bluuebunny/crossref_metadata_2025 huggingface dataset:
Stretching and bending of line elements in random flows
Authors: Drummond I. T.
Published: April 2006

We study the stretching and bending of line elements transported in random flows with known Eulerian statistics in two and three dimensions. By making use of a cumulant expansion for the log-size of material elements we are able to analyse the exponential stretching they exhibit in random flows and identify conditions under which it will and will not occur. The results are confirmed by our numerical simulation.

We also examine the evolution of curvature in material elements and confirm by numerical simulation that it is governed by an appropriate version of the Pope equation. By modelling this equation as stochastic differential equation we are able to explain the appearance of a power-law tail in the probability distribution for large curvature observed by Pope, Yeung & Girimaji (1989) for surface elements. In two dimensions the appearance of the tail can indeed be attributed to the occurrence of events in which the material element undergoes contraction rather than stretching while subject to bending. In three dimensions the relationship between episodes of contraction and strong bending is less direct. This power-law tail allows us to reconcile the observed asymptotic stability, which we confirm here, of the powers and cumulants of the log-curvature with the unboundedness of powers of the curvature itself.
Sloshing frequencies of longitudinal modes for a liquid contained in a trough
Authors: McIver P. , McIver M.
Published: April 2006

The sloshing under gravity is considered for a liquid contained in a horizontal cylinder of uniform cross-section and symmetric about a vertical plane parallel to its generators. Much of the published work on this problem has been concerned with twodimensional, transverse oscillations of the fluid. Here, attention is paid to longitudinal modes with variation of the fluid motion along the cylinder. There are two known exact solutions for all modes; these are for cylinders whose cross-sections are either rectangular or triangular with a vertex semi-angle of ¼π. Numerical solutions are possible for an arbitrary geometry but few calculations are reported in the open literature. In the present work, some general aspects of the solutions for arbitrary geometries are investigated including the behaviour at low and high frequency of longitudinal modes. Further, simple methods are described for obtaining upper and lower bounds to the frequencies of both the lowest symmetric and lowest antisymmetric modes. Comparisons are made with numerical calculations from a boundary element method.
Surface waves on shear currents: solution of the boundary-value problem
Authors: Shrira Victor I.
Published: April 2006

We consider a classic boundary-value problem for deep-water gravity-capillary waves in a shear flow, composed of the Rayleigh equation and the standard linearized kinematic and dynamic inviscid boundary conditions at the free surface. We derived the exact solution for this problem in terms of an infinite series in powers of a certain parameter e, which characterizes the smallness of the deviation of the wave motion from the potential motion. For the existence and absolute convergence of the solution it is sufficient that e be less than unity.

The truncated sums of the series provide approximate solutions with a priori prescribed accuracy. In particular, for the short-wave instability, which can be interpreted as the Miles critical-layer-type instability, the explicit approximate expressions for the growth rates are derived. The growth rates in a certain (very narrow) range of scales can exceed the Miles increments caused by the wind.

The effect of thin boundary layers on the dispersion relation was also investigated using an asymptotic procedure based on the smallness of the product of the layer thickness and wavenumber. The criterion specifying when and with what accuracy the boundary-layer influence can be neglected has been derived.
Overturning of nonlinear acoustic waves. Part 1 A general method
Authors: Hammerton P. W. , Crighton D. G.
Published: April 2006

We consider model nonlinear wave equations of the form ut + uux = [Hscr ](x,t;u, ux,…) arising in gasdynamics and other fields, [Hscr ] incorporating various linear mechanisms of dissipation and dispersion. If [Hscr ] includes a thermoviscous dissipation term ∈uxx, then it is generally believed that u(x,t) will remain single-valued for all t > 0 and all single-valued u(x, 0), for any ∈ > 0. The question addressed here is whether, if thermoviscous dissipation is excluded from [Hscr ], u(x, t) remains single-valued for all t > 0, or whether certain dissipative-dispersive mechanisms (such as relaxation processes) are in themselves insufficient to prevent wave overturning. To answer this we propose a numerical scheme based on the use of intrinsic coordinates ψ = ψ(s, t) to describe the waveform at each time. In this paper, the method is described and validated by comparisons with the exact solutions for certain [Hscr ] ([Hscr ] = 0, [Hscr ] = -αu, [Hscr ] = ∈uxx). These comparisons show that the scheme is free of numerical viscosity effects which preclude the solution of the problem by finite-difference or spectral methods applied to the signal u(x, t), that it can reliably distinguish between finite-time overturning and merely the formation of steep gradients, and that it can accurately predict the time of overturning when it does occur. Having established the validity of the method, attention can then be turned to those cases where criteria for overturning have not as yet been determined by conventional methods. In Part 2, harmonic wave propagation through a relaxing gas is investigated.
Unsteady confined buoyant plumes
Authors: Desrayaud Gilles , Lauriat Guy
Published: April 2006

