bluuebunny/crossref_metadata_2025 huggingface dataset:
The effect of Langmuir circulation on the distribution of submerged bubbles caused by breaking wind waves
Authors: Thorpe S. A.
Published: April 2006

Clouds of bubbles are generated at the sea surface by breaking wind waves or by heavy rain. Rows of subsurface bubble clouds have been detected by a bottom-mounted side-scan sonar, and are possibly formed by the effects of Langmuir circulation. A simple equation is devised to describe the effects of the turbulent diffusion of bubbles from the free surface, bubble rise and dissolution, and advection by Langmuir circulation. The equation is solved analytically using a series expansion in which advection is supposed small in comparison with diffusion. The solution provides a quantitative measure of the principal effects produced by the circulation, in particular the distortion of the bubble field, and estimates of the advective flux.

A random-walk numerical model, in which changes occurring in individual bubbles are followed, is tested against the analytical model in the range for which the latter is valid. There is good agreement. The numerical model is useful in extending the solutions to more complex cases which include a broad distribution of bubble sizes, and to ranges in which the analytic solution is invalid. The model is used to quantify the effect of the circulation on the acoustic scattering cross-section of the bubble clouds and to explore differences between the conclusions of earlier models and observations by Johnson & Cooke (1979).

In the appendices an estimate is made of the depth to which bubbles can be carried by the vertical velocities observed below wind rows, and this is found to agree reasonably well with the maximum depth to which bubbles are observed to penetrate. Estimates of mean vertical diffusion coefficients based on observations of bubbles are compared with some calculated solely from the advective flux in Langmuir circulation. The latter are, as expected, smaller than those representing the sum of all the contributions to the flux, but are a significant fraction, of the order of 0.2–0.4. A method of deriving the vertical diffusion coefficient from observations of the vertical distribution of the acoustic scattering cross-section of bubbles appears not to be very sensitive to circulation and may provide estimates within about 25% of the actual values.
Time-dependent critical layers in shear flows on the beta-plane
Authors: Hickernell Fred J.
Published: April 2006

The problem of a finite-amplitude free disturbance of an inviscid shear flow on the beta-plane is studied. Perturbation theory and matched asymptotics are used to derive an evolution equation for the amplitude of a singular neutral mode of the Kuo equation. The effects of time-dependence, nonlinearity and viscosity are included in the analysis of the critical-layer flow. Nonlinear effects inside the critical layer rather than outside the critical layer determine the evolution of the disturbance. The nonlinear term in the evolution equation is some type of convolution integral rather than a simple polynomial. This makes the evolution equation significantly different from those commonly encountered in fluid wave and stability problems.
Three-dimensional stability of an elliptical vortex in a straining field
Authors: Robinson A. C. , Saffman P. G.
Published: April 2006

The three-dimensional linear stability of a rectilinear vortex of elliptical cross-section existing as a steady state in an irrotational straining field is studied numerically in the case of finite strain. It is shown that the instability predicted analytically for weak strain persists for finite strain and that the weak-strain results continue to be quantitatively valid for finite strain. The dependence of the growth rates of the unstable modes on the strain and the axial-disturbance wavelength is discussed. It is also shown that a three-dimensional instability is always more unstable than a two-dimensional instability in the range of parameters of most interest.
Stirring by chaotic advection
Authors: Aref Hassan
Published: April 2006

In the Lagrangian representation, the problem of advection of a passive marker particle by a prescribed flow defines a dynamical system. For two-dimensional incompressible flow this system is Hamiltonian and has just one degree of freedom. For unsteady flow the system is non-autonomous and one must in general expect to observe chaotic particle motion. These ideas are developed and subsequently corroborated through the study of a very simple model which provides an idealization of a stirred tank. In the model the fluid is assumed incompressible and inviscid and its motion wholly two-dimensional. The agitator is modelled as a point vortex, which, together with its image(s) in the bounding contour, provides a source of unsteady potential flow. The motion of a particle in this model device is computed numerically. It is shown that the deciding factor for integrable or chaotic particle motion is the nature of the motion of the agitator. With the agitator held at a fixed position, integrable marker motion ensues, and the model device does not stir very efficiently. If, on the other hand, the agitator is moved in such a way that the potential flow is unsteady, chaotic marker motion can be produced. This leads to efficient stirring. A certain case of the general model, for which the differential equations can be integrated for a finite time to produce an explicitly given, invertible, area-preserving mapping, is used for the calculations. The paper contains discussion of several issues that put this regime of chaotic advection in perspective relative to both the subject of turbulent advection and to recent work on critical points in the advection patterns of steady laminar flows. Extensions of the model, and the notion of chaotic advection, to more realistic flow situations are commented upon.
Turbulent boundary-layer interaction with a shock wave at a compression corner
Authors: Agrawal S. , Messiter A. F.
Published: April 2006

