases in the case of a drop. This fundamental difference comes from the fact that the electric field outside the bubble exerts a suppressive surface force while the electric field outside the conducting drop exerts a pulling force on the surface.

Description: Motivated by the study of blood flow in the coronary arteries, this paper examines the flow of an incompressible Newtonian fluid in a tube of time-dependent curvature. The flow is driven by an oscillatory pressure gradient with the same dimensionless frequency, α, as the curvature variation. The dimensionless governing parameters of the flow are α, the curvature ratio δ0 a secondary streaming Reynolds number Rs and a parameter Rt representing the time-dependence of curvature. We consider the parameter regime δ0 ≪ Rt ≪ 1 (Rs and α remain O(1) initially) in which the effect of introducing time-dependent curvature is to perturb the flow driven by an oscillatory pressure gradient in a fixed curved tube. Flows driven by low- and high-frequency pressure gradients are then considered. At low frequency (δ0 ≪ Rt ≪ α ≪ 1) the flow is determined by using a sequence of power series expansions (Rs = O(1)). At high frequency (δ0 ≪ Rt ≪ 1/α2 ≪ 1) the solution is obtained using matched asymptotic expansions for the region near the wall (Stokes layer) and the region away from the wall in the interior of the pipe. The behaviour of the flow in the interior is then determined at both small and intermediate values of Rs. For both the low and high frequency cases, we find the principal corrections introduced by the time-varying curvature to the primary and secondary flows, and hence to the wall shear stress. The physiological application to flow in the coronary arteries is discussed.

Description: Vortex connections al the surface are fundamental and proouncnc features to free-surface vortical flows. To understand the detailed mechanism of such connection, we consider, as a canonical problem, the laminar vortex connections at a free surface when an oblique vortex ring impinges upon thai surface. We perform numerical simulations of the Navier Stokes equations with viscous free-surface boundary conditions. It is found that the key to understanding the mechanism of vortex connection at a free surface is the surface layers: a viscous layer resulting from the dynamic zero-stress boundary conditions at the free surface, and a thicker blockage layer which is due to the kinematic boundary condition at the surface. In the blockage layer, the vertical vorticity component increases due to vortex stretching and vortex turning (from the transverse vorticity component). The vertical vorticity is then transported to the free surface through viscous diffusion and vortex stretching in the viscous layer leading to increased surface-normal vorticity. These mechanisms take place at the aft-shoulder regions of the vortex ring. Connection at the free surface is different from that at a free-slip wall owing to the generation of surface secondary vorticity. We study the components of this surface vorticity in detail and find that the presence of a free surface accelerates the connection process. We investigate the connection time scale and its dependence on initial incidence angle, Froude and Reynolds numbers. It is found that a criterion based on the streamline topology provides a precise definition for connection time, and may be preferred over existing definitions, e.g. those based on free-surface elevation or net circulation.

Description: Direct numerical simulations are performed of a confined three-dimensional, temporally developing, initially isothermal gas mixing layer with one stream laden with as many as 7.3×105 evaporating hydrocarbon droplets, at moderate gas temperature and subsonic Mach number. Complete two-way phase couplings of mass, momentum and energy are incorporated which are based on a thermodynamically self-consistent specification of the vapour enthalpy, internal energy and latent heal of vaporization. Effects of the initial liquid mass loading ratio (ML), initial Stokes number (St0), initial droplet temperature and flow three-dimensionality on the mixing layer growth and development are discussed. The dominant parameter governing flow modulation is found to be the liquid mass loading ratio. Variations in the initial Stokes number over the range 0.5 ≤ St0 ≤ 2.0 do not cause significant modulations of either first- or second-order gas phase statistics. The mixing layer growth rate and kinetic energy are increasingly attenuated for increasing liquid loadings in the range 0 ≤ ML ≤ 0.35. The laden stream becomes saturated before evaporation is completed for all but the smallest liquid loadings owing to: (i) latent heat effects which reduce the gas temperature, and (ii) build up of the evaporated vapour mass fraction. However, droplets continue to be entrained into the layer where they evaporate owing to contact with the relatively higher-temperature vapour-free gas stream. The droplets within the layer are observed to be centrifuged out of high-vorticity regions and lo migrate towards high-strain regions of the flow. This results in the formation of concentration streaks in spanwise braid regions which are wrapped around the periphery of secondary streamwise vortices. Persistent regions of positive and negative slip velocity and slip temperature are identified. The velocity component variances in both the streamwise and spanwise directions are found to be larger for the droplets than for the gas phase on the unladen stream side of the layer; however, the cross-stream velocity and temperature variances are larger for the gas. Finally, both the mean streamwise gas velocity and droplet number density profiles are observed to coincide for all ML when the cross-stream coordinate is normalized by the instantaneous vorticity thickness; however, first-order thermodynamic profiles do not coincide.

