bluuebunny/crossref_metadata_2025 huggingface dataset:
Internal waves of permanent form in fluids of great depth
Authors: Benjamin T. Brooke
Published: March 2006

This paper presents a general theoretical treatment of a new class of long stationary waves with finite amplitude. As the property in common amongst physical systems capable of manifesting these waves, the density of the (incompressible) fluid varies only within a layer whose thickness h is much smaller than the total depth, and it is h rather than the total depth that must be considered as the fundamental scale against which wave amplitude and length are to be measured. Internal-wave motions supported by the oceanic thermocline appear to be the most promising field of practical application for the theory, although applications to atmospheric studies are also a possibility.

The waves in question differ in important respects from long waves of more familiar kinds, and in § 1 their character is discussed on the basis of a comparison with solitary-wave and cnoidal-wave theories on customary lines, such as apply to internal waves in fluids of limited depth. A summary of some simple experiments is included at the end of § 1. Then, in § 2, the comparatively easy example of two-fluid systems is examined, again to illustrate principles and to prepare the way for the main analysis in § 3. This proceeds to a second stage of approximation in powers of wave amplitude, and its leading result is an equation (3·51) determining, for arbitrary specifications of the density distribution, the form of the streamlines in the layer of heterogeneous fluid. Periodic solutions of this equation are obtained, and their properties are discussed with regard to the interpretation of internal bores and wave-resistance phenomena. Solutions representing solitary waves are then obtained in the form
\[
f(x) = a\lambda^2/(x^2+\lambda^2),
\]
where xis the horizontal co-ordinate and where aΛ = O(h2). (The latter relation between wave amplitude and length scale contrasts with the customary one, aΛ2 = O(h3)). The main analysis is developed with particular reference to systems in which the heterogeneous layer lies on a rigid horizontal bottom below an infinite expanse of homogeneous fluid; but in § 4 ways are given to apply the results to various other systems, including ones in which the heterogeneous layer is uppermost and is bounded by a free surface. Finally, in §5, three specific examples are treated: the density variation with depth is taken, respectively, to have a discontinuous, an exponential and a ‘tanh’ profile.
Solitary internal waves in deep water
Authors: Davis Russ E. , Acrivos Andreas
Published: March 2006

A new type of solitary wave motion in incompressible fluids of non-uniform density has been investigated experimentally and theoretically. If a fluid is stratified in such a manner that there are two layers of different density joined by a thin region in which the density varies continuously, this type of wave propagates along the density gradient region without change of shape. In contrast to previously known solitary waves, these disturbances can exist even if the fluid depth is infinite. The waves are described by an approximate solution of the inviscid equations of motion. The analysis, which is based on the assumption that the wavelength of the disturbance is large compared with the thickness, L, of the region in which the density is not constant, indicates that the propagation velocity, U, is characterized by the dimensionless group (gL/U2) In (ρ1/ρ2), where g is the gravitational acceleration and ρ is the density. The value of this group, which is dependent on the wave amplitude and the form of the density gradient, is of the order one. Experimentally determined propagation velocities and wave shapes serve to verify the theoretical model.
A unified linear theory of homogeneous and stratified rotating fluids
Authors: Barcilon V. , Pedlosky J.
Published: March 2006

A unified picture of the linear dynamics of rotating fluids with given arbitrary stratification is presented. The range of stratification which lies outside the region of validity of both the theories of homogeneous fluids, $\sigma S < E^{\frac{2}{3}}$ and the strongly stratified fluids, σS > E½, is studied, where σS = vαgΔT/κΩ2L and E =v/ΩL2. The transition from one dynamics to the other is elucidated by a detailed study of the intermediate region E2/3 < σS < E½. It is shown that, within this intermediate stratification range, the dynamics differs from that of either extreme case, except in the neighbourhood of horizontal boundaries where Ekman layers are present. In particular the side wall boundary layer exhibits a triple structure and is made up of (i) a buoyancy sublayer of thickness (σS)−1/4E½ in which the viscous and buoyancy forces balance, (ii) an intermediate hydrostatic, baroclinic layer of thickness (σS)½ and (iii) an outer E¼-layer which is analogous to the one occurring in a homogeneous fluid. In the interior, the dynamics is mainly controlled by Ekman-layer suction, but displays hybrid features; in particular the dynamical fields can be decomposed into a ‘homogeneous component’ which satisfies the Taylor-Proudman theorem, and into a ‘stratified component’ which is baroclinic and which satisfies a thermal wind relation. In all regions the structure of the flow is displayed in detail.
The turbulence structure of equilibrium boundary layers
Authors: Bradshaw P.
Published: March 2006

