ALBERT

Description: The paper is concerned with formulation of the gas dynamic conservation equations for the individual species in a non-equilibrium partially ionized gas mixture. As an example, the conservation equations for the electrons and the overall conservation equations are developed for a three component plasma consisting of electrons, singly-ionized positive ions and neutral atoms. Non-elastic collisions are represented by the collisional-radiative decay mechanism of Bates, Kingston & McWhirter (1962a, b). Maxwellian velocity distributions are assumed, but the electrons are allowed to have a temperature different from the heavier particles and to drift relative to them. Particular attention is given to the electron energy balance equation which differs from that used by other investigators. © 1964, Cambridge University Press. All rights reserved.

Description: First- and second-order boundary-layer theory are examined in detail for some specific flow cases of practical interest. These cases are for flows over blunt axisymmetric bodies in hypersonic high-altitude (or low density) flow where second-order boundary-layer quantities may become important. These cases consist of flow over a hyperboloid and a paraboloid both with free-stream Mach number infinity and flow over a sphere at free-stream Mach number 10. The method employed in finding the solutions is an implicit finite-difference scheme. It is found to exhibit both stability and accuracy in the examples computed. The method consists of starting near the stagnation-point of a blunt body and marching downstream along the body surface. Several interesting properties of the boundary layer are pointed out, such as the nature of some second-order boundary-layer quantities far downstream in the flow past a sphere and the effect of strong vorticity interaction on the second-order boundary layer in the flow past a hyperboloid. In several of the flow cases, results are compared with other theories and experiments. © 1964, Cambridge University Press. All rights reserved.

Description: A characteristic feature of a steady trailing line vortex from one side of a wing, and of other types of line vortex, is the existence of strong axial currents near the axis of symmetry. The purpose of this paper is to account in general terms for this axial flow in trailing line vortices. the link between the azimuthal and axial components of motion in a steady line vortex is provided by the pressure; the radial pressure gradient balances the centrifugal force, and any change in the azimuthal motion with distance x downstream produces an axial pressure gradient and consequently axial acceleration. It is suggested, in a discussion of the evolution of an axisymmetric line vortex out of the vortex sheet shed from one side of a wing, that the two processes of rolling-up of the sheet and of concentration of the vorticity into a smaller cross-section should be distinguished; the former always occurs, whereas the latter seems not to be inevitable. In § 4 there is given a similarity solution for the flow in a trailing vortex far downstream where the departure of the axial velocity from the free stream speed is small. The continual slowing-down of the azimuthal motion by viscosity leads to a positive axial pressure gradient and consequently to continual loss of axial momentum, the asymptotic variation of the axial velocity defect at the centre being as x−1 log x. The concept of the drag associated with the core of a trailing vortex is introduced, and the drag is expressed as an integral over a transverse plane which is independent of x. This drag is related to the arbitrary constant appearing in the above similarity solution. © 1964, Cambridge University Press. All rights reserved.

Description: Random fluctuations in sea level, ζ, in the frequency range 0·1-60 cycles per hour were measured along the coast near Oceanside, California, where the coastline and bottom contours are fairly straight and parallel for 30 km. The two-dimensional covariance $R(eta, au) = langle zeta (y,t) zeta (y + eta, t+ au) angle$ was computed for points separated by various distances η along the coast. The Fourier transform $S(f,n) = int int R(eta, au)exp [2pi i (n eta + f au)]d eta d au$ gives the contribution towards the ‘energy’ $langle zeta ^2 angle$ per unit temporal frequency f per unit spacial frequency (long-shore component) n. It is found that most of the energy is confined to a few narrow bands in (f, n) space, and these observed bands correspond very closely to the gravest trapped modes (or edge waves) computed for the actual depth profile. The bands are 0·02 cycles per km wide, which equals the theoretical resolution of the 30 km array. Very roughly S(f,n) ≈ S(f, -n), corresponding to equal partition of energy between waves travelling up and down the coast. Theory predicts ‘Coriolis splitting’ between the lines f± (n) corresponding to these oppositely travelling waves, but this effect is below the limit of detection. The principal conclusion is that most of the low-frequency wave energy is trapped.

Description: The problem of flow over a circular cone inclined slightly to a uniform stream is solved using the technique of matched asymptotic expansions. The outer expansion is equivalent to Stone's solution of the problem. The inner expansion, valid in a thin layer near the body, represents Ferri's vortical layer. The solution to first order in angle of attack so obtained is uniformly valid everywhere in the flow field. In the second-order expansion an additional non-uniformity appears near the leeward ray. This defect is removed by inspection. The first-order solution is in agreement with that of Cheng, Woods, Bulakh and Sapunkov. Formulas are given that may be used to render Kopal's numerical result uniformly valid to second order in angle of attack. © 1964, Cambridge University Press. All rights reserved.

Description: We consider here flow past an obstacle on the lower of two rotating horizontal planes which bound a viscous fluid. It is found that when the Taylor number is large viscous effects are confined to Ekman boundary layers on the solid surfaces and to a free shear layer coincident with the vertical cylinder circumscribing the bottom obstacle. The flow in the main body of the fluid outside the cylinder proves to be two-dimensional with zero relative vorticity, while inside the cylinder the fluid is stagnant in the rotating frame. The shear layer provides a continuous transition between the exterior and interior of the cylinder, and in addition provides a means by which fluid from the Ekman layers on the horizontal planes is exchanged with fluid outside the Ekman layers and exterior to the circumscribing cylinder. The predicted flow proves to be in agreement with many of the experimental results. © 1964, Cambridge University Press. All rights reserved.

Description: This paper considers the transient motion of a viscous fluid in a container rotating with constant angular velocity. The principal objective is to study the manner in which an arbitrary initial state of motion becomes a rigid rotation. In order to concentrate on the effects of viscosity, only the spherical container is studied here in great detail. A general theory will be presented in a subsequent publication.Several sources of non-uniform behaviour make the analysis difficult and complex. In particular, there are three important time scales, viscous boundary layers, boundary-layer resonances at critical latitudes and intricate side-wall effects. The basic aproach consists of an expansion procedure by means of which the general inviscid solution is corrected for viscous effects and is made uniformly valid in time through the critical spin-up phase. Uniform validity is effected through the elimination of secular terms with unacceptable growth rates arising from the asymptotic perturbation series.The interior (inviscid) motion leads to a non-self-adjoint partial differential equation eigenvalue problem with many intriguing properties. The general expansion theorem, orthogonality relationships, and viscous decay factors are deduced and used to solve the arbitrary intial-value problem. It is shown that the depth averaged circulation about circular contours, x2 + y2 = r2, is extracted from the fluid in the spin-up time scale T = L(Ωv)½. This is accomplished by a secondary non-oscillatory convective motion produced by suction into the Ekman layer. The excess circulation not eliminated in this way excites inviscid inertial oscillations which are also caused to decay by the boundary layers in the same time scale. Some very small residual effects decay in the ordinary viscous diffusion time, but all the essential processes are concluded in the much shorter interval. All modal oscillations in a sphere are determined and several specific calculations of frequency and decay rate are made and compared with experimental data. Perhaps the most important of these concerns the mode corresponding to rigid internal motion about another axis which can be produced by impulsively changing the rotation axis of the container. Agreement between theory and experiment is very good in all cases compared thus far.

Description: The wall-jet is the flow of fluid emanating from a narrow slot and blowing over a rigid wall. The configuration of the turbulent wall-jet is that of a very narrow, plane, turbulent half-jet investigated by Liepmann & Laufer (1947). The width of the slot is of the order of the boundary layer on the infinite wall and the jet, in this case, mixes with a stream moving with constant velocity. This flow has drawn considerable basic and applied interest in the past few years for it has the characteristics of both a boundary-layer and a free-mixing flow. © 1964, Cambridge University Press. All rights reserved.