The local interaction of an oblique shock wave with an unseparated turbulent boundary layer at a shallow two-dimensional compression corner is described by asymptotic expansions for small values of the non-dimensional friction velocity and the flow turning angle. It is assumed that the velocity-defect law and the law of the wall, adapted for compressible flow, provide an asymptotic representation of the mean velocity profile in the undisturbed boundary layer. Analytical solutions for the local mean-velocity and pressure distributions are derived in supersonic, hypersonic and transonic small-disturbance limits, with additional intermediate limits required at distances from the corner that are small in comparison with the boundary-layer thickness. The solutions describe small perturbations in an inviscid rotational flow, and show good agreement with available experimental data in most cases where effects of separation can be neglected. Calculation of the wall shear stress requires solution of the boundary-layer momentum equation in a sublayer which plays the role of a new thinner boundary layer but which is still much thicker than the wall layer. An analytical solution is derived with a mixing-length approximation, and is in qualitative agreement with one set of measured values.
Roll-pattern evolution in finite-amplitude Rayleigh–Beńard convection in a two-dimensional fluid layer bounded by distant sidewalls
Authors: Daniels P. G.
Published: April 2006

This paper considers the temporal evolution of two-dimensional Rayleigh–Bénard convection in a shallow fluid layer of aspect ratio 2L ([Gt ] 1) confined laterally by rigid sidewalls. Recent studies by Cross et al. (1980, 1983) have shown that for Rayleigh numbers in the range R = R0 + O(L−1) (where R0 is the critical Rayleigh number for the corresponding infinite layer) there exists a class of finite-amplitude steady-state ‘phase-winding’ solutions which correspond physically to the possibility of an adjustment in the number of rolls in the container as the local value of the Rayleigh number is varied. It has been shown (Daniels 1981) that in the temporal evolution of the system the final lateral positioning of the rolls occurs on the long timescale t = O(L2) when the phase function which determines the number of rolls in the system satisfies a one-dimensional diffusion equation but with novel boundary conditions that represent the effect of the sidewalls. In the present paper this system is solved numerically in order to determine the precise way in which the roll pattern adjusts after a change in the Rayleigh number of the system. There is an interesting balance between, on the one hand, a tendency for the number of rolls to change by the least number possible and, on the other, a tendency for the even or odd nature of the initial configuration to be preserved during the transition. In some cases this second property renders the natural evolution susceptible to arbitrarily small external disturbances, which then dictate the form of the final roll pattern.

The complete transition involves an analysis of the motion on three timescales, a conductive scale t = O(1), a convective growth scale t = O(L) and a convective diffusion scale t = O(L2).
Structure of a high-Reynolds-number turbulent wake in supersonic flow
Authors: Bonnet J. P. , Jayaraman V. , Roquefort T. Alziary De
Published: April 2006

An experimental study of a high-Reynolds-number turbulent wake in supersonic flow is performed using space and space–time correlation measurements by means of hot-wire anemometry. The correlations for the streamwise component of the mass-flux fluctuations are given for six stations starting from the trailing edge down to the asymptotic part. The validity of the Taylor's hypothesis is tested, the convection velocities are determined and the downstream evolution of the optimum space–time correlation is given; the frequency spectra are discussed and the integral lengths are analysed. Finally, the three-dimensional isocorrelation surfaces are given at the six test stations and discussed in relation to classical incompressible-flow results. The downstream evolution of the correlations shows that the two sides of the wake are statistically independent near the trailing edge, and a statistical link is gradually established during the wake development. A three-zonal description of wakes generated by fully developed turbulent boundary layers applies as well for mean quantities (velocity, width) as for turbulence correlations. In the near-wake region the overall structure of the isocorrelation curves is close to that observed in turbulent boundary layers in incompressible flows; some low-frequency phenomena are observed in this region. In the latest part of the wake, an asymptotic state is reached for all the correlation characteristics; the final state reached is not explained by the double-roller-eddy model established for lower-Reynolds-number wakes; it appears that wakes generated by fully turbulent boundary layers behave quite differently from initially laminar wakes, and new turbulent structure models for high-Reynolds-number wakes are to be devised.
Monte Carlo simulation of coagulation in discrete particle-size distributions. Part 1. Brownian motion and fluid shearing
Authors: Pearson H. J. , Valioulis I. A. , List E. J.
Published: April 2006

A method for the Monte Carlo simulation, by digital computer, of the evolution of a colliding and coagulating population of suspended particles is described. Collision mechanisms studied both separately and in combination are: Brownian motion of the particles, and laminar and isotropic turbulent shearing motions of the suspending fluid. Steady-state distributions are obtained by adding unit-size particles at a constant rate and removing all particles once they reach a preset maximum volume. The resulting size distributions are found to agree with those obtained by dimensional analysis (Hunt 1982).
Flow past wing-body junctions
Authors: Smith F. T. , Gajjar J.
Published: April 2006