Two-dimensional time-dependent buoyancy-induced flows above a horizontal line heat source inside rectangular vessels, with adiabatic sidewalls and top and bottom walls maintained at uniform temperature, are studied numerically. Transitions to unsteady flows are performed by direct simulations for various depths of immersion of the source in the central vertical plane of air-filled vessels. For a square vessel and a line source near the bottom wall, the numerical solutions exhibit a sequence of instabilities, called natural swaying motion of confined plumes, beginning with a periodic regime having a high fundamental frequency followed by a two-frequency locked regime. Then, broadband components appearing in the spectra indicate chaotic behaviour and a weakly turbulent motion arises via an intermittent route to chaos. For rectangular vessels of aspect ratio greater than 2 and depths of immersion greater than the width, the flow undergoes a pitchfork bifurcation. This symmetry breaking is driven by the destabilization of an upper unstable layer of stagnant fluid above the plume. Then a subcritical Hopf bifurcation occurs. On the other hand, if the depth of immersion is lower than the width of the vessel, a stable layer of fluid is at rest below the line source. Then penetrative convection sets the whole air-filled vessel in motion and an oscillatory motion of very low frequency arises through supercritical Hopf bifurcation followed by a two-frequency locked state.
Collapse and winding of an asymmetric annulus of vorticity
Authors: Peurrung A. J. , Notte J. , Fajans J.
Published: April 2006

The dynamics of an asymmetric annulus of vorticity in an incompressible, inviscid twodimensional fluid are experimentally studied using a pure electron plasma. A strict fluid analogy requires that the plasma system behave like an ideal fluid in a frictionless cylindrical container. For certain parameters the asymmetric annulus undergoes a complex evolution which is quite different from that of a symmetric annulus. During the first ‘active’, phase the symmetries grow until the annulus collapses, leaving a large vortex at the device centre. In the second, ‘passive’, phase the remainder of the annulus winds around this central vortex into an ever tighter spiral. Finally, slow shear instabilities destroy the structure of the highly evolved spiral.
On the evolution of thermally driven shallow cavity flows
Authors: Daniels P. G. , Wang P.
Published: April 2006

The temporal evolution of thermally driven flow in a shallow laterally heated cavity is studied for the nonlinear regime where the Rayleigh numberRbased on cavity height is of the same order of magnitude as the aspect ratioL(length/height). The horizontal surfaces of the cavity are assumed to be thermally insulating. For a certain class of initial conditions the evolution is found to occur over two non-dimensional timescales, of order one and of orderL2. Analytical solutions for the motion throughout most of the cavity are found for each of these timescales and numerical solutions are obtained for the nonlinear time-dependent motion in end regions near each lateral wall. This provides a complete picture of the evolution of the steady-state flow in the cavity for cases where instability in the form of multicellular convection does not occur. The final steady state evolves on a dimensional timescale proportional tol2/κ, wherelis the length of the cavity,κis the thermal diffusivity of the fluid and the constant of proportionality depends on the ratioR/L.
Edge waves forced by short-wave groups
Authors: Schäuffer Hemming A.
Published: April 2006

On the basis of the theory for infragravity waves induced by short-wave groups developed by Schäffer (1993), three-dimensional infragravity waves are analysed. The theory relies on the linearized depth-integrated conservation equations for mass and momentum combined to give a second-order long-wave equation with forcing expressions in terms of the radiation stress. This forcing gives a dynamic set-up originating from oscillations of the break-point position and a dynamic set-down bound to the short-wave groups. For small angles of incidence leaky-mode solutions are found while trapped modes appear when the primary waves are sufficiently oblique. In the latter case resonant edge-wave excitation may occur. A semi-analytical steady-state solution for the infragravity motion is presented. The solution is restricted to periodicity along a plane beach connected to a shelf and valid only for small primary-wave modulations.
Wall shear stress and velocity in a turbulent axisymmetric boundary layer
Authors: Wietrzak Anthony , Lueptow Richard M.
Published: April 2006

Instantaneous streamwise fluctuations of the wall shear stress have been measured using a hot-element probe in a thick axisymmetric turbulent boundary layer on a cylinder aligned parallel to the flow. The measurements were made at a momentum-thickness Reynolds number Rθ = 3050 and a ratio of boundary-layer thickness to cylinder radius of δ/a = 5.7. The ratio of the r.m.s. of the fluctuation to the mean value of the wall shear stress, $\tau_{rms}/\bar{\tau}$, is about 0.32, a value slightly lower than that for recent measurements for flow over a flat plate. The probability density function of the wall shear stress is similar to that for planar wall-bounded flows within experimental error. The power spectral density of the wall shear stress shows that a cylindrical boundary layer contains less energy at lower frequencies and more energy at higher frequencies than other wall-bounded flows. Analysis of simultaneous measurement of the streamwise wall shear stress and the streamwise velocity using VITA and peak detection suggests that transverse curvature has little effect on the near-wall burst–sweep cycle compared to planar wall-bounded flows. The angle of inclination of the structures is similar to that measured for large-scale structures in planar wall-bounded flows. However, measurements of the cross-correlation between the shear stress and the velocity suggest the existence of smaller structures yawed to the axis of the cylinder. The coherence between shear stress and velocity shows a low frequency associated with the inclined structures and a higher frequency associated with the yawed structures. The yawed structures could have an arrowhead or half-arrowhead shape and may be associated with fluid from the outer flow washing over the cylinder.
The role of coherent structures in bubble transport by turbulent shear flows
Authors: Sene K. J. , Hunt J. C. R. , Thomas N. H.
Published: April 2006