Description: An experimental investigation of the fluid dynamic forces on spheres suspended in a Poiseuille flow was performed. Small spheres of polystyrene, nylon, and Lucite, having diameters ranging from 0.061 in. to 0.126 in. were suspended in Poiseuille flows in a 0.419 in. diameter tube. Variations in particle size and density, the fluid properties, and the angle of inclination of the tube, resulted in a sphere Reynolds number (based on particle diameter and approach velocity) ranging from 80 to 250. The results are presented as curves which include the coefficients of lift and drag, and the dimensionless rotation speed plotted versus Reynolds number and a dimensionless shear parameter. © 1964, Cambridge University Press. All rights reserved.

Description: A theoretical analysis is given of the unsteady flow of a liquid within a cylinder of finite length started suddenly so as to spin about its axis. It is found that a secondary flow, caused by the end walls of the cylindrical container, has a strong effect on the generation of spin in the liquid. In the vicinity of the end walls the fluid motion is characterized by a boundary-layer flow, which can be either laminar or turbulent. The fluid within the boundary layers rotates faster than that at a large distance from the end walls, and therefore is thrown, by centrifugal forces, radially outwards. The radial outflow in the boundary layer creates a slow secondary motion within the spinning liquid. Due to the secondary flow, the transport of angular momentum from the walls to the interior is accomplished by convection rather than diffusion. A treatment is given for both laminar and turbulent end-wall boundary layers. The theoretical results are compared with experimental observations and good agreement is found. © 1964, Cambridge University Press. All rights reserved.

Description: The viscous hypersonic flow past an axisymmetric blunt body is analysed based upon the Navier-Stokes equations. It is assumed that the fluid is a perfect gas having constant specific heats, a constant Prandtl number, P, whose numerical value is of order one, and a viscosity coefficient varying as a power, Ω, of the absolute temperature. Limiting forms of solutions are studied as the free-stream Mach number, M, and the free-stream Reynolds number based on the body nose radius, R, go to infinity, and ε = (γ − 1)/(γ + 1), where γ is the ratio of the specific heats, and δ = 1/(γ − 1) M 2go to zero. © 1964, Cambridge University Press. All rights reserved.

Description: When the angular amplitude of oscillation of a sphere in an infinite mass of elastico-viscous fluid is fairly small, so that velocity and stress components may be expanded as power series in this amplitude, the purely periodic primary motion has associated with it a secondary flow which has a steady component as well as a component of double the primary frequency. An expression for the stream function of the steady secondary flow is obtained for all possible frequencies and the results are illustrated by considering in detail a particular fluid. It is shown that the streamline projections on a plane containing the axis of rotation are strongly dependent on the parameters measuring the elasticity of the fluid and on the frequency. The circulatory secondary flow can be in the opposite sense to that in a Newtonian fluid, in either the whole or part of the elastico-viscous fluid. © 1964, Cambridge University Press. All rights reserved.

Description: The mathematical basis for resonance is investigated using a model equation describing one-dimensional dispersive waves interacting weakly through a quadratic term. If suitable time-invariant boundary conditions are imposed, possible oscillations of infinitesimal amplitude are restricted to a discrete set of wave-numbers. An asymptotic expansion valid for small amplitude shows that oscillations of different wave-number interact primarily in independent resonant trios. Energy is redistributed between members of a trio over a characteristic time inversely proportional to the amplitude of the oscillations in a periodic manner. The period depends on the initial conditions but is in general finite. Cubic interactions through resonant quartets are also discussed. The methods used are valid for a fairly wide class of equations describing weakly non-linear dispersive waves, but the expansion procedure used here fails for a continuous spectrum. © 1964, Cambridge University Press. All rights reserved.

Description: The problem of defining an effective velocity of convection for turbulent fluctuations in a shear flow is considered, and the definitions adopted by various workers are discussed. An experiment in the shear layer of a circular jet has shown that the usual definitions, based on peaks of the space-time correlations of the fluctuations, yield convection velocities whose magnitudes depend on the value of space or time separation chosen. An alternative approach shows that, by considering the turbulent field as a superposition of harmonic travelling waves, a wave-number/velocity spectrum can be defined that lends itself to the definition of a wave-number-dependent convection velocity and an overall convection velocity, both of which have real physical significance. An experimental technique is described for obtaining the spectrum, and results are presented for one position in the shear layer of the jet. © 1964, Cambridge University Press. All rights reserved.

Description: The classical problem of the motion of a one-dimensional unsteady shock generated by a piston moving with velocity v p = ct n is extended to take into account thermal radiation effects by the similarity method of Taylor and Sedov. Gray gas and local thermodynamic equilibrium are assumed and a modification of the Schuster-Schwarzschild differential equation for the heat flux is adopted. The optical thickness is not restricted to be thin or thick, and the absorption coefficient is assumed to vary with the density and temperature. Numerical results indicate that the pressure and velocity are not affected much by the radiation, but the density, temperature and radiant heat flux are changed considerably. © 1964, Cambridge University Press. All rights reserved.

Description: When a smooth jet of water falls vertically on to a horizontal plane, it spreads out radially in a thin layer bounded by a circular hydraulic jump, outside which the depth is much greater. The motion in the layer is studied here by means of boundary-layer theory, both for laminar and for turbulent flow, and relations are obtained for the radius of the hydraulic jump. These relations are compared with experimental results. The analogous problems of two-dimensional flow are also treated. © 1964, Cambridge University Press. All rights reserved.

Description: In this paper the turbulent flow characteristics of viscoelastic fluids are investigated quantitatively. The outstanding property of these flow fields is seen to be a very pronounced suppression of turbulence, accompanied by major reductions in the turbulent drag coefficients. Careful measurements of the rheological properties of the several fluids used suggest that the observed turbulence suppression is a function of the ratio of the elastic to the viscous forces developed in the fluid. An empirical correlation of the results, based upon this observation, is proposed; the present data, while indicative, are not sufficiently extensive to verify conclusively the existence of a quantitative correlation. In a number of respects, the observed reduction in drag is similar to that which may be obtained through promotion of ‘slip’ at the tube wall or by addition of particulate matter. It is shown that slip phenomena are clearly distinct from those studied in the present work but that particulate effects (albeit of much greater magnitude than observed heretofore) cannot be ruled out as contributory mechanisms. Further studies are thus required to determine the relative importance of continuum (viscoelastic) and particulate effects. © 1964, Cambridge University Press. All rights reserved.

Description: A study of the stability of flow between concentric cylinders, with the inner one rotating, distinguished three kinds of instabilities: the familiar axisymmetric mode, an azimuthal mode with the predicted exp i(θ—Ωt) angular dependence, and a completely non-symmetric instability which apparently arises from the interaction of the other two. The effect of small axial flow upon all of these modes was to give an approximately parabolic dependence of the critical Taylor number on the axial flow rate. In the case of the axisymmetric mode, agreement with the theory of Krueger & Di Prima (1964) was found to be excellent. © 1964, Cambridge University Press. All rights reserved.

Description: Goldstein (1931) has considered the stability of a shear layer within which the velocity and the density vary linearly and outside which they are constant. Rayleigh (1880, 1887) had found that the corresponding, homogeneous shear flow is unstable in and only in a finite band of wave-numbers. Goldstein concluded that a small density gradient renders the flow unstable for all wave-numbers. This conclusion appears to depend on the acceptance of all possible branches of a multi-valued eigenvalue equation, and it is shown that the principal branch of this eigenvalue equation yields one and only one unstable mode if and only if the wave-number lies in a band that decreases from Rayleigh's band to zero as the Richardson number increases from 0 to ¼. © 1964, Cambridge University Press. All rights reserved.