The three-dimensional laminar flow past a junction formed by a thin wing protruding normally from a locally flat body surface is considered for wings of finite span but short or long chord. The Reynolds number is taken to be large. The leading-edge interaction for a long wing has the triple-deck form, with the pressure due to the wing thickness forcing a three-dimensional flow response on the body surface alone. The same interaction describes the flow past an entire short wing. Linearized solutions are presented and discussed for long and short two-dimensional wings and for certain three-dimensional wings of interest. The trailing-edge interaction for a long wing is different, however, in that the three-dimensional motions on the wing and on the body are coupled together and in general the coupling is nonlinear. Linearized properties are retrieved only for reduced chord lengths. The overall flow structure for a long wing is also discussed, including the traditional three-dimensional corner layer, which is shown to have an unusual singular starting form near the leading edge. Qualitative comparisons with experiments are made.
Flow in a rotating non-aligned straight pipe
Authors: Berman J. , Mockros L. F.
Published: April 2006

A third-order regular perturbation solution is developed for laminar flow through a straight pipe that is rotating about an axis not aligned with the pipe axis. Coriolis accelerations produce transverse secondary velocities (similar to those in flow through coiled tubes) and modify the axial-velocity profile. The effects of rotation on the velocity fields are shown to depend on two parameters: (i) the product of axial and rotational Reynolds numbers, and (ii) the square of the rotational Reynolds number itself. Even though their strength increases with increases in parameter magnitudes, transverse circulations are qualitatively insensitive to parametric values. The axial profile, on the other hand, can be significantly modified by the rotation; the zeroth-order parabolic axial profile can be skewed toward the outside, dimpled in the centre with maximums on either side of the centreline, or both, depending on the values of the two parameters. The modification of the axial-velocity profile has important ramifications in the design of heat/mass-transfer devices.
On the stability of the similarity solutions for swirling flow above an infinite rotating disk
Authors: Bodonyi R. J. , Ng B. S.
Published: April 2006

A stability theory for the steady swirling flow above an infinite rotating disk immersed in an otherwise unbounded rigidly rotating fluid is developed in order to corroborate the various numerical computations considered for this problem. An analysis of the initial-value problem for linearized time-dependent perturbations on the steady-state similarity solutions shows that the disturbance equations have a stable continuum spectrum which, under certain conditions, exhibits only algebraic decay in time. In addition, a numerical analysis on the discrete spectrum shows that there are unstable eigenvalues for certain rotational rates of the disk relative to the fluid at infinity. The results obtained are in good agreement with the large-time behaviour of the corresponding solutions of the unsteady similarity equations.
The interference of thermal fields from line sources in grid turbulence
Authors: Warhaft Z.
Published: April 2006

The interference of passive thermal fields produced by two (and more) line sources in decaying grid turbulence is studied by using the inference method described by Warhaft (1981) to determine the cross-correlation coefficient ρ between the temperature fluctuations produced by the sources. The evolution of ρ as a function of downstream distance, for 0.075 <d/l< 10, wheredis the wire spacing andlis the integral lengthscale of the turbulence, is determined for a pair of sources located at various distances from the grid. It is found that ρ may be positive or negative (thereby enhancing or diminishing the total temperature variance) depending on the line-source spacing, their location from the grid and the position of measurement. It is also shown that the effects of a mandoline (Warhaft & Lumley 1978) may be idealized as the interference of thermal fields produced by a number of line sources. Thus new light is shed on the rate of decay of scalar-variance dissipation. The thermal field of a single line source is also examined in detail, and these results are compared with other recent measurements.
Surface-wave propagation over sinusoidally varying topography
Authors: Davies A. G. , Heathershaw A. D.
Published: April 2006

Surface waves travelling in water of finite depth may be scattered by a region of undulating bottom topography. The present study is concerned with the idealized two-dimensional situation in which long-crested surface waves are incident upon a patch of long-crested regular bottom ripples. The principal question examined concerns the amount of incident wave energy that is reflected by the ripple patch. Linear perturbation theory is used to show that the reflection coefficient is both oscillatory in the quotient of the length of the patch and the surface wavelength, and also strongly dependent upon the quotient of the surface and bed wavelengths. In particular, there is a Bragg resonance between the surface waves and the ripples, which is associated with the reflection of incident wave energy. A secondary question concerns the nature of the wave field in the immediate vicinity of the ripple patch. In resonant cases, it is shown how the partially standing wave on the upwave side of the ripple patch gives way, in an almost linear manner over the patch itself, to a progressive transmitted wave on the downwave side. The theoretical predictions are compared with an extensive set of laboratory observations made in a wave tank. Comparisons relating both to the reflection coefficient, and also to the wave field over the ripple patch, are shown to give consistently good agreement. Finally, the implications of the results for sediment transport on an erodible bed are examined.
Asymmetric flow between parallel rotating disks
Authors: Lai C.-Y. , Rajagopal K. R. , Szeri A. Z.
Published: April 2006

Flows occurring between parallel rotating disks have recently been generalized by Parter & Rajagopal (1984) to include solutions that are not axisymmetric. They prove existence, whereas in the present paper we report, for the first time, numerical results for two cases: (i) rotation about a common axis, and (ii) rotation about distinct axes. Calculations were performed for two values of the Ekman number E = v/d2ω at the relative disk rotations of s = 0·8, s = 0 and s = −0·25, where s = ω2/ω1.
Nonlinear Faraday resonance
Authors: Miles John W.
Published: April 2006

