bluuebunny/crossref_metadata_2025 huggingface dataset:
The existence of steady flow in a collapsed tube
Authors: Jensen O. E. , Pedley T. J.
Published: April 2006

Self-excited oscillations arise during flow through a pressurized segment of collapsible tube, for a range of values of the time-independent controlling pressures. They come about either because there is an (unstable) steady flow corresponding to these pressures, or because no steady flow exists. We investigate the existence of steady flow in a one-dimensional collapsible-tube model, which takes account of both longitudinal tension and jet energy loss E downstream of the narrowest point. For a given tube, the governing parameters are flow-rate Q, and transmural pressure P at the downstream end of the collapsible segment. If E = 0, there exists a range of (Q, P)-values for which no solutions exist; when E ≠ 0 a solution is always found. For the case E ≠ 0, predictions are made of pressure drop along the collapsible tube; these solutions are compared with experiment.
The effects of temperature-dependent viscosity and the instabilities in convection rolls of a layer of fluid-saturated porous medium
Authors: Or A. C.
Published: April 2006

Convection of two-dimensional rolls in an infinite horizontal layer of fluid-saturated porous medium heated from below is studied numerically. Several important finite-amplitude states are isolated, and their bifurcation properties are shown. Effects of the temperature-dependent viscosity are included. The stability of these states is investigated with respect to the class of disturbances that have a ½π phase shift relative to the basic state. In particular, the oscillatory mechanism and the mean-flow generating mechanism through the variable viscosity are discussed.
Nonlinear evolution of interacting oblique waves on two-dimensional shear layers
Authors: Goldstein M. E. , Choi S.-W.
Published: April 2006

We consider the effects of critical-layer nonlinearity on spatially growing oblique instability waves on nominally two-dimensional shear layers between parallel streams. The analysis shows that three-dimensional effects cause nonlinearity to occur at much smaller amplitudes than it does in two-dimensional flows. The nonlinear instability wave amplitude is determined by an integro-differential equation with cubic-type nonlinearity. The numerical solutions to this equation are worked out and discussed in some detail. We show that they always end in a singularity at a finite downstream distance.
The effect of large-eddy breakup devices on flow noise
Authors: Dowling A. P.
Published: April 2006

The installation of large-eddy breakup devices (LEBUs) or ‘flow manipulators’ in a turbulent boundary layer over a rigid plane surface is known to lead to reductions in skin-friction coefficient, turbulence intensity and fluctuating Reynolds stress. We investigate the effect of such devices on the surface pressure spectrum and the far-field sound radiation. A model problem, in which a two-dimensional elliptical vortex is convected past a LEBU, is solved analytically in the low-Mach-number limit. The main noise source mechanisms are identified in this idealized problem and we go on to obtain scaling laws for the sound produced by a turbulent boundary-layer flow over a LEBU. The introduction of a LEBU reduces the strength of the Lighthill quadrupole source terms, but it produces an additional dipole source. However, the pressure fluctuations in this dipole field decay rapidly with distance from a LEBU, and we find that an array of LEBU's could have a beneficial effect on the flow noise for radian frequencies which are large in comparison with c/30Δ, where c is the sound speed and Δ denotes the boundary-layer thickness. At lower frequencies the LEBUs are predicted to increase the flow noise.
Inviscid drops with internal circulation
Authors: Pozrikidis C.
Published: April 2006

The shape of a moving inviscid axisymmetric drop is considered as a function of surface tension and of the intensity of the internal circulation. In a frame of reference moving with the drop, the drop is modelled as a region of diffused vorticity which is bounded by a vortex sheet, and is imbedded in streaming flow. First, an asymptotic analysis is performed for a slightly non-spherical drop whose circulation is very close to that required for the spherical shape. The results indicate that steady drop shapes may exist at all but a number of distinct values of the Weber number, the lowest two of which are 4.41 and 6.15. For highly deformed drops, the problem is formulated as an integral equation for the shape of the drop, and for the strength of the bounding vortex sheet. A numerical procedure is developed for solving this equation, and numerical calculations are performed for Weber numbers between 0 and 4.41. Limiting members in the computed family of solutions contain spherical drops, and inviscid bubbles with vanishing circulation. Computed new shapes include saucer-like shapes with a rounded main body and an elongated tip. The relationship between inviscid drops and drops moving at large Reynolds numbers is discussed.
Unsteady disturbances of streaming motions around bodies
Authors: Atassi H. M. , Grzedzinski J.
Published: April 2006

For small-amplitude vortical and entropic unsteady disturbances of potential flows, Goldstein proposed a partial splitting of the velocity field into a vortical part u(I) that is a known function of the imposed upstream disturbance and a potential part ∇ϕ satisfying a linear inhomogeneous wave equation with a dipole-type source term. The present paper deals with flows around bodies with a stagnation point. It is shown that for such flows u(I) becomes singular along the entire body surface and its wake and as a result ∇ϕ will also be singular along the entire body surface. The paper proposes a modified splitting of the velocity field into a vortical part u(R) that has zero streamwise and normal components along the body surface, an entropy-dependent part and a regular part ∇ϕ* that satisfies a linear inhomogeneous wave equation with a modified source term.