Description: Radial flow takes place in a heterogeneous porous formation of random and stationary log-conductivity Y (x), characterized by the mean (Y), the variance σY 2 and the two-point autocorrelation ρY which in turn has finde and different horizontal and vertical integral scales, I and Ir, respectively. The steady flow is driven by a head difference between a fully penetrating well and an outer boundary, the mean velocity U being radial. A tracer is injected for a short time through the well envelope and the thin plume spreads due to advection by the random velocity field and to pore-scale dispersion. Transport is characterized by the mean front r = R(t) and by the second spatial moment of the plume Srr. Under ergodic conditions, i.e. for a well length much larger than the vertical integral scale, Srr is equal to the radial fluid trajectory variance Xrr. The aim of the study is to determine Xrr(t) for a given heterogeneous structure and for given pore-scale dispersivities The problem is more complex than the similar one for mean uniform flow. To simplify it, the well is replaced by a line source, the domain is assumed to be infinite and a first-order approximation in σY 2 is adopted. The solution is still difficult, being expressed with the aid of a few quadratures. It is found, however, that it can be derived quite accurately for a sufficiently small anisotropy ratio e = Iv/I by retaining only one term of the velocity two-point covariance. This major simplification leads to simple calculations and even to analytical solutions in the absence of pore-scale dispersion. To compare the results with those prevailing in homogeneous media, apparent and equivalent macrodispersivities are defined for convenience. The major difference between transport in radial and uniform flow is that the asymptotic, large-time, apparent macrodispersivity in the former is smaller by a factor of 3 than in the latter. For a three-dimensional point source the reduction is by a factor of 5. This effect is explained by the rapid change of the mean velocity during the period in which the velocities of two particles injected at the source become uncorrelated. In contrast, the equivalent macrodispersivity tends to its value in uniform flow far from the well, where the flow is slowly varying in space.

Description: The sound generated by vortex pairing in axisymmetric jets is determined by direct solution of the compressible Navier-Stokes equations on a computational grid that includes both the near field and a portion of the acoustic far field. At low Mach number, the far-field sound has distinct angles of extinction in the range of 60°-70° from the jet's downstream axis which can be understood by analogy to axisymmetric, compact quadrupoles. As the Mach number is increased, the far-field sound takes on a superdirective character with the dominant sound directed at shallow angles to the jet's downstream axis. The directly computed sound is compared to predictions obtained from Lighthill's equation and the kirchhoff surface method. These predictions are in good agreement with the directly computed data. The Lighthill source terms have a large spatial distribution in the axial direction necessitating the introduction of a model to describe the source terms in the region downstream of the last vortex pairing. The axial non-compactness of the quadrupole sources must be adequately treated in the prediction method.

Description: The spreading characteristics of jets from several asymmetric nozzles, and a set of rectangular orifices are compared, covering a jet Mach number range of 0.3-2.0. The effect of 'tabs' for a rectangular and a round nozzle is also included in the comparison. Compared to a round jet, the jets from the asymmetric nozzles spread only slightly more at subsonic conditions whereas at supersonic conditions, when 'screech' occurs, they spread much more. The dynamics of the azimuthal vortical structures of the jet. organized and intensified under the screeching condition, are thought to be responsible for the observed effect at supersonic conditions. Curiously, the jet from a 'lobed' nozzle spreads much less at supersonic condition compared to all other cases; this is due to the absence of screech with this nozzle. Screech stages inducing flapping, rather than varicose or helical, flow oscillation cause a more pronounced jet spreading. At subsonic conditions, only a slight increase in jet spreading with the asymmetric nozzles contrasts previous observations by others. The present results show that the spreading of most asymmetric jets is not much different from that of a round jet. This inference is further supported by data from the rectangular orifices. In fact, jets from the orifices with small aspect ratio (AR) exhibit virtually no increase in the spreading. A noticeable increase commences only when AR is larger than about 10. Thus, 'shear layer perimeter stretching', achieved with a larger AR for a given cross-sectional area of the orifice, by itself, proves to be a relatively inefficient mechanism for increasing jet spreading. In contrast, the presence of stream wise vortices or 'natural excitation' can cause a significant increase - effects that might explain the observations in the previous investigations. Thus far, the biggest increase in jet spreading is observed with the tabs. This is true in the subsonic regime, as well as in the supersonic regime, in spite of the fact that screech is eliminated by the tabs. The characteristic spreading of the tabbed jets is explained by the induced motion of the tab-generated streamwise vortex pairs. The tabs, however, incur thrust loss; the flow blockage and loss in thrust coefficient, vis-à-vis the spreading increase, are evaluated for various configurations.