A cylinder containing liquid with a free surface is subjected to a vertical oscillation of amplitude εg/ω2and frequency 2ω, where ω is within O(εω) of the natural frequency of a particular(primary)mode in the surface-wave spectrum and 0 < ε 1. A Lagrangian formulation, which includes terms of second and fourth order in the primary mode and second order in the secondary modes (which are excited by the primary mode), together with the method of averaging, leads to a Hamiltonian system for the slowly varying amplitudes of the primary mode. The fixed points (which correspond to harmonic motions) and phase-plane trajectories and their perturbations due to linear damping are determined. It is shown that ε > δ, where δ is the damping ratio (actual/critical) of the primary mode, is a necessary condition for subharmonic response of that mode. Explicit results are given for the dominant axisymmetric and antisymmetric modes in a circular cylinder. Internal resonance, in which a pair of modes have frequencies that approximate ω and 2ω, is discussed separately, and the fixed points and their stability for the special case ω2= 2ω1are determined. Internal resonance for ω2= ω1is discussed in an appendix.
Mass transport in a turbulent boundary layer under a progressive water wave
Authors: Jacobs S. J.
Published: April 2006

The bottom boundary layer under a progressive water wave is studied using Saffman's turbulence model. Saffman's equations are analysed asymptotically for the case Re [Gt ] 1, where Re is a Reynolds number based on a characteristic magnitude of the orbital velocity and a characteristic orbital displacement. Approximate solutions for the mass-transport velocity at the edge of the boundary layer and for the bottom stress are obtained, and Taylor's formula for the rate of energy dissipation is verified. The theoretical results are found to agree well with observations for sufficiently large Reynolds numbers.
Uniqueness in linearized two-dimensional water-wave problems
Authors: Simon M. J. , Ursell F.
Published: April 2006

A geometrical condition sufficient for uniqueness in two-dimensional time-periodic linear water-wave problems is derived, and some examples are given. The technique can be seen as a generalization of work by John (1950), who established uniqueness for surface-piercing bodies that have the property that vertical lines down from the free surface do not intersect the body.

The paper is in two parts. Part 1 provides a review of current knowledge on the topic of uniqueness, and gives a simple form of John's proof. Part 2 describes the recent progress, in which the ideas of John's proof are extended so that uniqueness can be demonstrated for surface-piercing bodies that do not satisfy John's geometrical criterion. In fact the new technique proves uniqueness for a large class of problems involving floating bodies, submerged bodies and multiple-body systems. However, the present work still does not constitute a general proof of uniqueness for all configurations.
Motion of a particle generated by chemical gradients. Part 2. Electrolytes
Authors: Prieve D. C. , Anderson J. L. , Ebel J. P. , Lowell M. E.
Published: April 2006

When immersed in a non-uniform electrolyte solution, a rigid charged sphere migrates toward higher or lower concentration of the electrolyte depending on the relative ionic mobilities and the charge borne by the sphere. This motion has a twofold origin: first, a macroscopic electrolyte gradient produces an electric field which acts on the charged sphere (electrophoresis); secondly, the electrolyte gradient polarizes the cloud of counterions surrounding the charged sphere by making the cloud thinner on the high-concentration side (chemiphoresis). In this paper, we compute the terminal velocity of a non-conductive sphere through a slightly non-uniform solution of a symmetrically charged binary electrolyte. The analysis proceeds through an expansion in the small parameter λ (defined as the ratio of the counterion-cloud thickness to the particle radius). Results to O(λ) are presented. The only property of the sphere's surface that affects the velocity is its zeta potential ζ when the electrolyte gradient vanishes; no information concerning the dependence of ζ upon ionic strength is needed. While the chemiphoretic effect always directs the particle toward higher electrolyte concentration, the electrophoretic contribution can move the particle in either direction depending on the sign of βζ, where β is a normalized difference in mobilities between cation and anion of the elecytrolyte; thus particle movement could be directed toward either higher or lower electrolyte concentration depending on the physical properties of the system. With slight algebraic rearrangement, our results are also applicable to conventional electrophoresis (particle motion in an applied electric field) and show excellent agreement with the numerical calculations of O'Brien & White (1978).
Drag on a sphere moving horizontally through a stratified liquid
Authors: Lofquist Karl E. B. , Purtell L. Patrick
Published: April 2006

The drag on a sphere moving horizontally through stably stratified salt water is measured in laboratory experiments. The increment ΔCD in drag coefficient due to the stratification is obtained as a function of a stratification parameter κ and, in principle, the usual Reynolds number R. In these experiments, where R ranges from 150 to 5000, ΔCD is insensitive to R. But, as a function of κ, ΔCD has both positive and negative values attributable respectively to lee-wave drag and to suppression of turbulence in the wake. An observed delay in flow separation also apparently results from the lee-wave drag.
Parametrically excited solitary waves
Authors: Miles John W.
Published: April 2006

