bluuebunny/crossref_metadata_2025 huggingface dataset:
The spreading of insoluble surfactant at the free surface of a deep fluid layer
Authors: Jensen O. E.
Published: April 2006

The unsteady spreading of an insoluble monolayer containing a fixed mass of surface-active material over the initially horizontal free surface of a viscous fluid layer is investigated. A flow driving the spreading is induced by gradients in surface tension, which arise from the nonuniform surfactant distribution. Distinct phases in the flow's dynamics are distinguished by a time T = H02/v, where H0 is the fluid depth and v its viscosity. For times t [Lt ] T, i.e. before the lower boundary has any significant influence on the flow, a laminar sub-surface boundary-layer flow is generated. The effects of gravity, capillarity, surface diffusion or surface contamination may be weak enough for the flow to drive a substantial unsteady displacement of the free surface, upward behind the monolayer's leading edge and downward towards its centre. Similarity solutions are identified describing the spreading of a localized planar monolayer strip (which spreads like t1/2) or an axisymmetric drop (which spreads like t3/8); using the Prandtl transformation, the associated boundary-layer problems are solved numerically. Quasi-steady sub-layers are shown to exist at the centre and at the leading edge of the monolayer; that due to surface contamination, for example, may eventually grow to dominate the flow, in which case spreading proceeds like t3/4. Once t = O(T), vorticity created at the free surface has diffused down to the lower boundary and the flow changes character, slowing appreciably. The dynamics of this stage are modelled by reducing the problem to a single nonlinear diffusion equation. For a spreading monolayer strip or drop, the transition from an inertia-dominated (boundary-layer) flow to a viscosity-dominated (thin-film) flow is predicted to be largely complete once t ≈ 85 T.
A new model to determine dynamic surface tension and elongational viscosity using oscillating jet measurements
Authors: Bechtel S.E. , Cooper J.A. , Forest M.G. , Petersson N.A. , Reichard D.L. , Saleh A. , Venkataramanan V.
Published: April 2006

We present an integro-differential equation model which, combined with experimental measurements of an oscillating free surface jet, calculates dynamic surface tension and elongational viscosity of a fluid. Our model builds upon previous models due to Rayleigh and Bohr in that it self-consistently incorporates the effects of viscosity and gravity. Further, surface tension and viscosity are allowed to be non-constant. The principal result of this paper is a technique for the measurement of surface tension of newly forming surfaces on the millisecond timescale relevant for agricultural spray mixtures. Coincidentally, our model independently yields the elongational viscosity of the fluid, although our present experimental apparatus limits the accuracy of measurement of this material property.

In this paper we take measurements from physical jet experiments and implement our inverse model to deduce these material properties. The model is first benchmarked against standard techniques on a well-characterized fluid with constant surface tension and Newtonian viscosity. We then apply our method to an agricultural spray mixture, with non-constant surface tension and non-Newtonian rheology. We measure (i) the rapid decay of surface tension from the newly formed surface (aged less than a millisecond) to the much lower equilibrium value, and (ii) the rate dependence of elongational viscosity.
Pressure structure functions and spectra for locally isotropic turbulence
Authors: Hill Reginald J. , Wilczak James M.
Published: April 2006

Beginning with the known relationship between the pressure structure function and the fourth-order two-point correlation of velocity derivatives, we obtain a new theory relating the pressure structure function and spectrum to fourth-order velocity structure functions. This new theory is valid for all Reynolds numbers and for all spatial separations and wavenumbers. We do not use the joint Gaussian assumption that was used in previous theory. The only assumptions are local homogeneity, local isotropy, incompressibility, and use of the Navier–Stokes equation. Specific formulae are given for the mean-squared pressure gradient, the correlation of pressure gradients, the viscous range of the pressure structure function, and the pressure variance. Of course, pressure variance is a descriptor of the energy-containing range. Therefore, for any Reynolds number, the formula for pressure variance requires the more restrictive assumption of isotropy. For the case of large Reynolds numbers, formulae are given for the inertial range of the pressure structure function and spectrum and of the pressure-gradient correlation; these are valid on the basis of local isotropy, as are the formulae for mean-squared pressure gradient and the viscous range of the pressure structure function. Using the experimentally verified extension to fourth-order velocity structure functions of Kolmogorov's theory, we obtainr4/3andk−7/3laws for the inertial range of the pressure structure function and spectrum. The modifications of these power laws to account for the effects of turbulence intermittency are also given. New universal constants are defined; these require experimental evaluation. The pressure structure function is sensitive to slight departures from local isotropy, implying stringent conditions on experimental data, but applicability of the previous theory is likewise constrained. The results are also sensitive to compressibility.
Nonlinear symmetric stability of planetary atmospheres
Authors: Bowman John C. , Shepherd Theodore G.
Published: April 2006