Description: Large-eddy simulation is used to investigate fully developed turbulent flow in a neutral channel wherein the lower wall is sinusoidal. The numerical results are compared with experimental observations for wave slopes ranging from 0 to 0.628. Particular emphasis is placed on the separated now induced by a large-amplitude wave. A detailed comparison with the data of Buckles, Hanratty & Adrian (1984) shows generally good agreement. Large-eddy simulation surface pressures are integrated to calculate form drag as a function of wave slope. Drag is found to increase quadratically with slope for small-amplitude waves, with a somewhat slower increase for larger amplitudes. However, comparison with experimental measurements is confounded by uncertainties with the values reported in the literature. An interesting feature characteristic of all wavy-surface simulations is an increase in transverse velocity fluctuations on the wave upslope. Although the precise mechanism responsible is not known, analysis shows it to be associated with temporally persistent vortex-like structures localized near the surface. The magnitude of the fluctuation increase appears to scale quadratically with slope for small-amplitude waves, in contrast to the streamwise fluctuations, which increase linearly.

Description: Helical vortices in swirl flow are studied theoretically and experimentally. A theory of helical vortices has been developed. It includes the following results: an analytical solution describing an elementary helical vortex structure - an infinitely thin filament; a solution for axisymmetrical vortices accounting for the helical shape of vortex lines and different laws of vorticity distribution; a formula for calculation of the self-induced velocity of helical vortex rotation (precession) in a cylindrical tube; an explanation of the zone with reverse flow (recirculation zone) arising in swirl flows; and the classification of vortex structures. The experimental study of helical vortices was carried out in a vertical hydrodynamical vortex chamber with a tangential supply of liquid through turning nozzles. Various vortex structures were formed owing to changing boundary conditions on the bottom and at the exit section of the chamber. The hypothesis of helical symmetry is confirmed for various types of swirl flow. The stationary helical vortex structures are described (most of them for the first time) the features of which agree with the results and predictions of the theoretical model developed. They are the following: a rectilinear vortex; a composite columnar vortex; helical vortices screwed on the right or on the left; a vortex with changing helical symmetry; a double helix - two entangled vortex filaments of the same sign.

Description: Inertial waves are a ubiquitous feature of rapidly rotating fluids. Although much is known about their initial excitation, little is understood about their stability. Experiments indicate that they are generically unstable and in many cases catastrophically so, quickly causing the whole flow to collapse to small-scale disorder The linear stability of two three-dimensional inertial waves observed to break down in the laboratory is considered here at experimentally small but finite Ekman numbers of ≤ 10-4. Surprisingly small threshold amplitudes for instability are found. The results support the conjecture that triad resonances are the generic mechanism for secondary instability in rapidly rotating fluids but also highlight the ability of geostrpphic flows to derive energy through a finite-amplitude inertial wave. This latter finding may go some way to explaining the significant mean circulations typically observed in inertial wave experiments.

Description: This paper describes laboratory experiments on the self-organizing character of geophysical turbulence The experiments were carried out in a linearly stratified fluid, forced horizontally with sources and sinks around a horizontal ring. The flow was visualized with small particles illuminated by a horizontal light sheet recorded with a camera and analysed with an advanced particle tracking system Qualitative and quantitative data, such as flow patterns, velocity and vorticity fields, were obtained. In the experiments, the inverse energy cascade was clearly observed: the flow organized into a single quasi-steady, coherent vortex structure of the largest available scale. This vortex is maintained against diffusion of momentum by entrainment of vorticity from its exterior. In this process, patches of vorticity of the same sign as the core of this large structure intermittently cross the vorticity barrier around the vortex. Patches of opposite vorticity were observed to be effectively blocked by the barrier. The direction of rotation of the vortex was set by a slight bias in the experimental apparatus and could be changed by imposing a small initial circulation in the opposite sense, the magnitude of which suggested a measure for the bias. A detailed study of the effects of changing forcing parameters was carried out. The number of sinks (which play only a passive role) does not affect the flow, whereas the number of sources sets the lengthscale of the forcing and thereby determines the size of the vortices that are created close to the ring, as well as that of the large central vortex that emerges. However, after longer times of forcing, the vortex size also depends on the strength of the forcing. The velocities in the large vortex structure scale with the mean velocity from the sources, and with the square root of their number. Measurements were also taken of the decay of the vortex. After switching off the forcing it quickly becomes axisymmetric and a linear functional relationship is established between the vorticity and streamfunction. The spin-down time was observed to be much shorter than can be accounted for by vertical viscous diffusion alone: initially the short horizontal scale of the vorticity barrier causes a relatively fast decay, whereas at later times the size of the vortex as a whole is important.

Description: The onset of transition in a boundary layer is dependent on the initialization and interaction of disturbances in a laminar flow. Here, theory and full Navier-Stokes simulations focus on the transient period just after disturbances enter the boundary layer. The temporal evolution of disturbances within a boundary layer is investigated by examining a series of initial value problems. In each instance, the complete spectra (i.e. the discrete and the continuum) are included so that the solutions can be completely arbitrary. Both numerical and analytical solutions of the linearized Navier-Stokes equations subject to the arbitrary initial conditions are presented. The temporal evolution of disturbances during the transient period are compared with the spatial evolution of the same disturbances and a strong correlation between the two approaches is demonstrated indicating that the theory may be used for the transient period of disturbance evolution. The theory and simulations demonstrate that strong amplification of the disturbances can occur as a result of the inclusion of the continuum in the prediction of disturbance evolution. The results further show that any approach proposed for use in bypass boundary layer transition must include the transient growth that results from the continuum. Finally, although a connection between temporal and spatial evolution in the transient period has been demonstrated, a theoretical basis as an explanation for this connection remains the focus of additional study.