Measurements in three boundary layers, one with constant free-stream velocity and two with power-law variations of free-stream velocity giving ‘moderate’ and ‘strong’ adverse pressure gradients, are presented and discussed. Several unifying features of the turbulent motion, expected to appear in all boundary layers not too far from equilibrium, are identified. The intensity spectra at higher wavenumbers follow the Kolmogorov inertial-subrange law, although the Reynolds number is not particularly high even by laboratory standards: in addition the smaller-scale motion in the outer layer is determined entirely by the local shear stress and the boundary-layer thickness. The large eddy motion increases in strength relative to the general turbulence level as the general turbulence level increases, and the limited evidence available suggests that the large eddies are similar to those in the free mixing layer. In all cases the large eddies contribute a significant proportion of the shear stress in the outer layer.
On the inviscid instability of certain two-dimensional vortex-type flows
Authors: Michalke Alfons , Timme Adalbert
Published: March 2006

As a contribution to the breakdown phenomenon of vortices in a two-dimensional free boundary layer, this paper deals with the question whether a single cylindrical (i.e. two-dimensional) vortex can become unstable. For this reason a single vortex, as it occurs in a free boundary layer, is approximated by an axisymmetrical vortex model. The inviscid stability theory of rotating fluids is then applied to this vortex model. By general stability criteria it was found that a vortex consisting of vorticity of one sign only is stable according to the Rayleigh criterion, but, if the vorticity has an extremum value outside the axis, may become unstable with respect to cylindrical disturbances. Furthermore, stability calculations for three special types of vortex were performed. It was found that they were more unstable with respect to cylindrical disturbances than to three-dimensional ones.
Experiments on the flow past an inclined disk
Authors: Calvert J. R.
Published: March 2006

The wake of a disk at an angle to a stream contains marked periodic motions which arise from the regular shedding of vortices from the trailing edge. The vortices are in the form of a chain of irregular rings, each one linked to the succeeding one, and they move downstream at about 0·6 of free-stream velocity. The prominence of the vortex shedding increases as the angle of incidence (measured from the normal) increases up to at least 50°. The shedding frequency increases with the angle of incidence, but by a suitable choice of reference velocity and length scale, may be described by a wake Strouhal number which has the constant value 0·21 for all angles of incidence above zero, up to at least 40°.

Axially-symmetric bodies at zero incidence shed vortices in a similar manner, except that the orientation of the plane of vortex shedding is not fixed and varies from time to time.
The perturbed region behind a diffracting shock wave
Authors: Skews B. W.
Published: March 2006

The results of an experimental study of the diffraction of shock waves on plane-walled convex corners are given for a Mach number range from 1·0 to 5·0. The behaviour of the disturbances produced in the region perturbed by the corner are discussed. It is shown that the position of the slipstream and tail of the Prandtl-Meyer fan, and the velocities of the contact surface and second shock become independent of corner angle for angles greater than 75°. Comparisons with theoretical predictions of Jones, Martin & Thornhill (1951) and Parks (1952) are included. In most cases fair agreement is obtained.
Boundary-layer instability experiment with localized disturbance
Authors: Vasudeva B. R.
Published: March 2006

Laminar boundary layer on a flat plate with zero pressure gradient was excited by a solitary three-dimensional disturbance. The growth of the disturbance was studied using a 10-channel linearized d.c.-coupled hot-wire system. The development of the disturbance flow field was studied and compared with theory by Criminale & Kovasznay (1962). The experimental results indicate a departure from the theory on account of the simplifying assumptions made in the theory. Certain new results were also obtained.
On the trapping of wave energy round islands
Authors: Longuet-Higgins M. S.
Published: March 2006

It is shown that islands can trap long-wave energy in a way similar to the capture of a particle by an atomic nucleus. The frequencies of the captured waves form a discrete set, being determined by the shape of the island and the contours of the surrounding sea bed. If the depth at great distances tends to a constant value, the trapped modes must leak some energy to infinity, though the consequent rate of decay may be exceedingly small. The initial energy of the trapped modes may be absorbed from incident radiation of the same frequency or from a sharp pulse. The particular example of a rectilinear pulse incident on a circular island is discussed in some detail.

The effect of the rotation of the Earth is to split the frequencies of a pair of waves progressing in opposite directions round the island. The splitting of the frequencies produces slow beats in the waves as seen at any fixed point. Slight asymmetry in the island induces a slow exchange of energy between each pair of progressive modes.

The present investigation was suggested by the occurrence of regular oscillations having a period of 6 min and a beat period of about 3 h in long-wave records taken at Macquarie Island, in the Southern Ocean.
Mass transfer from an axial source in a turbulent radial wall jet
Authors: Fletcher B.
Published: March 2006

This paper considers the mixing of a turbulent radial wall jet with a secondary fluid introduced into the impingement area of the wall jet so as to form a steady axisymmetric state. Similarity of the concentration profiles perpendicular to the wall has been assumed and, by solving the momentum and mass flow equations, the concentration distribution through the layer and a similarity exponent giving the variation of concentration along the wall have been determined.
On convection under an air surface
Authors: Koschmieder E. L.
Published: March 2006

