bluuebunny/crossref_metadata_2025 huggingface dataset:
Laminar incompressible flow past a semi-infinite flat plate
Authors: Davis R. T.
Published: March 2006

Laminar incompressible flow past a semi-infinite flat plate is examined by using the method of series truncation (or local similarity) on the full Navier-Stokes equations. The first and second truncations are calculated at points on the plate away from the leading edge, while only the first truncation is calculated at the leading edge. The solutions are compared with the results from other approximate methods.
On the slow motion of a sphere parallel to a nearby plane wall
Authors: O'Neill M. E. , Stewartson K.
Published: March 2006

A new method using a matched asymptotic expansions technique is presented for obtaining the Stokes flow solution for a rigid sphere of radius a moving uniformly in a direction parallel to a fixed infinite plane wall when the minimum clearance ea between the sphere and the plane is very much less than a. An ‘inner’ solution is constructed valid for the region in the neighbourhood of the nearest points of the sphere and the plane where the velocity gradients and pressure are large; in this region the leading term of the asymptotic expansion of the solution satisfies the equations of lubrication theory. A matching ‘outer’ solution is constructed which is valid in the remainder of the fluid where velocity gradients are moderate but it is possible to assume that ε = 0. The forces and couples acting on the sphere and the plane are shown to be of the form (α0+α1ε) log ε + β0 + O(ε) where α0, α1 and β0 are constants which have been determined explicitly.
Oscillations of a vapour cavity in a rotating cylindrical tank
Authors: Pao Sui-Kwong P. , Siekmann J.
Published: March 2006

The oscillations of a curved interface are considered, neglecting the effects of gravity. The system under consideration consists of a right, circular, cylindrical tank of finite length, partially filled with an inviscid, incompressible, wetting liquid. When the container spins about its axis of revolution, the large-scale vapour cavity takes an elongated spheroid-like shape, symmetric about the axis of rotation. The fluid—vapour interface will oscillate about the equilibrium configuration if disturbing forces are present. The case where the vapour cavity touches the walls of the tank is not included in this investigation.

The equations of motion are linearized. However, the resulting eigenvalue problem is non-linear. Surface tension and rotation are taken into account only to the extent allowed by a linearized stability theory.

The self-sustained oscillations are governed by a partial differential equation of elliptic type, the field equation of the perturbation pressure. According to the results obtained from theory, all eigenfrequencies for this case are greater than twice the angular speed of the tank. The first two eigenfrequencies can be computed with high accuracy. The relation between the bubble shape and the eigen-frequency is shown in a graph for a specific example.

The governing differential equation is hyperbolic for forced oscillations induced by a small force field of constant magnitude and direction in an inertial frame of reference. A solution for this problem exists only in case of a cylindrical tank of infinite length. Discontinuities in the velocity components occur in the flow field. A numerical example has been carried out.
Gravity wave damping of hydrostatic oscillations for a buoyant disk
Authors: Warren F. W. G.
Published: March 2006

A disk (i.e. a body whose maximum thickness is small compared with its lateral dimensions) floats with its central plane of symmetry upright. Its hydrostatic oscillations are lightly damped by the reaction of the gravity waves generated. A damping coefficient is obtained. It is shown that superimposed upon these oscillations is a small displacement which decays with the timetliket−4ort−5.
Numerical solution of an integral equation for flow from a circular orifice
Authors: Hunt Bruce W.
Published: March 2006

This study was begun as an attempt to either confirm or disprove conflicting results of previous research upon a classical problem in potential theory. An integral equation resulting from a surface distribution of vorticity is used to solve numerically for the flow through a circular orifice. Free-surface profiles and contraction coefficients are determined for four different ratios of orifice area to pipe area, and a comparison is made between the numerical and experimental results of both this study and previous studies.
Secondary flow in a Hele-Shaw cell
Authors: Thompson B. W.
Published: March 2006

Riegels (1938) investigated the breakdown of Hele-Shaw flow in a Hele-Shaw cell with unusually large separation distance 2h* between the walls. A theoretical outer expansion for the velocity was constructed in the case where the obstacle is a circular cylinder, using an intuitive inner boundary condition that seems to be correct in the limit h* → 0, but without explicit matching with the inner expansion.