For periodic disturbances, explicit expressions for u(R) are given for three-dimensional flows past a single obstacle and for two-dimensional mean flows past a linear cascade. For weakly sheared flows, it is shown that if the mean flow has only a finite number of isolated stagnation points, u(R) will be finite along the body surface. On the other hand, if the mean flow has a stagnation line along the body surface such as in two-dimensional flows then the component of u(R) in this direction will have a logarithmic singularity.

For incompressible flows, the boundary-value problem for ϕ* is formulated in terms of an integral equation of the Fredholm type. The theory is applied to a typical bluff body. Detailed calculations are carried out to show the velocity and pressure fields in response to incident harmonic disturbances.
Growth and translation of a liquid-vapour compound drop in a second liquid. Part 1. Fluid mechanics
Authors: Vuong S. T. , Sadhal S. S.
Published: April 2006

The fluid dynamics associated with a compound drop consisting of a vapour bubble, partly surrounded by its own liquid in another immiscible liquid is considered. The fluid motion is analysed in the limit of Stokes flow and at the same time the surface tension forces are considered to be large enough to allow the interfaces to have uniform curvature. The flow field consists of translation and growth that can arise from change of phase.

An exact analytical solution for the axisymmetric flow field is obtained. The important results of physical interest are the drag force and the flow behaviour. In the case without growth, the drag force lies between the bubble and the solid-sphere limits for a sphere of the same volume as the total liquid and vapour dispersed phase. The maximum drag force is observed when the liquid and vapour volumes are nearly the same. This is the effect of weak circulation due to the smaller available space as compared with a spherical drop. With growth this effect appears to be enhanced. The flow streamlines exhibit secondary vortices in the dispersed phase when there is growth. The velocity field and the drag results here are applied to the heat transfer problem for the compound drop in Part 2 of this two-part series.
Thermal dispersion from a line source in the shearless turbulence mixing layer
Authors: Veeravalli S. , Warhaft Z.
Published: April 2006

We experimentally investigate dispersion from a heated line source placed in the central region of a turbulence mixing layer. Recently described by Veeravalli & Warhaft (1989) the mixing layer has no mean shear and consists of gradients in the velocity variance and scale; it is formed from a composite grid of constant solidity from which two distinct velocity scales are formed, one on either side of the stream. Mixing is effected by intermittent turbulent penetration and diffusion. The dispersion measurements were carried out in the convective regime where both plume flapping and fine-scale turbulent mixing play a role, the latter becoming more dominant as the plume evolves. The mean and variance temperature profiles are strongly skewed (with larger tails on the low turbulence side of the flow) in the earlier stages of the plume development. Here, in the convective range, the median and peak of the mean plume are deflected toward the large-scale region. As the flow evolves the profiles become more symmetrical but as the plume enters the turbulent diffusive stage there is evidence that the profiles again became asymmetric but now with longer tails in the high turbulence side of the flow (owing to the higher diffusivity). The temperature variance and heat flux budgets are highly asymmetric but tend to exhibit many of the characteristics of the budget of a line source in decaying homogeneous grid turbulence which is also presented here. However, a distinct region of negative production (counter-gradient heat flux) is found in the temperature variance budget and this is shown to be a consequence of the asymmetry of the transverse velocity probability density function in the mixing layer. Temperature spectra, both of the time series and of the intermittency function, across the plume are described. They are shown to peak at high wavenumbers in the centre and edge of the plume and at lower wavenumbers in the intermediate region. Their form is shown to change as the plume develops fine-scale structure and flapping becomes less important.
A sphere in shear flow at finite Reynolds number: effect of shear on particle lift, drag, and heat transfer
Authors: Dandy David S. , Dwyer Harry A.
Published: April 2006

Three-dimensional numerical solutions have been obtained for steady, linear shear flow past a fixed, heated spherical particle over a wide range of Reynolds number (0.1 [les ] R [les ] 100) and dimensionless shear rates (0.005 [les ] α [les ] 0.4). The results indicate that at a fixed shear rate, the dimensionless lift coefficient is approximately constant over a wide range of intermediate Reynolds numbers, and the drag coefficient also remains constant when normalized by the known values of drag for a sphere in uniform flow. At lower values of the Reynolds number, the lift and drag coefficients increase sharply with decreasing R, with the lift coefficient being directly proportional to R−½. For the range of shear rates studied here, the rate of heat transfer to the particle surface was found to depend only on the Reynolds number, that is, it was insensitive to the shear rate. The dimensionless rate of heat transfer, the Nussel number Nu, was seen to increase monotonically with R.
Similarity solutions of the two-dimensional unsteady boundary-layer equations
Authors: Ma Philip K. H. , Hui W. H.
Published: April 2006