A shallow layer of silicone oil on a plane, circular copper plate was uniformly heated from below. The air on its surface was kept at constant temperature by a uniformly cooled glass plate close to the oil, which at the same time inhibited air motions. Motions began with concentric circular rolls which, after the centre ring had formed, broke down into a hexagonal pattern. A rather accurate determination of the wavelength of the motions was possible. The wavelength was found to be variable with the depth of the fluid layer in qualitative accordance with the theory of Nield. Supercritical motions are briefly discussed.
A note on disturbances in slightly supercritical plane Poiseuille flow
Authors: Tam Christopher K. W.
Published: March 2006

The evolution of disturbances after a laminar, slightly supercritical flow between parallel planes is disturbed is considered as an initial-value problem. An asymptotic solution of the disturbances for large time possesses the same characteristic features as the turbulent spots observed by Emmons (1951).
Theoretical and experimental investigations of the reflexion of normal shock waves with vibrational relaxation
Authors: Johannesen N. H. , Bird G. A. , Zienkiewicz H. K.
Published: March 2006

The one-dimensional problem of shock-wave reflexion with relaxation is treated numerically by combining the shock-wave, characteristic, and Rayleigh-line equations. The theoretical results are compared with pressure and density measurements in CO2, and the agreement is found to be excellent.
The annihilation of a two-dimensional jet by a transverse magnetic field
Authors: Moffatt H. K. , Toomre J.
Published: March 2006

The effect of an applied transverse magnetic field on the development of a two-dimensional jet of incompressible fluid is examined. The jet is prescribed in terms of its mass flux ρQ0 and its lateral scale d at an initial section x = 0. The three dimensionless numbers characterizing the problem are a Reynolds number R = Q0/ν, a magnetic Reynolds number Rm = μσQ0, and a magnetic interaction parameter N = σB20d2/ρQ0, where ρ represents density, σ conductivity, μ permeability and B0 applied field strength, and it is assumed that
\[
R_m \ll 1,\quad R\gg 1,\quad N\ll 1.
\]
It is shown that when M2 = RN [Gt ] 1, an inviscid treatment is appropriate, and that the effect of the magnetic field is then to destroy the jet momentum within a distance of order N−1 in the downstream direction. A general solution for inviscid development is obtained, and it is shown that a large class of velocity profiles (though not all of them) are self-preserving.

When M2 [Lt ] 1, it is shown that the viscous similarity solution obtained by Moreau (1963a, b) is relevant. This solution is re-derived and re-interpreted; it implies that the jet momentum is destroyed within a distance of order $R^{\frac{1}{4}}N^{-\frac{3}{4}}$ in the downstream direction.

Some further aspects of the jet annihilation problem are qualitatively discussed in § 4, viz. the nature of the overall flow field, the effect of the presence of distant boundaries, the effect of increasing Rm to order unity and greater, and the effect of oblique injection. Finally the development of a jet of conducting fluid into a nonconducting environment is considered; in this case the jet is not stopped by the magnetic field unless a return path outside the fluid for the induced current is available.
The magnetohydrodynamic thrust bearing
Authors: Maki E. Roland , Kuzma Dennis C. , Donnelly Russell J.
Published: March 2006

The magnetohydrodynamic lubrication flow in a step-type thrust bearing has been investigated both theoretically and experimentally. The experimental configuration consisted of nine identical segments arranged on a circular plate. Flow was induced by rotating the upper plate. Pressures, voltages and torques were recorded. Provision was made for passing current radially through the segments, and it was shown that the torque could be reduced to zero and even reversed by this means. Good agreement has been obtained between the analysis and the experimental results.
The stability of rotating flows with a cylindrical free surface
Authors: Pedley T. J.
Published: March 2006

The stability to small inviscid disturbances of a rotating flow, whose velocity components in cylindrical polars (r, 0, z) are (0,V(r), 0), is investigated when one boundary of the flow (r=b) is a free surface under the action of surface tension (γ), and the other is either at infinity, or a rigid cylinder (r=a≠b), or at the axis (r= 0). The free surface may be the inner or the outer boundary. A necessary and sufficient condition for stability to axisymmetric disturbances is derived, which requires that Rayleigh's criterion of increasing circulation be satisfied, and otherwise depends only onb,V(b), γ and the density of the swirling liquid. This condition may be extended to include non-axisymmetric disturbances whenV∝ 1/rand whenV∝ralthough in the latter case it is no longer a necessary one. It is shown that, in the caseV∝r, as well asV∝ 1/r, the ‘most unstable’ disturbance on a rotating column of fluid will be non-axisymmetric if the rotation speed at the surface is sufficiently great. Several applications of the theory are suggested, and a possible experiment to test it is described.
Finite amplitude cellular convection induced by surface tension
Authors: Scanlon J. W. , Segel L. A.
Published: March 2006

A non-linear analysis of cellular convection driven by surface tension in a semi-infinite liquid layer heated from below has been made. The purpose is to determine whether or not one can predict the emergence of the hexagonal flow pattern from the interaction of a certain large class of important disturbances. The principal conclusion is that, compared with gravity driven convection, there is generally a much greater band of imposed temperature difference associated with hexagonal convective patterns. Partial results for the more realistic assumption of finite depth support this conclusion.
The flow field near the centre of a rolled-up vortex sheet
Authors: Mangler K. W. , Weber J.
Published: March 2006