A modulated cross-wave of resonant frequencyω1, carrier frequencyω =ω1 {1 + O(ε)}, slowly varying complex amplitude O(ε½b), longitudinal scale b/ε½ and timescale 1/εω is induced in a long channel of breadth b that contains water of depth d and is subjected to a vertical oscillation of amplitude O(εb) and frequency 2ω, where 0 < ε [Lt ] 1. The complex amplitude satisfies a cubic Schrödinger equation, generalized to incorporate weak damping and the parametric excitation. A solution is obtained that describes the standing solitary wave observed by Wu, Keolian & Rudnick (1984). The results depend on both d/b and l*/b, where l* is the capillary length (l* = 2.7 mm for clean water), and solitary waves are impossible if d/b < 0.325 for l*/b = 0 or if l*/b > 0.045 for d/b [gsim ] 1. The corresponding cnoidal waves (of which the solitary wave is a limiting case) are considered in an appendix.
Natural convection in porous media
Authors: Prasad V. , Kulacki F. A. , Keyhani M.
Published: April 2006

Experimental results on free convection in a vertical annulus filled with a saturated porous medium are reported for height-to-gap ratios of 1.46, 1 and 0.545, and radius ratio of 5.338. In these experiments, the inner and outer walls are maintained at constant temperatures. The use of several fluid–solid combinations indicates a divergence in the Nusselt-number–Rayleigh-number relation, as also reported by previous investigators for horizontal layers and vertical cavities. The reason for this divergence is the use of the stagnant thermal conductivity of the fluid-filled solid matrix. A simple model is presented to obtain an effective thermal conductivity as a function of the convective state, and thereby eliminate the aforementioned divergence. A reasonable agreement between experimentally and theoretically determined Nusselt numbers is then achieved for the present and previous experimental results. It is thus concluded that a unique relationship exists between the Nusselt and Rayleigh numbers unless Darcy's law is inapplicable. The factors that influence the breakdown of Darcian behaviour are characterized and their effects on heat-transfer rates are explained. It is observed that, once the relation between the Nusselt and Rayleigh numbers branches out from that obtained via the mathematical formulation based on Darcy's law, its slope approaches that for a fluid-filled enclosure of the same geometry when the Rayleigh number is large enough. An iterative scheme is also presented for estimation of effective thermal conductivity of a saturated porous medium by using the existing results for overall heat transfer.
Fluid motion inside a spinning nutating cylinder
Authors: Vaughn Harold R. , Oberkampf William L. , Wolfe Walter P.
Published: April 2006

The incompressible three-dimensional Navier–Stokes equations are solved numerically for a fluid-filled cylindrical cannister that is spinning and nutating. The motion of the cannister is characteristic of that experienced by spin-stabilized artillery projectiles. Equations for the internal fluid motion are derived in a non-inertial aeroballistic coordinate system. Steady-state numerical solutions are obtained by an iterative finite-difference procedure. Flow fields and liquid induced moments have been calculated for viscosities in the range of 0.9 × 104−1 × 109 cSt. The nature of the three-dimensional fluid motion inside the cylinder is discussed, and the moments generated by the fluid are explained. The calculated moments generally agree with experimental measurements.
Wave generation by an oscillating surface-pressure and its application in wave-energy extraction
Authors: Sarmento A. J. N. A. , Falcão A. F. de O.
Published: April 2006

A two-dimensional analysis, based on linear surface-wave theory, is developed for an oscillating-water-column wave-energy device in water of arbitrary constant depth. The immersed part of the structure is assumed of shallow draught except for a submerged vertical reflecting wall. Both the cases of linear and nonlinear power take-off are considered. The results show that air compressibility can be important in practice, and its effects may in general be satisfactorily represented by linearization. The analysis indicates that using a turbine whose characteristic exhibits a phase difference between pressure and flow rate may be a method of strongly reducing the chamber length and turbine size with little change in the capability of energy extraction from regular waves. It was found in two examples of devices with strongly nonlinear power take-off that the maximum efficiency is only marginally inferior to what can be achieved in the linear case.
A model for Stokes flow in a two-dimensional bifurcating channel
Authors: Ranger K. B. , Brenner H.
Published: April 2006

A model is presented for steady two-dimensional Stokes flow in a bifurcating channel. The geometry of the bifurcation is symmetric, but the flow rates through the two branches may be unequal. It is found that separation takes place on the outer boundary of the channel for either a smooth or a sharp flow divider, but only when the flow rates through the two branches of the channel are sufficiently different.
Observation and analysis of the Mach reflection of weak uniform plane shock waves. Part 1. Observations
Authors: Dewey J. M. , McMillin D. J.
Published: April 2006