The method of Lie group transformations is used to derive all group-invariant similarity solutions of the unsteady two-dimensional laminar boundary-layer equations. A new method of nonlinear superposition is then used to generate further similarity solutions from a group-invariant solution. Our results are shown to include all the existing solutions as special cases. A detailed analysis is given to several classes of solutions which are also solutions to the full Navier–Stokes equations and which exhibit flow separation.
The effects of truncation on surface-wave Hamiltonians
Authors: Milder D. Michael
Published: April 2006

The Hamiltonian form of the equations for surface waves can generate very nonlinear, realistic-looking solutions even when the Hamiltonian is truncated to low order – two or three terms – in its slope expansion. A perturbation analysis of these equations shows that most of the basic fluid behaviour is retained in the low-order terms; however, the lowest-order nonlinear equations become dramatically unstable at wavenumbers greater than g/w2, where w is the local vertical surface velocity. One more term in the Hamiltonian mitigates this instability, extending the regime of stable slopes and wavenumbers.
A spherical particle straddling a fluid/gas interface in an axisymmetric straining flow
Authors: Stoos J. A. , Leal L. G.
Published: April 2006

Numerical solutions, obtained via the boundary-integral technique, are used to consider the effect of a linear axisymmetric straining flow on the existence of steady-state configurations in which a neutrally buoyant spherical particle straddles a gas–liquid interface. The problem is directly applicable to predictions of the stability of particle capture in flotation processes, and is also of interest in the context of contact angle and surface tension measurements. A primary goal of the present study is a determination of the critical capillary number, Cac, beyond which an initially captured particle is pulled from the interface by the flow, and the dependence of Cac on the equilibrium contact angle θc. We also present equilibrium configurations for a wide range of contact angles and subcritical capillary numbers.
Inviscid spatial stability of a compressible mixing layer. Part 2. The flame sheet model
Authors: Jackson T. L. , Grosch C. E.
Published: April 2006

We report the results of an inviscid spatial stability calculation for a compressible reacting mixing layer. The limit of infinite Damkohler number is taken and the diffusion flame is approximated by a flame sheet. Results are reported for the phase speeds of the neutral waves and maximum growth rates of the unstable waves as a function of the parameters of the problem: the ratio of the temperature of the stationary stream to that of the fast stream, the Mach number of the fast stream, the heat release per unit mass fraction of the reactant, the equivalence ratio of the reaction, and the frequency of the disturbance. These results are compared to the phase speeds and growth rates of the corresponding non-reacting mixing layer. We show that the addition of combustion has important, and complex, effects on the flow stability. In particular, we show that the flow can become absolutely unstable with a sufficient amount of heat release.
Vortex formation around an oscillating and translating airfoil at large incidences
Authors: Ohmi Kazuo , Coutanceau Madeleine , Loc Ta Phuoc , Dulieu Annie
Published: April 2006

The starting flows past a two-dimensional oscillating and translating airfoil are investigated by visualization experiments and numerical calculations. The airfoil, elliptic in cross-section, is set in motion impulsively and subjected simultaneously to a steady translation and a harmonic oscillation in pitch. The incidence of the airfoil is variable between 0° and 45° and the Reynolds number based on the chord length is between 1500 and 10000. The main object of the present study is to reveal some marked characteristics of the unsteady vortices produced from the oscillating airfoil set at large incidences in excess of the static stall angle. Another purpose is to examine, in some detail, the respective and combined effects of the major experimental parameters on the vortex wake development. It is shown that, in general, the dominant parameter of the flow is the reduced frequency not only when the airfoil oscillates at incidences close to the static stall angle but also at larger incidences. It is also demonstrated that, as the pitching frequency is increased, the patterns of the vortex wake are dependent on the product of the reduced frequency and the amplitude rather than on the frequency itself. It is noted that the combined effect of a high reduced frequency and a large amplitude can give rise to cyclic superposition of leading-edge vortices from which a gradually expanding standing vortex is developed on the upper surface.
The compressible vortex pair
Authors: Heister S. D. , Mcdonough J. M. , Karagozian A. R. , Jenkins D. W.
Published: April 2006