Most of the existing methods for calculating the inviscid flow past a delta wing with leading-edge vortices are based on slender-body theory. When these vortices are represented by rolled-up vortex sheets in an otherwise irrotational flow, some of the assumptions of slender-body theory are violated near the centres of the spirals. The aim of the present report is to describe for the vortex core an alternative method in which only the assumption of a conical velocity field is made. An asymptotic solution valid near the centre of a rolled-up vortex sheet is derived for incompressible flow. Further asymptotic solutions are determined for two-dimensional flow fields with vortex sheets which vary with time in such a manner that the sheets remain similar in shape. A particular two-dimensional flow corresponds to the slender theory approximation for conical sheets.
Boundary conditions at a naturally permeable wall
Authors: Beavers Gordon S. , Joseph Daniel D.
Published: March 2006

Experiments giving the mass efflux of a Poiseuille flow over a naturally permeable block are reported. The efflux is greatly enhanced over the value it would have if the block were impermeable, indicating the presence of a boundary layer in the block. The velocity presumably changes across this layer from its (statistically average) Darcy value to some slip value immediately outside the permeable block. A simple theory based on replacing the effect of the boundary layer with a slip velocity proportional to the exterior velocity gradient is proposed and shown to be in reasonable agreement with experimental results.
Stability of the developing laminar flow in a parallel-plate channel
Authors: Chen T. S. , Sparrow E. M.
Published: March 2006

The hydrodynamic stability of the developing laminar flow in the entrance region of a parallel-plate channel is investigated using the theory of small disturbances. The stability of the fully developed flow is also re-examined. A wide range of analytical (i.e. asymptotic) and numerical methods are employed in the stability investigation. Among the asymptotic methods, each of three viscous solutions (singular, regular and composite) is used along with the inviscid solution to provide critical Reynolds numbers and complete neutral stability curves. Two numerical methods, finite differences and stepwise integration, are applied to calculate critical Reynolds numbers. The basic flow in the development region is treated from two stand-points: as a channel velocity profile and as a boundary-layer velocity profile. Extensive comparisons among the various methods and flow models disclose their various strengths and ranges of applicability. As a general result, it is found that the critical Reynolds number decreases monotonically with increasing distance from the channel entrance, approaching the fully developed value as a limit.
‘Inactive’ motion and pressure fluctuations in turbulent boundary layers
Authors: Bradshaw P.
Published: March 2006

Townsend's (1961) hypothesis that the turbulent motion in the inner region of a boundary layer consists of (i) an ‘active’ part which produces the shear stress τ and whose statistical properties are universal functions of τ and y, and (ii) an ‘inactive’ and effectively irrotational part determined by the turbulence in the outer layer, is supported in the present paper by measurements of frequency spectra in a strongly retarded boundary layer, in which the ‘inactive’ motion is particularly intense. The only noticeable effect of the inactive motion is an increased dissipation of kinetic energy into heat in the viscous sublayer, supplied by turbulent energy diffusion from the outer layer towards the surface. The required diffusion is of the right order of magnitude to explain the non-universal values of the triple products measured near the surface, which can therefore be reconciled with universality of the ‘active’ motion.

Dimensional analysis shows that the contribution of the ‘active’ inner layer motion to the one-dimensional wave-number spectrum of the surface pressure fluctuations varies as τ2w/k1 up to a wave-number inversely proportional to the thickness of the viscous sublayer. This result is strongly supported by the recent measurements of Hodgson (1967), made with a much smaller ratio of microphone diameter to boundary-layer thickness than has been achieved previously. The disagreement of the result with most other measurements is attributed to inadequate transducer resolution in the other experiments.
On the light-scatter technique for the study of turbulence and mixing
Authors: Becker H. A. , Hottel H. C. , Williams G. C.
Published: March 2006

The light-scatter technique facilitates study of the concentration field when one of the streams entering a mixing region is marked with colloidal particles. The present paper provides basic information for the effective application of the technique. The system response is analysed theoretically. The problems of noise and spatial resolution are examined and experimental data are presented from which the response parameters are evaluated numerically. The capabilities of the technique for characterizing turbulent concentration fluctuations are described. The technique is excellent in the convection region of the turbulence spectrum but fails where molecular diffusion is important.
The nozzle-fluid concentration field of the round, turbulent, free jet
Authors: Becker H. A. , Hottel H. C. , Williams G. C.
Published: March 2006

The light-scatter technique has been used to study the nozzle-fluid concentration field in an isothermal, turbulent, axisymmetric air/air free jet with the nozzle air marked by an oil smoke. The data on the mean concentration field appear to be the most accurate yet obtained, due to the peculiar advantages of the technique. The turbulent concentration fluctuations have been characterized as to intensity, spectral distribution, and two-point correlation. The intermittency factor has been measured and the properties of the turbulent fluid computed. Comparison with the results of other investigators who used heat to mark the nozzle fluid indicates a close similarity between the concentration and temperature fields.
The two-dimensional wave pattern produced by a disturbance moving in an arbitrary direction in a density stratified liquid
Authors: Rarity B. S. H.
Published: March 2006