An inner expansion has now been found, and it shows that the solution in the inner layer forces terms into the outer expansion that are larger than those found by Riegels whenever h* is finite and not zero.
A uniformly valid solution for the hypersonic flow past blunted bodies
Authors: Schneider W.
Published: March 2006

The plane and axisymmetric hypersonic flow past blunted bodies is investigated as an inverse problem (shock shape given). The fluid may behave as a real gas in local thermodynamic equilibrium. Viscosity and heat conduction are neglected. An analytical solution uniformly valid in the whole flow field (from the stagnation region up to large distances from the body nose) is given. The solution is based on two main assumptions: (i) the density ratio ε across the shock is very small, (ii) the pressure at a pointPof the disturbed flow field isnotvery small compared with the pressure immediately behind the shock in the intersection point of the shock surface with its normal throughP.TermsO(ε) are neglected in comparison with 1, but it is not necessary for the shock layer to be thin. The change of velocity along streamlines is taken into account. In order to calculate the flow quantities one has to evaluate only two integrals (equations (49) and (53) together with the boundary values (5) and (10)). The application of the solution is illustrated and the accuracy is tested in some examples.
Vortex evolution in a round jet
Authors: Becker H. A. , Massaro T. A.
Published: March 2006

A study has been made of the varicose instability of an axisymmetrical jet with a velocity distribution radially uniform at the nozzle mouth except for a laminar boundary layer at the wall. The evolutionary phenomena of instability, such as the rolling up of the cylindrical vortex layer into ring vortices, the coalescence of ring vortex pairs, and the eventual disintegration into turbulent eddies, have been investigated as a function of the Reynolds number using smoke photography, stroboscopic observation, and the light-scatter technique.

Emphasis has been placed on the wavelength with maximum growth rate. The jet is highly sensitive to sound and the effects of several types of acoustic excitation, including pure tones, have been determined.
Effect of waves at a gas—liquid interface on a turbulent air flow
Authors: Cohen Leonard S. , Hanratty Thomas J.
Published: March 2006

Air flowing over a liquid surface encounters an increased resistance if waves are present. The relation of this increased resistance to the properties of the waves has been studied. Air and a liquid flowed co-currently in an enclosed channel which is 12 in. wide and 1 in. high and which is long enough so that flow in the air and the liquid and the interfacial structure are fully developed. The drag on interfaces with three-dimensional wave structures was found to increase with the square of the gas velocity and to depend more on the height of the waves than on other parameters characterizing the interface. The ratio of the equivalent sand roughness to the root-mean-square of the fluctuations in the height of the liquid film is approximately equal to 3 √2. The velocity profiles in the gas were found to be different from what has been reported for flows over sand roughened surfaces.
Two-dimensional flow under gravity in a jet of viscous liquid
Authors: Clarke N. S.
Published: March 2006

This paper is concerned with the steady, symmetric, two-dimensional flow of a viscous, incompressible fluid issuing from an orifice and falling freely under gravity. A Reynolds number is defined and considered to be small. Due to the apparent intractability of the problem in the neighbourhood of the orifice, interest is confined to the flow region below the orifice, where the jet is bounded by two free streamlines. It is assumed that the influence of the orifice conditions will decay exponentially, and so the asymptotic solutions sought have no dependence upon the nature of the flow at the orifice. In the region just downstream of the orifice, it is expected that the inertia effects will be of secondary importance. Accordingly the Stokes solution is sought and a perturbation scheme is developed from it to take account of the inertia effects. It was found possible only to express the Stokes solution and its perturbations in the form of co-ordinate expansions. This perturbation scheme is found to be singular far downstream due to the increasing importance of the inertia effects. Far downstream the jet is expected to be very thin and the velocity and stress variations across it to be small. These assumptions are used as a basis in deriving an asymptotic expansion for small Reynolds numbers, which is valid far downstream. This expansion also has the appearance of being valid very far downstream, even for Reynolds numbers which are not necessarily small. The method of matched asymptotic expansions is used to link the asymptotic solutions in the two regions. An extension of the method deriving the expansion far downstream, to cover the case of an axially-symmetric jet, is given in an appendix.
Experimental and theoretical study of the stability of plane shock waves reflected normally from perturbed flat walls
Authors: Briscoe M. G. , Kovitz A. A.
Published: March 2006

The rate of damping of perturbations on a shock wave reflected from a perturbed flat wall was measured in a shock tube. Incident shock wave Mach numbers of 1·45 and 1·09 in air together with sinusoidal and Gaussian wall perturbations were employed. These measurements were compared with a modified form of a linearized theory due to Zaidel (1960). The linearization was performed about the basic solution of a plane shock wave reflected normally from a flat wall.

The rate of decay and the frequency and phase of oscillations agreed very well with the theoretical predictions; the amplitudes of the oscillations were some-what larger than predicted. The reflected shock shape was initially in good agreement with theory, but higher frequency perturbations on the reflected shock front caused deviations from the predicted shape after the shock front had travelled about one wall-wavelength away from the wall.