A numerical solution for the flow field associated with a compressible pair of counter-rotating vortices is developed. The compressible, two-dimensional potential equation is solved utilizing the numerical method of Osher et al. (1985) for flow regions in which a non-zero density exists. Close to the vortex centres, vacuum ‘cores’ develop owing to the existence of a maximum achievable flow speed in a compressible flow field. A special treatment is required to represent these vacuum cores. Typical streamline patterns and core boundaries are obtained for upstream Mach numbers as high as 0.3, and the formation of weak shocks, predicted by Moore & Pullin (1987), is observed.
Unsteady flow past a rotating circular cylinder at Reynolds numbers 103 and 104
Authors: Badr H. M. , Coutanceau M. , Dennis S. C. R. , Ménard C.
Published: April 2006

The unsteady flow past a circular cylinder which starts translating and rotating impulsively from rest in a viscous fluid is investigated both theoretically and experimentally in the Reynolds number range 103 [les ] R [les ] 104 and for rotational to translational surface speed ratios between 0.5 and 3. The theoretical study is based on numerical solutions of the two-dimensional unsteady Navier–Stokes equations while the experimental investigation is based on visualization of the flow using very fine suspended particles. The object of the study is to examine the effect of increase of rotation on the flow structure. There is excellent agreement between the numerical and experimental results for all speed ratios considered, except in the case of the highest rotation rate. Here three-dimensional effects become more pronounced in the experiments and the laminar flow breaks down, while the calculated flow starts to approach a steady state. For lower rotation rates a periodic structure of vortex evolution and shedding develops in the calculations which is repeated exactly as time advances. Another feature of the calculations is the discrepancy in the lift and drag forces at high Reynolds numbers resulting from solving the boundary-layer limit of the equations of motion rather than the full Navier–Stokes equations. Typical results are given for selected values of the Reynolds number and rotation rate.
Axisymmetric draining of a cylindrical tank with a free surface
Authors: Zhou Qiao-Nian , Graebel W. P.
Published: April 2006

The withdrawal of layered fluids from an open tank through a hole centred on the bottom is investigated numerically under the assumption of potential flow. A fully cubic-spline nonlinear axisymmetric boundary-integral-method scheme with the built-in boundary conditions, which effectively reduces the numerical errors at the intersection lines where the tank wall and the density interfaces meet, is used. Two cases are studied: (i) the tank contains only one fluid with a free surface; (ii) the tank contains two fluids having different densities with a distinct interface and a free surface

The numerical results show two different phenomena, depending upon the drain rate and initial conditions. When the tank is rapidly drained, a dip forms at the centre of the lower interface and extends into the hole very quickly, as observed by Lubin & Springer (1967). For a slowly draining tank, a jet forms in the centre of the depression region. This jet can either shoot up or move down, depending on the initial conditions.
The transient settling of stable and flocculated dispersions
Authors: Auzerais FranÇois M. , Jackson R. , Russel W. B. , Murphy W. F.
Published: April 2006

An experimental investigation of the sedimentation of monodisperse colloidal silica spheres with grafted octadecyl chains with three different interaction potentials is presented. Small particles (0.27 μm) behaved as hard spheres in cyclohexane, but larger ones (0.60 and 0.94 μm) are weakly flocculated by van der Waals attractions. The smallest particles (0.08 μm) in hexadecane are strongly flocculated by attractions between the octadecyl layers. A medical computer tomography (CT) scanner provided an accurate and absolute density measurement without disrupting the process. For the hard spheres and the weakly flocculated systems, the kinetics of sedimentation for the dispersed phase could readily be predicted utilizing the flux curve. For flocculated networks, we found a power-law relationship between compressive yield stresses and solids fractions comparable with other experimental systems.
Chaos in small-amplitude surface gravity waves over slowly varying bathymetry
Authors: Brown Michael G. , Tappert Frederick D. , Sundaram Sekhar E. R. B.
Published: April 2006

We consider the motion of small-amplitude surface gravity waves over variable bathymetry. Although the governing equations of motion are linear, for general bathymetric variations they are non-separable and cannot be solved exactly. For slowly varying bathymetry, however, approximate solutions based on geometric (ray) techniques may be used. The ray equations are a set of coupled nonlinear ordinary differential equations with Hamiltonian form. It is argued that for general bathymetric variations, solutions to these equations - ray trajectories - should exhibit chaotic motion, i.e. extreme sensitivity to initial and environmental conditions. These ideas are illustrated using a simple model of bottom bathymetry, h(x,y) = h0(1 + εcos (2πx/L) cos (2πy/L)). The expectation of chaotic ray trajectories is confirmed via the construction of Poincaré sections and the calculation of Lyapunov exponents. The complexity of chaotic geometric wavefields is illustrated by considering the temporal evolution of (mostly) chaotic wavecrests. Some practical implications of chaotic ray trajectories are discussed.
Faraday resonance in rectangular geometry
Authors: Umeki Makoto
Published: April 2006