The two-dimensional internal wave pattern produced by a small disturbance moving with constant velocity along a line of arbitrary inclination in an inviscid, density stratified liquid is studied. It is shown how the far field wave pattern evolves as the inclination changes from the extreme of horizontal motion to that of vertical motion. It is found that, in general, waves propagate ahead of the disturbance.
Heat transfer from a circular cylinder by acoustic streaming
Authors: Richardson Peter D.
Published: March 2006

An analysis is described for convection from a circular cylinder subjected to transverse oscillations relative to the fluid in which it is immersed. The analysis is based upon use of the acoustic streaming flow field. It is assumed that the frequency involved is sufficiently small that the acoustic wavelength in the fluid is much larger than the cylinder diameter, and that there is no externally imposed mean flow across or along the cylinder. Solutions are presented which are appropriate for a wide range of Prandtl number, and the cases of small and of large streaming Reynolds number are distinguished. The analysis compares favourably with experiments when the influence of natural convection is small.
On slow transverse motion of a sphere through a rotating fluid
Authors: Stewartson K.
Published: March 2006

The fluid is assumed to be inviscid and to be confined within two parallel planes, each perpendicular to the axis of rotation. A sphere is set moving, relative to the rotating fluid, in a straight line with uniform velocity and the temporal development of the flow structure examined. It is found that ultimately the flow has different properties inside and outside the cylinder [Cscr ], circumscribing the sphere and having its generators parallel to the axis of rotation. Inside [Cscr ] the fluid moves with the sphere as if solid; in early experiments of Taylor (1923) this phenomenon was observed. Outside [Cscr ] the motion is a two-dimensional potential flow past [Cscr ] as if it were solid. Then the asymmetry observed by Taylor and predicted in an earlier theory of the author for an unbounded fluid (1953) is not borne out. A partial explanation is offered.
On expansion waves generated by the refraction of a plane shock at a gas interface
Authors: Henderson L. F.
Published: March 2006

The paper deals with the regular refraction of a plane shock at a gas interface for the particular case where the reflected wave is an expansion fan. Numerical results are presented for the air–CH4 and air–CO2 gas combinations which are respectively examples of ‘slow–fast’ and ‘fast–slow’ refractions. It is found that a previously unreported condition exists in which the reflected wave solutions may be multi-valued. The hodograph mapping theory predicts a new type of regular–irregular transition for a refraction in this condition. The continuous expansion wave type of irregular refraction is also examined. The existence of this wave system is found to depend on the flow being self-similar. By contrast the expansion wave becomes centred when the flow becomes steady. Transitions within the ordered set of regular solutions are examined and it is shown that they may be either continuous or discontinuous. The continuous types appear to be associated with fixed boundaries and the discontinuous types with movable boundaries. Finally, a number of almost linear relations between the wave strengths are noted.
Direct measurement of aerodynamic pressure above a simple progressive gravity wave
Authors: Shemdin Omar H. , Hsu En Yun
Published: March 2006

Measurements of the aerodynamic pressure distribution at the interface between air and simple progressive water waves are obtained with the use of a pressure sensor that follows the water surface. The theory of Miles (1957, 1959) and Benjamin (1959) on shear flows past a wavy boundary predicts a phase shift between the pressure distribution along the boundary and the boundary itself. An experimental verification of this theory is sought especially. A wind–wave facility 115 ft. long, 6 ft. high and 3 ft. wide was used. The facility is equipped with an oscil-lating-plate wave-generator which is capable of generating sinusoidal or arbitrary wave-forms, and a suction fan which can produce wind velocities up to 80 ft./sec when the water is at a nominal depth of 3 ft. The pressure sensor used for the measurements of pressure, was mounted on an oscillating device such that the sensor could be maintained at a fixed small distance (within 1/4 in.) above a propagating wavy surface at all times. The perturbation pressure over progressive waves is extracted from recorded data sensed by the moving sensor. The results compare favourably with the theoretical predictions of Miles (1959).
An asymptotic solution of the tidal equations
Authors: Jacobs Stanley J.
Published: March 2006

The Cauchy problem for the β-plane form of the tidal equations is solved for both oscillatory and delta function initial data. The radius of deformation is assumed to be much less than the radius of the earth, and in accord with this assumption a ray approximation is employed.