The generally satisfactory agreement between theory and experiment supports the use of linearized analysis in predicting shock wave stability.
On a non-linear theory of thin jets. Part 1
Authors: Ackerberg Robert C.
Published: March 2006

The injection of a two-dimensional jet into a uniform stream is considered, the fluids being assumed inviscid and incompressible. When the total head of the jet is much larger than that of the uniform flow, the motion is characterized by two disparate length scales, and uniformly valid asymptotic solutions can be found by the method of matched expansions. Inner and outer expansions are developed for the jet and the external flow. The first-order outer solution in the jet is the usual thin jet approximation which fails in the neighbourhood of the jet exit except for 90° injection, when it is uniformly valid. The basic non-linearity introduced by the pressure condition along the vortex sheet separating the jet from the external flow appears as a non-linear boundary condition for the first-order outer solution in the external flow. A novel feature of the analysis is the necessity of imposing a logarithmic singularity as an ‘inner’ boundary condition for the outer solution in the external flow. The first-order fluid speed and streamline deflection angle are shown to be given correctly toO(1) uniformly in the external flow (for all injection angles) by the first-order outer solution.
An experimental study of reverse transition in two-dimensional channel flow
Authors: Narayanan M. A. Badri
Published: March 2006

An experimental investigation on reverse transition from turbulent to laminar flow in a two-dimensional channel was carried out. The reverse transition occurred when Reynolds number of an initially turbulent flow was reduced below a certain value by widening the duct in the lateral direction. The experiments were conducted at Reynolds numbers of 625, 865, 980 and 1250 based on half the height of the channel and the average of the mean velocity. At all these Reynolds numbers the initially turbulent mean velocity profiles tend to become parabolic. The longitudinal and vertical velocity fluctuations ($\overline{u^{\prime 2}}$and$\overline{v^{\prime 2}}$) averaged over the height of the channel decrease exponentially with distance downstream, but$\overline{u^{\prime}v^{\prime}} $tends to become zero at a reasonably well-defined point. During reverse transition$\overline{u^{\prime}}\overline{v^{\prime}}/\sqrt{\overline{u^{\prime 2}}}\sqrt{\overline{v^{\prime 2}}}$also decreases as the flow moves downstream and Lissajous figures taken withu’ andv’ signals confirm this trend. There is approximate similarly between$\overline{u^{\prime 2}} $profiles if the value of$\overline{u^{\prime 2}_{\max}} $and the distance from the wall at which it occurs are taken as the reference scales. The spectrum of$\overline{u^{\prime 2}} $is almost similar at all stations and the non-dimensional spectrum is exponential in wave-number. All the turbulent quantities, when plotted in appropriate co-ordinates, indicate that there is a definite critical Reynolds number of 1400±50 for reverse transition.
The rise of a body through a rotating fluid in a container of finite length
Authors: Moore D. W , Saffman P. G.
Published: March 2006

The drag on an axisymmetric body rising through a rotating fluid of small viscosity rotating about a vertical axis is calculated on the assumption that there is a Taylor column ahead of and behind the body, in which the geostrophic flow is determined by compatibility conditions on the Ekman boundary-layers on the body and the end surfaces. It is assumed that inertia effects may be neglected. Estimates are given of the conditions for which the theory should be valid.
The observed motion of a sphere through a short, rotating cylinder of fluid
Authors: Maxworthy T.
Published: March 2006

The drag on a sphere has been measured as it moves through a slightly viscous fluid contained in a rotating cylinder that is short compared to the length of ‘Taylor columns’ created by the sphere motion. These results suggest that only when inertia effects are very much smaller than both viscous and Coriolis forces are the results of Moore & Saffman (1968) approached. Flow field observations show qualitative agreement with many of the features described in their paper but differ sufficiently to warrant a further, fairly extensive discussion. These differences are characterized by a marked fore and aft asymmetry in the shear layer and boundary-layer flows for all values of the parameters covered by this study.
The structure of normal shock waves in a binary gas mixture
Authors: Bird G. A.
Published: March 2006

A previous study of normal shock waves through numerical experiments with a simulated gas on a digital computer is extended to binary gas mixtures. The mixture is simulated by two sets of rigid elastic sphere molecules with the appropriate mass and diameter ratios. Velocity profile results for medium strength waves in a mixture of equal parts argon and helium are in qualitative agreement with the continuum calculations of Sherman (1960), but there is no initial acceleration of the argon in mixtures containing a very small initial mole fraction of this gas. The temperature profiles are similar to those for the velocity in that the argon profile lags behind the helium profile. However, when there is a small proportion of heavy gas, the profiles cross-over and the temperature of the heavy gas overshoots the Rankine-Hugoniot downstream value. For very strong shock waves, the overall shock thickness expressed in upstream mean free paths becomes larger, but the profiles are generally similar to those for the medium strength waves.
Report on the International Conference on stratified fluids
Authors: Graebel W. P. , Long R. R. , Wu T. Y. T.
Published: March 2006

