bluuebunny/crossref_metadata_2025_split huggingface dataset:
Gravity currents and related phenomena
Authors: Benjamin T. Brooke 
Published: March 2006

This paper presents a broad investigation into the properties of steady gravity currents, in so far as they can be represented by perfect-fluid theory and simple extensions of it (like the classical theory of hydraulic jumps) that give a rudimentary account of dissipation. As usually understood, a gravity current consists of a wedge of heavy fluid (e.g. salt water, cold air) intruding into an expanse of lighter fluid (fresh water, warm air); but it is pointed out in § 1 that, if the effects of viscosity and mixing of the fluids at the interface are ignored, the hydrodynamical problem is formally the same as that for an empty cavity advancing along the upper boundary of a liquid. Being simplest in detail, the latter problem is treated as a prototype for the class of physical problems under study: most of the analysis is related to it specifically, but the results thus obtained are immediately applicable to gravity currents by scaling the gravitational constant according to a simple rule.
In § 2 the possible states of steady flow in the present category between fixed horizontal boundaries are examined on the assumption that the interface becomes horizontal far downstream. A certain range of flows appears to be possible when energy is dissipated; but in the absence of dissipation only one flow is possible, in which the asymptotic level of the interface is midway between the plane boundaries. The corresponding flow in a tube of circular cross-section is found in § 3, and the theory is shown to be in excellent agreement with the results of recent experiments by Zukoski. A discussion of the effects of surface tension is included in § 3. The two-dimensional energy-conserving flow is investigated further in § 4, and finally a close approximation to the shape of the interface is obtained. In § 5 the discussion turns to the question whether flows characterized by periodic wavetrains are realizable, and it appears that none is possible without a large loss of energy occurring. In § 6 the case of infinite total depth is considered, relating to deeply submerged gravity currents. It is shown that the flow must always feature a breaking ‘head wave’, and various properties of the resulting wake are demonstrated. Reasonable agreement is established with experimental results obtained by Keulegan and others.
Magnetohydrodynamic flow along cylindrical pipes under non-uniform transverse magnetic fields
Authors: Todd L. 
Published: March 2006

The unidirectional flow of an incompressible, electrically conducting, viscous fluid along cylindrical pipes is considered. An external magnetic field, B0, which lies in the plane transverse to the flow is applied. It is shown that the governing equations, written in the co-ordinate system traced out by B0, are mathematically very similar to those for a uniform field.The paper deals mainly with ducts whose walls are insulators. Though exact solutions (valid for all values of the Hartmann number) are derived, the limit of high Hartmann number is taken for detailed discussion. Transition layers (or, loosely, ‘wakes’) can arise which are centred on curved field lines. In some cases, reversed flow occurs in part of the core (‘radial-type’ fields). Situations also arise where the magnitude (and sign) of the velocity remains the same as for B0 = 0, whatever the strength of the applied, transverse (azimuthal) magnetic field.
Penetrative convection
Authors: Musman Steven 
Published: March 2006

A penetrative convection problem which depends on the density maximum of water near 4 °C is solved numerically. When a layer of water has its lower boundary maintained at 0 °C and its upper boundary at some temperature above 4 °C, the layer will be divided into a lower convectively unstable region and an upper convectively stable region. Steady-state finite-amplitude solutions to this problem are obtained using the mean field approximation and free boundaries.Convective mixing alters the temperature structure of the layer so that the temperature of a large fraction of the layer is slightly below the temperature of maximum density, in agreement with laboratory measurements. The largest motions are found in a principal convective cell which extends from the lower boundary to a temperature of 7° or 8 °C. Above the principal cell one or more counter cells may form, depending on the temperature of the upper boundary. The velocities in the counter cells are substantially less than those in the principal cell and fall off rapidly going upward. When the temperature of the upper boundary is 7 °C or higher, convection first takes place at a finite amplitude and at a Rayleigh number less than that predicted by the linear theory. When the temperature of the upper boundary is 10 °C or higher, the upper boundary is no longer important dynamically to the system.As the Rayleigh number is increased above critical stability the velocities in the principal cell, heat transport and the distortion of the temperature field all increase. In addition, the principal cell becomes more slender and fills a greater fraction of the layer. Also, as the Rayleigh number increases the counter cells become more flattened, fill a smaller fraction of the layer, and the velocities in the counter cells decrease relative to those in the principal cell.The most important penetration of convective motions takes place in the form of nearly horizontal motions in the lowest part of the stable region, corresponding to the upper part of the principal cell. The velocities of these motions are a large fraction of the largest vertical velocities in the unstable region.
On the trapping of waves along a discontinuity of depth in a rotating ocean
Authors: Longuet-Higgins M. S. 
Published: March 2006