Shock fronts and fluid-particle trajectories throughout a two-dimensional shock wave flow have been measured by multiple schlieren photography in a detailed study of the Mach reflection from a 10° wedge of plane uniform shocks with Mach numbers of 1.105, 1.240 and 1.415. Correction of optical distortions throughout the field of view permitted the positions and shapes of the shock fronts and the magnitudes and directions of the particle velocities to be measured with a high degree of accuracy. No departure from self-similarity of the flow fields could be detected. The cross-sections of the reflected shocks were found to be circular and centred on a point which moved with the velocity of the flow behind the incident shock. The triple-point trajectories were linear. The velocity of the curved Mach stem shock was found to be constant at any one height above the wedge surface and to decrease monotonically with height. A deviation from perpendicularity was noticed where the Mach stems met the surface of the wedge, the shocks having a slight forward inclination of as much as 1°. The experimental results cannot be completely explained using the classical three-shock theory and an alternative model for weak Mach reflection is developed in Part 2 of this paper.
Nonlinear thermal convection with finite conducting boundaries
Authors: Riahi N.
Published: April 2006

Finite-amplitude thermal convection in a horizontal layer with finite conducting boundaries is investigated. The nonlinear steady problem is solved by a perturbation technique, and the preferred mode of convection is determined by a stability analysis. Square cells are found to be the preferred form of convection in a semi-infinite three-dimensional region Ω in the (γb,γt, P)-space (γb and γt are the ratios of the thermal conductivities of the lower and upper boundaries to that of the fluid and P is the Prandtl number). Two-dimensional rolls are found to be the preferred convection pattern outside Ω. The dependence on γb, γt and P of the heat transported by convection is computed for the various solutions analysed in the paper.
A Lagrangian solution for internal waves
Authors: Sanderson Brian
Published: April 2006

A perturbation procedure is used to obtain first- and second-order solutions for small-amplitude internal waves in a Lagrangian coordinate system. The first-order Lagrangian equations are formally accurate to the same order as the first-order Eulerian equations; however, they are different and the Lagrangian solution gives a more realistic wave shape. First-order Lagrangian solutions for internal waves in uniformly stratified fluid have a shape similar to that found in the second-order Eulerian solution. Wave profiles in uniformly stratified fluid exhibit broad crests and narrow troughs near the surface, a sinusoidal shape at mid-depth, and narrow crests and broad troughs near the bottom. The difference between the shape of crests and troughs grows as the wave amplitude is increased. Solutions obtained in a uniformly stratified fluid with a small bottom slope yield plausible shapes for breaking waves.
Lifting of a large object from a porous seabed
Authors: Mei Chiang C. , Yeung Ronald W. , Liu Ko-Fei
Published: April 2006

The time required to pull a large object from a sandy seabed is estimated by assuming that the seabed is porous but rigid. The phenomenon of breakout (i.e., sudden release) is shown to occur without the assumption of elasticity of the soil skeleton (Foda 1982). A new case of wedge-shaped gap is also studied, and compared to a uniform gap, Laboratory experiments are shown to support the theory.
Surface waves in basins of variable depth
Authors: Miles John
Published: April 2006

The linearized boundary-value problem for surface waves of frequency ω in a closed basin of variable depth is reduced to a non-self-adjoint partial differential equation in the plane of the free surface. The corresponding variational form (which does not provide a definite upper or lower bound) for the eigenvalue κ = ω2/gis constructed. A self-adjoint partial differential equation, for which the variational form is the Rayleigh quotient (which provides an upper bound to κ), also is constructed; it offers significant advantages vis-à-vis the non-self-adjoint formulation, but at the expense of a more complicated operator. Three relatively simple variational approximations are constructed, two for a class of basins with sloping sides and the third for basins for which the variation of the depth relative to its mean is small. These general results are illustrated by comparison with Rayleigh's (1899) results for a semicircular channel, Sen's (1927) inverse results for a family of circular basins, and Lamb's (1932) results for a shallow circular paraboloid. The eigenvalue for the dominant mode in the paraboloid is determined throughO(δ5), where δ = depth/radius.
Nonlinear Rayleigh–Bénard convection with square planform
Authors: Arter Wayne
Published: April 2006

Fully three-dimensional numerical solutions are presented for Rayleigh-Bénard convection subject to stress-free boundary conditions. A motion with square planform is studied for varying Rayleigh number R and Prandtl number σ. It may be understood partly in terms of a truncated modal representation (after Lorenz 1963). Thermal layers of unusual structure are found at high R. For small σ, steady solutions exist, but are not of ‘flywheel’ type, and the heat transport depends strongly on σ. The study also verifies that laminar convective flows may be ergodic.
Torque measurements on a stationary axially positioned sphere partially and fully submerged beneath the free surface of a slowly rotating viscous fluid
Authors: Kunesh J. G. , Brenner H. , O'Neill M. E. , Falade A.
Published: April 2006