Description: An experimental and theoretical investigation has been carried out to understand the tonal noise generation mechanism on aerofoils at moderate Reynolds number. Experiments were conducted on a NACA0012 aerofoil section in a low-turbulence closed working section wind tunnel. Narrow band acoustic tones were observed up to 40 dB above background noise. The ladder structure of these tones was eliminated by modifying the tunnel to approximate to anechoic conditions. High-resolution flow velocity measurements have been made with a three-component laser-Doppler anemometer (LDA) which have revealed the presence of strongly amplified boundary-layer instabilities in a region of separated shear flow just upstream of the pressure surface trailing edge, which match the frequency of the acoustic tones. Flow visualization experiments have shown these instabilities to roll up to form a regular Kármán-type vortex street. A new mechanism for tonal noise generation has been proposed, based on the growth of Tollmien-Schlichting (T-S) instability waves strongly amplified by inflectional profiles in the separating laminar shear layer on the pressure surface of the aerofoil. The growth of fixed frequency, spatially growing boundary-layer instability waves propagating over the aerofoil pressure surface has been calculated using experimentally obtained boundary-layer characteristics. The effect of boundary-layer separation has been incorporated into the model. Frequency selection and prediction of T-S waves are in remarkably good agreement with experimental data.

Description: Sound transmission through straight circular ducts with a uniform inviscid mean flow and a constant acoustic lining (impedance wall) is classically described by a modal expansion. A natural extension for ducts with axially slowly varying properties (diameter and mean flow, wall impedance) is a multiple-scales solution. It is shown in the present paper that a consistent approximation of boundary condition and isentropic mean flow allows the multiple-scales problem to have an exact solution. Since the calculational complexities are no greater than for the classical straight duct model, the present solution provides an attractive alternative to a full numerical solution if diameter variation is relevant. A unique feature of the present solution is that it provides a systematic approximation to the hollow-to-annular cylinder transition problem in the turbofan engine inlet duct.

Description: We discuss laboratory experiments with a continuous source or sink of fluid in a two-layer rotating environment which produces anticyclonic and cyclonic vortices, respectively. Experiments were carried out with a sloping bottom in order to simulate the β-effect and they were conducted for different values for the source/sink flow rate Q and the Coriolis parameter F. The Rossby number Ro of these vortices was small but finite and the flow was expected to be quasi-geostrophic. The qualitative behaviour of the anticyclonic and cyclonic vortices was generally similar, but it depended on the flow rate. For low flow rates, a single vortex formed at the source and extended to the west. At higher flow rates, the vortex broke free from the source and moved to the west; this vortex was then followed sequentially by other vortices behaving similarly. The westward velocity U of these vortices was calculated and compared with the speed Us of a linear topographic Rossby wave. For multiple vortices the westward velocities were greater than Us while for a single vortex produced by a low flow rate the velocity was less than Us. Significant asymmetry between the anticyclonic and cyclonic vortices was observed in the transition zone from single to multiple vortices which implies that ageostrophic effects were still present in the flow.

Description: Effects of the Lewis number and radiative heat loss on flame bifurcations and extinction of CH4/O2-N2-He flames are investigated numerically with detailed chemistry. Attention is paid to the interaction between radiation heat loss and the Lewis number effect. The Planck mean absorption coefficients of CO, CO2, and H2O are calculated using the statistical narrow-band model and compared with the data given by Tien. The use of Tien's Planck mean absorption coefficients overpredicts radiative heat loss by nearly 30% in a counterflow configuration. The new Planck mean absorption coefficients are then used to calculate the extinction limits of the planar propagating flame and the counterflow flame when the Lewis number changes from 0.967 to 1.8. The interaction between radiation heat loss and the Lewis number effect greatly enriches the phenomenon of flame bifurcation. The existence of multiple flames is shown to be a physically intrinsic phenomenon of radiating counterflow flames. Eight kinds of typical patterns of flame bifurcation are identified. The competition between radiation heat loss and the Lewis number effect results in two distinct phenomena, depending on if the Lewis number is greater or less than a critical value. Comparisons between the standard limits of the unstrained flames and the flammability limits of the counterflow flames indicate that the flammability limit of the counterflow flame is lower than the standard limit when the Lewis number is less than the critical value and is equal to the standard limit when the Lewis number is higher than this critical value. Finally, a G-shaped curve and a K-shaped curve which respectively represent the flammable regions of the multiple flames for Lewis numbers lower and higher than the critical value are obtained. The G- and K-shaped curves show a clear relationship between the stretched counterflow flame and the unstrained planar flame. The present results provide a good explanation of the physics revealed experimentally in microgravity.