Using Auton's force law for the unsteady motion of a spherical bubble in inhomogeneous unsteady flow, two key dimensionless groups are deduced which determine whether isolated vortices or shear-layer vortices can trap bubbles. These groups represent the ratio of inertial to buoyancy forces as a relaxation parameter [tcy ] = ΔU2/2gx and a trapping parameter [Gcy ] = ΔU/VT where ΔU is the velocity difference across the vortex or the shear layer, x is streamwise distance measured from the effective origin of the mixing layer and VT is the terminal slip speed of the bubble or particle. It is shown here that whilst buoyancy and drag forces can lead to bubbles moving in closed orbits in the vortex flows (either free or forced), only inertial forces result in convergent trajectories. Bubbles converge on the downflow side of the vortex at a location that depends on the inertial and lift forces. It is important to note that the latter have been omitted from many earlier studies.

A discrete-vortex model is used to simulate the large-scale unsteady flows within horizontal and vertical mixing layers between streams with velocity difference ΔU. Trajectories of non-interacting small bubbles are computed using the general force law. In the horizontal mixing layer it is found that Γ needs to have a value of about 3 to trap about 50% of the bubbles if Π is about 0.5 and greater if Π is less. The pairing of vortices actually enhances their trapping of bubbles. In the vertical mixing layer bubbles are trapped mainly within the growing vortices but bubbles are concentrated on the downflow side of the vortices as Γ and Π increase. In a companion paper we show that lateral dispersion of bubbles can be approximately described by an advective diffusion equation with the diffusivity about equal to the eddy viscosity, i.e. rather less than the diffusivity of heat or other passive scalars.
Stretched vortices – the sinews of turbulence; large-Reynolds-number asymptotics
Authors: Moffatt H. K. , Kida S. , Ohkitani K.
Published: April 2006

A large-Reynolds-number asymptotic theory is presented for the problem of a vortex tube of finite circulation [Gcy ] subjected to uniform non-axisymmetric irrotational strain, and aligned along an axis of positive rate of strain. It is shown that at leading order the vorticity field is determined by a solvability condition at first-order in ε = 1/R[Gcy ] where R[gcy ] = [gcy ]/ν. The first-order problem is solved completely, and contours of constant rate of energy dissipation are obtained and compared with the family of contour maps obtained in a previous numerical study of the problem. It is found that the region of large dissipation does not overlap the region of large enstrophy; in fact, the dissipation rate is maximal at a distance from the vortex axis at which the enstrophy has fallen to only 2.8% of its maximum value. The correlation between enstrophy and dissipation fields is found to be 0.19 + O(ε2). The solution reveals that the stretched vortex can survive for a long time even when two of the principal rates of strain are positive, provided R[gcy ] is large enough. The manner in which the theory may be extended to higher orders in ε is indicated. The results are discussed in relation to the high-vorticity regions (here described as ‘sinews’) observed in many direct numerical simulations of turbulence.
Thermals with background rotation and stratification
Authors: Helfrich Karl R.
Published: April 2006

Scaling analysis and experiments are used to study the evolution of thermals in the presence of background rotation. When the ambient environment is homogeneous, the thermal rises and expands until it reaches a critical height where the Rossby number becomes ∼ 1. The thermal then stops expanding and rises in a column. Both the critical height and column radius scale with (F0f-2)1/4. F0 is the initial thermal buoyancy and f is the Coriolis frequency. The thermal vertical velocity is independent of f. When the background is stratified with buoyancy frequency N, the thermal rises to a neutral buoyancy level which scales with (F0N-2)1/4. For N/f [Lt ] 0.6 column formation occurs before the thermal reaches the neutral level. For N/f [Gt ] 0.6 the thermal reaches the neutral level before rotation is important. In both regimes, geostrophic adjustment eventually causes the formation of a baroclinic vortex consisting of an anticyclonic lens of thermal fluid at the neutral level and cyclonic circulation below. The lens has Nh/fl ∼ 1. The lens thickness 2h and the radius l obey relations of the form (F0N-2)1/4 (N/f)m. However, the exponents m are different in the two regimes. The relevance of these results to deep-ocean convection and hydrothermal venting is discussed.
Chaotic transitions of convection rolls in a rapidly rotating annulus
Authors: Or A. C.
Published: April 2006