The energy–Casimir method is applied to the problem of symmetric stability in the context of a compressible, hydrostatic planetary atmosphere with a general equation of state. Formal stability criteria for symmetric disturbances to a zonally symmetric baroclinic flow are obtained. In the special case of a perfect gas the results of Stevens (1983) are recovered. Finite-amplitude stability conditions are also obtained that provide an upper bound on a certain positive-definite measure of disturbance amplitude.
Experimental and computational investigations of hypersonic flow about compression ramps
Authors: Simeonides G. , Haase W.
Published: April 2006

Comprehensive results of a joint experimental and computational study of the two-dimensional flow field over flat plate/compression ramp configurations at Mach 14 are presented. These geometries are aimed to simulate, in a simplified manner, the region around deflected control surfaces of hypersonic re-entry vehicles. The test cases considered cover a range of realistic flow conditions with Reynolds numbers to the hinge line varying between 4.5 × 105 and 2.6 × 106 (with a reference length taken as the distance between the leading edge and the hinge line) and a wall-to-total-temperature ratio of 0.12. The combination of flow and geometric parameters gives rise to fully laminar strong shock wave/boundary layer interactions with extensive separation, and transitional interactions with transition occurring near the reattachment point. A fully turbulent interaction is also considered which, however, was only approximately achieved in the experiments by means of excessive tripping of the oncoming hypersonic laminar boundary layer. Emphasis has been placed upon the quality and level of confidence of both experiments and computations, including a discussion on the laminar-turbulent transition process and the associated striation phenomenon. The favourable comparison between the experimental and computational results has profided the grounds for an enhanced understanding of the relevant flow processes and their modelling. Particularly in relation to transitional shock wave/boundary layer interactions, where laminar-turbulent transition is promoted by the adverse pressure gradient and flow concavity in the reattachment region, a method is proposed to compute extreme adverse effects in the interaction region avoiding such inhibiting requirements as transition modelling or turbulence modelling over separated regions.
Impact of drops on solid surfaces: self-similar capillary waves, and splashing as a new type of kinematic discontinuity
Authors: Yarin A. L. , Weiss D. A.
Published: April 2006

The impact of drops impinging one by one on a solid surface is studied experimentally and theoretically. The impact process is observed by means of a charge-coupled-device camera, its pictures processed by computer. Low-velocity impact results in spreading and in propagation of capillary waves, whereas at higher velocities splashing (i.e. the emergence of a cloud of small secondary droplets, absent in the former case) sets in. Capillary waves are studied in some detail in separate experiments. The dynamics of the extension of liquid lamellae produced by an impact in the case of splashing is recorded. The secondary-droplet size distributions and the total volume of these droplets are measured, and the splashing threshold is found as a function of the impact parameters.

The pattern of the capillary waves is predicted to be self-similar. The calculated wave profile agrees well with the experimental data. It is shown theoretically that the splashing threshold corresponds to the onset of a velocity discontinuity propagating over the liquid layer on the wall. This discontinuity shows several aspects of a shock. In an incompressible liquid such a discontinuity can only exist in the presence of a sink at its front. The latter results in the emergence of a circular crown-like sheet virtually normal to the wall and propagating with the discontinuity. It is predicted theoretically and recorded in the experiment. The crown is unstable owing to the formation of cusps at the free rim at its top edge, which results in the splashing effect. The onset velocity of splashing and the rate of propagation of the kinematic discontinuity are calculated and the theoretical results agree fairly well with the experimental data. The structure of the discontinuity is shown to match the outer solution.
Primary and secondary instabilities of the asymptotic suction boundary layer on a curved plate
Authors: Park Daniel S. , Huerre Patrick
Published: April 2006

The temporal growth of Görtler vortices and the associated secondary instability mechanisms are investigated numerically in the case of an asymptotic suction boundary layer on a curved plate. Highly inflectional velocity profiles are generated in both the spanwise and vertical directions. The inflectional velocity profile develops earlier in the spanwise direction. There exist two distinct modes of instability that eventually lead to the breakdown of Görtler vortices: the sinuous mode and the varicose mode. The temporal secondary instability analysis of the three-dimensional inflectional velocity profile reveals that the sinuous mode becomes unstable earlier than the varicose mode. The sinuous mode is shown to be primarily related to shear in the spanwise direction, ∂U/∂z, and the varicose mode to shear in the vertical direction, ∂U/∂y.
Sedimentation and sediment flow in settling tanks with inclined walls
Authors: Kapoor B. , Acrivos A.
Published: April 2006

