bluuebunny/crossref_metadata_2025 huggingface dataset:
Maximal two-layer exchange over a sill and through the combination of a sill and contraction with barotropic flow
Authors: Farmer D. M. , Armi L.
Published: April 2006

The analysis of two-layer exchange flow through contractions with a barotropic component treated by Armi & Farmer (1986) is extended to include exchange flows over sills and through a combination of a sill and contraction. It is shown that exchange over a sill is fundamentally different from exchange through a contraction. Control at the sill crest acts primarily through the deeper layer into which the sill projects and only indirectly controls the surface layer. This asymmetry in the control results in asymmetrical flows. The interface depth above the crest is not one half the total depth, as assumed in other studies by analogy with flow through contractions, but is somewhat deeper; the maximal exchange rate is less than for flow through a contraction of equal depth. When both a sill and a contraction are present, the contraction influences control at the sill crest only if it lies between the sill and the source of denser water. The response to barotropic flow is also asymmetrical: the transition to single-layer flow occurs at much lower speeds for a barotropic component in one direction than the other.

Results of the analysis are applied to exchange flow through the Strait of Gibraltar, which includes both a sill and a contraction. It is shown that maximal exchange conditions apply throughout part of the tidal cycle, and observations illustrate several of the analytical predictions for barotropic flows, including the formation of fronts, single-layer flow, submaximal exchange and reverse flow.
Weakly nonlinear acoustic and shock-wave theory of the noise of advanced high-speed turbopropellers
Authors: Tam Christopher K. W. , Salikuddin M.
Published: April 2006

An acoustic and shock-wave theory of the noise generated by advanced turbo-propellers operating at supersonic tip helical velocity and high-subsonic cruise Mach number is developed. The theory includes the thickness and loading noise of the highly swept propeller blades. When operating at their design conditions these propellers radiate extremely intense sound waves. Because of the weakly nonlinear propagation effects these high-intensity acoustic disturbances steepen up quickly to form shock waves. In the present theory advantage is taken of the fact that in the blade fixed-rotating-coordinate system the acoustic and shock-wave fields are time independent. The problem is formulated in this coordinate system as a boundary-value problem. Weakly nonlinear propagation effects are incorporated into the solution following Whitham's nonlinearization procedure (Whitham 1974). The change in the disturbance-propagation velocity due to fluid-particle motion as well as the change in the speed of sound resulting from compression and rarefaction are all taken into account. It is found that the equal-area rule of Whitham's shock-fitting method is also applicable to the present problem. This method permits easy construction of the three-dimensional shock surfaces associated with the acoustic disturbances of these high-speed turbopropellers. Numerical results of the present theory are compared with the measurements of the JETSTAr flight experiment and the United Technology Research Center low-cruise Mach number open-wind-tunnel data. Very favourable overall agreements are found. The comparisons indicate clearly that, when these supersonic turbopropellers are operated at their high subsonic design-cruise Mach number, weakly nonlinear propagation effects must be included in the theory if an accurate prediction of the waveform of the sound wave incident on the design aircraft fuselage is to be obtained. This is especially true for noise radiated in the upstream or forward directions. In the forward directions the effective propagation velocity of the acoustic disturbances is greatly reduced by the convection velocity of the ambient flow. This allows more time for the cumulative nonlinear propagation effects to exert their influence, leading to severe distortion of the waveform and the formation of shock waves.
Melt motion in a Czochralski crystal puller with an axial magnetic field: isothermal motion
Authors: Hjellming L. N. , Walker J. S.
Published: April 2006

A magnetic field suppresses turbulence and thermal convection in a Czochralski crystal puller. The amounts and distributions of dopants and oxygen in the crystal are determined by the motion of the molten silicon during crystal growth. This paper presents analytical solutions for the melt motion in a Czochralski puller with a strong, uniform, axial magnetic field. The relatively small electrical conductivity of the crystal plays a key role in determining the flow. Certain combinations of crystal and crucible rotation rates lead to flow patterns with a large volume of almost stagnant fluid under most of the crystal face. The values of these rotation rates depend on the magnetic field strength.
Oscillations over basins of variable depth
Authors: Ghanimati G. R. , Naghdi P. M.
Published: April 2006

This paper is concerned with oscillations of a body of water in basins of variable depth, employing a system of linearized equations which can be obtained from the theory of a directed fluid sheet for an incompressible, homogeneous, inviscid fluid (Green & Naghdi 1976a, 1977). For free oscillations over a level bottom, an assessment of the range of validity of the linearized theory is made by an appropriate comparison with a corresponding well-known exact solution (Lamb 1932). This assessment indicates an ‘intermediate’ range of validity for the linearized theory not covered by usual classical approximations for long waves. Encouraged by this assessment, we apply the linear theory of a directed fluid sheet to basins of variable depth; and, in particular, consider a class of basin profiles whose equilibrium depth (along its width) varies in one direction only. By a method of asymptotic integration, a general solution is obtained which is relatively simple and accounts for the effect of vertical inertia. The solution is sinusoidal in time, periodic along the breadth direction and involves Bessel functions of the first order in the width direction. For two special basin profiles, detailed comparisons are made between the predictions of the asymptotic solution (i.e. the frequencies in the lowest modes of oscillations) with corresponding results obtained by other procedures.
Energy computations for evolution of class I and II instabilities of Stokes waves
Authors: Stiassnie Michael , Shemer Lev
Published: April 2006