Drifting convection rolls in a rapidly rotating cylindrical annulus with conical endwalls exhibit different transitional modes to chaotic flows at different Prandtl numbers. Three transition sequences for Prandtl numbers 0.3, 1.0 and 7.0 are studied for a moderately large Coriolis parameter and a wavenumber near the critical value using an initial-value code. As the Rayleigh number increases, each transition sequence first leads to a vacillating flow, and then to an aperiodic flow, the route of which is Prandtl-number dependent. From the low Prandtl number to the high Prandtl number, the transitions take different routes of torus folding, period doubling, and mode-locking intermittency.
Existence theorems for trapped modes
Authors: Evans D. V. , Levitin M. , Vassiliev D.
Published: April 2006

A two-dimensional acoustic waveguide of infinite extent described by two parallel lines contains an obstruction of fairly general shape which is symmetric about the centreline of the waveguide. It is proved that there exists at least one mode of oscillation, antisymmetric about the centreline, that corresponds to a local oscillation at a particular frequency, in the absence of excitation, which decays with distance down the waveguide away from the obstruction. Mathematically, this trapped mode is related to an eigenvalue of the Laplace operator in the waveguide. The proof makes use of an extension of the idea of the Rayleight quotient to characterize the lowest eigenvalue of a differential operator on an infinite domain.
Observations of parametric instability and breaking waves in an oscillating tilted tube
Authors: Thorpe S. A.
Published: April 2006

Experiments are described in which a rectangular tube filled with a stratified fluid and tilted at an angle α (about 12°) is rocked at the critical frequency of waves on a slope, σ = N sin α, where N is the uniform buoyancy frequency of the fluid in the central section of the tube. Localized overturns with axes transverse to the flow are observed with a scale comparable with the tube height, producing convective motions and mixing. The overturns have a periodic structure along the tube and, although occurring on each forcing cycle, they alternate in position, so that they reoccur at a given position only every two cycles, that is at the frequency of the first subharmonic of the forcing frequency. The wavelength and vertical structure of the disturbance are consistent with the presence of an internal wave mode with a frequency half that of the forcing, and this is indicative of a parametric instability. The parameters of the regions where static instability occurs show that, as observed, the fluid is more likely to be unstable to convective motions than in earlier experiments (Thorpe 1994b) on standing waves.
An estimate of the Kelvin impulse of a transient cavity
Authors: Best J. P. , Blake J. R.
Published: April 2006

The Lagally theorem is used to obtain an expression for the Bjerknes force acting on a bubble in terms of the singularities of the fluid velocity potential, defined within the bubble by analytic continuation. This expression is applied to transient cavity collapse in the neighbourhood of boundaries, allowing analytical estimates to be made of the Kelvin impulse of the cavity. The known result for collapse near a horizontal rigid boundary is recovered, and the Kelvin impulse of a cavity collapsing in the neighbourhood of a submerged and partially submerged sphere is estimated. A numerical method is developed to deal with more general body shapes and in particular, bodies of revolution. Noting that the direction of the impulse at the end of the collapse phase generally indicates the direction of the liquid jet that may form, the behaviour of transient cavities in these geometries is predicted. In these examples the concept of a zone of attraction is introduced. This is a region around the body, within which the Kelvin impulse at the time of collapse, and consequent jet formation, is expected to be directed towards the body. Outside this zone the converse is true.
Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid Part 1. Sedimentation
Authors: Feng J. , Hu H. H. , Joseph D. D.
Published: April 2006

This paper reports the result of direct simulations of fluid–particle motions in two dimensions. We solve the initial value problem for the sedimentation of circular and elliptical particles in a vertical channel. The fluid motion is computed from the Navier–Stokes equations for moderate Reynolds numbers in the hundreds. The particles are moved according to the equations of motion of a rigid body under the action of gravity and hydrodynamic forces arising from the motion of the fluid. The solutions are as exact as our finite-element calculations will allow. As the Reynolds number is increased to 600, a circular particle can be said to experience five different regimes of motion: steady motion with and without overshoot and weak, strong and irregular oscillations. An elliptic particle always turn its long axis perpendicular to the fall, and drifts to the centreline of the channel during sedimentation. Steady drift, damped oscillation and periodic oscillation of the particle are observed for different ranges of the Reynolds number. For two particles which interact while settling, a steady staggered structure, a periodic wake-action regime and an active drafting–kissing–tumbling scenario are realized at increasing Reynolds numbers. The non-linear effects of particle–fluid, particle–wall and interparticle interactions are analysed, and the mechanisms controlling the simulated flows are shown to be lubrication, turning couples on long bodies, steady and unsteady wakes and wake interactions. The results are compared to experimental and theoretical results previously published.
Drop formation during coating of vertical fibres
Authors: Kalliadasis Serafim , Chang Hsueh-Chia
Published: April 2006