Description: The effect of compressibility is included here in a study of wakes created in two-dimensional, steady, aligned-fields, magnetogasdynamic flow past obstacles. The gas is assumed to be viscous, resistive, and thermally conducting. With the Oseen type of approximation as well as the magnetogasdynamic boundary-layer approximation, a great simplification in the formulation of wakes results. The boundary-layer equations, although linearized, still retain the coupling between the velocity, the magnetic, and the temperature fields. The solution of the magnetogasdynamic wake, in general, is a superposition of three individual-wake components, each satisfying a diffusion type of equation. Only one of them is capable of extending upstream. Hence the wake picture is generally characterized by a conventional downstream wake with also the possibility of the existence of anupstream one. To be sure, the general features of the magnetogasdynamic wakes in aligned-fields flows are similar to those of an incompressible fluid; however, the flow now, instead of just being subalfvénic, must be subcritical for the upstream wake to occur. That is, the flow condition corresponding to the occurrence of the upstream wake is such that the sum of the square of the Mach number M∞ and the square of the Alfvén number A∞ is less than unity. Evidently it is the mechanism of magneto-sonic-wave propagation that modifies the transition of wakes from the Alfvénic point A∞ = 1 (for an incompressible fluid) to the subcritical are [Formula omited] in the Taniuti-Resler diagram. © 1964, Cambridge University Press. All rights reserved.

Description: The heat transfer in a radial liquid jet is investigated. In the region where a similarity solution of the momentum equation is available solutions of the energy equation describing the effects of viscous dissipation, initial heating and wall heating are obtained in closed form. Two examples illustrating the work are discussed. In the second of these an approximate method, based on the heat flux equation, is used to describe the initial development of the thermal boundary layer. © 1964, Cambridge University Press. All rights reserved.

Description: A compact cluster of 3 to 6 rigid equal spheres is falling under gravity in a viscous liquid. The small effects of intertia on a horizontal regular polygonal configuration are that the polygon expands as it falls and small perturbations from this configuration die out, although when the polygon is large enough it becomes weakly unstable. This is an extension of the analysis of Hocking (1964, which was applied to the experiments of Jayaweera, Mason & Slack (1964). © 1964, Cambridge University Press. All rights reserved.

Description: General solutions for the linearized three-dimensional unsteady motion of a relaxing gas, including the effects of direct heat addition, are obtained using a Green's function technique. The possibility of direct heat addition into each separate mode of energy storage must be catered for, and the solutions have some bearing on the question of the heating of a gas by radiation. Examples are given of initial-value and heat-source solutions in an unbounded domain and the boundary-value problem is exemplified by a study of the motion created by a spherical piston. © 1964, Cambridge University Press. All rights reserved.

Description: The previous works on the motions of visco-plastic fluids in porous media are generalized in order to include the phenomenon of hysteresis. Only slowly varying motions are studied to any extent, when all inertial terms can be neglected without appreciable error. A general equation is deduced and some features of the motion are discussed. Simple particular cases without free or seepage surfaces are considered, when the fluid moves in the whole porous medium. A plane example points out the existence of different regions of motion during the variation of the heads in upper and lower water reservoirs. The necessity of further research on the flow of visco-plastic fluids through porous media is emphasized. © 1964, Cambridge University Press. All rights reserved.

Description: The logical design method of Owen & Zienkiewicz (1957), for the generation of linear shear flow, is successfully applied to the case of symmetrical velocity profiles in a two-dimensional duct. Shear parameters as high as 0·8 (corresponding to [formula omited] are obtained experimentally. Some slight modification to the theoretical grid spacing is required because of the non-uniformity of the upstream flow. The decay of the profiles is confined to that due to the developing boundary layer and to the breakdown of the physically impossible condition at the centre, where symmetry implies a change in sign of the uniform shear. © 1964, Cambridge University Press. All rights reserved.

Description: An approximate analytical solution is derived for the inclined non-inertial sinkage of a flat plate onto a smooth surface, for the case of small angles. The result is a modification to the existing solutions for parallel sinkage, in the form of a polynomial in the dimensionless angle of inclination. Even the smallest angle of inclination of plate to surface produces a large increase in the sinkage rate otherwise obtained. The results are confirmed experimentally. © 1964, Cambridge University Press. All rights reserved.

Description: Experimental confirmation of some streamline patterns suggested by Taylor (1961) is described. It is shown, also, that when a viscous fluid is expelled from a tube by an inviscid fluid, the interface has a ‘localized’ effect on the fluid ahead of the interface between the two liquids, that is, the usual Hagen–Poiseuille law is obeyd in the viscous fluid except for the region behind a point about 1½ tube diameters ahead of the bubble. Use of a recently developed tungsten-iodide lamp is also described. © 1964, Cambridge University Press. All rights reserved.

Description: An experimental investigation was made of the stability of a two-dimensional jet at low Reynolds numbers with extremely small residual disturbances both in and around the jet. The velocity distribution of a laminar jet is in agreement with Bickley's theoretical result. The stability and transition of a laminar jet are characterized by the Reynolds number based on the slit width and the maximum velocity of the jet. When the Reynolds number is less than 10, the whole jet is laminar. When the Reynolds number is between 10 and around 50, periodic velocity fluctuations are found in the jet. They die out as they travel downstream without developing into irregular fluctuations. When the Reynolds number exceeds about 50, periodic fluctuations develop into irregular, turbulent fluctuations. The frequency of the periodic fluctuation is roughly proportional to the square of the jet velocity. The stability of the jet against an artificially imposed disturbance was also investigated. Sound was used as an artificial disturbance. The disturbance is either amplified or damped in the jet depending on its frequency. The conventional stability theory was modified by considering the streamwise increase of Reynolds number. The experimental results are in agreement with the theoretical results. © 1964, Cambridge University Press. All rights reserved.

Description: An analytical investigation is carried out to determine the conditions for instability in a viscous fluid contained between rotating coaxial cylinders of arbitrary radius ratio. A solution method is outlined and then applied to cylinders having radius ratios ranging from 0·95 to 0·1. Consideration is given to both cases wherein the cylinders are rotating in the same direction and in opposite directions. Results are reported for the Taylor numbers and wave-numbers which mark the onset of instability. The present results are also employed to delineate the range of applicability of the closed-form instability predictions of Taylor and of Meksyn, which were derived for narrow-gap conditions. © 1964, Cambridge University Press. All rights reserved.

Description: The sedimentation of small clusters of uniform spheres, falling freely through a viscous liquid, has been studied with Reynolds numbers (based on diameter of the sphere and its velocity of free fall in the unbounded fluid) of individual spheres ranging from 10−4to 10. The fall velocity of a cluster is, in all cases, greater than that of individual spheres, the more so when the spheres are closer together. Two spheres falling side-by-side rotate inwards and separate as they fall if Re 〉 0·05, but no rotation nor separation is observed for Re 〈 0·03. When equal-sized spheres of Re 〉 1 fall vertically one behind the other, the rear sphere is accelerated into the wake of the leader, rotates, round it and separates from it when the line of centres is horizontal. If two spheres of unequal size but the same individual terminal velocity fall together, the smaller always travels faster than the larger. When three similar equally spaced spheres are dropped in a horizontal line, they interchange positions but do not separate when 0·06 〈 Re 〈 0·16. But, if 0·16 〈 Re 〈 3, one sphere is always left behind; which sphere depends critically upon the initial spacing. If three to six equal spheres, of 0·06 〈 Re 〈 7, start falling as a compact cluster, they eventually draw level and arrange themselves in the same horizontal plane at the vertices of a regular polygon. The polygon expands at a decreasing rate during fall. When three spheres are arranged initially in a horizontal isosceles triangle, the spheres oscillate about their equilibrium positions but eventually the spheres form a stable equil triagnle. If Re 〉 7, or the cluster contains 7 or more equal spheres, it shows no tendency to form a regular polygon but breaks up into two or more groups. A regular heptagon, and a hexagon with an additional sphere at its centre, are also unstable. © 1964, Cambridge University Press. All rights reserved.

Description: The theory is presented for the generation of secondary vorticity in a stratified fluid. The analysis is a generalization of original work of Hawthorne (1951). It is shown that for an incompressible, inviscid and non-diffusive flow, a flowwise vorticity component will be generated in a curved stream when a density gradient exists in the direction of the bi-normal to the streamline. © 1964, Cambridge University Press. All rights reserved.