The modified Zakharov equation is used to study the coupled evolution of class I and class II instabilities of surface gravity waves on infinitely deep water. In contrast to single class (I or II) evolution, the coupled behaviour is non-periodic. Except for the very steep waves a dominance of class I modes over those of class II is observed. Energy calculations show that the Hamiltonian of the wave field considered is nearly constant. Thus the Zakharov and the modified Zakharov equations represent consistent approximations of the original water-wave problem.
Convection in a rotating cylindrical annulus. Part 2. Transitions to asymmetric and vacillating flow
Authors: Or A. C. , Busse F. H.
Published: April 2006

The instabilities of convection columns (also called thermal Rossby waves) in a cylindrical annulus rotating about its axis and heated from the outside are investigated as a function of the Prandtl number P and the Coriolis parameter η*. When this latter parameter is sufficiently large, it is found that the primary solution observed at the onset of convection becomes unstable when the Rayleigh number exceeds its critical value by a relatively small amount. Transitions occur to columnar convection which is non-symmetric with respect to the mid-plane of the small-gap annular layer. Further transitions introduce convection flows that vacillate in time or tend to split the row of columns into an inner and an outer row of separately propagating waves. Of special interest is the regime of non-symmetric convection, which exhibits decreasing Nusselt number with increasing Rayleigh number, and the indication of a period doubling sequence associated with vacillating convection.
Multiple solutions and flow limitation for steady flow through a collapsible tube held open at the ends
Authors: Reyn J. W.
Published: April 2006

Equations for the steady flow of an incompressible, inviscid fluid through a collapsible tube under longitudinal tension are derived by treating the tube longitudinally as a membrane, and taking the collapsibility of the tube into account in an approximate way by replacing in the equation for an axisymmetric membrane a term representing the resistance of the tube to area change by the tube law for collapsible tubes. The flow is assumed to be uniform in a cross-section. A nonlinear differential equation is obtained for the shape of the tube for given values of total pressure p0, flow rate q, longitudinal tension τ and tube law P = P(ρ); where ρ = (A/πR2)½ is the equivalent radius of the tube (A = area of a cross-section, R = radius of the unloaded, then circular tube). The equation can be integrated and analysed in the phase plane. Equilibrium points correspond to uniform flow through cylindrical tubes; saddle points correspond to subcritical flow (S < 1), centrepoints to supercritical (S > 1) and a higher-order point to critical flow (S = 1). Here S is the speed index, the ratio of the flow speed to the speed of long waves. Near centrepoints there are solutions, that represent area-periodic tubes. For a finite tube, held open at the ends, the steady flow is formulated as a two-point boundary-value problem. On the basis of numerical calculations, and a bifurcation analysis using the method of Lyapunov–Schmidt, the existence and multiplicity of the solutions of this problem are discussed and the process of flow limitation studied. For negative total pressures two collapsed solutions are found that disappear at the flow-limitation value of the flow rate. For positive total pressures a distinction is made between subcritical, critical and supercritical total pressures. In all these cases there is a multiplicity, proportional to the ratio of the tube length to [Lscr ]1(0), the wavelength of the collapsed periodic solution for vanishing flow rate, and having maximum radius ρ = 1. For subcritical total pressures increase of the flow rate leads to a gradual loss of all solutions in higher-order flow limitations until final flow limitation occurs by the mergence of two collapsed solutions. For supercritical total pressures increase of the flow rate also leads to a gradual loss of all solutions in higher-order flow limitations in a process which now also depends upon the ratio of the tube length to the wavelength L of periodic solutions with vanishing amplitude and ρ ≡ 1.
The influence of a peripheral layer of different viscosity on peristaltic pumping with Newtonian fluids
Authors: Brasseur James G. , Corrsin Stanley , Lu Nan Q.
Published: April 2006

The analysis by Shapiro et al. (1969) of a two-dimensional peristaltic pump at small Reynolds number and with long wavelengths is extended to include a Newtonian peripheral layer adjacent to the wall to simulate the effect of a coating in physiological flows. An earlier analysis by Shukla et al. (1980) violates mass conservation because of an incorrect deduction of the interface shape. We present a detailed analysis of the effect of the peripheral layer on the fluid motions, the pumping characteristics, and the phenomena of reflux and trapping. For prescribed wall motion, a peripheral layer more viscous than the inner fluid improves pumping performance, while a less-viscous outer layer degrades performance. Even a very thin peripheral layer may substantially reduce pumping if the viscosity in this layer is very low relative to the inner region. The effects of the peripheral layer on reflux and trapping depend on the conditions which are held fixed while making the comparison. However, the general trend with decreasing peripheral-layer viscosity is towards an overall decrease in trapping, a decrease in reflux with fixed total volume flow rate, but an increase in reflux with fixed pressure head.
Laminar separation in buoyant channel flows
Authors: Modi Vijay , Moore F. K.
Published: April 2006