When the coating film around a vertical fibre exceeds a critical thickness hc, the interfacial disturbances triggered by Rayleigh instability can undergo accelerated growth such that localized drops much larger in dimension than the film thickness appear. We associate the initial period of this strongly nonlinear drop formation phenomenon with a self-similar intermediate asymptotic blow-up solution to the long-wave evolution equation which describes how static capillary forces drain fluid into the drop. Below hc, we show that strongly nonlinear coupling between the mean flow and axial curvature produces a finite-amplitude solitary wave solution which prevents local finite-time blow up and hence disallows further growth into drops. We thus estimate hc by determining the existence of solitary wave solutions. This is accomplished by a matched asymptotic analysis which joins the capillary outer region of a large solitary wave to the thin-film inner region. Our estimate of hc = 1.68R3H–2, where R is the fibre radius and H is the capillary length H = (σ/ρg)½, is favourably compared to experimental data.
The evolution of capillary fountains
Authors: Schulkes R. M. S. M.
Published: April 2006

In this paper we consider the dynamics of a growing capillary fountain. We assume that at some time t = 0 an inviscid incompressible fluid is ejected vertically upwards through a circular nozzle. The subsequent dynamics of the resulting fountain is studied numerically using the boundary-element technique. The rate at which fluid is discharged from the nozzle and the Bond number (a measure of gravitational and surface tension forces) are the parameters that govern the dynamics of the fountain. For small discharge rates the fountain assumes the form of a slowly growing sessile drop, with dynamic effects not modifying the shape of the drop significantly. For large discharge rates we find that close to the symmetry axis a region develops with a high curvature. This strongly curved region results in a physical instability which can take on one of two forms: either a liquid drop is ejected from the free surface or the capillary surface entrains a bubble. For intermediate values of the discharge rate we find that fluid lobes develop which fall over the side of the nozzle. A number of experimental results are also presented showing the evolution of water fountains for different Bond numbers and discharge rates. Some of our numerical predictions are confirmed by the experimental results.
Interaction of a turbulent round jet with the free surface
Authors: Madnia C. K. , Bernal L. P.
Published: April 2006

The interaction of a turbulent round jet with the free surface was investigated experimentally. Flow visualization, free-surface curvature measurements and hot-film velocity measurements were used to study this flow. It is shown that surface waves are generated by the large-scale vortical structures in the jet flow as they interact with the free surface. These waves propagate at an angle with respect to the flow direction which increases as the Froude number is increased. Propagation of the waves in the flow direction is suppressed by the surface current produced by the jet. Farther downstream the surface motions are caused by the large-scale vortical structures. Characteristic dark circular features are observed in shadowgraph images associated with concentrated vorticity normal to the free surface. The normal vorticity is believed to be the result of vortex line reconnection processes in the turbulent flow. Measurements of the mean velocity and turbulence intensity are reported. Owing to the confinement by the free surface, the decay rate of the maximum mean velocity is reduced by a factor of √2 compared to an unconfined jet.
A new computational method for the solution of flow problems of microstructured fluids. Part 2. Inhomogeneous shear flow of a suspension
Authors: Szeri Andrew J. , Leal L. Gary
Published: April 2006

A numerical investigation is conducted into the flow of a dilute suspension of rigid rod-like particles between parallel flat plates, driven by a uniform pressure gradient. The particles are assumed to be small relative to lengthscales of the flow with the effect that particle orientations evolve according to the local velocity gradient; the particles are also assumed to be small in an absolute sense, with the consequence that Brownian motions are of consequence. The calculations are performed using a novel approach, with a theoretical basis that has been developed previously in a companion paper (Szeri & Leal 1992). The new approach permits one to solve flow problems of microstructured fluids (such as suspensions, liquid crystals, polymer solutions and melts) without ‘pre-averaging’ or closure approximations. In the present work, the new approach is used to expose previously unknown pathological, non-physical predictions in various constitutive models derived using closure approximations. This appears to have passed unnoticed in prior work. In addition, the new approach is shown to possess several computational advantages. The determination of the orientation distribution of particles is self-adaptive; this leads, in effect, to a very efficient solution of the associated Smoluchowski (or Fokker–Planck) equation. Moreover, the new approach is highly suited to parallel (and vector) implementation on modern computers. These issues are explored in detail in the context of the example flow.
On the subharmonic instabilities of steady three-dimensional deep water waves
Authors: Ioualalen Mansour , Kharif Christian
Published: April 2006

A numerical procedure has been developed to study the linear stability of nonlinear three-dimensional progressive gravity waves on deep water. The three-dimensional patterns considered herein are short-crested waves which may be produced by two progressive plane waves propagating at an oblique angle, γ, to each other. It is shown that for moderate wave steepness the dominant resonances are sideband-type instabilities in the direction of propagation and, depending on the value of γ, also in the transverse direction. It is also shown that three-dimensional progressive gravity waves are less unstable than two-dimensional progressive gravity waves.
A bifurcation study of viscous flow through a rotating curved duct
Authors: Selmi M. , Nandakumar K. , Finlay W. H.
Published: April 2006