Description: The problem of solidification or melting under the action of a forced hydrodynamic flow is considered. In the appropriate parameter régime, the problem admits a formulation in terms of analytic functions. It is shown that a crystal with a parabolic tip propagates without change of shape at a steady velocity. Some novel explicit solutions are presented.

Description: If a sill-enclosed basin, connected to a large reservoir, is suddenly subjected to a de-stabilizing surface buoyancy flux, it will first mix vertically by turbulent convection before the resulting lateral buoyancy gradient generates a horizontal exchange flow across the sill. We present a study which examines the unsteady adjustment of such a basin under continued steady forcing. It is shown, through theoretical development and laboratory experimentation, that two consecutive unsteady regimes characterized by different dynamic balances are traversed as the flow approaches a steady state. Once established the exchange flow is controlled at the sill crest where it is hydraulically critical. In the absence of a lateral contraction, the single control at the sill crest allows a range of submaximal exchange states with the flow at the sill being dependent not only on the forcing and geometrical parameters but also on mixing conditions within the basin which are, in turn, dependent on the sill exchange. The sill-basin system is therefore strongly coupled although it remains isolated from the external reservoir conditions by a region of internally supercritical flow. Results from the laboratory experiments are used to demonstrate the link between the forcing and the exchange flow at the sill. Steady-state measurements of the interior mean velocity and buoyancy fields are also compared with previous analytical models.

Description: This paper reports on a series of large-eddy simulations of a round jet issuing normally into a crossflow. Simulations were performed at two jet-to-crossflow velocity ratios, 2.0 and 3.3, and two Reynolds numbers, 1050 and 2100, based on crossflow velocity and jet diameter. Mean and turbulent statistics computed from the simulations match experimental measurements reasonably well. Large-scale coherent structures observed in experimental flow visualizations are reproduced by the simulations, and the mechanisms by which these structures form are described. The effects of coherent structures upon the evolution of mean velocities, resolved Reynolds stresses, and turbulent kinetic energy along the centreplane are discussed. In this paper, the ubiquitous far-field counter-rotating vortex pair is shown to originate from a pair of quasi-steady 'hanging' vortices. These vortices form in the skewed mixing layer that develops between jet and crossflow fluid on the lateral edges of the jet. Axial flow through the hanging vortex transports vortical fluid from the near-wall boundary layer of the incoming pipe flow to the back side of the jet. There, the hanging vortex encounters an adverse pressure gradient and breaks down. As this breakdown occurs, the vortex diameter expands dramatically, and a weak counter-rotating vortex pair is formed that is aligned with the jet trajectory.

Description: The linear stability of Burgers and Lamb-Oseen vortices is addressed when the vortex of circulation Γ and radius δ is subjected to an additional strain field of rate s perpendicular to the vorticity axis. The resulting non-axisymmetric vortex is analysed in the limit of large Reynolds number RΓ = Γ/ν and small strain s ≪ Γ/δ2 by considering the approximations obtained by Moffatt et al. (1994) and Jiménez et al. (1996) for each case respectively. For both vortices, the TWMS instability (Tsai & Widnall 1976; Moore & Saffman 1975) is shown to be active, i.e. stationary helical Kelvin waves of azimuthal wavenumbers m = 1 and m = -1 resonate and are amplified by the external strain in the neighbourhood of critical axial wavenumbers which are computed. The additional effects of diffusion for the Lamb-Oseen vortex and stretching for the Burgers vortex are proved to limit in time the resonance. The transient growth of the helical waves is analysed in detail for the distinguished scaling s ∼ Γ/(δ2RΓ1/2). An amplitude equation describing the resonance is obtained and the maximum gain of the wave amplitudes is calculated. The effect of the vorticity profile on the instability characteristic as well as of a time-varying stretching rate are analysed. In particular the stretching rate maximizing the instability is calculated. The results are also discussed in the light of recent observations in experiments and numerical simulations. It is argued that the Kelvin waves resonance mechanism could explain various dynamical behaviours of vortex filaments in turbulence.

Description: The surface profile histories of gentle spilling breakers generated mechanically with a dispersive focusing technique are studied experimentally. Froude-scaled generation conditions are used to produce waves with three average frequencies: f0 = 1.42, 1.26, and 1.15 Hz. At each frequency, the strength of the breaker is varied by varying the overall amplitude of the wavemaker motion. It is found that in all cases the beginning of the breaking process is marked by the formation of a bulge in the profile at the crest on the forward face of the wave. The leading edge of this bulge is called the toe. As the breaking process continues, the bulge becomes more pronounced while the toe remains in nearly a fixed position relative to the crest. Capillary waves form ahead of the toe. At a time of about 0.1/f0 after the bulge first becomes visible, the toe begins to move down the face of the wave and very quickly accelerates to a constant velocity which scales with the wave crest speed. During this phase of the breaker evolution, the surface profile between the toe and the crest develops ripples which eventually are left behind the wave crest. It is found that the height of the toe above the mean water level scales with the nominal wavelength λ0 = g/(2πf20) of the breaker, while the size and shape of the bulge and the length of the capillary waves ahead of the toe are independent of f0.