A slow moving flow in a duct emerging into a quiescent negatively buoyant environment may separate from its inner wall prior to the lip. Buoyancy accelerates the flow, curving the streamlines within the duct away from the walls. The resulting deceleration at the wall may be sufficient to provoke separation. The problem of the location of this separation point in a two-dimensional channel is studied. A potential-flow model is examined first to explore the large-Reynolds-number behaviour. The form of the potential-flow description in the vicinity of the assumed location of separation is characterized by the presence of a square-root singularity in the pressure gradient at the wall. This permits use of the ideas of viscous-inviscid interaction, proposed by Sychev (1972), to determine the separation location as a function of Froude and Reynolds numbers. Results obtained in the high-Reynoldsnumber limit show that the channel flow separates at shorter distances from the entrance as Froude number is reduced.
On the role of subharmonic perturbations in the far wake
Authors: Meiburg E.
Published: April 2006

The possibility of an excitation of individual subharmonic perturbations in each of the shear layers forming the far wake is investigated numerically. Principal considerations allow for the existence of two equivalent subharmonic modes which by opposite routes can lead to a doubling of the wavelength in the wake. Since vortical disturbances in the far wake are amplified only convectively, the simultaneous existence of both modes in the flow field is possible, which could provide an explanation for the group structure observed experimentally in the far wake. These considerations also provide a logical explanation of the finding of a very regular vortex pairing process in forced wakes.

Two-dimensional numerical simulations assuming incompressible flow and almost inviscid dynamics illustrate the opposite developments of regions dominated by the two different modes and also confirm the possibility of a resulting group structure. As an important result it is demonstrated that, if vortex pairing plays an important role in the growth of the far-wake structure, this does not have to be related to the excitation of the subharmonic peak in the frequency spectrum. Quite the contrary, it is to be expected that the subharmonic itself is of minor importance and that instead a small frequency and its multiples related to the group structure of the flow dominate the spectrum. In the light of these considerations measurements by Cimbala (1984) are discussed and frequency spectra recorded by him are analysed more closely. Various properties of these spectra seem to indicate that vortex pairing might be significant with respect to the evolution of the far-wake structure.
Measurement of shear-induced self-diffusion in concentrated suspensions of spheres
Authors: Leighton David , Acrivos Andreas
Published: April 2006

A novel technique is presented for determining the coefficient of shear-induced particle self-diffusion in concentrated suspensions of solid spheres, which relies on the fact that this coefficient can be computed from the measured variations in the time taken by a single marked particle in the suspension to complete successive circuits in a Couette device. Since this method does not involve the direct measurement of the lateral position of the marked particle, it requires a much simpler experiment than that used by Eckstein, Bailey & Shapiro (1977) which is shown to be constrained by wall effects at high particle concentration. The diffusion coefficient thus determined was found to be proportional to the product γa2, where γ is the shear rate and a the particle radius, and to have the asymptotic form 0.5γa2ϕ2 in the dilute limit when the particle concentration ϕ → 0.
The motion of a cylinder of fluid released from rest in a cross-flow
Authors: Rottman James W. , Simpson John E. , Stansby Peter K.
Published: April 2006

The two-dimensional motion of a cylinder of fluid released from rest into a flow that is uniform far upstream of the cylinder is studied. We consider cases where the cylinder is initially of circular cross-section and the fluid is either inviscid or viscous. For the inviscid fluid, we use analytical methods to determine the motion for small and large times after release and three numerical methods, the vortex-sheet method, the vortex-blob method and the vortex-in-cell method, to determine the intermediate-time motion. For the viscous-fluid problem we use the vortex-in-cell method with random walks to compute both the initial flow around the cylinder and the motion of the released fluid at a Reynolds number of 484. In the inviscid case, the released fluid deforms into a structure that resembles a vortex pair that propagates down stream at a speed less than the onset flow speed. In the viscous case, after a wake representative of a Kármán vortex street has developed, the released fluid usually deforms into an elongated horseshoe shape that travels downstream at a speed greater or less than the incident flow speed (depending on when in the vortex-shedding cycle the cylinder is released). The results of the numerical calculations are compared with some simple experiments in a water channel.
An experimental study of the turbulence structure in smooth- and rough-wall boundary layers
Authors: Perry A. E. , Lim K. L. , Henbest S. M.
Published: April 2006

The turbulence structure in zero-pressure-gradient boundary layers above smooth, rough and wavy surfaces was investigated. The mean flow, turbulence intensity and spectral data for both smooth and rough surfaces show support for the attached eddy hypothesis of Townsend (1976), the model for wall turbulence proposed by Perry & Chong (1982) and the extended version developed by Perry, Henbest & Chong (1986). Anomalies in hot-wire behaviour when measuring in the turbulent wall region of the flow were discovered and some of these have been resolved.
A boundary-layer analysis of Rayleigh-Bénard convection at large Rayleigh number
Authors: Jimenez Javier , Zufiria Juan A.
Published: April 2006