The combined effects of system rotation (Coriolis force) and curvature (centrifugal force) on the bifurcation structure of two-dimensional flows in a toroidal geometry of rectangular cross-section are examined. The problem depends on the Rossby number, Ro = U/bΩ, the Ekman number, Ek = ν/b2Ω, the aspect ratio, γ = b/h and the radius ratio, η = ri/ro; here U is the velocity scale, b is the channel width in the spanwise direction, Ω is the rotational speed, (ri, ro) are the inner and outer radii of the duct, h = ro – ri is the channel gap in the radial direction and v is the kinematic viscosity of the fluid. A pseudospectral method is devised to discretize the two-dimensional Navier–Stokes equation in stream-function form. Continuation schemes are used to track the solution paths with Rossby number as the control parameter. Extended systems are used to determine the precise location of the singular points of the discretized system. The loci of such singular points are tracked with respect to curvature of the duct. Unlike the findings of Miyazaki (1973) on the same problem, curvature is found to have profound effects on the solution structure; flow mutations take place through a tilted cusp at (Ro = 7.122, η = 0.678) and a transcritical bifurcation point at (Ro = 1.357, η = 0.349).
Scale-dependent merging of baroclinic vortices
Authors: Verron J. , Valcke S.
Published: April 2006

The influence of stratification on the merging of like-sign vortices of equal intensity and shape is investigated by numerical simulations in a quasi-geostrophic, two-layer stratified model. Two different types of vortices are considered: vortices defined as circular patches of uniform potential vorticity in the upper layer but no PV anomaly in the lower layer (referred to as PVI vortices), and vortices defined as circular patches of uniform relative vorticity in the upper layer but no motion in the lower layer (referred to as RVI vortices). In particular, it is found that, in the RVI case, the merging behaviour depends strongly on the magnitude of the stratification (i.e. the ratio of internal Rossby radius and vortex radius). The critical point here appears to be whether or not the initial eddies have a deep flow signature in terms of PV.

The specific phenomenon of scale-dependent merging observed is interpreted in terms of the competitive effects of hetonic interaction and vortex shape. In the case of weaker stratification, the baroclinic structure of the eddies can be seen as dominated by a mechanism of hetonic interaction in which bottom flow appears to counteract the tendency of surface eddies to merge. In the case of larger stratification, the eddy interaction mechanism is shown to be barotropically dominated, although interface deformation still determines the actual eddy vorticity profile during the initialization stage. Repulsion (hetonic) effect therefore oppose attraction (barotropic shape) effects in a competitive process dependent on the relationship between the original eddy lengthscale and the first internal Rossby radius.

A concluding discussion considers the implications of such analysis for real situations, in the ocean or in the laboratory.
Wall imperfections as a triggering mechanism for Stokes-layer transition
Authors: Blondeaux P. , Vittori G.
Published: April 2006

The boundary layer generated by the harmonic oscillations of a wavy wall in a fluid otherwise at rest is studied. First the wall waviness is assumed to be of small amplitude and large values of the Reynolds number are considered. The results obtained by means of a linear analysis, where the time variable appears only as a parameter, show that resonance may occur. Indeed it is found that when the Reynolds number is larger than a critical value, an instant within the decelerating part of the cycle exists such that a waviness of infinitesimal amplitude induces unbounded perturbations of the flow in the Stokes layer. The passage through resonance is then studied by means of a multiple-timescale approach, taking into account the damping effect of local acceleration within a small time range around resonance. The asymptotic approach fails beyond a threshold value of the Reynolds number, because the damping effect of the local acceleration terms spreads over the whole cycle. The problem is then tackled by means of an approach that takes into account the above damping effect throughout the whole cycle. Finally, a numerical procedure is used that also allows the inclusion of nonlinear terms and the study of the interactions among forced and free modes. The numerical approach reveals that, even for relatively large values of the amplitude of the wall waviness, nonlinear effects are negligible and the damping of resonance is mainly due to local acceleration effects. The relevance of the results to the understanding of transition to turbulence in Stokes layers is discussed.
Nonlinear stability of multilayer quasi-geostrophic flow
Authors: Mu Mu , Qingcun Zeng , Shepherd Theodore G. , Yongming Liu
Published: April 2006