Experimental measurements are presented for the hydrodynamic torque exerted on a stationary sphere situated at the axis of a slowly rotating viscous liquid at small rotary sphere Reynolds numbers (Re < 0.1) as a function of depth of submersion of the sphere below the free surface. Effects of free-surface proximity on the torque furnished the impetus for the study. Experiments were performed for different depths of sphere immersion beneath the free surface, varying from full to partial submersion. Rotation rates were maintained sufficiently low to approximate a planar interface. Torque measurements agreed well with existing theoretical predictions for both the interface-straddling and fully submerged sphere cases. In particular, the predicted continuity of the torque and its derivative at the interface-penetration point (where the sphere first starts to protrude through the free surface) was observed. Free-surface curvature as well as meniscus-curvature effects upon the torque were found to be negligible in the experiments, including even the extreme case where the sphere was in the almost fully withdrawn configuration.
Decay of momentum flux in submerged jets
Authors: Schneider Wilhelm
Published: April 2006

Slender laminar and turbulent, plane and axisymmetric jets emerging from orifices in plane or conical walls are studied at large distances from the orifices. The entrainment of momentum coupled with the entrainment of volume into a jet is determined, and its effect on the flow field is analysed by combining inner and outer expansions with a multiple scaling approach.

In turbulent (plane or axisymmetric) jets, the axial velocity decreases more rapidly than predicted by classical boundary-layer solutions, and the momentum flux vanishes as the distance from the orifice tends to infinity. The analysis unveils a source of discrepancies in previous experimental data on turbulent jets.

In a laminar plane jet, the momentum flux changes but little. In a laminar axisymmetric jet, the momentum flux changes slowly, yet considerably. When a critical distance from the orifice is approached, the momentum flux in the jet becomes very small, the jet diameter very large, and a toroidal viscous eddy is predicted. The structure of the flow field is briefly discussed.
Stokes wall effects for particles moving near cylindrical boundaries
Authors: Falade A. , Brenner H.
Published: April 2006

An asymptotic scheme is derived for calculating values of the ‘reflected’ Stokeslet-field velocity dyadic V and its gradient ${\boldmath \nabla V}$ back at the Stokeslet location for situations in which this singular point lies in close proximity to the wall of an infinitely long circular cylinder. The asymptotic formulas furnished by this scheme permit calculations of first- and second-order wall effects in the non-dimensional parameter κ = a/R0 (a ≡ characteristic particle radius, R0 ≡ cylinder radius) upon the Stokes resistance of a particle of arbitrary shape, location and orientation when translating and/or rotating near the wall of an otherwise quiescent fluid-filled or fluid-surrounded circular cylinder. This reflection-type calculation is applicable for circumstances in which the inequalities κ [Lt ] 1 and 1 − β [Lt ] 1 are each separately satisfied, while simultaneously K/(1 − β) [Lt ] 1. (Here β = b/R0 is the fractionally eccentric Stokeslet location, or equivalently the centre of reaction of the particle, with b its distance from the tube axis.) The main result of this paper is the development of the pair of asymptotic wall-correction formulas
\[
W_{jk}\sim {}_0C_{jk}(1-\beta)^{-1}+{}_1C_{jk}+{}_2C_{jk}(1-\beta)+O(1-\beta)^2
\]
and
\[
W_{jk,\,l}\sim {}_0D_{jkl}(1-\beta)^{-2}+{}_1D_{jkl}(1-\beta)^{-1}+{}_2D_{jkl}+O(1-\beta)
\]
to particle resistance, with W ≡ Wjk and ${\boldmath \nabla W}\equiv W_{kl,j}$ respectively the non-dimensional normalized wall-effect dyadic and its gradient at the Stokeslet location β. The numerical, rational fraction, β-independent, nCjk and nDjkl coefficients (n = 0, 1, 2) appearing above are evaluated by solving a recursive sequence of Stokes-flow boundary-value problems in the semi-infinite fluid domain bounded by a plane wall. These simple asymptotic formulas are shown to agree excellently in the range near β = 1 with existing values derived from the exact solution of the original circular-cylinder boundary-value problem, involving tedious infinite-series summations of complicated Bessel-function integrands extended over infinite integration domains. Generalizations of the scheme to particle motion in the space external to a circular cylinder is briefly sketched, as too is the case of cylinders of non-circular cross-section.
Modelling of porous media by renormalization of the Stokes equations
Authors: Kim Sangtae , Russel William B.
Published: April 2006

The permeability of a random array of fixed spheres has been calculated over the range of volume fractions from dilute to almost closest packing, by assuming pairwise-additive (low-Reynolds-number) hydrodynamic interactions within an effective medium. Non-convergent pair interactions arising from the long-range decay of the Stokeslet were removed by renormalizing the Stokes equation to determine the permeability of the effective medium, i.e. to include the mean screening effect of the other spheres. Pair interactions in this Brinkman medium were calculated by the method of reflections in the far field and boundary collocation in the near field.

The permeability predicted by the theory asymptotes correctly to established results for dilute arrays, and compares favourably (within 15%) with the Carman correlation for volume fractions between 0.3 and 0.5. The magnitude also falls within the range of exact results for periodic arrays at the higher concentrations, but our model does not reproduce the dependence on structure.