Description: It is shown that the solution of the problem of scattering of long waves by a finite barrier in a rotating system may be obtained directly from the solution of the electromagnetic scattering problem for a finite strip. It is shown that the effect of rotation is to produce on both sides of the barrier, but away from the ends, a wave which is propagated without attenuation parallel to the barrier. © 1964, Cambridge University Press. All rights reserved.

Description: We have made a theoretical and experimental study of the effect of radiative transfer on the onset of convection by means of comparative experiments in dry air and ammonia contained between horizontal aluminium plates maintained at different temperatures. The first effect of radiative transfer is to distort the linear, diffusive profile in the initial, pre-convective state. A non-grey theory is developed and is shown to predict the non-linear profiles profiles with satisfactory accuracy. The change of radiative flux with pressure of ammonia is also predicted and confirmed. This constitutes the first successful attempt to confirm the methods used in atmospheric studies in controlled experiment. A non-grey theory of the onset of convection is developed, separating effects associated with the distortion of the static state and the dissipative effect of radiative transfer on temperature fluctuations. Experiments indicate that the critical Rayleigh number in ammonia can be greatly increaesd over the best value for air (Ra c = 1786); the maximum observed being Ra c = 4870. Theory and experiment are shown to agree within reasonable limits. This work confirms the validity of methods which are of great importance to astrophysics and atmospheric physics. © 1964, Cambridge University Press. All rights reserved.

Description: The instability to small two-dimensional disturbances of an electrically conducting fluid of variable density is investigated. The viscous fluid is bounded between two vertical parallel planes normal to which a magnetic field of constant intensity is applied. Significant parameters upon which the behaviour of the Rayleigh number at neutral stability depends are the Hartmann number M and the wave-number α which is associated with a periodic disturbance with periodicity in the unbounded horizontal direction. The solution may be sought by considering basic disturbances which are either symmetric or antisymmetric about the median plane parallel to the boundary planes. It is found that for a given magnetic field strength the critical Rayleigh number governing stability is associated with an antisymmetric disturbance of zero wave-number. The least stable symmetric disturbance which arises when the wave-number is zero is less easily excited. This trend is seen again in the purely hydrodynamic case (M = 0) where, corresponding to a finite wave-numbe value, the more unstable mode at neutral stability is found to be an antisymmetric one. The most unstable situation occurs when both the Hartmann number and the wave number zero. In this case the result of Wooding (1960) that the minimum critical Rayleigh number is zero and is associated with a symmetric disturbance is reobtained. © 1964, Cambridge University Press. All rights reserved.

Description: This paper studies the effect of profile thickness on the propulsive forces generated by the swimming of a two-dimensional fish. Comparison of numerical calculations with reported experimental data shows good agreement and demonstrates a decrease of thrust with increasing thickness. Previous two-dimensional linearized theories on fish propulsion dealing with the motion of an infinitesimally thin hydrofoil are included in the present contribution as special cases. © 1964, Cambridge University Press. All rights reserved.

Description: Derived herein is a set of partial differential equations governing the propagation of an arbitrary, long-wave disturbance of small, but finite amplitude. The equations reduce to that of Boussinesq (1872) when the assumption is made that the disturbance is propagating in one direction only. The equations are hyperbolic with characteristic curves of constant slope. The initial-value problem can be solved very readily by numerical integration along characteristics. A few examples are included. © 1964, Cambridge University Press. All rights reserved.

Description: A simple numerical method is presented for solving the eigenvalue problem which governs the stability of Couette flow. The method is particularly useful in obtaining the eigenfunctions associated with the various modes of instability. When the cylinders rotate in opposite directions, these eigenfunctions exhibit an exponentially damped oscillatory behaviour for sufficiently large values of − μ, where μ = Ω2/Ω1. In terms of the stream function which describes the motion in planes through the axis of the cylinders, this means that weak, viscously driven cells appear in the outer layes of the fluid which, according to Rayleigh's criterion, are dynamically stable. For μ = − 3, for example, four cells are present, the amplitudes of which are in the ratios 1·0:0·0172:0·013:0·00125. © 1964, Cambridge University Press. All rights reserved.

Description: The problem of the flow near a stagnation point when the main stream outside the boundary layer fluctuates in magnitude but not in direction about a steady mean is discussed. The velocity distribution is found in the two limiting cases of small and large values of the frequency of the oscillation. The corresponding two approximate solutions give similar results in an overlapping range of frequency. © 1964, Cambridge University Press. All rights reserved.

Description: The interaction between a turbulent wind and the motion of uniform saltating grains of sand or soil, so massive as to fail to enter into suspension, is examined on the basis of two complementary hypotheses. The first asserts that the effect of the moving grains on the fluid outside the region to which saltation is confined is similar to that of solid roughness of height comparable with the depth of the saltation layer. The second requires the concentration of particles engaging in the saltation to adjust itself so that the shear stress exerted by the wind on the ground—different from that acting on the fluid outside the saltation layer by an amount accountable to the change in horizontal momentum suffered by the particles in their passage through the fluid—is just sufficient to maintain the sand-strewn surface in a mobile state. Existing experimental data on the wind profiles outside the saltation region and the horizontal flux of particles through it are shown to be consistent with these hypotheses. The second hypothesis implies a self-balancing mechanism for controlling the concentration of saltating particles. For if the concentration is too low the shear stress at the surface rises above the value required merely to secure mobility and more particles are encouraged to leave the surface; conversely, too large a concentration depresses the surface stress, and the consequent loss of surface mobility inhibits saltation and reduces th concentration of particles until equilibrium is restored. © 1964, Cambridge University Press. All rights reserved.

Description: The stability of Couette flow is discussed in the case in which the cylindes rotate in opposite directions by an asymptotic method in which the Taylor number is treated as a large parameter. On assuming the principle of exchange of stabilities to hold, the problem is then governed by a sixth-order differential equation with a simple turning point. It is shown how the solutions of this equation can be represented asymptotically in terms of the solutions of the comparison equation y vi= xy. The solutions of this comparison equation have recently been tabulated and we thus have an explicit representation of the solution of the stability problem in terms of tabulate functions. Detailed results for the critical Taylor number and wave-number at the onset of instability and the associated eigenfunctions are given for the limiting case μ → − ∞, where μ = Ω2/Ω1, and Ω1 and Ω2 are the angular velocities of the inner and outer cylinders respectively. In this limiting case it is found that there exists and infinite number of cells between the cylinders, but that the amplitude of the secondary motion in all but the innermost cell is small. © 1964, Cambridge University Press. All rights reserved.

Description: A new instrument for measuring the velocities of particles suspended in a flowing fluid is described. The instrument is linear and is therefore capable of measuring the mean velocity in a fluctuating stream, even when these fluctuations are greater than this mean value. This particular instrument was developed for free convection work where the velocities to be measured were in the range + k 0.2 in./sec to − 0.2 in./sec, but there seems to be no reason why this range could not be considerably extended. © 1964, Cambridge University Press. All rights reserved.

Description: A theoretical study is made of the behaviour of clusters of spheres falling in a viscous fluid under the assumptions that: (a) intertial effects are negligible, (b) the distance between any two spheres is larg compared with their radii. The equations of motion are derived and solved for a number of particular cases and the results compared with the experimental observations of the same motions reported in the preceding paper (Jayaweera, Mason & Slack 1964). For three or four spheres, initially in a horizontal line, the theory is in general agreement with the experiments. Three spheres forming an isosceles triangle are shown to oscillate about the horizontal and about the equlateral shape, so that this theory is unable to explain the observed tendency for three to six spheres to form a regular horizontal polygon. The stability of the steady configuration of n spheres at the vertices of a regular horizontal polygon is examined and it is found that the configuration is only stable for n 〈 7, which explains why this configuration is not observed for more than six spheres. © 1964, Cambridge University Press. All rights reserved.

Description: The Hartmann air jet generator (Hartmann 1939) is sound wave generator in which the sound is generated by oscillations of a shock developed in an over-expanded air jet by means of a blunt body: a resonator. A theory for the instability mechanism is advanced, and it is found that the result calculated from the theory are in good agreement with the experimental observations. © 1964, Cambridge University Press. All rights reserved.

