ALBERT

Description: The decay of small perturbations on a plane shock wave propagating along a two-dimensional channel into a fluid at rest is investigated mathematically. The perturbations arise from small departures of the walls from uniform parallel shape or, physically, by placing small obstacles on the otherwise plane parallel walls. An expression for the pressure on a shock wave entering a uniformly, but slowly, diverging channel already exists (given by Chester 1953) as a deduction from the Lighthill (1949) linearized small disturbance theory of flow behind nearly plane shock waves. Using this result, an expression for the pressure distribution produced by the obstacles upon the shock wave is built up as an integral of Fourier type. From this, the shock shape, ξ, is deduced and the decay of the perturbations obtained from an expansion (valid after the disturbances have been reflected many times between the walls) for ξ in descending power of the distance, ζ, travelled by the shock wave. It is shown that the stability properties of the shock wave are qualitatively similar to those discussed in a previous paper (Freeman 1955); the perturbations dying out in an oscillatory manner like ζ−3/2. As before, a Mach number of maximum stability (1·15) exists, the disturbances to the shock wave decaying most rapidly at this Mach number. A modified, but more complicated, expansion for the perturbations, for use when the shock wave Mach number is large, is given in §4.In particular, the results are derived for the case of symmetrical ‘roof top’ obstacles. These predictions are compared with data obtained from experiments with similar obstacles on the walls of a shock tube.

Description: This paper describes the results of an experimental investigation of the turbulent boundary layer on a flat plate with zero pressure gradient, carried out at the Laboratoire de Mécanique de l'Atmosphère (I. M. F. Marseille). Transition to turbulent flow was obtained either by increasing the preturbulence upstream of the plate by means of a grid, or by a series of emery paper roughness elements beginning at the leading edge. The measurements were of the space-time double correlations, i.e. double velocity correlations with both spatial separation and time delay, from which isocorrelation lines for optimum delay can be drawn, and of the spectrum function, all for the longitudinal component of the velocity fluctuation.

Description: This paper concerns a narrow basin of uniform cross-section open to the sea at one end and closed at the other. An incident long wave of prescribed general form is supposed to enter from the sea and to represent the combination of tide and surge as generated in the sea. The solution of the linear terms of the equations of continuity and motion gives the reflection of this wave at the head of the estuary. This paper gives the next approximation when the non-linear terms are retained, the second-order motion being made determinate by the condition that, at the mouth, it reduces to a progression towards the sea.The chief results relate to the surface elevation at the head of the estuary. When the first order elevation there increases steadily to a maximum, the effect of the ‘shallow water terms’ is to make high water higher and earlier, while the effect of the ‘frictional term’ is to make high water lower and later. For a short estuary, the interaction of the tide on a surge, due to a given sequence of meteorological conditions over the sea, is to make it higher when its maximum occurs at the time of tidal high water than when its maximum occurs at the time of tidal low water. This is directly opposite to the corresponding result when the estuary is of infinite length.

Description: This paper is concerned with the problem of obtaining higher approximations to the flow past a sphere and a circular cylinder than those represented by the well-known solutions of Stokes and Oseen. Since the perturbation theory arising from the consideration of small non-zero Reynolds numbers is a singular one, the problem is largely that of devising suitable techniques for taking this singularity into account when expanding the solution for small Reynolds numbers.The technique adopted is as follows. Separate, locally valid (in general), expansions of the stream function are developed for the regions close to, and far from, the obstacle. Reasons are presented for believing that these ‘Stokes’ and ‘Oseen’ expansions are, respectively, of the forms $Sigma ;f_n(R) psi_n(r, heta)$ and $Sigma ; F_n(R) Psi_n(R_r, heta)$ where (r, θ) are spherical or cylindrical polar coordinates made dimensionless with the radius of the obstacle, R is the Reynolds number, and $f_{(n+1)}|f_n$ and $F_{n+1}|F_n$ vanish with R. Substitution of these expansions in the Navier-Stokes equation then yields a set of differential equations for the coefficients ψn and Ψn, but only one set of physical boundary conditions is applicable to each expansion (the no-slip conditions for the Stokes expansion, and the uniform-stream condition for the Oseen expansion) so that unique solutions cannot be derived immediately. However, the fact that the two expansions are (in principle) both derived from the same exact solution leads to a ‘matching’ procedure which yields further boundary conditions for each expansion. It is thus possible to determine alternately successive terms in each expansion.The leading terms of the expansions are shown to be closely related to the original solutions of Stokes and Oseen, and detailed results for some further terms are obtained.