A conference on ‘Stratified Fluids’ was held in Ann Arbor, Michigan, 11–14 April 1967, under the sponsorship of the National Science Foundation. The meeting was organized by Professor C.-S. Yih (University of Michigan) with the assistance of the other members of the scientific committee: T. B. Benjamin, W. R. Debler, L. N. Howard, R. R. Long, J. W. Miles, W. H. Munk and O. M. Phillips. Thirty papers were delivered on subjects involving the application of stratified fluid problems to geophysical phenomena, waves in stratified fluids, experimental and observational investigations, and stability. We give here a brief account of the proceedings. The papers delivered will not be published in a formal volume; references to where they can be found are given at the end of this article.
On the solution of the Boltzmann equation for an unsteady cylindrically symmetric expansion of a monatomic gas into a vacuum
Authors: Freeman N. C. , Grundy R. E.
Published: March 2006

The problem of an unsteady axisymmetric expansion of a monatomic gas into a vacuum is considered in the limit of small source Knudsen number. It is shown that a solution of the Boltzmann equation for Maxwell molecules valid for large time can be constructed, which matches with the known equilibrium solution for an inviscid expansion of a fixed mass of gas into a vacuum provided that the region near the zero density front is excluded. This solution is formally the same as that obtained for the similar problem of steady spherical expansion into a vacuum—the variations along each particle path of the unsteady flow being the same as that in the steady flow.

Near the front, the expansion procedure breaks down and the equations require a different scaling. A modified form of the Boltzmann equation is obtained which leads to a corresponding set of moment equations. Unfortunately, the set of moment equations is no longer closed and no essential simplification has been made.
The prediction of turbulent boundary layer development in compressible flow
Authors: Green J. E.
Published: March 2006

Starting from Head's semi-empirical method for incompressible flow, two approaches to the prediction of turbulent boundary-layer development in compressible flow are explored. The first uses Head's incompressible method in conjunction with a compressibility transformation similar to Stewartson's transformation for laminar flow; the second carries over Head's physical arguments to treat the compressible flow directly. Measurements in supersonic flow, both on flat plates and downstream of an abrupt pressure rise, show broad agreement with the predictions of the second method but do not support the compressibility transformation. In particular, measurements on flat plates reveal that as Mach number increases the entrainment rate decreases to a lesser extent than the skin-friction coefficient. Whilst this result is consistent with the second treatment in this paper, it is difficult to reconcile with any of the compressibility transformations discussed, and the validity of these transformations in turbulent flow is therefore questioned.
On the resonant interaction of neutral disturbances in two inviscid shear flows
Authors: Kelly R. E.
Published: March 2006

The second-order resonant interaction of two disturbances which are neutrally stable on a linear basis is investigated for cases when the mean flow is, first, an inviscid, homogeneous jet and, secondly, a stably stratified, antisymmetric shear layer for which the linear eigen-solutions are regular. For the former case, the periodic nature of the neutral disturbances is unaffected by the interaction. For the latter, the interaction can lead to an O(ε½) temporal growth rate of one disturbance, where ε is a characteristic disturbance amplitude.
Low Reynolds number heat transfer from a circular cylinder
Authors: Hieber C. A. , Gebhart B.
Published: March 2006

Theoretical results are obtained for forced heat convection from a circular cylinder at low Reynolds numbers. Consideration is given to the cases of a moderate and a large Prandtl number, the analysis in each case being based upon the method of matched asymptotic expansions. Comparison between the moderate Prandtl number theory and known experimental results indicates excellent agreement; no relevant experimental work has been found for comparison with the large Prandtl number theory.
On the propulsion efficiency of swimming flexible hydrofoils of finite thickness
Authors: Uldrick J. P.
Published: March 2006

This paper presents some recent theoretical results on the energy exchange between a swimming flexible two-dimensional hydrofoil of finite profile thickness and the inviscid incompressible fluid in which the body swims. The rate at which kinetic energy is transferred to the fluid by the undulating hydrofoil, the power required to maintain the prescribed motion, and the resulting power available for propulsion are calculated in terms of the thickness to chord ratio and the displacement and rate of displacement of the hydrofoil. With a small unsteady perturbation theory, the analysis is decomposed to show separately the effects of the circulatory and non-circulatory flows, both depending on the first-order terms of the unsteady perturbation velocity components. In addition, an analysis is presented showing the effect of the non-linear unsteady pressure distribution on the surface of the hydrofoil. Contrary to what might be expected, this latter effect is of the same order of magnitude for a thick rounded-nose profile as for the flat plate where the effect is concentrated at the sharp leading edge and is related to the so-called suction force. However, except for small values of the reduced frequency, the non-linear contribution is negligible in comparison with the linear contribution.

New functions associated with the retarded flow in the wake are introduced and special techniques for their solution are presented, these being related to Theodorsen's function of unsteady airfoil theory for the special case of the undulating flat plate.