Use of the Brinkman equation with an effective viscosity leads to an apparent slip velocity at the boundary of a porous medium. Our calculation of the bulk stress via volume averaging determines the effective viscosity and hence the slip coefficient unambiguously for dilute porous media. However, at concentrations corresponding to the available experimental results the lengthscale characterizing pressure or velocity gradients becomes comparable to the interparticle spacing, and the averaging technique fails. Indeed the Brinkman equation itself is no longer valid.
Finger-interaction mechanisms in stratified Hele-Shaw flow
Authors: Tryggvason Grétar , Aref Hassan
Published: April 2006

Interactions between a few fingers in sharply stratified Hele-Shaw flow are investigated by numerical integration of the initial-value problem. It is shown that fingers evolving from an initial perturbation of an unstable interface consisting of a single wave are rather insensitive to variations of the control parameters governing the flow. Initial perturbations with at least two waves, on the other hand, lead to important finger-interaction and selection mechanisms at finite amplitude. On the basis of the results reported here many features of an earlier numerical study of the ‘statistical-fingering’ regime can be rationalized.
On the structure of jets in a crossflow
Authors: Andreopoulos J.
Published: April 2006

Spectral analysis and flow visualization are presented for various velocity ratios and Reynolds numbers of a jet issuing perpendicularly from a developing pipe flow into a crossflow. The results are complete with conditional averages of various turbulent quantities for one jet-to-cross-flow velocity ratio R of 0.5. A unique conditional-sampling technique separated the contributions from the turbulent jet flow, the irrotational jet flow, the turbulent crossflow and the irrotational crossflow by using two conditioning functions simultaneously. The intermittency factor profiles indicate that irrotational cross-flow intrudes into the pipe but does not contribute to the average turbulent quantities, while the jet-pipe irrotational flow contributes significantly to them in the region above the exit where the interaction between the boundary-layer eddies and those of the pipe starts to take place. Further downstream, the contributions of the oncoming boundary-layer eddies to the statistical averages reduce significantly. The downstream development depends mainly on the average relative eddy sizes of the interacting turbulent fields.
Nonlinear Kelvin–Helmholtz instability of a finite vortex layer
Authors: Pozrikidis C. , Higdon J. J. L.
Published: April 2006

The nonlinear growth of periodic disturbances on a finite vortex layer is examined. Under the assumption of constant vorticity, the evolution of the layer may be analysed by following the contour of the vortex region. A numerical procedure is introduced which leads to higher-order accuracy than previous methods with negligible increase in computational effort. The response of the vortex layer is studied as a function of layer thickness and the amplitude and form of the initial disturbance. For small initial disturbances, all unstable layers form a large rotating vortex core of nearly elliptical shape. The growth rate of the disturbances is strongly affected by the layer thickness; however, the final amplitude of the disturbance is relatively insensitive to the thickness and reaches a maximum value of approximately 20% of the wavelength. In the fully developed layers, the amplitude shows a small oscillation owing to the rotation of the vortex core. For finite-amplitude initial disturbances, the evolution of the layer is a function of the initial amplitude. For thin layers with thickness less than 3% of the wavelength, three different patterns were observed in the vortex-core region: a compact elliptic core, an elongated S-shaped core and a bifurcation into two orbiting cores. For thicker layers, stationary elliptic cores may develop if the thickness exceeds 15% of the wavelength. The spacing and eccentricity of these cores is in good agreement with previously discovered steady-state solutions. The growth rate of interfacial area (or length of the vortex contour) is calculated and is found to approach a constant value in well-developed vortex layers.
A separated-flow model for collapsible-tube oscillations
Authors: Cancelli Claudio , Pedley T. J.
Published: April 2006

A new model is presented to describe flow in segments of collapsible tube mounted between two rigid tubes and surrounded by a pressurized container. The new features of the model are the inclusion of (a) longitudinal wall tension and (b) energy loss in the separated flow downstream of the time-dependent constriction in a collapsing tube, in a manner which is consistent with the one-dimensional equations of motion. As well as accurately simulating steady-state collapse, the model predicts self-excited oscillations whose amplitude is large enough to be observable only if the flow in the collapsible tube becomes supercritical somewhere (fluid speed exceeding long-wave propagation speed). The dynamics of the oscillations is dominated by longitudinal movement of the point of flow separation, in response to the adverse pressure gradient associated with waves propagating backwards and forwards between the (moving) narrowest point of the constriction and the tube outlet.
Shallow rotating flow over an isolated obstacle
Authors: Lamb V. R. , Janowitz G. S.
Published: April 2006

The deflection of flow around an isolated obstacle in a rotating homogeneous fluid is investigated. Criteria for the onset of closed streamlines over an isolated obstacle are reviewed. In the flow regime where no closed streamlines exist, steady solutions for the stream function are obtained for both quasigeostrophic and finite-Rossby-number flows. A measure is proposed to allow quantitative evaluation of the flow patterns, and the dependence of deflection on obstacle volume and aspect ratio is examined. In the regime where closed streamlines can exist, the presence of a trapped vortex to the right (looking downstream) of the obstacle is investigated by means of time integration of the shallow-water equations. The significance of the trapped vortex for a real fluid is then tested through the addition of the frictional effect of Eckman pumping.