Description: Viscous liquid drops undergoing forced oscillations have been shown to exhibit hysteretic deformation under certain conditions both in experiments and by solution of simplified model equations that can only provide a qualitative description of their true response. The first hysteretic deformation results for oscillating pendant drops obtained by solving the full transient, nonlinear Navier-Stokes system are presented herein using a sweep procedure in which either the forcing amplitude A or frequency Ω is first increased and then decreased over a given range. The results show the emergence of turning-point bifurcations in the parameter space of drop deformation versus the swept parameter. For example, when a sweep is carried out by varying Ω while holding A fixed, the first turning point occurs at Ω ≡ Ωu as Ω is being increased and the second one occurs at Ω ≡ Ωl 〈 Ωu as Ω is being decreased. The two turning points shift further from each other and toward lower values of the swept parameter as Reynolds number Re is increased. These turning points mark the ends of a hysteresis range within which the drop may attain either of two stable steady oscillatory states -limit cycles - as identified by two distinct solution branches. In the hysteresis range, one solution branch, referred to as the upper solution branch, is characterized by drops having larger maximum deformations compared to those on the other branch, referred to as the lower solution branch. Over the range Ωl ≤ Ω ≤ Ωu, the sweep procedure enables detection of the upper solution branch which cannot be found if initially static drops are set into oscillation as in previous studies of forced oscillations of supported and captive drops, or liquid bridges. The locations of the turning points and the associated jumps in drop response amplitudes observed at them are studied over the parameter ranges 0.05 ≤ A ≤ 0.125, 20 ≤ Re ≤ 40, and gravitational Bond number 0 ≤ G ≤ 1. Critical forcing amplitudes for onset of hysteresis are also determined for these Re values. The new findings have important ramifications in several practical applications. First, that Ωu - Ωl increases as Re increases overcomes the limitation which is inherent to the current practice of inferring the surface tension and/or viscosity of a bridge/drop liquid from measurement of its resonance frequencies (Chen & Tsamopoulos 1993; Mollot et al 1993). Moreover, that the value of A for onset of hysteresis can be as low as 5% of the drop radius, or lower, has important implications for other free-surface flows such as coating flows.

Description: This paper considers the structure of steady-state solutions of the Swift-Hohenberg equation describing convection in shallow rectangular-planform containers heated from below. The lateral dimensions of the planform are assumed to be much larger than the characteristic wavelength of convection. Results are restricted to patterns composed of rolls orthogonal to the sides of the rectangle in which case convection sets in at a critical value of the Rayleigh number in the form of rolls parallel to the shorter sides. This primary bifurcation from the conductive state of no motion produces a solution which subsequently undergoes a secondary bifurcation in which the low-amplitude motion near the shorter sides is replaced locally by cross-rolls perpendicular to the sides. This results in the formation of grain boundaries (or domain boundaries) within the fluid which mark the division between the different roll orientations. With increasing Rayleigh number the grain boundaries approach the sides of the rectangle and a boundary-layer structure is formed. In the present paper the method of matched asymptotic expansions is used to determine this boundary-layer structure and to predict the location of the grain boundaries. An interesting feature of the solution is that the grain boundaries develop significant curvature and bend into the corners of the rectangle, where the local solution is also determined. The results are compared with numerical computations of the secondary solution branch and with previous numerical and experimental work.

Description: Unsteady circular jets are treated experimentally and numerically. The time evolution of circular pulse jets is investigated systematically for a wide range of jet strength, with the focus on the jet evolution, in particular the formation processes of Mach disks in the middle stage and of shock-cell structures in the later stage. It is shown that unsteady second shocks are realized for all sonic underexpanded jets and they either breed conical shocks for lower pressure ratios or truncated cones (Mach disk and reflected shock) for higher pressure ratios. The vortex ring produced near the nozzle lip plays an important role in the formation of the shock-cell structure. In particular, interactions between the vortex ring and the Mach disk connected with a strong second shock affect remarkably the formation process of the first shock cell. Different formation processes of the first cell structure are found. It is also made clear that the Kelvin-Helmholtz instability along slip surfaces originating from the triple point at the outer edge of the Mach disk is responsible for the generation of large second vortices which entrain the first vortex. This results in strong mixing between the primary jet and surrounding gas for higher pressure ratios. Numerical simulations with a TVD-scheme for the Euler equations are also performed and the numerical results are compared with the experimental ones to understand and predict the flow characteristics of the pulse jets.