A boundary-layer analysis is presented for the two-dimensional nonlinear convection of an infinite-Prandtl-number fluid in a rectangular enclosure, in the limit of large Rayleigh numbers. Particular emphasis is given to the analysis of the periodic boundary layers, and on the removal of the singularities that appear near the corners of the cell. It is argued that this later step is necessary to ensure the correctness of the boundary-layer assumptions. Numerical values are obtained for the heat transfer and stress characteristics of the flow.
Coalescing of geostrophic vortices
Authors: Griffiths R. W. , Hopfinger E. J.
Published: April 2006

Close interactions between pairs of two-dimensional vortices of like sign were investigated in experiments with barotropic vortices and baroclinic vortices. The vortices were generated by sources or sinks in a rotating fluid which, respectively, was homogeneous or contained a two-layer density stratification. For two identical anticyclonic, unstratified vortices there was a critical separation distance beyond which the vortices coalesced to form a single larger anticyclone. The critical distance d*, scaled by the radius R of a core having non-zero relative vorticity, was d*/R = 3.3 ± 0.2. This value is in agreement with results of previous numerical simulations for finite-area vortices in non-rotating flows. The effects on vortex structure of Ekman pumping due to the presence of a rigid boundary caused cyclonic vortices to coalesee from larger distances. Baroclinic vortices in a two-layer stratification were also found to coalesce despite a potential-energy barrier. However, the critical separation distance depended on the internal Rossby radius. When the Rossby radius was large compared with the core radius, vortices coalesced from distances much greater than the critical distance for barotropic vortices. Coalescing of two vortices of equal size and strength led to two symmetric entwined spirals of water, while close interaction of unequal vortices caused the weaker vortex to be wrapped around the outer edge of the stronger. Implications of these results are discussed for ocean eddies and intense atmospheric cyclones.
A two-equation model for contaminant dispersion in natural streams
Authors: Smith Ronald
Published: April 2006

A simple two-equation model is derived which has the properties that the total contaminant exposure, the mean time of arrival, the temporal spread, and the skewness, are asymptotically correct at large distances downstream of a discharge. The role of changes in the breadth of a river upon the dispersion process is investigated by a means of an illustrative example. This reveals cubic dependence upon the breadth, and hence the great importance of wide reaches of rivers as regards contaminant dispersion.
The wave structure of turbulent spots in plane Poiseuille flow
Authors: Henningson Dan S. , Alfredsson P. Henrik
Published: April 2006

The wave packets located at the wingtips of turbulent spots in plane Poiseuille flow have been investigated by hot-film anemometry. The streamwise velocity disturbances associated with the waves were found to be antisymmetric with respect to the channel centreline. The amplitude of the waves had a maximum close to the wall that was about 4% of the centreline velocity. The modified velocity field outside the spot was measured and linear stability analysis of the measured velocity profiles showed that the flow field was less stable than the undisturbed flow. The phase velocity and amplitude distribution of the waves were in reasonable agreement with the theory, which together with the symmetry properties indicate that the wave packet consisted of the locally least stable Tollmien-Schlichting mode.
Rolls versus squares in thermal convection of fluids with temperature-dependent viscosity
Authors: Jenkins D. R.
Published: April 2006

We consider finite-amplitude thermal convection, in a horizontal fluid layer. The viscosity of the fluid is dependent upon its temperature. Using a weakly nonlinear expansion procedure, we examine the stability of two-dimensional roll and three-dimensional square planforms, in order to determine which should be preferred in convection experiments. The analysis shows that the roll planform is preferred for low values of the ratio of the viscosities at the top and bottom boundaries, but the square planform is preferred for larger values of the ratio. At still larger values, subcritical convection is predicted. We also include the effects of boundaries having finite thermal conductivity, which enables favourable comparison to be made with experimental studies. A discrepancy between the present work and a previous study of this problem (Busse & Frick 1985) is discussed.
The flow in industrial cyclones
Authors: Bloor M. I. G. , Ingham D. B.
Published: April 2006

A simple mathematical model for the flow in a conical cyclone is developed which allows solutions to be obtained in closed form. The flow in the main body of the cyclone is regarded as inviscid but the nature of the fluid entry to the device and the conical geometry ensure that secondary flows develop which make the flow highly rotational. The results of the theory are compared with data from two quite different experimental investigations, and good agreement is obtained.
Shock wave structure in gas mixtures with large mass disparity
Authors: Fernández-Feria R. , Mora J. Fernández De La
Published: April 2006