The motion of subharmonic resonant modes of surface waves in a rectangular container subjected to vertical periodic oscillation is studied based on the weakly nonlinear model equations derived by both the average Lagrangian and the two-timescale method. Explicit estimates of the nonlinearity of some specific modes are given. The bifurcations of stationary states including a Hopf bifurcation are examined. Numerical calculations of the dissipative dynamical equations show periodic and chaotic attractors. Theoretical parameter-space diagrams and numerical results are compared in detail with Simonelli & Gollub's (1989) surface-wave modecompetition experiments. It is shown that the average Hamiltonian system for the present 2:1:1 external-internal resonance with suitable coefficients has homoclinic chaos, which was mathematically proven by Holmes (1986) for the specific case of 2:1:2 external-internal resonance.
Instability of stationary unbounded stratified fluid
Authors: Batchelor G. K. , Nitsche J. M.
Published: April 2006

Suppose that the density of stationary unbounded viscous fluid is a sinusoidal function of the vertical position coordinate z. Is this body of fluid gravitationally unstable to small disturbances, and, if so, under what conditions, and to what type of disturbance? These questions are considered herein, and the answers are that the fluid is indeed unstable, for any non-zero value of the amplitude of the sine wave, to disturbances with large horizontal wavelength. These disturbances have approximately vertical velocity everywhere and tilt the alternate layers of heavier and of lighter fluid, causing the fluid in the former to slide down and that in the latter to slide up, leading to a sinusoidal variation of the vertically averaged density and thereby to reinforcement of the vertical motion. The identification of this novel and efficient global instability mechanism prompts a consideration of the stability of other cases of unbounded fluid stratified in layers. Two other types of undisturbed density distribution, the first an isolated central layer of heavier or lighter fluid, with density varying say as a Gaussian function, and the second an isolated layer of fluid in which the density varies as the derivative of a Gaussian function, are found to be unstable, at all values of the magnitude of the density variation, to disturbances having the same global character. For the first of these two types of density distribution, the behaviour of a disturbance with long horizontal wavelength depends only on the net excess mass of unit area of the central layer, and for the second it depends only on the first moment of the density in the central layer. For the second type there arises another global instability mechanism in which light fluid is stripped away from one side of the layer and heavy fluid from the other without any tilting. In all cases the properties of a neutral disturbance are determined numerically, and the growth rate is found as a function of the Rayleigh number, the Prandtl number, and the horizontal wavenumber of the disturbance. An energy argument gives results easily for the inviscid non-diffusive limit, when all disturbances grow, and reveals the tilting-sliding mechanism of the instability of a disturbance with large horizontal wavelength in its simplest form.
Elliptical vortices and integrable Hamiltonian dynamics of the rotating shallow-water equations
Authors: Holm Darryl D.
Published: April 2006

The problem of the dynamics of elliptical-vortex solutions of the rotating shallow-water equations is solved in Lagrangian coordinates using methods of Hamiltonian mechanics. All such solutions are shown to be quasi-periodic by reducing the problem to quadratures in terms of physically meaningful variables. All of the relative equilibria - including the well-known rodon solution - are shown to be orbitally Lyapunov stable to perturbations in the class of elliptical-vortex solutions.
Horizontal jets and vortex dipoles in a stratified fluid
Authors: Voropayev S. I. , Afanasyev Ya. D. , Filippov I. A.
Published: April 2006

When a horizontal force is applied locally to some volume of a viscous densitystratified fluid, flows with high concentration of vertically oriented vorticity (vortex dipoles) are generated. The processes of generation and evolution with time of these unsteady flows in a stratified fluid are studied. A convenient way to produce and study these flows in the laboratory is to use a submerged horizontal jet as a ‘point’ source of momentum. The main governing parameter (the ‘force’) is easily controlled in this case. Two regimes were studied: starting jets with dipolar vortex fronts (the force acts continuously) and impulsive vortex dipoles (the force acts for a short period of time). A conductivity microprobe, aluminium powder, shadowgraph, thymol-blue and other techniques have been used to measure the velocity and density distributions in the flows. It is found that in both regimes the flows are self-similar: the lengthscale of the flows increases with time as t½ for starting jets and as t1/3 for vortex dipoles. Detailed information about the generation mechanism, kinematics and dynamics of the flows is obtained. On the basis of similarity principles a theoretical explanation of the experimental results is given. The theory is in good agreement with the results obtained.
Experimental study of directional solidification of aqueous ammonium chloride solution
Authors: Chen C. F. , Chen Falin
Published: April 2006