Description: Motivated by observations of interleaving in the equatorial Pacific, we consider the linear stability of a basic state on an equatorial β-plane which is susceptible to both double-diffusive interleaving, driven by a meridional salinity gradient, and inertial instability driven by meridional shear. In a parameter regime compatible with the observations strong interaction can occur between the two processes, indicating that the stability of the system is dependent on the meridional gradients of both salinity and zonal velocity. Meridional shear is found to enhance the interleaving motion even for values of shear well below the cutoff for inertial instability. In the presence of diffusion inertial instability can also be excited by vertical shear, but only if the shear is comparable to the buoyancy frequency. When double-diffusive driving is weak relative to inertial driving the growth can be oscillatory, in which case the mechanism for instability is viscous-diffusive. In this case interleaving layers can slope downwards towards the fresh side of the front in the fingering regime, inhibiting their own growth.

Description: In this paper we study the acoustic scattering between two flat-plate cascades, with the aim of investigating the possible existence of trapped modes. In practical terms this question is related to the phenomenon of acoustic resonance in turbomachinery, whereby such resonant modes are excited to large amplitude by unsteady processes such as vortex shedding. We use the Wiener-Hopf technique to analyse the scattering of the various wave fields by the cascade blades, and by considering the fields between adjacent blades, as well as between the cascades, we are able to take full account of the genuinely finite blade chords. Analytic expressions for the various scattering matrices are derived, and an infinite-dimensional matrix equation is formed, which is then investigated numerically for singularity. One advantage of this formulation is that it allows the constituent parts of the system to be analysed individually, so that for instance the behaviour of the gap between the blade rows alone can be investigated by omitting the finite-chord terms in the equations. We demonstrate that the system exhibits two types of resonance, at a wide range of parameter values. First, there is a cut-on/cut-off resonance associated with the gap between the rows, and corresponding to modes propagating parallel to the front face of the cascades. Second, there is a resonance of the downstream row, akin to a Parker mode, driven at low frequencies by a vorticity wave produced by trapped duct modes in the upstream row, and at higher frequencies by radiation modes (and the vorticity wave) between the blade rows. The predictions for this second set of resonances are shown to be in excellent agreement with previous experimental data. The resonant frequencies are also seen to be real for this twin cascade system, indicating that the resonances correspond to genuine trapped modes. The analysis in this paper is completed with non-zero axial flow but with zero relative rotation between the cascades - in Part 2 (Woodley & Peake 1999) we will show how non-zero rotation of the upstream row can be included.

Description: The investigation undertaken deals with the development of disturbances in a supersonic wake (free viscous layer and regular wake) behind a flat plate both in its linear and nonlinear stages. The influence of a number of factors (Mach and Reynolds numbers, temperature factor, thickness of the plate, length of its stern) on the wake stability and transition was studied. The development of the artificial disturbances in a wake at Mach number M∞ = 2 was investigated also. It was found that compressibility of the flow (increasing Mach number) stabilizes the wake disturbances - their amplification rates decrease, and the transition point moves away from the model plate. Cooling of the model surface at M∞ ∼ 7 has a destabilizing influence on the development of disturbances in the wake. With increase of unit Reynolds number the beginning of transition in the wake moves forward to a rear critical point. It was confirmed that a distinctive maximum in the spectral distribution of fluctuations appears, corresponding to Strouhal number (based on frequency of this maximum) of 0.3. With the growth of the model thickness the disturbance amplification rates in the wake increase, which results in earlier transition of a laminar wake into turbulent one. With the growth of length of the plate stern, the position of the wake transition moves back accordingly, while the wake stability increases a little (though very unsignificantly). In the nonlinear stage of development of disturbances, the occurrence of a triad of waves, satisfying the resonant correlation of frequencies, and the growth of harmonics are observed. A monochromatic packet of waves of Tollmien-Schlichting type, rather narrow (in the transversal coordinate) in the boundary layers on a flat plate with an opposite wedge at the stern, was found to extend in the wake. The wake disturbances have a complex wave structure. At the Mach number of free flow 2.0, the three-dimensional disturbances are the most unstable in the wake.

Description: We have performed a series of experiments on the dynamics of sedimenting, surface gravity currents. The physical situation concerns a current, with total density ρC, evolving at the surface of a fluid of greater density, ρA. In turn ρC is made up of interstitial fluid of density ρI and heavy particles with a concentration by weight c and a density ρP. Only the case of the release of a constant volume of particles and interstitial fluid has been considered in detail. It has been found that the sedimentation of the particles, plus some of the interstitial fluid, through the interface between the two fluids has a profound effect upon the motion of the current. When the rejected mixture of particles and upper- and lower-layer fluids reaches the bottom of the experimental tank it generates a secondary gravity current which in turn interacts with the primary current to further modify its behaviour. Using simple models we have been able to rationalize the observations and reveal the dynamical balances which appear to be important. A subsidiary experiment and analysis on the flux characteristics of the interface have been performed in order to further clarify the important effects of the particle motion through that region.