It is shown that, owing to the rapid rate of propagation of inertio-gravity waves, the motion in its initial development tends towards geostrophic balance. However, the solution given by the ray approximation is singular on certain surfaces in space and time, the envelopes of the rays. A local boundary-layer theory is employed to correct this deficiency. The existence of these caustics implies that the process of geostrophic adjustment is more complicated than hitherto imagined.
Propagation of internal gravity waves in fluids with shear flow and rotation
Authors: Jones Walter L.
Published: March 2006

In a rotating system, the vertical transport of angular momentum by internal gravity waves is independent of height, except at critical levels where the Doppler-shifted wave frequency is equal to plus or minus the Coriolis frequency. If slow rotation is ignored in studying the propagation of internal gravity waves through shear flows, the resulting solutions are in error only at levels where the Doppler-shifted and Coriolis frequencies are comparable.
Convection in a box: linear theory
Authors: Davis Stephen H.
Published: March 2006

The linear stability of a quiescent, three-dimensional rectangular box of fluid heated from below is considered. It is found that finite rolls (cells with two non-zero velocity components dependent on all three spatial variables) with axes parallel to the shorter side are predicted. When the depth is the shortest dimension, the cross-sections of these finite rolls are near-square, but otherwise (in wafer-shaped boxes) narrower cells appear. The value of the critical Rayleigh number and preferred wave-number (number of finite rolls) for a given size box is determined for boxes with horizontal dimensionsh, ¼ ≤h/d≤ 6, wheredis the depth.
The velocity distribution function within a shock wave
Authors: Bird G. A.
Published: March 2006

The structure of normal shock waves in a gas composed of rigid sphere molecules is investigated by numerical experiments with a simulated gas on a digital computer. The non-equilibrium between the temperatures based on the longitudinal and lateral velocity components is studied and the results compared with the theory of Yen (1966). Details of the velocity distribution function are presented for a shock of Mach number 10. The distribution functions for both the longitudinal and lateral velocity components are plotted for a number of locations in the shock profile and are compared with the equilibrium distribution.
Nonlinear theory of open-channel steady flow past a solid surface of finite-wave-group shape
Authors: Howe M. S.
Published: March 2006

This paper deals with a detailed application of the Whitham theory of finite amplitude wave propagation to open-channel steady flow of water of infinite depth past a slowly modulated wavy wall. A numerical procedure first used by Garabedian & Lieberstein (1958) for the solution of ‘stably-posed’ elliptic Cauchy problems is employed to obtain a map of the wave pattern on the free surface of the water. The appearance of a ‘shock’ in this solution is discussed in terms of previous analytical and experimental results.
The pipeline flow of capsules. Part 9
Authors: Kruyer Jan , Redberger P. J. , Ellis H. S.
Published: March 2006

The velocities and the associated pressure gradients of infinitely long liquid-borne cylinders flowing freely in pipes are related analytically to their radial positions. These velocities and pressure gradients are compared with those of liquids in cylinder-free pipes and expressed as ratios. A digital computer was used to evaluate the resultant equations for values of the cylinder/pipe diameter ratio between 0·25 and 0·97, with radial positions varying from the fully eccentric to the fully concentric position. As the clearance between the pipe and the bottom of the cylinder increases, the pressure ratio (RP) decreases and the velocity ratio (RV) increases. The relationship between RP and RV is independent of liquid viscosity and density, capsule density and pipe diameter, and is shown to be nearly linear for the larger diameter ratios. The R, R relationships are compared with data from three experimental capsule pipelines with pipe diameters from ½ to 4 in., involving a variety of diameter ratios, cylinder lengths and densities, and oil viscosities. The experimental results for single capsules of finite length are shown to be in close agreement with the predictions for infinitely long cylinders.

The relevance of the analysis to capsule pipelining is indicated by relating experimental values of capsule velocity over a wide range of densities to the theoretical clearance of the capsules. A general relationship of this type would permit the optimization of the power requirements of any particular throughput of a given commodity.
Two-dimensional turbulent wakes
Authors: Gartshore Ian S.
Published: March 2006

The equations of mean motion indicate that two-dimensional turbulent wakes, when subjected to appropriately tailored adverse pressure gradients, can be self-preserving. An experimental examination of two nearly self-preserving wakes is reported here. Mean velocity, longitudinal and lateral turbulence intensity, inter-mittency and shear stress distributions have been measured and are compared with Townsend's data from the small-deficit undistorted wake. In comparison with the undistorted case, the present wakes have slightly lower turbulent intensities and significantly lower shear stresses, all quantities being non-dimensionalized by a local velocity scale taken as the maximum mean velocity deficit. A consideration of the reasons for the shear stress reduction leads to an expression from which the shear stresses in any symmetrical free equilibrium shear flow can be found. This relationship is used to calculate the rate of growth in the measured wakes, with reasonable success.
Finite amplitude convection cells
Authors: Robinson J. L.
Published: March 2006

In this paper we consider two-dimensional steady cellular motion in a fluid heated from below at large Rayleigh number and Prandtl number of order unity. This is a boundary-layer problem and has been considered by Weinbaum (1964) for the case of rigid boundaries and circular cross-section. Here we consider cells of rectangular cross-section with three sets of velocity boundary conditions: all boundaries free, rigid horizontal boundaries and free vertical boundaries (referred to here as periodic rigid boundary conditions), and all boundaries rigid; the vertical boundaries of the cells are insulated. It is shown that the geometry of the cell cross-section is important, such steady motion being not possible in the case of free boundaries and circular cross-section; also that the dependence of the variables of the problem on the Rayleigh number is determined by the balances in the vertical boundary layers.