It is shown that, according to the linearized theory of long waves in a rotating, unbounded sea, if there is a discontinuity in depth along a straight line separating two regions each of uniform depth, then wave motions may exist which are propagated along the discontinuity and whose amplitude falls off exponentially to either side. Thus the discontinuity acts as a kind of wave-guide.The period of the waves is always greater than the inertial period. The wave period also exceeds the period of Kelvin waves in the deeper medium. As the ratio of the depth tends to infinity, the wave period tends to the inertial period or to the Kelvin wave period, whichever is the greater. On the other hand as the wavelength decreases (within the limits of shallow-water theory) so the waves tend to the non-divergent planetary waves found recently by Rhines.In an infinite ocean of uniform depth free waves with period greater than a pendulum-day cannot normally be propagated without attenuation (if the Coriolis parameter is constant). But non-uniformities of depth provide a means whereby such energy may be channelled over great distances with little attenuation.It is suggested that a gradually diminishing discontinuity will act as a chromatograph, each position along the discontinuity being marked by waves of a particular period.
Note on the numerical solution for unsteady viscous flow past a circular cylinder
Authors: Ingham D. B. 
Published: March 2006

The starting flow of a viscous fluid past a circular cylinder at Reynolds numbers 40 and 100 has been obtained by a numerical method. The method used is that developed by Payne (1957) but it has been extended here to cover a larger time interval.At Reynolds number 40 Payne's result for the drag coefficient at time t = 6 is in reasonable agreement with Kawaguti's (1953) result for the steady case but if Payne's computation, is extended to time t ≈ 24, the result is in better agreement with Apelt's (1961) result for the steady case. Also, a further investigation into the case R = 100 shows that Payne's mesh size is too crude. Similar observations can be made concerning the size of the standing vortices downstream of the circular cylinder and how they grow in time.
On high-frequency oscillatory viscous flows
Authors: Wang Chang-Yi 
Published: March 2006

The equations governing high-frequency oscillatory viscous flows are investigated through the separation of the steady and the unsteady parts. All Reynolds number ranges are studied and the orders of magnitude of the steady streaming produced by the Reynolds stresses are established.The oscillating circular cylinder at low Reynolds numbers is studied through the method of inner and outer expansions. Steady recirculating cells exist near the cylinder. The results compare very well with experiments. Analytic expressions for the streamfunction and the drag coefficient are obtained.The oscillating flow towards an infinite plate is investigated in detail. The steady streaming is caused by the steady component of the Reynolds stress. The pressure gradient always causes reverse flow near the solid boundary.
Use of the Wiener—Hermite expansion for nearly normal turbulence
Authors: Meecham William C. , Jeng Dah-Teng 
Published: March 2006

The turbulence problem is formulated using the Wiener stochastic expansion. The expansion is useful for processes which are in some sense nearly normal, and can be used for non-linear non-Gaussian processes such as many turbulent fluid flows. Here we present the general formulation for statistically inhomogeneous and anistropic processes.The transfer term in the energy equation, or equivalently the third-order velocity correlation, forms a sensitive measure of the amount of non-Gaussianity present in real fluid flows. Experimental evidence shows that in many flows this component is small compared with the Gaussian part. It is shown that a homogeneous and isotropic flow which has but a small non-Gaussian part possesses a distribution at one point which is Gaussian to terms of second order. The experiments suggest that immediately behind a grid in a wind tunnel the flow is very nearly normal. The non-Gaussian part grows at a moderate rate, at least within the range of distance downstream (or decay time) available in the usual experiments. This growth is probably due to the relative increase in the amount of energy in the smallest eddies, which are non-normal.A necessary criterion for the validity of the zero-fourth-cumulant approximation is suggested: the transfer term in dimensionless form should be small. It is shown that calculations using the zero-fourth-cumulant approximation have given negative energy spectra when this condition is violated, probably for the reason that the process is no longer nearly Gaussian. However, even when this condition is fulfilled, it is shown that that approximation must be corrected.It is suggested that the present theory is valid for quite large times of decay if initial energy spectra are chosen which are not too far from the actual physical values for fluid in turbulent flow. Equations are given for the next-higher-order term in a nearly normal approximation. The expansion is also used in § 6 to describe turbulent mixing problems and is compared with the zero-fourth-cumulant approximation for these problems. Computational results are presented in § 7 and compared with experiments by Stewart and Townsend.
On slow transverse flow past obstacles in a rapidly rotating fluid
Authors: Hide R. , Ibbetson A. , Lighthill M. J. 
Published: March 2006