Directional solidification experiments have been carried out using the analogue casting system of NH4Cl-H2O solution by cooling it from below with a constant-temperature surface ranging from - 31.5°C to + 11.9 °C. The NH4Cl concentration was 26% in all solutions, with a liquidus temperature of 15 °C. It was found that finger convection occurred in the fluid region just above the mushy layer in all experiments. Plume convection with associated chimneys in the mush occurred in experiments with bottom temperatures as high as + 11.0 °C. However, when the bottom temperature was raised to + 11.9 °C, no plume convection was observed, although finger convection continued as usual. A method has been devised to determine the porosity of the mush by computed tomography. Using the mean value of the porosity across the mush layer and the permeability calculated by the Kozeny-Carman relationship, the critical solute Rayleigh number across the mush layer for onset of plume convection was estimated to be between 200 and 250.
Locally axisymmetric turbulence
Authors: George William K. , Hussein Hussein J.
Published: April 2006

The failure of local isotropy to describe the experimentally obtained derivative moments in turbulent shear flows has previously been well-documented, but is briefly reviewed. The same data are then used to evaluate the hypothesis that the turbulence is locally axisymmetric. Locally axisymmetric turbulence is defined herein as turbulence which is locally invariant to rotations about a preferred axis.

The derivative moment relations are derived from the general form of the two-point velocity correlation tensor near the origin for axisymmetric turbulence. These are used to derive relations for the rate of dissipation of kinetic energy, the mean-square vorticity, and the components of each. Almost all of the experimental derivative moment data are shown to be consistent with these equations, and thus with local axisymmetry.
A study of baroclinic instability in a cylindrical annulus with the temperature gradient imposed on the lower surface
Authors: Miller Timothy L. , Reynolds Nathaniel D.
Published: April 2006

Laboratory experiments and numerical modelling studies have been performed for a rotating, thermally driven fluid system in a cylindrical annulus with a vertical rotation vector and axis of symmetry. The thermal forcing was through the imposition of an axisymmetric temperature gradient on a thermally conducting lower boundary, with additional heating through the outer sidewall. The upper and inner walls were nominally insulating. Flow patterns were observed in the experiments through the use of small, reflective flakes (Kalliroscope) in the working fluid, which was water. The rotation rate and temperature difference were varied to construct a regime diagram in thermal Rossby number–-Taylor number space. The curve separating axisymmetric flow from wave flow is ‘knee-shaped’, similar to the side-heated and -cooled baroclinic annulus which has been extensively investigated previously. Very near the transition curve, the initial wavenumber persists indefinitely, but well into the wave regime the initial wavenumber is higher than the equilibrated value. Far enough into the wave regime, the initial waves have wavenumbers several times that of the equilibrated value, and the initial disturbances form near the outer wall very early in the experiment. Numerical studies indicate that these waves are effective in distributing heat and that they occur in a region of positive static stability. These waves rapidly grow inward to fill the annulus and reduce in number as weaker waves are absorbed by the stronger ones. The period of transition between these waves and the equilibrated long-wave pattern is characterized by irregular flow. Closer to the transition curve, the temporal transition to longer waves as the flow equilibrates is simpler, with initial waves filling the annulus. In that case, the transition is characterized by a slow process of individual waves weakening and merging with adjacent waves.
On the shape of small sessile and pendant drops by singular perturbation techniques
Authors: O'Brien S. B. G.
Published: April 2006

The problem of obtaining asymptotic expressions describing the shape of small sessile and pendant drops is revisited. Both cases display boundary-layer behaviour and the method of matched asymptotic expansions is used to obtain solutions. These give good agreement when compared with numerical results. The sessile solutions are relatively straightforward, while the pendant drop displays a behaviour which is both rich and interesting.
Asymmetric oscillations in thermosolutal convection
Authors: Moore D. R. , Weiss N. O. , Wilkins J. M.
Published: April 2006

Thermosolutal convection provides a testbed for applications of nonlinear dynamics to fluid motion. If the ratio of solutal to thermal diffusivity is small and the solutal Rayleigh number RS is large, instability sets in at a Hopf bifurcation as the thermal Rayleigh number RT is increased. For two-dimensional convection in a rectangular box the fundamental mode is a single roll with point symmetry about its axis. The symmetries of periodic and steady solutions form an eighth-order group with invariant subgroups that describe pure single-roll and multiroll solutions. A systematic numerical investigation reveals a rich variety of spatiotemporal behaviour in the regime where RS [Gt ] RT−RS > 0. Point symmetry is broken and there is a branch of spatially asymmetric periodic solutions. These mixed-mode oscillations lose their temporal symmetry in a subsequent bifurcation, followed eventually by a transition to chaos. The numerical experiments can be interpreted by relating the physical form of the solutions to an appropriate bifurcation structure.
Final equilibrium state of a two-dimensional shear layer
Authors: Sommeria J. , Staquet C. , Robert R.
Published: April 2006