The numerical results reveal that the kinetic energy imparted to the fluid, the power required to maintain the motion, and the resulting propulsive power, follow closely those of an infinitesimal model for small values of the reduced frequency of oscillation, but diverge somewhat from the classical thin plate analysis for large reduced frequencies. Of particular interest is the fact that a very large percentage of the power required to maintain the motion is used in the generation of the wake, whereas a very small percentage of the power available for propulsion comes from the wake. This indicates that, if some mechanism could be devised to control the wake, very high swimming efficiencies could be attained. Fish, in all probability, have been succeeding in doing this for millions of years.
On the stability of viscous flow between eccentric rotating cylinders
Authors: Ritchie G. S.
Published: March 2006

The stability of viscous flow between eccentric cylinders is analysed for the case in which the inner cylinder rotates while the outer cylinder remains stationary, and where the difference in radii of the cylinders is small in comparison with their mean radius. The linearised equations governing the marginal stability of axially periodic disturbances are derived in general for the case where the cylinders are infinitely long, and are solved approximately to give estimates of the critical Taylor number at which vortex flow occurs for a range of relative eccentricity of the cylinders.

The results give an upper bound to the stability boundary, and certain results of DiPrima are used to establish a lower bound, and consequently the stability boundary is well established for eccentricity ratios less than about 0·6. One important conclusion is that for a considerable range of eccentricity ratio the flow is less stable than when the cylinders are concentric.
Excitation of internal waves in a stably-stratified atmosphere with considerable wind-shear
Authors: Townsend A. A.
Published: March 2006

The rate of generation of internal waves by a thin turbulent boundary layer was calculated in a previous paper for a stably-stratified atmosphere with no significant wind-shear outside the boundary layer by considering the excitation of normal modes of wave propagation. By using the concept of wave-packets propagating upwards from the boundary layer, the effects of wind-shear can be included. Conditions for the validity of the approximation are given. In general, the spectral distribution of wave-energy at a particular height takes large values in two bands of horizontal wave-number, one band deriving from wave-packets undergoing internal reflexion near that height and the other from wave-packets of very small local frequency that accumulate there. The ‘reflexion’ wave-numbers are dominant if the wind increases with height and the ‘accumulation’ wave-numbers if the wind initially decreases with height. The spectral energy distributions and intensities of the wave-motion are discussed in more detail for an atmosphere of uniform stability and unidirectional wind-shear. The accumulation process may lead to instability or overturning of the waves, and estimates are made of the probable scale and intensity of the ‘clear-air’ turbulence produced. An interesting point is that the rate of energy loss from the boundary layer by radiation of internal waves turns out to be comparable with the rate of production in the outer nine-tenths of the layer, both for atmospheric boundary layers and for the surface layer of the ocean. It seems likely that radiation limits the layer thickness to some extent.
The development of three-dimensional wave packets in a boundary layer
Authors: Gaster M.
Published: March 2006

The formation and growth of three-dimensional wave packets in a laminar boundary layer is treated as a linear problem. The asymptotic form of the disturbed region developing from a point source is obtained in terms of parameters describing two-dimensional instabilities of the flow. It is shown that a wave caustic forms and limits the lateral spread of growing disturbances whenever the Reynolds number is √2 times the critical value. The analysis is applied to the boundary layer on a flat plate and shapes of the wave-envelope are calculated for various Reynolds numbers. These show that all growing disturbances are contained within a wedge-shaped region of approximately 10° semi-angle.
Selective withdrawal from a stably stratified fluid
Authors: Wood I. R.
Published: March 2006

In this paper a reservoir connected through a horizontal contraction to a channel is considered. Both the reservoir and the channel are considered to contain a stable multi-layered system of fluids. The conditions under which there is a flow in only one layer, and the depth in this flowing layer decreases continuously from its depth in the reservoir to its depth in the channel, give the maximum discharge that can be obtained with a flow only from this single layer. For this case the volume discharge calculations are carried out at a single section (the section of minimum width). Where there are velocities in only two layers and the depth in each of these layers decreases continuously from their depths in the reservoir to their depths in the channel, the theory involves computations at two sections in the flow. These are the section of minimum width and a section upstream of the position of minimum width (the virtual point of control). For this flow it is shown that the solution is the one in which the velocity and density distributions are self similar and that the depths of the layers at the point of maximum contraction are two-thirds of those far upstream. It is then shown that for any stable continuous or discrete density stratification in the reservoir a self similar solution will satisfy the conditions for the depths of the flowing layers to decrease smoothly from the reservoir to downstream of the contraction. Again the ratio of the depth at the contraction to that far upstream is two-thirds.