Description: The uniform, slow motion of a sphere in a viscous fluid is examined in the case where the undisturbed fluid rotates with constant angular velocity Ω and the axis of rotation is taken to coincide with the line of motion. The various modifications of the classical problem for small Reynolds numbers are discussed. The main analytical result is a correction to Stokes's drag formula, valid for small values of the Reynolds number and Taylor number and tending to the classical Oseen correction as the last parameter tends to zero. The rotation of a free sphere relative to the fluid at infinity is also deduced. © 1964, Cambridge University Press. All rights reserved.

Description: Using some recent results it is established that, for very general boundary conditions, time-independent subcritical instabilities do not exist for the non-linear thermoconvective stability problem. © 1964, Cambridge University Press. All rights reserved.

Description: Analytical solutions for the quasi-one-dimensional flow of a gas not in thermodynamic equilibrium are presented for two distinct types of rate equation, namely, the linear rate equation which governs vibrational relaxation, and the nonlinear rate equation which governs dissociation. The solutions are derived for the case when, to a first approximation, the rate equation is uncoupled from the remaining flow equations. There are, in general, three distinct regions in non-equilibrium nozzle flow. First, a so-called near equilibrium region where a perturbation solution is expected to hold. This region is followed by a narrow transition-layer in which there is a rapid departure from equilibrium. Finally, downstream of this layer, the energy in the lagging mode tends asymptotically to some constant ‘frozenout’ value. The solutions applicable to each of these three regions are derived for both rate equations, the boundary conditions for the transition-layer solution and the asymptotic solution are obtained from appropriate matching procedures. In particular the structures of the asymptotic solutions are discussed. Several approximate methods for determining the asymptotic frozen level of theenergy in the lagging mode have been proposed in the literature. For the present case, when there is only a small amount of energy in the lagging mode, it is shown that none of these approximate methods is mathematically correct. © 1964, Cambridge University Press. All rights reserved.

Description: Further consideration is given to the stability of the flow of an idealized elasticoviscous liquid contained in the narrow channel between two rotating coaxial cylinders. The work of Part 1 (Thomas & Walters 1964) is extended to include highly elastic liquids. To facilitate this, use is made of the orthogonal functions used by Reid (1958) in his discussion of the associated Dean-type stability problem. It is shown that the critical Taylor number T c decreases steadily as the amount of elasticity in the liquid increases, until a transition is reached after which the roots of the determinantal equation which determines the Taylor number T as a function of the wave-number ε become complex. It is concluded that the principle of exchange of stabilities may not hold for highly elastic liquids. © 1964, Cambridge University Press. All rights reserved.

Description: Under the assumption that the local acceleration and inertial terms are to be neglected in the equation of motion, a comparison is made between the unsteadystate movement of a suspension of particles and the same problem in reversed (or negative) time. The formalized discussion confirms Bretherton's (1962) conclusion that, if in a steady unidirectional shear flow at small Reynolds number a rotationally symmetric particle twice assumes a position with its axis of rotation in the plane perpendicular to the flow, its orbit must be periodic. The reversibility phenomena observed by S. G. Mason (1963) in dilute and concentrated suspensions are explained as well. © 1964, Cambridge University Press. All rights reserved.

Description: The Rayleigh stability equation of inviscid linearized stability theory was integrated numerically for amplified disturbances of the hyperbolic-tangent velocity profile. The evaluation of the eigenvalues and eigenfunctions is followed by a discussion of the streamline pattern of the disturbed flow. Here no qualitative distinction is found between an amplified and the neutral disturbance. But considering the vorticity distribution of the disturbed flow it is shown that in the case of amplified disturbances two concentrations of vorticity occur within a disturbance wavelength, while in the neutral case only one maximum of vorticity exists. The results are discussed with respect to the instability mechanism of free boundary-layer flow. © 1964, Cambridge University Press. All rights reserved.

Description: It is shown how a simple property of the spherical vortex model can be used to investigate the dynamics of a buoyant, expanding thermal. No details of the vortex motion are required, only the fact that the flow round the buoyant region is potential. The main result is a demonstration that there is a relation between the two constants C and α arising in the dimensional analysis (where in the usual notation w = C (Δr) ½) and r = αz), which have up till now been measured separately and treated as independent. The analysis has been extended to spheroidal thermals by calculating the virtual mass for the appropriate outline, and it has also been generalized to include thermals in which the total buoyancy is increasing with time. Using these risults, and an earlier experimental verification that the mean angles of spread are nearly the same under various conditions of stability, it is suggested that the whole of the mean behaviour of a thermal can be calculated in two nearly independent steps. first, the density difference or Δ as a function of height may be calculated using purely kinematic equations of conservation. Secondly, the velocity is obtained from the local values of Δ and radius r, using a mean value of C, since this has now also been shown to vary little over a wide range of conditions. © 1964, Cambridge University Press. All rights reserved.

Description: The stability of viscous flow between rotating cylinders with an axial flow has been investigated theoretically by Goldstein (1937), Chandrasekhar (1960, 1962), and Di Prima (1960); and experimentally by Cornish (1933), Fage (1938), Kaye & Elgar (1957), Donnelly & Fultz (1960) and Snyder (1962a). As was pointed out by Di Prima (1960) there were a number of discrepancies in the early work of the 1930's which were clarified in part by the papers of the 1960's. In turn, there appear to be certain small detailed differences in the more recent papers. In part it is these differences with which the present paper is concerned. In addition, the results of the previous theoretical investigations which are limited to the case in which the cylinders rotate in the same direction, are extended to the case of counter rotation.

Description: Let D 0 be the Stokes drag on an axially symmetric body moving parallel to its axis with velocity U0 through an unbounded fluid. The drag D experienced by the same body oscillating with velocity U = U0eiσtalong its axis in the unbounded fluid is given by the expression[formula omited]where a is any characteristic particle dimension and [formula omited]is a dimensionaless number. The part of this drag formula which gives the energy dissipation is calculated for bodies of various shapes. © 1964, Cambridge University Press. All rights reserved.

Description: A solution is obtained for steady, moderately sheared, three-dimensional flow past a wire grid of arbitrary resistance distribution which is placed normal to the axis of a duct of arbitrary but constant cross-section. The formulation presented is an extension of those given by Owen & Zienkiewicz (1957) and Elder (1959) for weakly shared, two-dimensional flow past wire grids. Unlike these earlier formulations, however, in the present study the equations of motion are solved without placing restrictions on the magnitude of variation of resistance across the grid. The resulting solution, taking account of streamline deflexions, is verified experimentally for moderately sheared flow past three grids constructed to produce three widely differing velocity distributions in a water tunnel of circular cross-section. © 1964, Cambridge University Press. All rights reserved.

Description: The flow in the noise-producing region of a circular jet is found to be dominated by a group of large eddies, containing nearly a quarter of the turbulent shear stress in the quasi-plane region of the shear layer: their contribution to the shear stress decreases as the effects of axisymmetry become noticeable at more than about two diameters downstream of the nozzle. These large eddies appear to be almost entirely responsible for the irrotational fluctuations near the nozzle, which, for this and other reasons, are larger relative to the reference dynamic pressure than in other shear flows. As a consequence of this, the convection velocity near the high- and low-velocity edges of the flow is biased towards the mean velocity in the high-intensity region. The dominance of the large eddies therefore explains the measurements of near-field pressure fluctuations by Franklin & Foxwell (1958), and of convection velocity by Davies, Barratt & Fisher (1963) and the present authors. The strength of these large eddies, compared with those in the boundary layer or wake, is remarkable.The large eddies appear to be mixing-jets similar to those found by Grant (1958) in the wake, but with their projection in the (y, z)-plane inclined at about 45° to the y (radial) axis instead of lying along the y-axis as in the wake.It is suggested that the augmentation of these large eddies by artificial means could be used to increase the mixing rate and permit the reduction of jet noise by means of acceptably short ejector shrouds.The medium-scale motion is found to be far from isotropic in scales, although the two scales associated with a given vorticity component are more nearly equal. This phenomenon is also noticeable in the wake.It is found that the departure from self-preservation, which starts when the shear layer thickness is no longer small compared with the nozzle radius, does not grossly affect the region of high turbulence intensity and maximum noise production until this region itself is no longer small compared with the radius. The maximum shear stress seven diameters downstream of the exit is still 70% of its value near the exit, and the non-dimensional mean velocity gradient is practically unchanged.