We test a new statistical theory of organized structures in two-dimensional turbulence by direct numerical stimulations of the Navier–Stokes equations, using a pseudo-spectral method. We apply the theory to the final equilibrium state of a shear layer evolving from a band of uniform vorticity: a relationship between vorticity and stream function is predicted by maximizing an entropy with the constraints due the constants of the motion. A partial differential equation for the stream function is then obtained. In the particular case of a very thin initial vorticity band, the Stuart's vortices appear to be a family of solutions for this equation. In more general cases we do not solve the equation, but we test the theory by inspecting the relationship between stream function and vorticity in the final equilibrium state of the numerical computation. An excellent agreement is obtained in regions with strong vorticity mixing. However, local equilibrium is obtained before a complete mixing can occur in the whole fluid domain.
Three-dimensional simulations of large eddies in the compressible mixing layer
Authors: Sandham N. D. , Reynolds W. C.
Published: April 2006

The effect of Mach number on the evolution of instabilities in the compressible mixing layer is investigated. The full time-dependent compressible Navier–Stokes equations are solved numerically for a temporally evolving mixing layer using a mixed spectral and high-order finite difference method. The convective Mach number Mc (the ratio of the velocity difference to the sum of the free-stream sound speeds) is used as the compressibility parameter. Simulations with random initial conditions confirm the prediction of linear stability theory that at high Mach numbers (Mc > 0.6) oblique waves grow more rapidly than two-dimensional waves. Simulations are then presented of the nonlinear temporal evolution of the most rapidly amplified linear instability waves. A change in the developed large-scale structure is observed as the Mach number is increased, with vortical regions oriented in a more oblique manner at the higher Mach numbers. At convective Mach numbers above unity the two-dimensional instability is found to have little effect on the flow development, which is dominated by the oblique instability waves. The nonlinear structure which develops from a pair of equal and opposite oblique instability waves is found to resemble a pair of inclined A-vortices which are staggered in the streamwise direction. A fully nonlinear computation with a random initial condition shows the development of large-scale structure similar to the simulations with forcing. It is concluded that there are strong compressibility effects on the structure of the mixing layer and that highly three-dimensional structures develop from the primary inflexional instability of the flow at high Mach numbers.
The interactive breakdown in supersonic ramp flow
Authors: Smith F. T. , Khorrami A. Farid
Published: April 2006

The separating flow induced on a ramp under a supersonic main stream is discussed, for high Reynolds numbers, according to the interactive-boundary-layer approach. There are two principal motivations for the study, apart from the recent general upsurge of interest in compressible boundary-layer separation and stall. The first is the need for more progress to be made in the numerical determination of strongly reversed flows than has proved possible in computations hitherto. This is tackled by means of a new computational scheme based in effect on a global Newton iteration procedure but coupled with linearized shooting, second-order accurate windward differencing and linear multi-sweeping. The specific case addressed is the triple-deck version for steady laminar two-dimensional motion, although the present scheme, like the theory described below, also has broader application, for example to subsonic, hypersonic and/or unsteady interactive flows. The second motivation is to compare closely with the recent theoretical prediction (Smith 1988a) of a local breakdown or stall occurring in any interactive boundary-layer solution at a finite value of the controlling parameter, a say, within the reversed-flow region; the breakdown produces a large adverse pressure gradient and minimum negative surface shear locally. The first quantitative comparisons are made between the theory and computational results, derived in this work at values of a, here the scaled ramp angle, greater than those obtainable before, but with a fixed outer boundary. The agreement, while not complete, seems to prove fairly affirmative overall and tends to support the suggestion (in Smith 1988a) that, contrary to most earlier expectations, in general there is a finite upper limit on the extent to which the interacting boundary-layer approach can be taken on its own. A similar conclusion holds for unsteady interactive boundary layers concerning a finite-time breakdown (Smith 19886) and boundary-layer transition, and in the present context the local nonlinear breakdown provides an explanation for the severe computational difficulties encountered previously as well as for a form of airfoil stall.
On the baroclinic instability of cold-core coupled density fronts on a sloping continental shelf
Authors: Swaters Gordon E.
Published: April 2006

A theory is presented to describe the linear baroclinic instability of coupled density fronts on a sloping continental shelf. The new baroclinic model equations used to study the instability process correspond to an ‘intermediate lengthscale’ dynamical balance. Specifically, the frontal dynamics, while geostrophic, is not quasigeostrophic because frontal height deflections are not small in comparison with the frontal scale height. The evolution of the frontal height is strongly coupled to the geostrophic pressure in the surrounding slope water through the hydrostatic balance which expresses the continuity of the dynamic pressures across the frontal interface. The deeper surrounding slope water evolves quasi-geostrophically and is coupled to the front by baroclinic vortex-tube stretching/compression associated with the perturbed density front (allowing the release of mean frontal potential energy) and the topographic vorticity gradient associated with the sloping bottom. It is shown that the baroclinic stability characteristics are principally determined by a so-called non-dimensional interaction parameter (denoted μ) which physically measures the ratio of the destabilizing baroclinic vortex-tube stretching/compression to the stabilizing topographic vorticity gradient. For a given along-front mode wavenumber it is shown that a minimum μ is required for instability. Several other general stability results are presented: necessary conditions for instability, growth rate and phase speed bounds, the existence of a high wavenumber cutoff, and a semicircle theorem for the unstable modes. The linear stability equations are solved exactly for a parabolic coupled density front and a detailed description of the spatial and temporal characteristics of the instabilities is given. For physically realistic parameter values the instabilities are manifested as amplifying topographic Rossby waves in the slope water, and on the density front the unstable perturbations take the form of amplifying anticyclones which have maximum amplitude on the offshore side.
Exotic containers for capillary surfaces
Authors: Concus Paul , Finn Robert
Published: April 2006