The flow of a sediment layer that forms on an inclined plate as a consequence of the steady sedimentation of spherical particles was investigated theoretically as well as experimentally. The theoretical analysis was based on the model proposed by Nir & Acrivos (1990), modified to include shear-induced diffusion due to gradients in the shear stress as well as a slip velocity along the wall due to the finite size of the particles. The resulting set of partial differential equations, which is amenable to a similarity-type solution both near the leading edge as well as far downstream, was solved numerically using a finite difference scheme thereby yielding theoretical predictions for the particle concentration and velocity profiles, plus the local sediment layer thickness, all along the plate. In addition, a new experimental technique based on laser Doppler anemometry was developed and was used to measure the particle velocity profiles in the highly concentrated sediment layer as well as the corresponding slip coefficient which relates the slip velocity to the velocity gradient adjacent to a wall. The thickness profile of the sediment layer was also measured experimentally by means of video imaging. It was found that the experimental results thus obtained for the particle velocity profile and for the local sediment layer thickness were in very good agreement with the corresponding theoretical predictions especially considering that the latter did not make use of any adjustable parameters.
Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics. Part 2. Pseudoenergy-based theory
Authors: Kushner Paul J. , Shepherd Theodore G.
Published: April 2006

This paper represents the second part of a study of semi-geostrophic (SG) geophysical fluid dynamics. SG dynamics shares certain attractive properties with the better known and more widely used quasi-geostrophic (QG) model, but is also a good prototype for balanced models that are more accurate than QG dynamics. The development of such balanced models is an area of great current interest. The goal of the present work is to extend a central body of QG theory, concerning the evolution of disturbances to prescribed basic states, to SG dynamics. Part 1 was based on the pseudomomentum; Part 2 is based on the pseudoenergy.

A pseudoenergy invariant is a conserved quantity, of second order in disturbance amplitude relative to a prescribed steady basic state, which is related to the time symmetry of the system. We derive such an invariant for the semi-geostrophic equations, and use it to obtain: (i) a linear stability theorem analogous to Arnol'd's ‘first theorem’; and (ii) a small-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit. The results are analogous to their quasi-geostrophic forms, and reduce to those forms in the limit of small Rossby number.

The results are derived for both the f-plane Boussinesq form of semi-geostrophic dynamics, and its extension to β-plane compressible flow by Magnusdottir & Schubert. Novel features particular to semi-geostrophic dynamics include apparently unnoticed lateral boundary stability criteria. Unlike the boundary stability criteria found in the first part of this study, however, these boundary criteria do not necessarily preclude the construction of provably stable basic states.

The interior semi-geostrophic dynamics has an underlying Hamiltonian structure, which guarantees that symmetries in the system correspond naturally to the system's invariants. This is an important motivation for the theoretical approach used in this study. The connection between symmetries and conservation laws is made explicit using Noether's theorem applied to the Eulerian form of the Hamiltonian description of the interior dynamics.
Strongly nonlinear interfacial dynamics in core–annular flows
Authors: Kerchman V.
Published: April 2006

Nonlinear stability of a pressure driven core-annular flow is analysed, and a study of the large-amplitude interfacial dynamics is reported in the limit of a small ratio β of the annular clearance to the radius. An asymptotic nonlinear evolution equation for the annular film thickness is derived as a general case which involves shear coupling with the core flow. We discuss the effects of the surface tension parameter and viscosity stratification of various orders in β. The governing equation is investigated by solving it on extended intervals. Long-term simulations in a wide range of parameters reveal rich dynamics of wave patterns and coherent structures. Only in a narrow window of the small control parameters can it be described by the weakly nonlinear dissipative-dispersive equation, exhibiting behaviour of strictly bounded solutions which varies from a spatiotemporal chaos to the quasi-steady wavetrains. For sufficiently high surface tension, some pulses (to which the primary instabilities saturate) can coalesce into stable larger structures. This leads to the formation of solitary humps via cascade absorption. Substantial thickness non-uniformities can cause collapse of the perfect CAFF owing to the lens formation or extreme film thinning. Our critical value of the control parameter is in good agreement with the experimental data by Aul & Olbricht. Under strong coupling of the core flow with a less viscous annular film the interfacial evolution settles to a train of inverted pulses. Long-time behaviour in the intermediate range of parameters is diversified from regular pulse trains, to the formation of wide multi-peak structures or blow-up, depending on the apparent involvement of the core.
On the equilibrium and stability of a row of point vortices
Authors: Aref Hassan
Published: April 2006

The equilibrium and stability of a single row of equidistantly spaced identical point vortices is a classical problem in vortex dynamics, which has been addressed by several investigators in different ways for at least a century. Aspects of the history and the essence of these treatments are traced, stating some in more accessible form, and pointing out interesting and apparently new connections between them. For example, it is shown that the stability problem for vortices in an infinite row and the stability problem for vortices arranged in a regular polygon are solved by the same eigenvalue problem for a certain symmetric matrix. This result also provides a more systematic enumeration of the basic instability modes. The less familiar theory of equilibria of a finite number of vortices situated on a line is also recalled.
An experimental study of a three-dimensional pressure-driven turbulent boundary layer
Authors: Ölçmen Semİh M. , Simpson Roger L.
Published: April 2006