Description: Dispersion of fluid particles in non-homogeneous turbulence was studied for fully developed flow in a channel. A point source at a distance of 40 wall units from the wall is considered. Data obtained by carrying out experiments in a direct numerical simulation (DNS) are used to test a stochastic model which utilized a modified Langevin equation. All of the parameters, with the exception of the time scales, are obtained from Eulerian statistics. Good agreement is obtained by making simple assumptions about the spatial variation of the time scales.

Description: A series of laboratory experiments and numerical simulations have been performed to investigate the rapid fluid-like flow of a finite mass of granular material down a chute with partial lateral confinement. The chute consists of a section inclined at 40° to the horizontal, which is connected to a plane run-out zone by a smooth transition. The flow is confined on the inclined section by a shallow parabolic cross-slope profile. Photogrammetric techniques have been used to determine the position of the evolving boundary during the flow, and the free-surface height of the stationary granular deposit in the run-out zone. The results of three experiments with different granular materials are presented and shown to be in very good agreement with numerical simulations based on the Savage-Hutter theory for granular avalanches. The basal topography over which the avalanche flows has a strong channelizing effect on the inclined section of the chute. As the avalanche reaches the run-out zone, where the lateral confinement ceases, the head spreads out to give the avalanche a characteristic 'tadpole' shape. Sharp gradients in the avalanche thickness and velocity began to develop at the interface between the nose and tail of the avalanche as it came to rest, indicating that a shock wave develops close to the end of the experiments.

Description: The motion of three interacting point vortices with zero net circulation in a periodic parallelogram defines an integrable dynamical system. A method for solving this system is presented. The relative motion of two of the vortices can be 'mapped' onto a problem of advection of a passive particle in 'phase space' by a certain set of stationary point vortices, which also has zero net circulation. The advection problem in phase space can be written in Hamiltonian form, and particle trajectories are given by level curves of the Hamiltonian. The motion of individual vortices in the original three-vortex problem then requires one additional quadrature. A complicated structure of the solution space emerges with a large number of qualitatively different regimes of motion. Bifurcations of the streamline pattern in phase space, which occur as the impulse of the original vortex system is changed, are traced. Representative cases are analysed in detail, and a general procedure is indicated for all cases. Although the problem is integrable, the trajectories of the vortices can be surprisingly complicated. The results are compared qualitatively to vortex paths found in large-scale numerical simulations of two-dimensional turbulence.

Description: Direct numerical simulations (DNS) of natural convection in a vertical channel by Versteegh & Nieuwstadt (1998) are used for assessing the budget of the turbulent heat flux θui, and the temperature variance θ2, and for modelling the transport equations governing these two properties. The analysis is confined to a simple fully developed situation in which the gravitational vector, as the sole driving force, is perpendicular to the only non-zero component of the mean temperature gradient. Despite its simplicity, the flow displays many interesting features and represents a generic case of the interaction of buoyancy-driven turbulent temperature and velocity fields. The paper discusses the near-wall variation of the second moments and their budgets, as well as possible scaling of θui, and θ2 both in the near-wall region and away from the wall. Various proposals for the Reynolds-averaged modelling are analysed and new models are proposed for these two transport equations using the term-by-term approach. An a priori test (using the DNS data for properties other than Oθui and θ2) reproduced very well all terms in the transport equations, as well as their near-wall behaviours and wall limits, without the use of any wall-topology-dependent parameters. The computational effort is still comparable to that for the 'basic model'. The new term-by-term model of the θui and θ2 equations was then used for a full simulation in conjunction with a low-Reynolds-number second-moment velocity closure, which was earlier found to reproduce satisfactorily a variety of isothermal wall flows. Despite excellent term-by-term reproduction of thermal turbulence, the predictions with the full model show less satisfactory agreement with the DNS data than a priori validation, indicating a further need for improvement of the modelling of buoyancy effects on mechanical turbulence.

Description: We consider the stability of a rectilinear liquid region whose boundary is composed of a solid cylindrical substrate of arbitrary shape and a free surface whose cross-section, in the absence of gravity, is a circular arc. The liquid-solid contact angle is a prescribed material property. A variational technique, using an energy functional, is developed that predicts the minimum wavelength for transverse instability under the action of capillarity. Conversely, certain configurations are absolutely stable and a simple stability criterion is derived. Stability is guaranteed if, for given substrate geometry and given contact angle, the unperturbed meniscus pressure is an increasing function of the liquid cross-sectional area. The analysis is applied to a variety of liquid/substrate configurations including (i) a liquid ridge with contact lines pinned to the sharp edges of a slot or groove, (ii) liquid ridges with free contact lines on flat and wedge-shaped substrates as well as substrates of circular or elliptical cross-section. Results are consistent with special cases previously treated including those that employ a slope-small-slope approximation.