This paper describes an experimental and theoretical study of the complicated disturbance (Taylor column) due to the slow relative motion between a spherical, or short cylindrical, rigid object and an incompressible fluid of low viscosity in which the object is immersed, when the motion of the object is that of steady revolution with angular speed Ω rad/see about an axis (the Z-axis) whose perpendicular distance from the centre of the object,$\overline{R}$, is much greater than a typical linear dimension of the object,L, and the undisturbed fluid motion is one of steady rotation about the same axis with angular speed (Ω+Uϑ/R) and zero relative vorticity (i.e.d(UϑR)/dR= 0). It extends earlier experimental work on Taylor columns to systems of sufficiently large axial dimensions for Z variations in the disturbance pattern to be perceptible. Over the ranges of Rossby and Ekman numbers (based onL) covered by the experiments, namely ε = 1·89 × 10−3to 2·36 ×10−1and γ = 1·30 × 10−3to 2·03 × 10−2respectively, the axis of the Taylor column is found to trail in the downstream direction at a small angle ϕ = tan−1(Kε) to the line parallel to the Z-axis through the centre of the object, whereK= (1·54 ± 0·04) for a sphere. The variation with Z of the amplitude of the disturbance is roughly linear and the scale-length of this variation, Zc, is close toL/γ¼over the limited range of γ covered by the experiments.The experimental value ofKis remarkably close to the theoretical value derived by Prof. Lighthill in the appendix, where he applies his general linear theory of waves generated in a dispersive system by travelling forcing effects to the problem of describing a Taylor column at large distances from the moving object when the fluid is inviscid and unbounded.
On turbulent boundary-layer separation
Authors: Sandborn V. A. , Liu C. Y. 
Published: March 2006

An experimental and analytical study of the separation of a turbulent boundary layer is reported. The turbulent boundary-layer separation model proposed by Sandborn & Kline (1961) is demonstrated to predict the experimental results. Two distinct turbulent separation regions, an intermittent and a steady separation, with correspondingly different velocity distributions are confirmed. The true zero wall shear stress turbulent separation point is determined by electronic means. The associated mean velocity profile is shown to belong to the same family of profiles as found for laminar separation. The velocity distribution at the point of reattachment of a turbulent boundary layer behind a step is also shown to belong to the laminar separation family.Prediction of the location of steady turbulent boundary-layer separation is made using the technique employed by Stratford (1959) for intermittent separation.
The motion of a spherical liquid drop at high Reynolds number
Authors: Harper J. F. , Moore D. W. 
Published: March 2006

The steady motion of a liquid drop in another liquid of comparable density and viscosity is studied theoretically. Both inside and outside the drop, the Reynolds number is taken to be large enough for boundary-layer theory to hold, but small enough for surface tension to keep the drop nearly spherical. Surface-active impurities are assumed absent. We investigate the boundary layers associated with the inviscid first approximation to the flow, which is shown to be Hill's spherical vortex inside, and potential flow outside. The boundary layers are shown to perturb the velocity field only slightly at high Reynolds numbers, and to obey linear equations which are used to find first and second approximations to the drag coefficient and the rate of internal circulation.Drag coefficients calculated from the theory agree quite well with experimental values for liquids which satisfy the conditions of the theory. There appear to be no experimental results available to test our prediction of the internal circulation.
The oscillations of a fluid droplet immersed in another fluid
Authors: Miller C. A. , Scriven L. E. 
Published: March 2006

From an analysis of small oscillations of a viscous fluid droplet immersed in another viscous fluid a general dispersion equation is derived by which frequency and rate of damping of oscillations can be calculated for arbitrary values of droplet size, physical properties of the fluids, and interfacial viscosity and elasticity coefficients. The equation is studied for two distinct extremes of interfacial characteristics: (i) a free interface between the two fluids in which only a constant, uniform interfacial tension acts; (ii) an ‘inextensible’ interface between the two fluids, that is, a highly condensed film or membrane which, to first order, cannot be locally expanded or contracted. Results obtained are compared with those previously published for various special cases.When the viscosities of both fluids are low, the primary contribution to the rate of damping of oscillations is generally the viscous dissipation in a boundary layer near the interface, in both the free and inextensible interface situations. For this reason inviscid velocity profiles, which do not account for the boundarylayer flow, do not lead to good approximations to the damping rate. The two exceptions in which the approximation based on inviscid profiles is adequate occur when the interface is free and either the interior or exterior fluid is a gas of negligible density and viscosity.
On standing internal gravity waves of finite amplitude
Authors: Thorpe S. A. 
Published: March 2006