New nonlinear stability theorems are derived for disturbances to steady basic flows in the context of the multilayer quasi-geostrophic equations. These theorems are analogues of Arnol’d's second stability theorem, the latter applying to the two-dimensional Euler equations. Explicit upper bounds are obtained on both the disturbance energy and disturbance potential enstrophy in terms of the initial disturbance fields. An important feature of the present analysis is that the disturbances are allowed to have non-zero circulation. While Arnol’d's stability method relies on the energy–Casimir invariant being sign-definite, the new criteria can be applied to cases where it is sign-indefinite because of the disturbance circulations. A version of Andrews’ theorem is established for this problem, and uniform potential vorticity flow is shown to be nonlinearly stable. The special case of two-layer flow is treated in detail, with particular attention paid to the Phillips model of baroclinic instability. It is found that the short-wave portion of the marginal stability curve found in linear theory is precisely captured by the new nonlinear stability criteria.
Dynamics of an isolated barotropic eddy on a beta-plane
Authors: Korotaev Gennady K. , Fedotov Alexander B.
Published: April 2006

The dynamics of a Gaussian isolated barotropic eddy on a β-plane is considered. The analytical solution of the evolution of an isolated vortex is constructed by analogy to the theory of a point vortex. The results of a numerical experiment are compared with the conclusions of the theory for the case of the Gaussian vortex. Characteristics of the vortex such as its radius, trajectory of movement, kinetic energy, residual vorticity, and the structure of the vortex are discussed. The analysis of the numerical results shows that the experimentally determined radius of the vortex, its energy, and residual vorticity are in good agreement with the theory. On the other hand there is a difference between analytical and experimental values of velocity components, and hence in the trajectory of the centre of the vortex. The location of the separatrix of the streak function and its saddle point are considered as important characteristics of the structure of the vortex. We consider the phenomenon of the generation of the vortex sheet connected with the separatrix location as a cause of the difference between the experimental and analytical estimates of the velocity of the vortex.
Experimental investigation of the flow field of an oscillating airfoil and estimation of lift from wake surveys
Authors: Panda J. , Zaman K. B. M. Q.
Published: April 2006

The flow field of an airfoil oscillated periodically over a reduced frequency range, 0 ≤ k ≤ 1.6, is studied experimentally at chord Reynolds numbers of Rc = 22000 and 44000. For most of the data, the NACA0012 airfoil is pitched sinusoidally about one quarter chord between angles of attack α of 5° and 25°. The cyclic variation of the near wake flow field is documented through flow visualization and phase-averaged vorticity measurements. In addition to the familiar dynamic stall vortex (DSV), an intense vortex of opposite sign is observed to originate from the trailing edge just when the DSV is shed. The two together take the shape of the cross-section of a large ‘mushroom’ while being convected away from the airfoil. The phase delay in the shedding of the DSV with increasing k, as observed by previous researchers, is documented for the full range of k. It is observed that the sum of the absolute values of all vorticity convected into the wake over a cycle is nearly constant and is independent of the reduced frequency and amplitude of oscillation but dependent on the mean α. The time varying component of the lift is estimated in a novel way from the shed vorticity flux. The analytical foundation of the method and the various approximations are discussed. The estimated lift hysteresis loops are found to be in reasonable agreement with available data from the literature as well as with limited force balance measurements. Comparison of the lift hysteresis loops with the corresponding vorticity fields clearly shows that major features of the lift variation are directly linked to the evolution of the large-scale vortical structures and the phase delay phenomenon.
Molecular mixing in Rayleigh–Taylor instability
Authors: Linden P. F. , Redondo J. M. , Youngs D. L.
Published: April 2006

Mixing produced by Rayleigh–Taylor instability at the interface between two layers is the subject of a comparative study between laboratory and numerical experiments. The laboratory experiments consist of a layer of brine initially at rest on top of a layer of fresh water. When a horizontal barrier separating the two layers is removed, the ensuing motion and the mixing that is produced is studied by a number of diagnostic techniques. This configuration is modelled numerically using a three-dimensional code, which solves the Euler equations on a 1803grid. A comparison of the numerical results and the experimental results is carried out with the aim of making a careful assessment of the ability of the code to reproduce the experiments. In particular, it is found that the motions are quite sensitive to the presence of large scales produced when the barrier is removed, but the amount and form of the mixing is not very sensitive to the initial conditions. The implications of this comparison for improvements in the experimental and numerical techniques are discussed.
The collision rate of small drops in linear flow fields
Authors: Wang Hua , Zinchenko Alexander Z. , Davis Robert H.
Published: April 2006

A dilute dispersion containing small, force-free drops of one fluid dispersed in a second, immiscible in a linear flow field is considered for small Reynolds numbers and large Péclet numbers under isothermal conditions. The emphasis of our analysis is on the effects of pairwise drop interactions on their collision rate, as described by the collision efficiency, using a trajectory analysis. Simple shear flow and uniaxial extensional or compressional flow are considered. For both flows, the collision efficiency decreases with increasing drop viscosity due to the effects of hydrodynamic interactions. It also decreases as the ratio of the smaller drop radius to the larger radius decreases. For uniaxial flow, finite collision rates are predicted in the absence of interdroplet forces for all finite values of the drop size ratio and the ratio of the viscosities of the drop and suspending medium. In contrast, several kinds of relative trajectories exist for a pair of drops in simple shear flow, including open trajectories, collision trajectories, and closed and semi-closed trajectories, in the absence of interdroplet forces. When the ratio of small to large drop diameters is smaller than a critical value, which increases with increasing drop viscosity, all of the relative trajectories that start with the two drops far apart remain open (no collisions), unless in the presence of attractive forces. Attractive van der Walls forces are shown to increase the collision rates.
The spin-up from rest of a fluid-filled torus
Authors: Madden F. N. , Mullin T.
Published: April 2006