In this paper we discuss ‘exotic’ rotationally symmetric containers that admit an entire continuum of distinct equilibrium capillary free surfaces. The paper extends earlier work to a larger class of parameters and clarifies and simplifies the governing differential equations, while expressing them in a parametric form appropriate for numerical integration. A unified presentation suitable for both zero and non-zero gravity is given. Solutions for the container shapes are depicted graphically along with members of the free-surface continuum, and comments are given concerning possible physical experiments.
Studies of the Eulerian–Lagrangian transformation in two-dimensional random flows
Authors: Lynov J. P. , Nielsen A. H. , Pécseli H. L. , Rasmussen J. Juul
Published: April 2006

Two-dimensional, incompressible flows are discussed by a generalization of the line-vortex model. A large number of structures are randomly distributed initially. Each individual structure is convected by the superposed flow field of all the others. The statistical properties of the resulting space–time varying random flow are studied. Analytical expressions for both Eulerian and Lagrangian correlation functions are obtained for the limit where the density of structures is large. The analytical results compare favourably with numerical simulations. The study serves as a special test on proposed relations between Eulerian and Lagrangian averages which can be generally valid, i.e. also for three-dimensional, turbulent flows.
Resonance in nonlinear bubble oscillations
Authors: Longuet-Higgins Michael S.
Published: April 2006

In two recent papers (Longuet-Higgins 1989a,b) the author showed that the shape oscillations of bubbles can emit sound like a monopole source, at second order in the distortion parameter ε. In the second paper (LH2) it was predicted that the emission would be amplified when the second harmonic frequency 2σn of the shape oscillation approaches the frequency ω of the breathing mode. This ‘resonance’ would however be drastically limited by damping due to acoustic radiation and thermal diffusion. The predictions were confirmed by further numerical calculations in Longuet-Higgins (1990a).

Ffowcs Williams & Guo (1991) have questioned the conclusions of LH2 on the grounds that near resonance there is a slow (secular) transfer of energy between the shape oscillation and the volumetric mode which tends to diminish the amplitude of the shape oscillation and hence falsify the perturbation analysis. They have also argued that the volumetric mode never grows sufficiently to produce sound of the stated order of magnitude. In the present paper we show that these assertions are unfounded. Ffowcs Williams & Guo considered only undamped oscillations. Here we show that when the appropriate damping is included in their analysis the secular transfer of energy becomes completely insignificant. The resulting pressure pulse (figure 5 below) is found to be essentially identical to that calculated in LH2, figure 3. Moreover, in the initial-value problem considered in LH2, the excitation of the volumetric mode takes place not by a secular energy transfer but by a resonance during the first few cycles of the shape oscillation. This accounts for the amplification near resonance found in Longuet-Higgins (1990a). Finally, it is pointed out that the initial energy of the shape oscillations is far greater than is required to produce the O(ε2) volume pulsations that were studied in LH2, and which were used for a comparison with field data. This acoustic radiation was not calculated by Ffowcs Williams & Guo.
Microwave backscatter and acoustic radiation from breaking waves
Authors: Loewen M. R. , Melville W. K.
Published: April 2006

An experimental study of the microwave backscatter and acoustic radiation from breaking waves is reported. It is found that the averaged microwave and acoustic measurements correlate with the dynamics of wave breaking. Both the mean-square acoustic pressure and the backscattered microwave power correlate with the wave slope and dissipation, for waves of moderate slope (S < 0.28). The backscattered power and the mean-square pressure are also found to correlate strongly with each other. As the slope and wavelength of the breaking wave packet is increased, both the backscattered power and the mean-square pressure increase. It is found that a large portion of the backscattered microwave power precedes the onset of sound production and visible breaking. This indicates that the unsteadiness of the breaking process is important and that the geometry of the wave prior to breaking may dominate the backscattering. It is observed that the amount of acoustic energy radiated by an individual breaking wave scaled with the amount of mechanical energy dissipated during breaking. These laboratory results are compared to the field experiments of Farmer & Vagle (1988), Crowther (1989) and Jessup et al. (1990).