We assume only those boundary layers necessary to satisfy the boundary conditions and obtain a Nusselt number dependence $N \sim R^{\frac{1}{3}}$ for free vertical boundaries. For the periodic rigid case, Pillow (1952) has assumed that the buoyancy torque is balanced by the shear stress on the horizontal boundaries; this is equivalent to assuming velocity boundary layers beside the vertical boundaries (rather than the vorticity boundary layers demanded by the boundary conditions) and leads to a Nusselt number dependence N ∼ R¼. If it is assumed that the flow will adjust itself to give the maximum heat flux possible the two models are found to be appropriate for different ranges of the Rayleigh number and there is good agreement with experiment.

An error in the application of Rayleigh's method in this paper is noted and the correct method for carrying the boundary-layer solutions round the corners is given. Estimates of the Nusselt numbers for the various boundary conditions are obtained, and these are compared with the computed results of Fromm (1965). The relevance of the present work to the theory of turbulent convection is discussed and it is suggested that neglect of the momentum convection term, as in the mean field equations, leads to a decrease in the heat flux at very high Rayleigh numbers. A physical argument is given to derive Gill's model for convection in a vertical slot from the Batchelor model, which is appropriate in the present work.
Particle migrations in suspension flows
Authors: Maude A. D. , Yearn J. A.
Published: March 2006

An apparatus is described whereby the distribution of particles in a neutrally buoyant suspension of spheres flowing through a tube may be measured. The discovery of a narrow region of high concentration near to the tube wall, and the behaviour of this ‘wall peak’ at various concentrations and positions along the length of the tube are described. An explanation for this behaviour is put forward in terms of the formation of the peak by interaction of spheres with the tube entrance, and of the formation within this peak of necklace-like strings of particles.
The stability of finite amplitude cellular convection and its relation to an extremum principle
Authors: Busse F. H.
Published: March 2006

The stability of cellular convection flow in a layer heated from below is discussed for Rayleigh number R close to the critical value Rc. It is shown that in this region the stable stationary solution is determined by a minimum of the integral
\[
\int_0^{H_0}R(H)\,dH,
\]
where R(H) is a functional of arbitrary convective velocity fields which satisfy the boundary conditions. For the stationary solutions R(H) is equal to the Rayleigh number. H0 is a given value of the convective heat transport. In a second part of the paper explicit results are derived for the convection problem with deviations from the Boussinesq approximation owing to the temperature dependence of the material properties.
Acoustic streaming in an elastico-viscous fluid
Authors: Frater K. R.
Published: March 2006

The acoustic streaming of an idealized elastico-viscous fluid near a cylindrical obstacle is considered using the boundary-layer type of approximation. It is shown that this streaming phenomena can be greatly influenced by the presence of elasticity in the fluid.
The reflexion of a shock wave at a rigid wall in the presence of a boundary layer
Authors: Henderson L. F.
Published: March 2006

The paper discusses the reflexion of a shock wave off a rigid wall in the presence of a boundary layer. The basic idea is to treat the problem not as a reflexion but as a refraction process. The structure of the wave system is deduced by a simple mapping procedure. It is found that a Mach stem is always present and that the bottom of this wave is bifurcated—called a lambda foot. The reflexion is said to be regular if the Mach stem and the lambda foot are confined to the boundary layer and irregular if either extends into the main stream. Two types of regular reflexion are found, one that has reflected compression waves and the other that has both reflected compression and expansion waves. Initial conditions are given that enable one to decide which type will appear. There are also two types of irregular reflexion, one that has a Mach stem present in the main stream and the other that is characterized by a four-wave confluence. Finally there are also two processes by which regular reflexions become irregular. One is due to the formation of a downstream shock wave that subsequently sweeps upstream to establish the irregular system and the other is due to boundary-layer separation which forces the lambda foot into the main stream.
The stability of oscillatory internal waves
Authors: Davis Russ E. , Acrivos Andreas
Published: March 2006

The stability of a periodic internal wave has been investigated experimentally and theoretically. From the analysis it is found that if a primary wave, with wave-number k0 and frequency ω0, is perturbed by two infinitesimal wave-like disturbances with wave-numbers k1 and k1 + k0 and frequencies ω1 and ω1 + ω0, exponential growth of these disturbances will take place under certain conditions. The analysis also indicates which resonantly interacting disturbances can induce an instability and, when viscous dissipation is accounted for, predicts the minimum amplitude for which a wave is unstable. Experimental results demonstrate that this type of instability can cause the breakdown of a first mode internal wave propagating in a fluid composed of two layers of uniform density separated by a thin region in which the density varies continuously.
A criterion for nonlinear wave stability
Authors: Hasselmann K.
Published: March 2006

Nonlinear resonant wave triads composed of one finite and two infinitesimal components are unstable for sum interactions and neutrally stable for difference interactions. A similar criterion holds for tertiary interactions.
The effect of viscosity and heat conduction on internal gravity waves at a critical level
Authors: Hazel Philip
Published: March 2006