Description: Experiments have been done on the effects of heat transfer on wall-pressure distributions through separated regions with both laminar and turbulent boundary layers at a free-stream Mach number of about 3. The temperature of the flat plate on which the boundary layer was formed could be varied from about − 35° C to + 75° C. According to theory, this variation should have produced appreciable alterations at a laminar separation point in either the pressure or the pressure gradient, but no sign of this appeared in the overall pressure distributions, which, for laminar layers, remained unaffected by wall temperature. A possible explanation is given for this apparent discrepancy between theory and experiment. With turbulent layers, the variations in wall temperature did produce small changes in the pressure distributions. However, for most practical purposes such changes could be ignored. Hence the convenient conclusion is suggested that in supersonic separating flow with either a laminar or a turbulent boundary layer the pressure distributions are not significantly affected by moderate variations in wall temperature.

Description: The damping of surface waves of small amplitude in liquid contained in cylinders has been calculated. Viscous dissipation in an assumed laminar boundary layer was taken to be the primary cause of damping. Experimental results were obtained for the logarithmic decrement as a function of the ratio of liquid height to cylinder radius for several water-filled cylinders. Theory and experiment were found to be in good agreement.

Description: An analytical expression for the fringe shift in the N-wave is derived from the improved linearized theory for a slender supersonic projectile. From this expression an approximate mapping function is found which gives a simple test for N-wave flow. The validity of the fringe-shift expression is quantitatively confirmed by measuring an interferogram of the flow around a sphere. N-wave flow is shown to exist around a small sphere for r greater than about 70 diameters. Measurements of the shock waves from a sphere and a cone-cylinder show that the shocks assume their asymptotic positions for r 〉 14 diameters for the sphere, and r 〉 7 diameters for the cone-cylinder.

Description: According to the simple hydraulic theory of jets, each particle of a jet moves independently along a parabolic trajectory. Therefore a steady jet has a parabolic shape. We wish to consider how these results are modified by surface tension. For simplicity we will consider a two-dimensional jet of incompressible fluid.

Description: The finite difference method due to Lax (1954) is used to solve the equations of motion for a cylindrically symmetric flow of a compressible fluid. In particular, a converging cylindrical shock is found to increase in strength in agreement with the formula of Chisnell (1957). The artificial diffusion introduced by the method causes the pressure to remain finite at the axis, but a reflected diverging shock is obtained. © 1957, Cambridge University Press. All rights reserved.

Description: Shock waves generated in a shock tube by use of hydrogen or helium as a driver gas and air, nitrogen, oxygen or argon as a driven gas have higher velocities than predicted by simple theory when sufficiently large diaphragm pressure ratios are used. Expected shock-tube performance curves have been constructed using the equilibrium Hugoniot for the driven gas, both for the usual model of shock tube flow, which assumes instantaneous diaphragm removal, and for a suggested model based on a finite rupture time for the diaphragm. Agreement between experiment and the latter model is in general good, and the differences are qualitatively accounted for by the pressure waves expected to result from mixing between driver and driven gases at the contact surface. These waves may be either compression or expansion waves, depending on the relative heat capacities of the two gases. The maximum shock strength observed as a shock goes down the tube was found to occur at a distance from the diaphragm which increases with the shock strength, and the strongest shocks were found to be still accelerating at the end of a 42 ft. long shock tube of 3 1/2 in. square cross-section. Diaphragm breaking time has been measured and found to be consistent with the observations on the shock formation distance. © 1958, Cambridge University Press. All rights reserved.

Description: A set of two-dimensional subsonic flows past certain cylinders is obtained using hodograph methods, in which the true pressure-volume relationship is replaced by various straight-line approximations. It is found that the approximation obtained by a least-squares method possibly gives best results. Comparison is made with values obtained by using the von Kármán-Tsien approximation and also with results obtained by the variational approach of Lush & Cherry (1956). © 1958, Cambridge University Press. All rights reserved.

Description: Gravity waves occur on the surface of a liquid such as water, and the manner in which they propagate depends upon its depth. Although this dependence is described in principle by the equations of the ‘exact linear theory’ of surface waves, these equations have not been solved except in some special cases. Therefore, oceanographers have been unable to use the theory to describe surface wave propagation in water whose depth varies in a general way. Instead they have employed a simplified geometrical optics theory for this purpose (see, for example, Sverdrup & Munk (1944)). It has been used very successfully, and consequently various attempts, only partially successful, have been made to deduce it from the exact linear theory. It is the purpose of this article to present a derivation which appears to be satisfactory and which also yields corrections to the geometrical optics theory.

Description: A mechanism is proposed by which cellular convective motion of the type observed by H. B énard, which hitherto has been attributed to the action of buoyancy forces, can also be induced by surface tension forces. Thus when a thin layer of fluid is heated from below, the temperature gradient is such that small variations in the surface temperature lead to surface tractions which cause the fluid to flow and thereby tend to maintain the original temperature variations. A small disturbance analysis, analogous to that carried out by Rayleigh and others for unstable density gradients, leads to a dimensionless number B which expresses the ratio of surface tension to viscous forces, and which must attain a certain minimum critical value for instability to occur. The results obtained are then applied to the original cells described by Bénard, and to the case of drying paint films. It is concluded that surface tension forces are responsible for cellular motion in many such cases where the criteria given in terms of buoyancy forces would not allow of instability. © 1958, Cambridge University Press. All rights reserved.