Description: Using identical equipment, an attempt was made to reproduce the unexpected flow reversals obtained by Sibulkin (1962) in his study of the vortex motion associated with draining a liquid from a vessel through an orifice in its bottom. Reversal of the initial direction of rotation was found only for counter-clockwise filling with large initial heads and settling times, unless during settling or draining of the vessel a shock was applied to the system, in which case the tendency to reversal was generally more pronounced with clockwise filling. Re-reversal of the motion was also observed in a number of cases. An alternative explanation is proposed, based on a consideration of the surface waves generated by a shock. © 1964, Cambridge University Press. All rights reserved.

Description: The problem of small angle depressions in a liquid surface due to an impinging two-dimensional potential jet is considered. Using conformal mapping methods and finite Hilbert transforms, the problem is formulated as a non-linear singular integral equation. The integral equation is approximated by a set of non-linear algebraic equations which are solved numerically by a method of repeated linear corrections. In addition, an asymptotic solution (for low jet velocity) is derived. From the numerical solutions of the integral equation, the liquid-surface profiles and the free streamlines of the jet are calculated for four cases. These results verify the appearance of lips on the liquid surface which have been observed experimentally by others. © 1964, Cambridge University Press. All rights reserved.

Description: In this paper the equations for Alfvén waves are examined and conditions necessary for the occurrence of resonance are determined. An experiment using liquid sodium inside a torus is described. Waves were generated by means of alternating current supplied to a coil around the torus. A strong resonance was observed at the fundamental frequency, and a weak resonance at three times this frequency. At the fundamental resonance the alternating magnetic field attained magnitudes more than 9 times the magnitude of the field that would have been generated in free space by the exciting current. The results were in good agreement with prediction. © 1964, Cambridge University Press. All rights reserved.

Description: The travelling-wave plasma converter is analyzed allowing for the compressible nature of the fluid. The study is restricted to conditions for which the magnetic Reynolds number is small. Steady and time-varying flow conditions are investigated using small perturbation theory. Relations for optimum conversion effectiveness are established and the effects of variable electrical conductivity are investigated. Conditions for electromechanical resonance are derived and it is shown that the conversion process introduces a damping effect. © 1964, Cambridge University Press. All rights reserved.

Description: Secondary flows in non-circular ducts are accompanied by a longitudinal component of vorticity. The equation of motion defining this component in a turbulent flow is composed of three terms giving the rates of production, diffusion and convection. Since the expression for production is the second derivative of Reynolds strees components, longitudinal vorticity cannot exist in laminar flow. For turbulent flow in a square duct the Reynolds stress tensor is examined in detail. Symmetry requirements alone provide relationships showing that the production is zero along all lines of symmetry. General characteristics of flow in circular pipes are sufficient to indicate where the production must be greatest. Experimental measurements verify this result and define the point density of production, diffusion and convection of vorticity. Data also indicate that the basic pattern of secondary flow is independent of Reynolds number, but that with increasing values of Reynolds number the flows penetrate the corners and approach the walls. A similar experimental investigation of a rectangular duct shows that the corner bisectors separate independent secondary flow circulation zones. Production of vorticity is again associated with the region near the bisector. However, there is some evidence that the secondary flow pattern is not so complex as inferred from the distortion of the main longitudinal flow. © 1964, Cambridge University Press. All rights reserved.

Description: The application of a simple discrete velocity model to low Mach number Couette and Rayleigh flow is investigated. In the model, the molecular velocities are restricted to a finite set and in this study only eight equal speed velocities are allowed. The Boltzmann equation is reduced by this approximation to a set of coupled differential equations which can be solved in closed form. The fluid velocity and shear stress in Couette flow are in approximate accord with those of Wang Chang & Uhlenbeck (1954) and of Lees (1959) over the complete range of Knudsen number. Similarly, the Rayleigh flow solution is remarkably like those found by other investigators using moment methods. © 1964, Cambridge University Press. All rights reserved.

Description: The constant-density inviscid rotational flow in the neighbourhood of a general stagnation point on a wall is investigated. In all but very special cases, the solution is non-analytic and the vorticity at the wall is infinite; the stagnation streamline is tangent to the wall at the stagnation point; stagnation points of saddlepoint type cannot exist. The boundary-layer equations corresponding to the inviscid solutions studied are presented. © 1964, Cambridge University Press. All rights reserved.

Description: Magnetohydrodynamic squeeze films are investigated theoretically and experimentally. The theory of magnetohydrodynamic lubrication as applied to squeeze films is extended to include fluid-inertia effects and buoyant forces. Excellent agreement is obtained between theory and experiment. © 1964, Cambridge University Press. All rights reserved.

Description: In a previous paper on cellular thermal convection (Palm 1960) the importance of the effect caused by temperature variation of kinematic viscosity was pointed out. It was demonstrated that this effect would, owing to non-linear interactions, lead to a tendency towards hexagonal cells. For mathematical simplicity, only the interaction of two wave-components was taken into account. Segel & Stuart (1962), working with the same equations, have examined the stability of the various equilibrium solutions. They arrive at the important conclusion that a necessary condition for the solution corresponding to hexagons to be stable is that the variation of viscosity with temperature be sufficiently great. In the present paper the problem is discussed from a somewhat more general point of view. First it is shown that, when the variation of viscosity with temperature is sufficiently great, the solution corresponding to hexagons is the only stable one if only two wave-components are taken into account. To examine if this result is also true when the motion consists of an arbitrary number of wave-components, the case of three wave-components is studied. It turns out that in this case also the only possible mode is the pattern consisting of hexagons. The validity of this result is easily extended to a more general class of wave-components. It is shown that the solution corresponding to hexagons is stable for all small disturbances which can possibly occur. To prove this it is necessary to take into account non-linear disturbance theory. A reasonable conclusion from the paper by Segel & Stuart and the present paper is that a hexagonal pattern is observed only when a condition of the form (6.9) is fulfilled. Experiments concerning this problem are, however, lacking. © 1964, Cambridge University Press. All rights reserved.

Description: In a two-layer liquid system non-linear resonant interactions between a pair of external (surface) waves can result in transfer of energy to an internal wave when appropriate resonance conditions are satisfied. This energy transfer is likely to be more powerful than similar transfers between external waves. The shallow water case is discussed in detail. © 1964, Cambridge University Press. All rights reserved.

Description: Two sets of published data on an area 2700ft. by 1800ft. of sea surface in the North Atlantic are analysed by an optical computer which gives directly the directional spectrum. The results are compared with (i) those of other investigators obtained laboriously by using a digital computer, (ii) the frequency spectrum, and (iii) an empirical prediction. © 1964, Cambridge University Press. All rights reserved.

Description: The onset of steady, cellular convection driven by surface tension gradients on a thin layer of liquid is examined in an extension of Pearson's (1958) stability analysis. By accounting for the possibility of shape deformations of the free surface it is found that there is no critical Marangoni number for the onset of stationary instability and that the limiting case of ‘zero wave-number’ is always unstable. Surface viscosity of a Newtonian interface is found to inhibit stationary instability. A simple criterion is found for distinguishing visually the dominant force, buoyancy or surface tension, in cellular convection in liquid pools. © 1964, Cambridge University Press. All rights reserved.