A three-dimensional, pressure-driven turbulent boundary layer created by an idealized wing–body junction flow was studied experimentally. The data presented include time-mean static pressure and directly measured skin-friction magnitude on the wall. The mean velocity and all Reynolds stresses from a three-velocity-component fibre-optic laser-Doppler anemometer are presented at several stations along a line determined by the mean velocity vector component parallel to the wall in the layer where the $\overline{u^2}$ kinematic normal stress is maximum (normal-stress coordinate system). This line was selected by intuitively reasoning that overlap of the near-wall flow and outer-region flow occurs at the location where $\overline{u^2}$ is maximum. Along this line the flow is subjected to a strong crossflow pressure gradient, which changes sign for the downstream stations. The shear-stress vector direction in the flow lags behind the flow gradient vector direction. The flow studied here differs from many other experimentally examined three-dimensional flows in that the mean flow variables depend on three spatial axes rather than two axes, such as flows in which the three-dimensionality of the flow has been generated either by a rotating cylinder or by a pressure gradient in one direction only throughout the flow.

The data show that the eddy viscosity of the flow is not isotropic. These and other selected data sets show that the ratio of spanwise to streamwise eddy viscosities in the wall-shear-stress coordinate system is less scattered and more constant (about 0.6) than in the local free-stream coordinate system or the normal stress coordinate system. For y+ > 50 and y/δ < 0.8, the ratio of the magnitude of the kinematic shear stress |τ/ρ| to the kinematic normal stress $\overline{v^2}$ is approximately a constant for three-dimensional flow stations of both shear-driven and pressure-driven three-dimensional flows. In the same region, the ratio of the kinematic shear stresses $-\overline{vw}/-\overline{uw}$ appears to be a function of y+ in wall-stress coordinates for three-dimensional pressure-driven flows.
Theoretical aspects of gravity–capillary waves in non-rectangular channels
Authors: Groves Mark D.
Published: April 2006

This article reports the results of theoretical research concerning linear waves propagating on the surface of water in a uniform horizontal channel of arbitrary crosssection. Three different versions of the problem are considered. The first is the hydrodynamic problem when surface tension is neglected. The second and third include capillary effects, necessitating the use of edge conditions at the points of contact of the free edges and the channel walls. Two sets of edge constraints are used: pinned edges, where the lines of contact are fixed, and free edges, where the surface meets locally vertical walls orthogonally. These choices are physically realistic and have certain advantages for mathematical analysis.

The hydrodynamic problems are shown to have a Hamiltonian structure in which the non-local operators inherent in the water-wave problem are explicitly exhibited. The existence, properties and applications of normal-mode solutions are discussed, and a qualitative comparison of those obtained for each problem is given. Explicit and numerical calculations of the dispersion relations for the normal modes are also carried out. A long-wave theory based upon a decomposition of the hydrodynamic problems in Fourier-transform space is developed. Finally a bifurcation theory for linear travelling waves is discussed, a potential application of which is the construction of an existence theory for periodic travelling-wave solutions of the corresponding nonlinear problems.
Three-dimensional disturbances to a mixing layer in the nonlinear critical-layer regime
Authors: Churilov S. M. , Shukhman I. G.
Published: April 2006

We consider the nonlinear spatial evolution in the streamwise direction of slightly three-dimensional disturbances in the form of oblique travelling waves (with spanwise wavenumber kz much less than the streamwise one kx) in a mixing layer vx = u(y) at large Reynolds numbers. A study is made of the transition (with the growth of amplitude) to the regime of a nonlinear critical layer (CL) from regimes of a viscous CL and an unsteady CL, which we have investigated earlier (Churilov & Shukhman 1994). We have found a new type of transition to the nonlinear CL regime that has no analogy in the two-dimensional case, namely the transition from a stage of ‘explosive’ development. A nonlinear evolution equation is obtained which describes the development of disturbances in a regime of a quasi-steady nonlinear CL. We show that unlike the two-dimensional case there are two stages of disturbance growth after transition. In the first stage (immediately after transition) the amplitude A increases as x. Later, at the second stage, the ‘classical’ law A ∼ x2/3 is reached, which is usual for two-dimensional disturbances. It is demonstrated that with the growth of kz the region of three-dimensional behaviour is expanded, in particular the amplitude threshold of transition to the nonlinear CL regime from a stage of ‘explosive’ development rises and therefore in the ‘strongly three-dimensional’ limit kz = O(kx) such a transition cannot be realized in the framework of weakly nonlinear theory.
Explosive resonant interaction of baroclinic Rossby waves and stability of multilayer quasi-geostrophic flow
Authors: Vanneste Jacques
Published: April 2006

The amplitude equations governing the nonlinear interaction among normal modes are derived for a multilayer quasi-geostrophic channel. The set of normal modes can represent any wavy disturbance to a parallel shear flow, which may be stable or unstable. Orthogonality in the sense of pseudomomentum or pseudoenergy is used to obtain the amplitude equations in a direct fashion, and pseudoenergy and pseudomomentum conservation laws permit the properties of the interaction coefficients to be deduced. Particular attention is paid to triads exhibiting explosive resonant interaction, as they lead to nonlinear instability of the basic flow. The relationship between this mechanism and the most recently discovered nonlinear stability conditions is discussed.