The structure of normal shock waves is considered when the ratio of molecular masses mp/m of a binary mixture of inert monatomic gases is large and the density ratio ρp/ρ is of order unity or below. Generalized hydrodynamic equations, valid for arbitrary intermolecular potentials, are obtained from a hypersonic closure of the kinetic equation for the heavy gas and a near-equilibrium closure for the light component. Because the Prandtl number of the light gas and the Schmidt number of the mixture are nearly constant, the only independent transport coefficient arising in the model is the viscosity μ of the light gas, which is absorbed into a new independent position variable s. Knowledge of μ as a function of temperature thus determines the shock structure independently from the details of the intermolecular potential, allowing comparison with experiments in the complete absence of free parameters. In terms of the ratio M (frozen Mach number) between the speed of propagation and the sound speed of the light gas in the unperturbed medium, one finds that: (i) When M > 1, the behaviour is similar to that of a ‘dusty gas’, with a broad relaxation layer (outer solution) following a sharp boundary layer through which the speed of the heavy gas is almost constant (a shock within a shock). (ii) When (1 + ρp/ρ)s−½ < M < 1, the boundary layer disappears, yielding a so-called ‘fully dispersed wave’. (iii) Because the internal energy of the heavy gas is negligible, the present problem differs from previous shock studies in that, for the first time, the structure of the relaxation region is obtained algebraically in phase space, thus permitting an exhaustive study of the behaviour. From it, the overshooting solution found by Sherman (1960) is related to the unphysical degenerate branch of the outer solution arising when M > 1, showing a failure of the Chapman–Enskog theory, even for weak shocks, when the heavy gas is dilute. Also, an algebraic explanation arises for the ‘double hump structure’ observed in He–Xe shocks. (iv) When M is nearly unity, the initial boundary layer spreads out, and the structure must be obtained by integration of a numerically unstable system of three differential equations. However, the reduction of order brought about by the weak variation of the light-gas entropy at the head of the shock, results in a stable system of equations that we integrate numerically. Excellent phase-space agreement with recent shock-tube experiments of Tarczynski, Herczynski & Walenta (1986) is found for both weak and strong shocks.
The damping of capillary–gravity waves at a rigid boundary
Authors: Hocking L. M.
Published: April 2006

The frequency and damping rate of surface capillary-gravity waves in a bounded region depend on the conditions imposed where the free surface makes contact with the boundary. Extreme cases are when the free surface meets the boundary orthogonally, as in the case of pure gravity waves, and when the contact line remains fixed throughout the motion. An edge condition that models to some extent the dynamics associated with moving contact lines, but not contact-angle hysteresis, is given by making the slope of the free surface at contact proportional to its velocity. This model, which includes the two extreme cases, is used to obtain the frequency and damping rate of a standing wave between two parallel vertical walls. The effect of viscosity in the boundary layers on the walls is included and it is shown that the dissipation associated with the surface forces can exceed that produced by viscosity. The results are compared with those obtained from a number of experimental investigations, in which damping rates too large to be attributed to viscous action have been measured.
Waves produced by a vertically oscillating plate
Authors: Hocking L. M.
Published: April 2006

The vertical oscillation of a plate partially immersed in a non-wetting fluid produces a radiated wavetrain when the contact line between the plate and the free surface of the fluid cannot move freely along the plate. Realistic conditions to apply at the contact line when capillarity is not negligible include the dynamic variation of the contact angle and contact-angle hysteresis. Both of these effects are included in this paper and the amplitude of the radiated waves and the energy dissipation at the contact line are calculated.
Turbulent energy dissipation in a wake
Authors: Browne L. W. B. , Antonia R. A. , Shah D. A.
Published: April 2006

The average turbulent energy dissipation is often estimated by assuming isotropy and measuring the temporal derivative of the longitudinal velocity fluctuation. In this paper, the nine major terms that make up the total dissipation have been measured in the self-preserving region of a cylinder wake for a small turbulence Reynolds number. The results indicate that local isotropy is not satisfied; the isotropic dissipation, computed by assuming isotropic relations, being smaller than the total dissipation by about 45% on the wake centreline and by about 80% near the wake edge. Indirect verification of the dissipation measurements is provided by the budget of the turbulent kinetic energy. This budget leads to a plausible distribution for the pressure diffusion term, obtained by difference.
A model of interfacial stress and spray generation by gas flowing over a deep, wavy pool
Authors: Sinai Y. L.
Published: April 2006

A wave Reynolds number is the controlling parameter in Wang & Street's (1978) correlation for spray generation in developing air flow over a pool. A novel feature of the present theory is the prediction of the time-mean interfacial stress which, together with a wave-height correlation, is proposed and tested as a tool for quantifying that Reynolds number. The validation exercise shows that the results are generally acceptable, although the technique tends to underestimate the interfacial stress and droplet flux at high gas speeds. The analysis of experimental data has revealed some differences in the correlations which should be resolved.
Vortex induction and mass entrainment in a small-aspect-ratio elliptic jet
Authors: Ho Chih-Ming , Gutmark Ephraim
Published: April 2006