When there is a very large density difference between the fluid in the lower dead water and that in the lowest flowing streamline then this streamline becomes horizontal and may be considered as a frictionless bed. The flow when the bed is not horizontal but where there is a small rise in the channel at the position of maximum contraction is considered for the case where two discrete layers flow under a volume of dead water. In this case the velocity and density profiles are not self similar.

Experiments have been carried out with a contraction in a flume for the withdrawal of two discrete layers from a three layer system and the withdrawal from a fluid with a linear density gradient. In both cases the reservoir and channel bed and hence the lowest streamline was effectively horizontal. These experiments confirmed the theoretical predictions.
Self-similar strong shocks in an exponential medium
Authors: Hayes Wallace D.
Published: March 2006

The self-similar one-dimensional propagation of a strong shock wave in a medium with exponentially varying density and ray-tube area is studied, using the Eulerian approach of Sedov. Conservation integrals analogous to Sedov's are obtained, with the expression for the Lagrangian variable. Calculated results are compared with the predictions of the CCW (Chisnell, Chester and Whitham) approximation. It was found that, in contrast to the implosion case, the propagation parameter from the CCW approximation is in error by 15% or more.
Smoke observations of the wake of a group of three cylinders at low Reynolds number
Authors: Zdravkovich M. M.
Published: March 2006

A smoke visualization technique was used to study the laminar wake behind a group of three cylinders. The main characteristic of the interaction between the three cylinders was the appearance of strong sinuous oscillations some distance downstream in the wake, which led to the formation of a new single vortex street. The mechanism of the formation process of this vortex street and the part played in it by the sinuous oscillations is demonstrated by a series of photographs. The effects of variation of the spacing of the configuration have been examined in detail.

In an interaction between three fully developed vortex streets, some of the rows of vortices crossed, and there was an extremely complicated re-arrangement of vorticity in the wake.

Transition to turbulence in such wakes was also studied. The most interesting result observed was an asymmetric transition. Wakes were sometimes observed which were laminar on one side and turbulent on the other.
On the mixing of angular momentum in a stirred rotating fluid
Authors: Bretherton F. P. , Turner J. S.
Published: March 2006

It has been suggested on theoretical grounds that a vortex could be initiated in a cylindrical region of fluid, originally in solid rotation, by the horizontal mixing of angular momentum produced by external stirring. In this paper various arguments for and against the mechanism are examined and their difficulties exposed. No firm conclusion is reached. A series of mathematical models of the mixing motions has been used, to bring out the differences between mechanical stirring and the agitation of a gas by random molecular motions. They suggest the introduction of a diffusion coefficient for angular momentum, to be determined empirically. These theoretical ideas are then applied to the interpretation of the results of a laboratory experiment which has been designed to test the proposed mechanism directly.

A wide, flat tank of liquid was set up on a rotating table and stirred with a vertically oscillated grid, whose elements were much smaller than the width of the tank. A neutrally buoyant particle was used as a tracer of fluid motions, to measure relative circulation velocities and the properties of the turbulence. The motion observed was dominated by the loss of angular momentum to the walls and the grid, an effect which has not been taken into account in previous theoretical assessments of the effects of mixing of angular momentum. The relative circulation present was not significantly different from zero, and the limits of error of the measurements imply that the rate of diffusion of angular momentum is less than 5% of that for fluid particles, with 95% probability.
Axially-symmetric eddies embedded in a rotational stream
Authors: Leibovich S.
Published: March 2006

The structure of eddies embedded in a swirling stream of an incompressible, inviscid fluid is examined. Laminar motion is assumed, as is axial symmetry of the flow field.

It is supposed that the eddies are formed either (i) as wakes behind solid obstacles placed in the stream, or (ii) as the products of vortex breakdown occurrences, in which case the eddies are completely surrounded by fluid of the outer stream. In either event, it is assumed that the presence of an eddy only slightly disturbs the motion of the outer stream, i.e. that the eddies are slender. In particular, this requires that r0Ω∞/W∞ [Lt ]1, where r0 is the maximum radial extent of the eddy, and Ω∞ and W∞ are defined below.

The analysis is carried to completion when the undisturbed outer stream is a wheel flow. Principal results derived are summarized below. (1) Eddy lengths are related directly and simply to the internal vorticity, providing that there is swirl in the outer flow. (2) A corollary is that very slender eddies have a maximum possible length. In the case (ii) above, this length is 2πW∞/Ω∞, where W∞ is the axial velocity and Ω∞ the angular velocity of the undisturbed outer flow. (3) The shapes of the free streamlines bounding the eddy are calculated. A feature is the cusped nature of an eddy free end. (4) The motion inside the eddy and the disturbance to the outer stream are calculated.