Situations in which the basic velocity is constant in each layer are treated in detail. A particular formulation of the stability condition is given that emphasizes the close connection between linear and nonlinear stability. It is established that this stability condition is also a necessary condition: when it is not satisfied, and when the flow is linearly stable, explosive resonant interaction of baroclinic Rossby waves acts as a destabilizing mechanism. Two- and three-layer models are specifically considered; their stability features are presented in the form of stability diagrams, and interaction coefficients are calculated in particular cases.
Chaotic transport by dipolar vortices on a β-plane
Authors: Velasco Fuentes O. U. , van Heijst G. J. F. , Cremers B. E.
Published: April 2006

During the meandering motion of a dipolar vortex on a β-plane mass is exchanged both between the dipole and the ambient fluid and between the two dipole halves. The mass exchange (as well as the meandering motion) is caused by variations of the relative vorticity of the vortices due to conservation of potential vorticity. Previous studies have shown that a modulated point-vortex model captures the essential features in the dipole evolution. For this model we write the equations of motion of passive tracers in the form of a periodically perturbed integrable Hamiltonian system and subsequently study transport using a ‘dynamical-systems theory’ approach. The amount of mass exchanged between different regions of the flow is evaluated as a function of two parameters: the gradient of ambient vorticity, β, and the initial direction of propagation of the dipole, α0. Mass exchange between the dipole and the surroundings increases with increasing both β and α0. The exchange rate (amount of mass exchanged per unit time) increases with β and has a maximum for a particular value of α0 (≈ 0.62π). Dipolar vortices in a rotating fluid (with a sloping bottom providing the ‘topographic’ β-effect) show, in addition to the relative vorticity variations, a second perturbation that leads to exchange of mass. The points where vorticity is extreme approach each other as the dipole moves to shallower parts of the fluid and separate as the couple moves to deeper parts. This mechanism is studied independently and it is shown to lead to a stronger exchange between the dipole halves and the ambient fluid but no exchange between the two dipole halves.
The nonlinear temporal evolution of a disturbance to a stratified mixing layer
Authors: Mallier Roland
Published: April 2006

Using a nonlinear critical layer analysis, Goldstein & Leib (1988) derived a set of nonlinear evolution equations governing the spatial growth of a two-dimensional instability wave on a homogeneous incompressible tanh y mixing layer. In this study, we extend this analysis to the temporal growth of the García model of an incompressible stratified shear layer. We consider the stage of the evolution in which the growth first becomes nonlinear, with the nonlinearity appearing inside the critical layer. The Reynolds number is assumed to be just large enough so that the unsteady, nonlinear and viscous terms all enter at the same order of magnitude inside the critical layer. The equations are solved numerically for the inviscid case.
Dissipation and ordering in capillary waves at high aspect ratios
Authors: Christiansen Bo , Alstrøm Preben , Levinsen Mogens T.
Published: April 2006

We present an experimental study of high-aspect-ratio Faraday waves. We have measured the dispersion relation and the damping rate, together with the critical amplitude for the primary instability for a wide range of frequencies. We find that our results are well explained by the linear theory, if damping from the moving contact line is considered in addition to the bulk damping. Just above the primary instability a seemingly disordered stationary state is observed. We argue that this state is a superposition of normal modes. Approximately 5% above the primary instability this state breaks down in favour of a quasi-crystalline state. This result is discussed, partly in the light of the recent third-order nonlinear theory.
The nonlinear evolution of magnetic instabilities in a rapidly rotating annulus
Authors: Hutcheson Kenneth A. , Fearn David R.
Published: April 2006

A numerical investigation of the stability of an axisymmetric magnetic field is discussed. The magnetic field permeates a finitely conducting fluid contained within a rapidly rotating cylindrical annulus. The fluid is incompressible and viscid. The evolution of a non-axisymmetric perturbation to the axisymmetric magnetic field is governed by the momentum and induction equations which are integrated using a spectral timestep method. We follow the growth of the perturbation to finite amplitude and find that the character of the solution is dominated by the most unstable axially dependent mode found from the linear theory.
The modified mild-slope equation
Authors: Chamberlain P. G. , Porter D.
Published: April 2006