A passive technique of increasing entrainment was found by using a small-aspect-ratio elliptic jet. The entrainment ratio of an elliptic jet was several times greater than that of a circular jet or a plane jet. The self-induction of the asymmetric coherent structure caused azimuthal distortions which were responsible for engulfing large amounts of surrounding fluid into the jet. In an elliptic jet, an interesting feature in the initial stability process is that the thickness of the shear layer varies around the nozzle. The data indicated that instability frequency was scaled with the thinnest initial momentum thickness which was associated with the maximum vorticity. Turbulence properties were also examined and were found to be significantly different in the major- and minor-axis planes.
The structure of a turbulent shear layer bounding a separation region
Authors: Castro I. P. , Haque A.
Published: April 2006

Detailed measurements within the separated shear layer behind a flat plate normal to an airflow are reported. A long, central splitter plate in the wake prevented vortex shedding and led to an extensive region of separated flow with mean reattachment some ten plate heights downstream. The Reynolds number based on plate height was in excess of 2 × 1044.

Extensive use of pulsed-wire anemometry allowed measurements of all the Reynolds stresses throughout the flow, along with some velocity autocorrelations and integral timescale data. The latter help to substantiate the results of other workers obtained in separated flows of related geometry, particularly in the identification of a very low-frequency motion with a timescale much longer than that associated with the large eddies in the shear layer. Wall-skin-friction measurements are consistent with the few similar data previously published and indicate that the thin boundary layer developing beneath the separated region has some ‘laminar-like’ features.

The Reynolds-stress measurements demonstrate that the turbulence structure of the separated shear layer differs from that of a plane mixing layer between two streams in a number of ways. In particular, the normal stresses all rise monotonically as reattachment is approached, are always considerably higher than the plane layer values and develop in quite different ways. Flow similarity is not a useful concept. A major conclusion is that any effects of stabilizing streamline curvature are weak compared with the effects of the re-entrainment at the low-velocity edge of the shear layer of turbulent fluid returned around reattachment. It is argued that the general features of the flow are likely to be similar to those that occur in a wide range of complex turbulent flows dominated by a shear layer bounding a large-scale recirculating region.
Capillary instability of an annular liquid jet
Authors: Meyer J. , Weihs D.
Published: April 2006

An analytical investigation of the stability of a viscous, annular liquid jet moving in an inviscid medium is presented. This problem is a generalization of the well-known cases of a round cylindrical jet (obtained here when the ratio of internal to external radii tends to zero) and the flat thin liquid sheet (when the ratio above tends to unity). A critical ‘penetration’ thicknessTis defined. When the annulus thickness is greater thanT, the annular jet behaves like a full liquid jet; the only unstable perturbations are axisymmetric, and their growth rate is independent of thickness. When the annulus thickness is less thanT, the jet behaves like a two-dimensional liquid sheet; the most unstable perturbations are antisymmetric and their growth rate increases as the jet thickness decreases. Therefore, an annular liquid jet with a sufficiently small ring thickness will disintegrate into spherical shells much faster than a full liquid jet disintegrates into drops, in accordance with existing experimental data. Non-dimensional expressions for the penetration thickness are given for both viscous and inviscid jets.
The salt-finger zone
Authors: Howard L. N. , Veronis G.
Published: April 2006

In order to investigate the stability of infinitely long fully developed salt fingers Stern (1975) has proposed a model in which the basic configuration is independent of the vertical and is sinusoidal in the horizontal direction, with constant background gradients of temperature and salinity. The present study deals with a model of finite vertical extent where τ, the ratio of the diffusivities of salt and heat, is small, and where the constant background salt gradient is replaced by a salt difference between the reservoirs above and below a salt-finger region of finite depth. Steady-state solutions in two and three dimensions are obtained for the zero-order (τ = 0) state in which rising (sinking) fingers have the salinity of the lower (upper) reservoir. For two-dimensional fingers the horizontal scale corresponding to maximum buoyancy flux turns out to be 1.7 times the buoyancy-layer scale associated with the background stable temperature gradient. Heat, salt and buoyancy fluxes are calculated. A boundary-layer analysis is given for the (salt) diffusive correction to the zero-order solution. The same set of calculations is carried out for salt fingers in a Hele-Shaw cell. An assessment of Schmitt's (1979a) model of a finger zone of finite depth shows that the parametric restrictions required by the model cannot be satisfied when Stern's idealization is used for the final state. The present model appears to be preferable for constructing a Schmitt-like theory for τ [Lt ] 1.
Stability of Bödewadt flow
Authors: Savaş ÖMer
Published: April 2006

Boundary-layer transition over a stationary disk in rotating flow is studied experimentally. Circular waves are observed in the boundary layer occurring on an end disk of a cylindrical cavity during impulsive spin-down to rest. The transient flow evolves into a quasi-steady regime that exhibits the properties of the Bödewadt flow. The circular waves develop in that flow. The critical Reynolds number Re = r(Ω/v)½ is determined from frequency and wavelength measurements to be about 25. The corresponding dimensionless wavenumber 2πr/ΛRe is about 0.6 and the frequency 2πf/ ΩiRe about 0.2.
Near-surface turbulence in a grid-stirred tank
Authors: Brumley Blair H. , Jirka Gerhard H.
Published: April 2006