Irrotational base flows are exceptional. Results (1) and (2) above cannot be found for a potential outer flow, but (3) and (4) remain.
The suppression of wind-generated waves by a surface film
Authors: Gottifredi J. C. , Jameson G. J.
Published: March 2006

In the presence of a surface-active agent waves generated on the surface of a liquid by a wind may be stabilized. This problem is explored, and in particular the critical wind velocity is calculated for wave inception in the presence of such a film.

The dominant parameter associated with the surface layer is the surface elasticity χ the ‘inextensible film’ of Lamb is obtained as a limit as χ → ∞. Such a value is hypothetical as for real films χ is not large, being in the range 0 to 80 dyne/cm approximately. Nevertheless, we show that damping exceeding that of the inextensible film can be obtained for short wavelengths, and that the critical wind speed for capillary ripples can be increased by a factor of ten in the presence of a film of experimentally attainable characteristics.

The reason for the effectiveness of films of low χ is that the damping in the liquid is related to χ and the wavelength λ in such a way that for small λ, damping is a maximum for small χ.
The axisymmetric convective regime for a rigidly bounded rotating annulus
Authors: Mcintyre Michael E.
Published: March 2006

The axisymmetric flow of liquid in a rigidly bounded annular container of heightH, rotating with angular velocity Ω and subjected to a temperature difference ΔTbetween its vertical cylindrical perfectly conducting side walls, whose distance apart isL, is analysed in the boundary-layer approximation for small Ekman numberv/2ΩL2, withgαΔTHv/4Ω2L2K∼ 1. The heat transfer across the annulus is then convection-dominated, as is characteristic of the experimentally observed ‘upper symmetric regime’. The Prandtl numberv/kis assumed large, andHis restricted to be less than about 2L. The side wall boundary-layer equations are the same as in (non-rotating) convection in a rectangular cavity. The horizontal boundary layers are Ekman layers and the four boundary layers, together with certain spatialaveragesin the interior, are determined independently of the interior flow details. The determination of the latter comprises a ‘secondary’ problem in which viscosity and heat conduction are important throughout the interior; the meridional streamlines are not necessarily parallel to the isotherms. The secondary problem is discussed qualitatively but not solved. The theory agrees fairly well with an available numerical experiment in the upper symmetric regime, forv/k[bumpe ] 7, after finite-Ekmannumber effects such as finite boundary-layer thickness are allowed for heuris-tically.
Inclined buoyant puffs
Authors: Richards J. M.
Published: March 2006

The motion of puffs which move in the direction of the buoyancy force was considered in a previous paper (Richards 1965). We now consider cases in which the directions of motion and of buoyancy are inclined or opposed. The main assumption is that the distributions of velocity inside the puffs are similar throughout the motion. Although this assumption is thought to be accurate only when the surroundings are neutral, we have reason to think that the same assumption may sometimes be used as an approximation to the internal distribution of velocity in other types of environment.

The particular case of motion when the buoyancy force is constant is calculated in detail. The calculations are supported by observations from new laboratory experiments.
Aerodynamic sound generation in a pipe
Authors: Davies H. G. , Williams J. E. Ffowcs
Published: March 2006

The paper deals with the problem of estimating the sound field generated by a limited region of turbulence in an infinitely long, straight, hard-walled pipe. The field is analysed in a co-ordinate system moving with the assumed uniform mean flow, and the possibility of eddy convection relative to that reference system is considered. Large-scale turbulence is shown to induce plane acoustic waves of intensity proportional to the sixth power of flow velocity. The same is true of small-scale turbulence of low characteristic frequency. In both cases convective effects increase the acoustic output and distribute the bulk of the energy in a mode propagating upstream against the mean flow. Small-scale turbulence of higher frequency excites more modes, the sound increasing with very nearly the eighth power of velocity (U7.7) as soon as the second mode is excited. In the limit, when more than about 20 modes are excited, the energy output is unaffected by the constraint of the pipe walls, increasing with the eighth power of velocity, and being substantially amplified by convective motion.
Phase jumps
Authors: Howe M. S.
Published: March 2006

An earlier paper (Howe 1967) considered a non-linear theory of open-channel steady flow of deep water past a slowly modulated wavy wall. The wave pattern on the free surface of the water was obtained as the solution of a stably posed elliptic Cauchy problem, the main feature of the solution being the appearance of a ‘shock’ across which there is an abrupt change of phase. Such phase jumps can occur in a wide range of similar problems, but the advantage of the present case is that it is rather well suited to experimental investigation. This paper is therefore a lead-in to the more general problem of phase jumps, and uses the principle of conservation of energy in conjunction with the earlier solution to predict the possible position of the discontinuity on the free surface of the water. The possible nature of the free surface in the vicinity of the phase jump is also discussed (figure 4). This is a region where the width of the wave troughs becomes dramatically shorter than that of the neighbouring troughs. An approximate method of determining the line along which the phase jump occurs, not depending on a knowledge of the solution of the Cauchy problem, is also presented.
On the stability of plane Poiseuille flow with a finite conductivity in an aligned magnetic field
Authors: Ko Sung-Hwan
Published: March 2006