A modified version of the mild-slope equation is derived and its predictions of wave scattering by two-dimensional topography compared with those of other equations and with experimental data. In particular, the modified mild-slope equation is shown to be capable of describing known scattering properties of singly and doubly periodic ripple beds, for which the mild-slope equation fails. The new equation compares favourably with other models of scattering which improve on the mild-slope equation, in that it is widely applicable and computationally cheap.
The dynamics of driven rotating flow in stadium-shaped domains
Authors: Kobine J. J. , Mullin T. , Price T. J.
Published: April 2006

Results are presented from an experimental investigation of the dynamics of driven rotating flows in stadium-shaped domains. The work was motivated by questions concerning the typicality of low-dimensional dynamical phenomena which are found in Taylor-Couette flow between rotating circular cylinders. In such a system, there is continuous azimuthal symmetry and travelling-wave solutions are found. In the present study, this symmetry is broken by replacing the stationary outer circular cylinder with one which has a stadium-shaped cross-section. Thus there is now only discrete symmetry in the azimuthal direction, and travelling waves are no longer observed. To begin with, the two-dimensional flow field was investigated using numerical techniques. This was followed by an experimental study of the dynamics of flow in systems with finite vertical extent. Configurations involving both right-circular and tapered inner cylinders were considered. Dynamics were observed which correspond to known mechanisms from the theory of finite-dimensional dynamical systems. However, flow behaviour was also observed which cannot be classified in this way. Thus it is concluded that while certain low-dimensional dynamical phenomena do persist with breaking of the continuous azimuthal symmetry embodied in the Taylor-Couette system, sufficient reduction of symmetry admits behaviour at moderately low Reynolds number which is without any low-dimensional characteristics.
Complementary approximations to wave scattering by vertical barriers
Authors: Porter R. , Evans D. V.
Published: April 2006

Scattering of waves by vertical barriers in infinite-depth water has received much attention due to the ability to solve many of these problems exactly. However, the same problems in finite depth require the use of approximation methods. In this paper we present an accurate method of solving these problems based on a Galerkin approximation. We will show how highly accurate complementary bounds can be computed with relative ease for many scattering problems involving vertical barriers in finite depth and also for a sloshing problem involving a vertical barrier in a rectangular tank.
Thermocapillary migration of a two-dimensional liquid droplet on a solid surface
Authors: Smith Marc K.
Published: April 2006

A two-dimensional liquid droplet placed on a non-uniformly heated solid surface will move towards the region of colder temperatures if the temperature gradient in the solid surface is large enough. Such behaviour is analysed for a thin viscous droplet using lubrication theory to develop an evolution equation for the shape of the droplet. For the small mobility capillary numbers examined in this work, the contact-line motion is controlled by a dynamic relationship posed between the contact-line speed and the apparent contact angle. Results are obtained numerically and also approximately using a perturbation technique for small heating. The initial spreading or shrinking of the droplet when placed on the heated solid is biased toward the direction of decreasing temperature on the solid. Possible steady-state responses are either a motionless droplet or one moving at a constant velocity down the temperature gradient without change in shape. These behaviours are the result of a thermocapillary recirculation cell inside the droplet that distorts the free surface and alters the apparent contact angles. This change in the apparent contact angles then modifies the contact-line speed.
Towards enhancing and delaying disturbances in free shear flows
Authors: Criminale W. O. , Jackson T. L. , Lasseigne D. G.
Published: April 2006

The family of shear flows comprising the jet, wake, and the mixing layer are subjected to perturbations in an inviscid incompressible fluid. By modelling the basic mean flows as parallel with piecewise linear variations for the velocities, complete and general solutions to the linearized equations of motion can be obtained in closed form as functions of all space variables and time when posed as an initial-value problem. The results show that there is a continuous spectrum as well as the discrete spectrum that is more familiar in stability theory and therefore there can be both algebraic and exponential growth of disturbances in time. These bases make it feasible to consider control of such flows. To this end, the possibility of enhancing the disturbances in the mixing layer and delaying the onset in the jet and wake is investigated. It is found that growth of perturbations can be delayed to a considerable degree for the jet and the wake but, by comparison, cannot be enhanced in the mixing layer. By using moving coordinates, a method for demonstrating the predominant early and long time behaviour of disturbances in these flows is given for continuous velocity profiles. It is shown that the early time transients are always algebraic whereas the asymptotic limit is that of an exponential normal mode. Numerical treatment of the new governing equations confirm the conclusions reached by use of the piecewise linear basic models. Although not pursued here, feedback mechanisms designed for control of the flow could be devised using the results of this work.
Transition modes of rotating shallow water waves in a channel
Authors: Iga Keita
Published: April 2006