Description: The uniqueness condition that was utilized by the author (Concus 1962) is considered. The condition, which excludes certain fluid depths, is shown to be physically unacceptable because it is essentially impossible to satisfy in practice. The resulting in validation of the perturbation method is discussed, and a revision is presented, which invokes the presence of viscosity and allows retention of the previously obtained solutions. The revision may also be applied to the work of other authors who utilized the same method to solve other standing-wave problems. © 1964, Cambridge University Press. All rights reserved.

Description: The position of the region of transition to turbulence and the manner in which turbulence develops are investigated using a hot-wire anemometer to study the character of the flow in the wake of a circular cylinder. In the range of Reynolds numbers greater than 200 in which turbulent motion is developed, the region of transition to turbulence moves towards the cylinder with increasing Reynolds number. The manner of transition to turbulence appears to undergo a basic change as the region of transition moves from the periodic wake into the region of flow immediately behind the cylinder where the separated layers have not rolled into vortices, that is, as the Reynolds number increases from Roshko's transition range to his irregular range. When transition occurs in the periodic wake it is a result of distortion due to large-scale three-dimensional effects. Turbulence, when it develops in the separated layers, is preceded by two-dimensional Tollmien-Schlichting waves which eventually degenerate to turbulence by the action of small-scale three-dimensionalities. © 1964, Cambridge University Press. All rights reserved.

Description: Experiments are described in which a spinning tube was initially filled with water and closed at both ends; when the water had acquired uniform angular velocity the tube was suddenly opened at one end and hence emptied by centrifugal action, so that a cavity progressed along it towards the far end. The velocity of the cavity was found to be steady and proportional to the speed of rotation over the range tested, which confirmed the supposition that gravity and viscosity had insignificant effects on the cavity motion. Contrary to expectation, since the cavity velocity seemed to be too large for it to occur, the ‘Taylor phenomenon’ was observed in the liquid ahead of the cavity; that is, the motion generated by the invasion of the cavity extended over a continually lengthening region beyond it. The theoretical discussion in § 4 explains several features of the experiments satisfactorily, although the complete analytical problem has so far proved insoluble. © 1964, Cambridge University Press. All rights reserved.

Description: The flow of an incompressible elastico-viscous fluid between two parallel, infinite disks is investigated when one disk is held at rest and the other performs rotary oscillations about their common axis. It is found that the purely periodic primary motion has associated with it a secondary steady velocity distribution, as well as a secondary periodic motion with twice the frequency of the primary. The steady component of the secondary flow is discussed in detail. © 1964, Cambridge University Press. All rights reserved.

Description: For flows in either rotating or stratified fluids, a technique is developed for solving initial-value problems using an Oseen approximation to the non-linear inertial terms in the equations of motion. The resulting equations for either application are similar. The solutions bear a strong qualitative resemblence to observed flows of both kinds, being characterized at small Rossby or Froude numbers by a blocked flow upstream of an obstacle and waves on the downstream side. © 1964, Cambridge University Press. All rights reserved.

Description: This paper concerns an investigation of turbulence in the density stratified shear flow of a specially designed wind tunnel in which the density gradient is created by differential heating of the air. The first three sections of the paper consist of a description of the apparatus and of the mean temperature and velocity gradients in the tunnel, together with a discussion of a method of measuring low wind speeds based on the periodic shedding of vortices by a circular cylinder. In the remaining sections details of the experimentally determined structure of the turbulence of the flow and of its eddy conductivity and viscosity are presented and their dependence on the over-all gradient form of Richardson number, [formula omitted] considered. © 1964, Cambridge University Press. All rights reserved.

Description: Large-amplitude atmospheric flows past mountain ridges are investigated. The flows are assumed to be steady and two-dimensional. Diffusive and viscous effects are neglected but static compressibility is taken into account. The larger part of the investigation is devoted to the study of waves in the lee of mountain ridges. The major contribution consists in the treatment of the large-amplitude motion. The flows are governed by an equation which turns out to be linear for certain upstream conditions. These conditions impose some restrictions on the wind profile and stratification of the entropy and specific energy far upstream. However, flow patterns representing realistic upstream conditions have been obtained. A comparison between a compressible flow and an incompressible flow with equivalent upstream conditions is included. © 1964, Cambridge University Press. All rights reserved.

Description: A body floting on the free surface of water is given a small vertical displacement from its equilibrium position and is then held fixed. When the fluid has again come to rest the body is released. The subsequent damped motion is investigated when viscosity and surface tension are neglected and the equations of motion are linearized. The method applies to bodies of arbitrary shape in two or three dimensions, and is described in detail here for the heaving motion of a horizontal half-immersed circular cylinder of radius a. The forced periodic motion of such a cylinder has been studied in earlier papers. In particular, the hydrodynamic forces exerted by the fluid on the body can be described by a dimensionless coefficient, [formula omitted], where Ω is the (real) angular frequency. The function ∧ can be found by convergent infinite processes, but not explicitly, and the difficulties of the problem are due to this. The free motion of the cylinder is solved in the present paper by Fourier methods. The motion is regarded as the superposition of simple harmonic motions, and the displacement y(t) is thus obtained in the form of a Fourier integral [formula omitted] It is seen that the integrand involves the force coefficient ∧ and is thus not strictly an explicit expression. The asymptotic behaviour for large times can be found explicitly when the depth is infinite:[formula omitted] A damped harmonic behaviour had been expected. The slow monotonic decay occurs because the function [formula omitted], when continued into the complex Ω-plane, can be shown to be many-valued near Ω = 0. No physical interpretation has yet been found for this property. The free motion of a cylinder set in motion by an applied force is also treated, with similar results. Reasons are given why there are no rapidly oscillatory terms in the asymptotic expression. For finite constant depth the function [formula omitted] is single-valued near Ω = 0, and the asymptotic expression for this case is not yet known. © 1964, Cambridge University Press. All rights reserved.

Description: In this paper flow-visualization techniques are used to study the flow instability which occurs at the orifice in the free convection open thermosyphon. The influence of the shape of the orifice on the measured heat transfer of the system is also studied under both laminar and turbulent conditions. The dominant mode of penetration of the hot stream by the cold fluid entering at the base of the reservoir is found to instigate a mixing region at the orifice. In turbulent flow this spreads into and eventually fills the tube as the Rayleigh number is increased. In such circumstances a sharp-edged orifice appears to give better overall heat transfer than a rounded orifice. © 1964, Cambridge University Press. All rights reserved.

Description: A remarkably stable type of revolving hollow water jet was generated in a vertical pipe, fixed in the base of a large tank, by first establishing Borda free flow, in which the jet springs clear of the pipe wall. Swirl was then imparted to the oncoming water, and an air core formed in the jet which was of varicose shape with alternate swellings and contractions of both its inner and outer surfaces. The observed wavelengths are compared with the theory, in which inertia and surface tension but not viscosity and gravity are taken into account. © 1964, Cambridge University Press. All rights reserved.

Description: A dispersion relation is set up for transverse waves in a plasma with an ellipsoidal velocity distribution. It is shown that the most unstable wave has its wavenormal parallel to the shortest axis of the ellipsoid and its vector potential parallel or anti-parallel to the longest axis. The maximum possible amplification rate is then calculated. Ellipsoidal velocity distributions arise in any flow with an anisotropic pressure tensor. If the resulting instability leads to a microscopic redistribution of particle velocities, then the effective transport coefficients of the plasma are changed. In Particular it is shown that the effective viscosity is decreased, and becomes dependent on the local gradient of the macroscopic velocity. © 1964, Cambridge University Press. All rights reserved.