Two-dimensional internal gravity waves in a rectangular container are examined theoretically and experimentally in (a) fluids which contain a single density discontinuity and (b) fluids in which the density gradient is everywhere continuous. The fractional density difference between the top and bottom of the fluid is small.Good agreement is found between the observed and calculated wave profiles in case (a). Unlike surface standing waves, which tend to sharpen at their crests as the wave amplitude increases, and which eventually break at the crests when fluid accelerations become equal to that of gravity, internal wave crests are found to be flat and exhibit no instability. In the case (a) breaking is found to occur at the nodes of the interfacial wave, where the current shear, generated by the wave itself, is greatest. For sufficiently large wave amplitudes, a disturbance with the form of a vortex but with direction of rotation reversing twice every cycle, grows at the wave node and causes mixing. This instability is found to be followed by the generation of cross-waves, of which two different forms are observed.Several modes of oscillation can be generated and are observed in a fluid with constant density gradient. The wave frequencies and shape are well predicted by theory. The experiments failed to establish any limitation of the possible wave amplitudes.
On the existence and structure of inertial boundary currents in a stratified ocean
Authors: Spiegel S. L. , Robinson A. R. 
Published: March 2006

An investigation is made of the properties of inertial boundary currents in a stably stratified, inviscid, non-diffusive ocean. The Boussinesq and β-plane approximations are adopted. The formalism developed by Robinson (1965) is used: the equations are transformed so that density replaces the vertical coordinate as an independent variable, and after a suitable non-dimensionalization of variables, the various fields are expanded as power series in the downstream co-ordinate η. The motion is shown to conserve potential vorticity. The equations and boundary conditions are obtained to order η2. Solutions are obtained in the region of formation of the coastal jet (i.e. the case of no mass flux through the plane η = 0) for several cases in which the potential vorticity function depends on stream function and density in a simple way. For these cases, it is found that in a constant depth ocean, a boundary current can exist only if the geostrophic drift at the boundary-layer edge is westward at all depths. This constraint is relaxed if the depth increases rapidly enough in the downstream (northward) direction. For slopes just in excess of the critical value, a deep onshore counter-current is predicted. Solutions of the first-order problem, using realistic values of the various parameters, have been computed and are found to be in qualitative agreement with observed features of the Florida Current.In an appendix, it is shown that the constraint of westward geostrophic drift at all levels must hold in a flat-bottomed ocean for arbitrary potential vorticity distributions consistent with stable stratification.
Convection in a box: on the dependence of preferred wave-number upon the Rayleigh number at finite amplitude
Authors: Davis Stephen H. 
Published: March 2006

Using Stuart's shape assumption and a condition of maximum heat transport it is found that the preferred number of finite roll cells present in Bénard convection in athree-dimensional rectangular boxtends todecreasewith increasing supercritical Rayleigh number in contradiction to the behaviour in an infinite layer but in accordance with experimental observation.This ‘end effect’ might explain the similar observation of wave-number decrease in the Taylor instability between rotating cylinders.
The distortion of turbulence by irrotational plane strain
Authors: Tucker H. J. , Reynolds A. J. 
Published: March 2006

The experiments reported here extend those of Townsend which form the basis of his model of free turbulence. Here straining is carried to a strain ratio of 6:1, while Townsend's straining went only to 4:1. Two kinds of distorting ducts are used to produce the uniform mean strain applied to initially nearly isotropic grid turbulence.The results differ from Townsend's in that: (i) a considerably higher degree of anisotropy is achieved, Townsend's measure of anisotropy attaining values up to 0·6, rather than the maximum of 0·42 he found; (ii) there is no evidence that an equilibrium structure is attained; and (iii) the strained turbulence rapidly becomes less anisotropic when the straining ceases.It is found to be possible to predict the variation of the total turbulence energy using rapid-distortion theory with a correction for decay. However, the individual components cannot be accurately predicted in this way.