In order to elucidate the turbulent structure below a shear-free gas-liquid interface, turbulence measurements were made in a 50 cm square by 40 cm deep tank stirred by a vertically oscillating grid well below the surface, using a split-film anemometer probe rotating in a horizontal circle. This instrument is able to measure both vertical and horizontal velocity fluctuations to within 0.4 mm of the surface, from which spatial spectra and profiles of r.m.s. velocity fluctuations and integral lengthscales can be calculated. The turbulent structure is affected by the presence of the surface within a ‘surface-influenced layer’ roughly one integral scale, or ten per cent of the distance from the surface to the centre of the grid stroke, in thickness. The shapes of the spectra and profiles within the surface-influenced layer are predicted to a good first approximation by the source theory of Hunt & Graham (1978), which treats the turbulent structure as the superposition of homogeneous turbulence with an irrotational velocity field driven by a source distribution at the surface which cancels the vertical velocity fluctuations there. The magnitudes (as opposed to the shapes) of the profiles scale according to the values that would otherwise occur in the vicinity of the surface-influenced layer were the surface not present. These magnitudes are adequately predicted by the bulk relations determined by Hopfinger & Toly (1976) and Thompson & Turner (1975), with no apparent dependence on turbulent Reynolds number. There are some minor discrepancies between the measured profiles and those of Hunt & Graham. A thin layer of reduced velocity fluctuations below what would be expected from the theory was observed near the surface. Also, anisotropy in the velocity spectra at depths within the surface-influenced layer extended well into the inertial subrange, whereas the Hunt & Graham theory predicts no anisotropy at high wavenumbers.
Mathematical aspects of trapping modes in the theory of surface waves
Authors: Ursell F.
Published: April 2006

A horizontal canal of infinite length and of constant width and depth contains inviscid fluid under gravity. The fluid is bounded internally by a submerged horizontal cylinder which extends right across the canal and has its generators normal to the sidewalls. Suppose that the fluid is set in motion by a surface pressure varying across the canal, then some of the energy is radiated to infinity while some of the energy is trapped in characteristic modes (bound states) near the cylinder. The existence of trapping modes in special cases was shown by Stokes (1846) and Ursell (1951); a general treatment, given by Jones (1953), is based on the theory of elliptic partial differential equations in unbounded domains. In the present paper a much simpler treatment is given which uses only the theory of bounded symmetric linear operators together with Kelvin's minimum-energy theorem of classical hydrodynamics.
Subharmonic resonance of nonlinear cross-waves
Authors: Lichter Seth , Chen Jerry
Published: April 2006

The evolution equation governing wavemaker-generated cross-waves near a cutoff frequency in an infinitely deep, infinitely long channel is shown to be the nonlinear Schrödinger equation with a homogeneous boundary condition at the wavemaker. With the inclusion of an empirically determined damping coefficient, numerical results for growth rate, slow modulation period, and wave amplitude show good agreement with previous experiments. The results also describe observations of trapped and propagating solutions.
Computation of vortex sheet roll-up in the Trefftz plane
Authors: Krasny Robert
Published: April 2006

Two vortex-sheet evolution problems arising in aerodynamics are studied numerically. The approach is based on desingularizing the Cauchy principal value integral which defines the sheet's velocity. Numerical evidence is presented which indicates that the approach converges with respect to refinement in the mesh-size and the smoothing parameter. For elliptic loading, the computed roll-up is in good agreement with Kaden's asymptotic spiral at early times. Some aspects of the solution's instability to short-wavelength perturbations, for a small value of the smoothing parameter, are inferred by comparing calculations performed with different levels of computer round-off error. The tip vortices’ deformation, due to their mutual interaction, is shown in a long-time calculation. Computations for a simulated fuselage-flap configuration show a complicated process of roll-up, deformation and interaction involving the tip vortex and the inboard neighbouring vortices.
Secondary instability of a temporally growing mixing layer
Authors: Metcalfe Ralph W. , Orszag Steven A. , Brachet Marc E. , Menon Suresh , Riley James J.
Published: April 2006

The three-dimensional stability of two-dimensional vortical states of planar mixing layers is studied by direct numerical integration of the Navier-Stokes equations. Small-scale instabilities are shown to exist for spanwise scales at which classical linear modes are stable. These modes grow on convective timescales, extract their energy from the mean flow and exist at moderately low Reynolds numbers. Their growth rates are comparable with the most rapidly growing inviscid instability and with the growth rates of two-dimensional subharmonic (pairing) modes. At high amplitudes, they can evolve into pairs of counter-rotating, streamwise vortices, connecting the primary spanwise vortices, which are very similar to the structures observed in laboratory experiments. The three-dimensional modes do not appear to saturate in quasi-steady states as do the purely two-dimensional fundamental and subharmonic modes in the absence of pairing. The subsequent evolution of the flow depends on the relative amplitudes of the pairing modes. Persistent pairings can inhibit three-dimensional instability and, hence, keep the flow predominantly two-dimensional. Conversely, suppression of the pairing process can drive the three-dimensional modes to more chaotic, turbulent-like states. An analysis of high-resolution simulations of fully turbulent mixing layers confirms the existence of rib-like structures and that their coherence depends strongly on the presence of the two-dimensional pairing modes.
Swirling recirculating flow an a liquid-metal column generated by a rotating magnetic field
Authors: Davidson P. A. , Hunt J. C. R.
Published: April 2006