We present the results of an experimental and numerical study of the spin-up from rest to solid-body rotation of a fluid-filled torus. In separate experiments, the rotation rate of the container is suddenly increased to a fixed value and the final rotation rate is used to define a non-dimensionalized control parameter,C. At low values ofC, the observed flows during the transient phase are axisymmetric and spin-up is achieved through viscous diffusion. This in turn is followed by significant secondary flow and the appearance of ‘fronts’ asCis increased. During the transient phase the fluid motion near the inner wall of the container is dynamically unstable according to Rayleigh's criterion. Thus at higher values ofCwave-like structures break the axisymmetry, non-uniqueness in the details of the process is found and finally, an innear wall instability is observed directly. A plot of the spin-up time versusCshows breaks in the slope at transition points between each of the above dynamical regimes but the overall trend is found to be insensitive to the details of the fluid motion. Further elucidation of the dynamical processes is provided by a novel variant of the now standard phase-space reconstruction techniques. The results show a systematic splitting of the phase paths asCis increased.

Finally, in the complementary numerical study, the time-dependent Navier–Stokes equations are solved for axisymmetric flows. Here, the flow is computed using a velocity–streamfunction–vorticity formulation in a two-dimensional plane with a velocity component normal to this plane. The quantitative and qualitative agreement between the numerical and experimental results is excellent for moderate values of the dynamical control parameterC.
Resonant capillary–gravity interfacial waves
Authors: Christodoulides P. , Dias F.
Published: April 2006

Two-dimensional space-periodic cabillary–gravity waves at the interface between two fluids of different densities are considered when the second harmonic and the fundamental mode are near resonance. A weakly nonlinear analysis provides the equations (normal form), correct to third order, that relate the wave frequency with the amplitudes of the fundamental mode and of the second harmonic for all waves with small energy. A study of the normal form for waves which are also periodic in time reveals three possible types of space- and time-periodic waves: the well-known travelling and standing waves as well as an unusual class of three-mode mixed waves. Mixed waves are found to provide a connection between standing and travelling waves. The branching behaviour of all types of waves is shown to depend strongly on the density ratio. For travelling waves the weakly nonlinear results are confirmed numerically and extended to finite-amplitude waves. When slow modulations in time of the amplitudes are considered, a powerful geometrical method is used to study the resulting normal form. Finally a discussion on modulational stability suggests that increasing the density ratio has a stabilizing effect.
Asymptotic analysis of the radiation by volume sources in supersonic rotor acoustics
Authors: Ardavan H.
Published: April 2006

The application of Lighthill's acoustic analogy to the generation of sound by rotating surfaces with supersonic speeds results in radiation integrals in which the stationary points of the phase function – that describes the sapce-time distance between each source point and a fixed observation point – have variable positions and coalesce at a caustic in the space of source points. Here, the leading term in the asymptotic expansion of the corresponding Green's function at this caustic is calculated, both in the time and the frequency domains, and it is shown that the radiation generated by volume sources, which are steady in the uniformly rotating blade-fixed frame, has an amplitude that does not obey the spherical spreading law, i.e. does not fall off like RP–1 with the radial distance RP away from the source. Within a finite solid angle, depending on the extent of the source distribution, the amplitude of this newly identified sound decays like RP–½, so that it is stronger in the far field than any previously studied element. That this is not incompatible with the conservation of energy is because the emission time intervals associated with the volume elements of the source distribution which contribute towards the non-spherically decaying component of the radiation are by a large (RP-dependent) factor greater than the time intervals during which the signals generated by these elements are received. The contributing source elements are those whose positions at the retarded time coincide witht the locus of singularities of the Green's function, i.e. with the one-dimensional locus of points, fixed in the rotating frame, which approach the observer with the wave speed and zero acceleration along the radiation direction. Because the signals received at two neighbouring instants in time arise from distinct, coherently radiating filamentary parts of the source which have both different extents and different strengths, the resulting overall waveform in the far zone consists of the superposition of a (continuous) set of narrow subpulses with uneven amplitudes. These subpulses are narrower the larger the distance from the source.

The differences between the new results and those of the earlier works in the literature are shown to arise from the error terms in the far-field and high-frequency approximations, approximations that are inappropriate for volume sources moving supersonically.