Description: The steady two-dimensional seepage flow by gravitational convection, of a fluid of density ρ1 surrounded by a fluid of density ρ0(≠ρ1) at rest, leads to a potential problem from which the shape of the interface can be determined. When the two fluids are slightly miscible the interface is replaced by a mixing layer, and it is shown that the first-order Prandtl equations for the flow in the layer possess an exact similarity solution. The profile across the layer is of the same form as the profile of the laminar incompressible half-jet with one fluid at rest, and a formula is obtained for the scale of mixing-layer thickness as a function of distance downstream. Three examples are discussed. (a) Flow of fluid of density ρ1 from a horizontal line source. When (ρ1 〉 ρ0) a stagnation point exists above the source, and the fluid ultimately descends in a vertical column of width proportional to the source strength. At the stagnation point, the mixing-layer thickness is finite and is proportional to the square root of the radius of curvature of the interface. At a sufficiently great distance downstream, the thickness increases as the square root of the distance, as in the straight laminar half-jet. These results have been tested experimentally in a Hele-Shaw cell. (b) Symmetrical flow of an ascending column of fluid (ρ1 〈 ρ0) about an obstacle in the form of a finite horizontal strip. The column reforms after passing the obstacle, and the mixing-layer thickness returns to the value corresponding to an unobstructed vertical half-jet. The flow has been produced experimentally. (c) Flow in a lens of fresh water overlying salt water, with inflow due to precipitation, as in a two-dimensional Ghyben-Herzberg lens. Here the potential flow solution is calculated approximately by means of Dupuit-Forchheimer theory. In the steady-state solution the thickness of the mixing layer between fresh and saline water is found to be finite and, as in (a), proportional to the square root of the radius of curvature of the lens. © 1964, Cambridge University Press. All rights reserved.

Description: A critical review is given of theories dealing with viscous effects in shock tubes for cases in which the boundary layer is thin compared with the diameter. In particular an examination of the radial variation of flow quantities is used to show that the approach of Mirels (1957) is based on firmer ground than that of Trimpi & Cohen (1955). It is also shown that the small perturbation theories of these and other authors lead to an equation for running time identical in form to that derived by Roshko (1960) from an asymptotic analysis of boundarylayer leakage through the interface. © 1964, Cambridge University Press. All rights reserved.

Description: This report is an extension of an earlier note in which a simple method of estimating the distribution of vibrational temperature along a hypersonic nozzle was described. Results were presented for hyperbolic, axisymmetric nozzles with reservoir conditions 1000⩽p0⩽ 4000 p.s.i.a., 1000⩽ T0 ⩽ 3000ºK. The problem was subsequently programmed for the Ferranti Mercury computer at the University of London computing centre, and the results of these computations are given here. The vibrational temperatures are compared with those of the previous simple method. The distributions of pressure and temperature through the nozzle are also given and a simple method of estimating the vibrational temperature is described. © 1964, Cambridge University Press. All rights reserved.

Description: The cells observed by Bénard (1901) when a horizontal layer of fluid is heated from below were explained by Rayleigh (1916) in terms of buoyancy, and by Pearson (1958) in terms of surface tension. These rival theories are now combined. Linear perturbation techniques are used to derive a sixth-order differential equation subject to six boundary conditions. A Fourier series method has been used to obtain the eigenvalue equation for the case where the lower boundary surface is a rigid conductor and the upper free surface is subject to a general thermal condition. Numerical results are presented. It was found that the two agencies causing instability reinforce one another and are tightly coupled. Cells formed by surface tension are approximately the same size as those formed by buoyancy. Bénard's experiments are briefly discussed. © 1964, Cambridge University Press. All rights reserved.

Description: The viscometer investigations of Donnelly & Ozima (1962) on the stability of flow between rotating cylinders in the presence of a magnetic field have been extended by making careful torque measurements in the subcritical range using improved mercury. The results show that a small difference in stability can be observed with conducting cylinders as compared with insulating cylinders. The results also suggest that a heretofore unnoticed instability begins well before the instability studied earlier. This relatively weak instability accounts for some of the anomalous effective viscosity observed by Donnelly & Ozima. © 1964, Cambridge University Press. All rights reserved.

Description: A thin, flexible hydrofoil has been used as a model to simulate the swimming of a two-dimensional fish in an ideal, incompressible fluid, as treated in recent theoretical papers. The apparatus is described in some detail and typical data are compared with the predictions of theory. An error analysis is given, showing that, within the expected errors and limits of validity of theory, experimental verification of theory is very good. © 1964, Cambridge University Press. All rights reserved.

Description: The steady separated flow past a circular cylinder was investigated experimentally. By artificially stabilizing the steady wake, this system was studied up to Reynolds numbers R considerably larger than any previously attained, thus providing a much clearer insight into the asymptotic character of such flows at high Reynolds numbers. Some of the experimental results were unexpected. It was found that the pressure coefficient at the rear of the cylinder remained unchanged for 25 ⩽ R ⩽ 177, that the circulation velocity within the wake approached a non-zero limit as the Reynolds number increased, and that the wake length increased in direct proportion to the Reynolds number. © 1964, Cambridge University Press. All rights reserved.

Description: Solutions for two types of problems involving the energy equation for flows with velocities described by the Blasius solution are presented. The first type arises in flows with arbitrary initial distributions of stagnation enthalpy and with surfaces downstream of the initial station either with constant wall enthalpy or with zero heat transfer. Exact solutions in these cases are obtained for constant ρμ, and Prandtl number of unity; they are given in terms of complete orthogonal sets of functions which can be used to obtain first- and higher-order corrections for the effects of variable ρμ, non-unity Prandtl number, and deviations of the velocity field from that described by the Blasius solution. The second type of problem pertains to flows with power-law descriptions of the wall enthalpy. Again the basic solutions are obtained for Prandtl number of unity and the effect of non-unity Prandtl number is treated as a perturbation. © 1964, Cambridge University Press. All rights reserved.

Description: The problem solved is that of the interaction between a laminar boundary layer on a semi-infinite flat plate and an oncoming shear flow of finite lateral dimensions bounded by uniform irrotational flow extending to infinity. The pressures along the plate and upstream of the same are deduced (to a linearized approximation) in the form of a Fourier integral based on the solution of a simpler periodic flow problem. It is found that while the assumption of an infinite, uniform shear flow gives asymptotically correct interaction pressure gradients on the plate near the leading edge, the pressure level even there (compared to upstream infinity) is strongly influenced by the boundedness of the external shear. At distances from the leading edge which are large compared to the lateral extent of the shear flow, the pressure gradients along the plate are shown to be vanishingly smaller than in the infinite shear case. © 1964, Cambridge University Press. All rights reserved.

Description: Propagation of a transverse electromagnetic wave along a d.c. magnetic field and interaction with a moving shock wave is investigated. The direction of propagation is normal to the shock front. Solutions to electromagnetic fields, gas velocities and Doppler shifts in frequency are found in both the ionized and un-ionized gases. A frequency is obtained at which electromagnetic reflexion from the shock front is minimized. In the ionized gas behind the shock front a fast and a slow electromagnetic wave result. By adjusting the shock velocity the frequency of the slow wave can be either raised or lowered. This frequency change, however, is not the Doppler shift and, consequently, can be made much larger than the Doppler shifts encountered at non-relativistic velocities. The slow wave attenuates much less than the ordinary fast wave, and applications to diagnostics and communications through plasmas and laser-beam frequency multiplication may be possible. © 1964, Cambridge University Press. All rights reserved.

Description: The boundary layers due to finite viscosity and magnetic diffusivity are studied in relation to two models of the flow of a conducting fluid past a body in an aligned magnetic field. In each case it is deduced that the growth of the boundary layer may have substantial effects, such as to raise doubts about the validity of the assumed basic flow patterns. © 1964, Cambridge University Press. All rights reserved.

Description: A general theory of unsteady cavitating flow past hydrofoils and other obstacles is given for the case of cavities of finite length L. If the circulation Γ, the cavity volume V and L are known as functions of time, the theory yields explicit formulae for the velocity over the wetted surface and for the cavitation number Σ. The theory is based on the approximation that the cavity is bounded by stream-lines, and so is valid only for slow rates of change of L, V and Γ. The possibility of allowing for the presence of a vortex sheet behind the cavity is discussed. The theory is extended to the case of a cascade of hydrofoils behind which extend growing cavities. Two examples of the theory are discussed, namely the unsteady flow past a symmetrical wedge, and the unsteady flow past a flat plate hydrofoil cavitating from the leading edge. © 1964, Cambridge University Press. All rights reserved.