In this paper we consider theoretical and experimental aspects of axisymmetric, swirling flow which is generated in a column of liquid metal by a rotating magnetic field. Two cases are discussed, one in which there is no axial variation in the stirring force, and one where the body force is restricted to a relatively short length of the column. The latter case is of considerable practical interest in continuous casting.

One-dimensional stirring, where the swirl is independent of z and θ, is well understood. The magnetic body force is balanced by shear, all inertial forces being zero (except for the centripetal acceleration). However, in two-dimensional axisymmetric stirring, the axial variation in swirl drives a strong secondary poloidal flow. The principal local force balance is between the magnetic torque and inertia. The body force spins up the fluid as it passes through the forced region and the secondary flow sweeps this angular momentum into the unforced region. Consequently, the size and distribution of the swirl is controlled by the secondary flow.

The role of wall friction is considered and shown to control the length of the recirculating eddy. An approximate solution of the inviscid equations of motion, based on the angular momentum integral, is derived for the flow in the forced region. This is compared with the results of numerical experiments.

The analysis predicts that the swirl velocity scales on {B(σ/ρω)½}ωR, has a maximum at the bottom of the driven region, and penetrates an axial distance of the order ℝR away from the forced region. (For turbulent flow the Reynolds number ℝ must be based on an effective eddy viscosity.) All these features were reproduced experimentally.
Uniformly valid analytical solution to the problem of a decaying shock wave
Authors: Sharma V. D. , Ram Rishi , Sachdev P. L.
Published: April 2006

An explicit representation of an analytical solution to the problem of decay of a plane shock wave of arbitrary strength is proposed. The solution satisfies the basic equations exactly. The approximation lies in the (approximate) satisfaction of two of the Rankine-Hugoniot conditions. The error incurred is shown to be very small even for strong shocks. This solution analyses the interaction of a shock of arbitrary strength with a centred simple wave overtaking it, and describes a complete history of decay with a remarkable accuracy even for strong shocks. For a weak shock, the limiting law of motion obtained from the solution is shown to be in complete agreement with the Friedrichs theory. The propagation law of the non-uniform shock wave is determined, and the equations for shock and particle paths in the (x, t)-plane are obtained. The analytic solution presented here is uniformly valid for the entire flow field behind the decaying shock wave.
Note on the direct calculation of mobility functions for two equal-sized spheres in Stokes flow
Authors: Yoon Byung Jun , Kim Sangtae
Published: April 2006

A simple, direct method is presented for the calculation of mobility functions for the translational and rotational velocities and stresslets of two equal-sized spheres in unbounded low-Reynolds-number flow when the ambient velocity field is a superposition of a uniform stream, a vorticity and a rate-of-strain field. Our numerical procedure furnishes accurate values for touching spheres and coefficients for the near-field asymptotic expansions. The singular behaviour of the mobility functions is clarified. These results have been used to determine accurately the coefficient of theO(c2) term in the expression for the bulk stress in a suspension of spheres (6.95c2instead of 7.6c2).
Time-dependent response of floating ice to a steadily moving load
Authors: Schulkes R. M. S. M. , Sneyd A. D.
Published: April 2006

When a steadily moving load is applied to a floating ice plate, the disturbance will generally approach a steady state (relative to the load) as time t → ∞. However, for certain ‘critical’ load speeds the disturbance may grow continuously with time, which represents some danger to vehicles driving on ice. To understand this phenomenon and the overall time development of the ice response, this paper analyses the problem of an impulsively applied, concentrated line load on a perfectly elastic homogeneous floating ice plate. An exact expression for the ice deflection is derived, and then interpreted by means of asymptotic expansions for large t in the vicinity of the source. The spatial development of the disturbance is analysed by considering asymptotic expansions as t → ∞ near an observer moving away from the load. Theoretical results are compared with field measurements, and some hitherto unexplained features can be understood.
Note on Bragg scattering of water waves by parallel bars on the seabed
Authors: Mei Chiang C. , Hara Tetsu , Naciri Mamoun
Published: April 2006

A recent theory of Bragg scattering of surface waves by sinusoidal sandbars on a seabed is applied to three cases not examined heretofore: (1) oblique incidence on a strip of infinitely long bars, (2) oblique incidence on the corner of a bar field and (3) seabed with a mean slope. While the Bragg mechanism has been studied previously for sandbars present on many shorelines, it can be a basis for breakwaters where the soil is not strong enough to support a single massive breakwater.