The differential equation for the vertical velocity of a gravity wave in an inviscid shear flow is singular at a level where the mean fluid velocity is equal to the horizontal phase velocity of the waves. It has been shown that a wave travelling through such a layer has its amplitude attenuated by a constant factor dependent on the local Richardson number. In this paper the results obtained by solving numerically the full sixth order differential equation, which is derived by including viscosity and heat conduction in the problem, (and is not singular) are discussed, and the same attenuation factor is found. Some experiments which confirm certain aspects of the theory are described in an appendix.
On normal vorticity and current density generated behind a magnetogasdynamic shock wave in unsteady flows
Authors: Ram Rishi
Published: March 2006

Two interesting linear relations between normal vorticity and current density generated behind a shock wave in three-dimensional unsteady flows of conducting gases have been investigated in order to determine them purely from dynamical considerations. Some interesting physical conclusions have also been made.
Stability of a visco-elastic liquid film flowing down an inclined plane
Authors: Gupta A. S.
Published: March 2006

The investigation concerns the stability of an incompressible second-order fluid (visco-elastic) film flowing down an inclined plane under gravity with respect to two-dimensional disturbances. When the elastic parameter is negative as in the case of a solution of polyisobutylene in cetane, surface disturbances (‘soft’ waves) are found to be unstable. The analysis in this case also reveals the existence of growing shear waves (‘hard’ waves) which are highly damped in ordinary Newtonian fluids.
Computational and experimental study of a captive annular eddy
Authors: Macagno Enzo O. , Hung Tin-Kan
Published: March 2006

Results of calculations and experiments on the flow of a viscous liquid through an axisymmetric conduit expansion are reported. The streamlines and vorticity contours are presented as functions of the Reynolds number of the flow. The dynamic interaction between the main flow and the captive eddy between it and the walls is analysed, and it is concluded that, for laminar flow, the main role of the eddy is that of shaping the flow with a rather small energy exchange.
Some developments in the theory of vortex breakdown
Authors: Benjamin T. Brooke
Published: March 2006

The primary aim of the analysis presented herein is to consolidate the ideas of the ‘conjugate-flow’ theory, which proposes that vortex breakdown is fundamentally a transition from a uniform state of swirling flow to one featuring stationary waves of finite amplitude. The original flow is assumed to be supercritical (i.e. incapable of bearing infinitesimal stationary waves), and the mechanism of the transition is explained on the basis of physical principles that are well established in relation to the analogous supercritical-flow phenomenon of the hydraulic jump or bore. In previous presentations of the theory the existence of appropriately descriptive solutions to the full equations of motion has only been inferred from these general principles, but here the solutions are demonstrated explicitly by means of a perturbation analysis. This has basically much in common with the classical theory of solitary and cnoidal waves, which is known to explain well the essential properties of weak bores.

In § 2 the basic equations of the problem are set out and the leading results of the original theoretical treatment are recalled. The new developments are mainly presented in § 3, where an analysis of finite-amplitude waves is completed by two different methods, each serving to illustrate points of interest. The effects of small energy losses and of small flow-force reductions (i.e. wave-resistance effects) are considered, and the analysis leads to a general classification of possible phenomena accompanying such changes of integral properties in either slightly supercritical or slightly subcritical vortex flows. The application to vortex breakdown remains the focus of attention, however, and § 3 includes a careful appraisal of some experimental observations on the phenomenon. In § 4 a summary is given of a variant on the previous methods which is required when the radial boundary of the flow is taken to infinity. The main analysis is developed without restriction to particular flow models, but in § 5 the results are applied to a specific example.
On Benjamin's theory of conjugate vortex flows
Authors: Fraenkel L. E.
Published: March 2006

Benjamin (1962) introduced the idea that a given ‘primary’ swirling flow, with cylindrical stream surfaces, may have associated with it ‘conjugate flows’, also swirling and cylindrical, which in a certain sense are equivalent to the primary one. He deduced that, in cases where such conjugate flows exist and where the primary flow cannot support standing waves of small amplitude, the conjugate flow nearest the primary one (a) can support such waves, and (b) has a ‘flow force’ greater than that of the primary flow. In the present paper these two results are proved rigorously by a method which differs from Benjamin's.
The effect of a containing cylindrical boundary on the velocity of a large gas bubble in a liquid
Authors: Collins R.
Published: March 2006

The velocity of a large gas bubble rising along the axis of a cylindrical container filled with liquid, is derived to a first approximation from a simple flow model of the system. The resulting expressions involve ratios of certain infinite series, which are evaluated for a few cases, and the theory is then seen to agree with previously accepted results at its asymptotes. Experiments are performed which confirm the first approximation and which allow the theory to be recast as a semi-empirical theory relating bubble volume with velocity. In this form it agrees with the results of Uno & Kintner (1956), but indicates that the relation between volume and curvature which they employed was in error.