Normal modes of shallow water waves in a channel wherein the Coriolis parameter and the depth vary in the spanwise direction are investigated based on the conservation of the number of zeros in an eigenfunction. As a result, it is generally shown that the condition for transition modes (Kelvin modes and mixed Rossby-gravity modes) to exist, besides Rossby and Poincaré modes, is determined only by boundary conditions. A Kelvin mode is interpreted as a modification of a Kelvin wave or a boundary wave along a closed boundary, and a mixed Rossby-gravity mode as a modification of an inertial oscillation or a boundary wave along an open boundary. Transition modes appearing in edge and continental-shelf waves, equatorial waves and free oscillations over a sphere are systematically understood by applying the theory in this paper.
Dynamics and stability of a vortex ring impacting a solid boundary
Authors: Swearingen J. D. , Crouch J. D. , Handler R. A.
Published: April 2006

Direct numerical simulations were used to study the dynamics of a vortex ring impacting a wall at normal incidence. The boundary layer formed as the ring approaches the wall undergoes separation and roll-up to form a secondary vortex ring. The secondary ring can develop azimuthal instabilities which grow rapidly owing to vortex stretching and tilting in the presence of the mean strain field generated by the primary vortex ring. The stability of the secondary ring was investigated through complementary numerical experiments and stability analysis. Both perturbed and unperturbed evolutions of the secondary ring were simulated at a Reynolds number of about 645, based on the initial primary-ring propagation velocity and ring diameter. The linear evolution of the secondary vortex-ring instability was modelled analytically by making use of a quasi-steady approximation. This allowed a localized stability analysis following Widnall & Sullivan's (1973) earlier treatment of an isolated vortex ring. Amplitude evolution and growth-rate predictions from this analysis are in good agreement with the simulation results. The analysis shows that the secondary vortex ring is unstable to long-wavelength perturbations, even though an isolated ring having similar characteristics would be stable.
Pressure-impulse theory for liquid impact problems
Authors: Cooker Mark J. , Peregrine D. H.
Published: April 2006

A mathematical model is presented for the high pressures and sudden velocity changes which may occur in the impact between a region of incompressible liquid and either a solid surface or a second liquid region. The theory rests upon the well-known idea of pressure impulse, for the sudden initiation of fluid motion in incompressible fluids. We consider the impulsive pressure field which occurs when a moving fluid region collides with a fixed target, such as when an ocean wave strikes a sea wall. The boundary conditions are given for modelling liquid-solid and liquid-liquid impact problems. For a given fluid domain, and a given velocity field just before impact, the theory gives information on the peak pressure distribution, and the velocity after impact. Solutions for problems in simple domains are presented, which give insight into the peak pressures exerted by a wave breaking against a sea wall, and a wave impacting in a confined space. An example of liquid-liquid impact is also examined. Results of particular interest include a relative insensitivity to the shape of the incident wave, and an increased pressure impulse when impact occurs in a confined space. The theory predicts that energy is lost from the bulk fluid motion and we suggest that this energy can be transferred to a thin jet of liquid which is projected away from the impact region.
Standing and travelling oscillatory blob convection
Authors: Clever R. M. , Busse F. H.
Published: April 2006

Results of numerical computations are presented of time-dependent three-dimensional convection flows in a horizontal layer heated from below which evolve from the oscillatory blob instability of steady two-dimensional rolls. It is shown that the heat transport is typically increased in the transition to blob convection. Oscillatory blob convection exists in the forms of standing or travelling blob convection. The latter type of solution represents the stable form bifurcating supercritically at the Rayleigh number RII for the onset of the oscillatory blob instability. In contrast to standing blob convection travelling blob convection exhibits a mean flow.
Refined similarity hypotheses for passive scalars mixed by turbulence
Authors: Stolovitzky G. , Kailasnath P. , Sreenivasan K. R.
Published: April 2006

In analogy with Kolmogorov's refined similarity hypotheses for the velocity field, two hypotheses are stated for passive scalar fields mixed by high-Reynolds-number turbulence. A ‘refined’ Yaglom equation is derived under the new assumption of local isotropy in pure ensembles, which is stronger than the usual assumption of local isotropy but weaker than the isotropy of the large scale. The new theoretical result is shown to be consistent with the hypotheses of refined similarity for passive scalars. These hypotheses are approximately verified by experimental data on temperature fluctuations obtained (in air) at moderate Reynolds numbers in the wake of a heated cylinder. The fact that the refined similarity hypotheses are stated for high Reynolds (and Péclet) numbers, but verified at moderate Reynolds (and Péclet) numbers suggests that these hypotheses are not sufficiently sensitive tests of universality. It is conjectured that possible departures from universality are hidden by the process of taking conditional expectations.
Flow focusing instability in a solidifying mushy layer
Authors: Chiareli A. O. P. , Worster M. Grae
Published: April 2006

The stability of the flow of interstitial liquid in a mushy layer driven by expansion or contraction upon solidification is analysed. The full perturbation equations are reduced in a particular aymptotic limit that allows the principal mechanisms controlling instability to be identified. Comparisons are made with the acid-etching instabilities in porous rocks. The full equations are then solved to determine the parametric dependences of the instability. It is found that, though the potential for instability exists, it is unlikely to occur in practice.
Edge waves along periodic coastlines. Part 2
Authors: Evans D. V. , Fernyhough M.
Published: April 2006