A study is made of the stability of a viscous, incompressible fluid with a finite conductivity flowing between parallel planes in a parallel magnetic field. The general form of the magnetohydrodynamic stability equation is a sixth-order differential equation. The complete sixth-order differential equation is solved numerically as an eigenvalue problem. Stability curves are obtained for a range of values of the magnetic Reynolds number Rm and the Alfvé n number A based on two-dimensional disturbances. It is found that the minimum critical Reynolds number is raised as Rm increases for a given A2 and as A2 increases for a given Rm, respectively. The stability curve closes and finally degenerates to a point which gives the critical value for Rm or A2. Results obtained for two-dimensional disturbances are modified to take into account three-dimensional disturbances. Then the minimum critical Reynolds number where three-dimensional disturbances become apparent is obtained, below which two-dimensional disturbances are the most unstable.
Numerical and experimental investigations of the shear layer between two parallel streams
Authors: Mills R. D.
Published: March 2006

The first-order laminar shear layer between two parallel streams has been calculated exactly for compressible and incompressible two-dimensional flow. Turbulent flow has been included by using Prandtl's hypothesis for the eddy viscosity. Results are presented for a wide range of the appropriate parameters. The problem is solved by writing the momentum and energy equations as a pair of coupled integral equations in Crocco variables; these are solved by the method of successive approximation after using a simple transformation to weaken the singularity at the outer edges of the boundary layer. This approach yields uniquely the shear stress and temperature as functions of the tangential velocityu. The so-called third boundary condition, derived by Ting (1959), is then readily satisfied in evaluating the transverse component of velocityvby quadrature. Experimental results are presented in the turbulent incompressible case. Good agreement with the exact theoretical results is obtained when one stream is much faster than the other, but this falls off as the speeds of the streams tend to equalize.
Double Kelvin waves with continuous depth profiles
Authors: Longuet-Higgins M. S.
Published: March 2006

The possibility of long waves in a rotating ocean being trapped along a straight discontinuity in depth was demonstrated in a recent paper (Longuet-Higgins 1968). The analysis is now extended to the situation where the depth varies continuously, in a zone separating two regions of different depths. The trapping of waves in the transition zone is investigated, taking full account of the horizontal divergence of the motion.

If the profile of the depth is assumed to be monotonic, then it is shown that the trapped waves always travel along the transition zone with the shallower water to their right in the northern hemisphere and to their left in the southern hemisphere. The wave period must always exceed a pendulum-day. The period is also bounded below by a quantity depending inversely on the maximum bottom gradient.

By allowing the width W of the transition zone to vary, asymptotic forms for the trapped modes are obtained, both as W → 0 and as W → ∞. In the limit as W → 0 the depth becomes discontinuous, and it is shown that the lowest mode then becomes a double Kelvin wave (Longuet-Higgins 1968) propagated along the discontinuity. The periods of the higher modes, on the other hand, all tend to infinity; these modes become steady currents.

Numerical calculations of the trapped modes are presented for two different laws of depth in the transition zone. It is found that as W → 0 the lowest mode is insensitive to the form of the depth profile. Higher modes depend on the details of the profile. Hence the lowest mode is the most likely to be observed in the real ocean.

The dispersion relation is also investigated. It is shown that the group-velocity of all modes must change sign at some point in the range of wave-numbers, if the divergence is taken into account. When the divergence was neglected the lowest mode appeared to be exceptional, in that the group-velocity was always in the same direction. This anomaly is now removed.
Free convection from a flat plate
Authors: Takhar H. S.
Published: March 2006

A numerical solution is presented for the development of free convection from a semi-infinite vertical flat plate which is uniformly heated up to a length l from the base and insulated for the rest of its length. At great heights above the heated part of the plate, the velocity and temperature distributions behave as if the heat were put in as a line source of heat at the base of the plate. Matching of the solutions for the heated and the insulated parts of the plate, by keeping the fluxes of heat and momentum continuous, determines the position of the effective origin of the similarity solution for the insulated plate in terms of the length, l, of the heated part of the plate. Graphs of the dimensionless velocity, temperature, heat flux and axial length parameters are given for different values of the Prandtl number.
Determination of the direction of surface velocity gradients in three-dimensional boundary layers
Authors: Karabelas Anastasius J. , Hanratty Thomas J.
Published: March 2006

An instrumental method is proposed to measure the direction of surface velocity gradients in three-dimensional boundary layers. It utilizes electrochemical techniques developed in this laboratory.