Numerical evidence of the existence of edge waves travelling along a periodic coastline consisting of a straight and vertical cliff face from which protrudes an infinite number of identical rectangular barriers, each extending throughout the water depth, is given based on a Galerkin approximation to an integral representation of the problem derived using the linear theory of water waves.
Analytical solutions for the actuator disk with variable radial distribution of load
Authors: Conway John T.
Published: April 2006

An analytical method somewhat analogous to finite wing theory has been developed which enables the flow induced by a linearized propeller actuator disk with variable radial distribution of load to be solved in closed form for the first time. Analytical solutions are given for various load distributions including the case of an arbitrary polynomial loading. As in finite wing theory, the case of elliptic loading is exceptionally simple and the induced velocities and stream function are simple expressions of elementary functions. Results are also given for a practical propeller load distribution with finite hub. The method can also be used to solve a wide range of analogous electromagnetic problems.
Time-dependent transport by convection and diffusion with exchange between two phases
Authors: Phillips C. G. , Kaye S. R. , Robinson C. D.
Published: April 2006

We consider the transport of a tracer substance through a system consisting of a tube containing flowing fluid surrounded by a wall layer in which the tracer is soluble. The fluid moves with either a Poiseuille or a uniform flow profile, and the outer boundary of the wall layer is either impermeable to tracer or absorbs it perfectly. The development of dispersive transport following the injection of tracer is described in terms of three time-dependent effective transport coefficients, viz. the fraction of tracer remaining in the system, the apparent convection velocity and the dispersion coefficient; the last two are defined in terms of the rates of change of the mean and variance of the axial tracer distribution. We assume that the timescale for tracer diffusion across the wall layer is much larger than that for diffusion across the flowing phase, and derive an asymptotic approximation corresponding to each timescale. Numerical results are given to illustrate sensitivity to the physical parameters of the system. It is shown that if the coefficients are based on tracer concentration in the fluid phase alone, as in previous work, paradoxical behaviour, such as negative apparent convection velocities, can result; we therefore base our results on averages of concentration over both phases. On the shorter timescale (the same timescale over which Taylor dispersion develops) at leading order it is found that the influence of the wall layer can be characterized by a single dimensionless parameter, and that conditions at the outer boundary have no effect. In many cases transport is also rather insensitive to the form of the flow profile. On the longer timescale, at leading order the influence of the wall layer is characterized by another dimensionless parameter, and unless uptake is very small diffusion within the layer is the rate-determining process; consequently transport is independent of the form of the flow profile. A further important conclusion is that the usual effective convection and dispersion coefficients, based on spatial moments, are of little use in predicting the time-varying concentration at a fixed position, because the spatial concentration profile becomes Gaussian only over the longer timescale.
Source–sink turbulence in a rotating stratified fluid
Authors: Linden P. F. , Boubnov B. M. , Dalziel S. B.
Published: April 2006

In a recent paper Boubnov, Dalziel & Linden (1994) described the response of a stratified fluid to forcing produced by an array of sources and sinks. The sources and sinks were located in a horizontal plane and the flow from the sources was directed horizontally so that fluid was withdrawn from, and re-injected at, its own density level. As a result vertical vorticity was imparted to the fluid with a minimum of vertical mixing. It was found that when the stratification was strong enough to suppress vertical motions an inverse energy cascade was observed leading to the establishment of a large-scale circulation in the fluid. Those experiments were restricted to eight source-sink pairs. The present paper extends this work in two ways. First, up to forty source-sink pairs are used to force the flow, thereby producing a much wider separation of scales between the forcing and the flow domain. An inverse cascade is again found, but in this case the energy transfer to large scales is more rapid. The basic pattern of the large-scale flow is independent of the number of sources but the detailed structure depends on the energy input scale. Second, the effects of rotation about a vertical axis are investigated. It is found that when the Rossby deformation radius exceeds the size of the flow domain, the inverse energy cascade still occurs. However, for smaller values of the deformation scale, which in these experiments are comparable to or smaller than the forcing scale, the inverse cascade is altered by baroclinic instability. When flow structures develop on a scale larger than the deformation scale, usually by the merging of vortices of like sign, these structures are observed to split into smaller vortices of a scale comparable to the deformation scale. The flow appears to evolve with a balance between an anticascade produced by the two-dimensionality of the flow and a cascade due to baroclinic instability. For Rossby radii much smaller than the domain size the flow evolves into finite clumps of vorticity and an asymmetry between anticyclones and cyclones develops. A predominance of coherent anticyclones is observed, and the cyclonic vorticity is contained in more diffuse structures.