Description: A formal solution to the initial value problem for a plane vortex sheet in an inviscid fluid is obtained by transform methods. The eigenvalue problem is investigated and the stability criterion determined. This criterion is found to be in agreement with that obtained previously by Landau (1944), Hatanaka (1949), and Pai (1954), all of whom had included spurious eigenvalues in their analyses. It is also established that supersonic disturbances may be unstable; related investigations in hydrodynamic stability have conjectured on this possibility, but the vortex sheet appears to afford the first definite example. Finally, an asymptotic approximation is developed for the displacement of a vortex sheet following a suddenly imposed, spatially periodic velocity. © 1958, Cambridge University Press. All rights reserved.

Description: A shock wave is generated by a uniform compressive piston motion and passes into a channel of slowly varying cross-section. A relation in closed form is obtained between shock strength and the area of the channel and is used to discuss converging cylindrical and spherical shocks. © 1958, Cambridge University Press. All rights reserved.

Description: This paper is a discussion of recent experiments in shock-wave refraction which have clarified a special type of shock outflow process appearing to have relevance to other shock interactions, and notably to shock reflection from an oblique wall. For certain incident shock strengths and angles of incidence α, the air/methane refraction problem simulates closely the situation in the trouble-some range of the reflection problem, in which α lies between the value αe at which the theoretical solutions terminate and the value α0 that marks the onset of Mach reflection, and in which the flow deflections cannot be reconciled with theoretically permissible reflected shock strengths. In the analogous refraction cases, the reflected shock is observed to increase in strength along its length to a maximum value at the intersection point, and to be followed by a subsonic rarefaction zone which also increases in severity near the intersection. In fact, this zone appers to coalesce into a subsonic discontinuity, just at the intersection point—a feature which would contradict one of the basic assumptions of the regular reflection and refraction theories. Other refraction experiments suggest that a similar process is relevant to the Mach reflection configuration, and may account for the discrepancies in the three-shock theory for weak incident shocks.

Description: In this paper transonic small-disturbance theory is applied to a simplified model of the flow near the front of a ducted body. The body is assumed to consist simply of two parallel flat plates which extend from the inlet station to infinity downstream. The velocity far upstream is sonic, and the velocity far downstream in the duct, which is assumed to be known, is slightly subsonic. Air is therefore ‘spilled’ around the intake edges. An analytic solution is found for the resulting flow field up to the ‘limiting Mach wave’, and asymptotic solutions are found for the supersonic flow and for the shock wave far from, and near, the intake edges. The pressure distribution along the outside walls is then known at both ends, and its computation is completed by an empirical procedure. Distributions of pressure along the centre-line and along the inside and outside walls are shown. These results may be used to compute the drag of sharp-edged intakes with a very small frontal area. © 1958, Cambridge University Press. All rights reserved.

Description: The subterranean mixing in permeable media of sea water and ground water is studied. The model for this mixing process which was suggested by C. K. Wentworth is adopted, but is soon discarded in favour of a more tractable formulation whose equivalence to the original model is established. The analysis is carried to the point where the determination of the salinity distribution of the ground water in a given subsoil requires only the solution of an elementary linear ordinary differential equation. © 1958, Cambridge University Press. All rights reserved.

Description: Two concentric spheres are supposed to rotate about the same axis with almost the same angular velocity, so that the viscous stresses over the surfaces of the spheres induce a flow which may be represented by a small perturbation superimposed upon a rigid body rotation of the fluid as a whole. The governing equations are therefore linearized in the magnitude of the perturbation, and it appears that the validity of this linearization is independent of the Reynolds number of the primary rotation. Attention is then restricted to the case in which the Reynolds number is large, the principal object of the note being to exemplify some of the properties of rotating systems at large Reynolds numbers in terms of a particularly simple mathematical model. It is found that the cylindrical surface that touches the inner sphere (the axis being the axis of rotation) is a singular surface in which velocity gradients are very large. Everywhere outside this cylinder, the fluid rotates as a rigid body with the same angular velocity as the outer sphere. Inside the cylinder, the velocity distribution in the central (inviscid) core of the motion is shown to be determined by the velocity distribution in the boundary layers over the spheres, and explicit solutions are obtained for all these velocity distributions. The mechanics of the cylindrical shear layer itself is also discussed, though no explicit solution is obtained in this case. © 1956, Cambridge University Press. All rights reserved.

Description: This paper considers the flow at high Mach number behind the curved shock formed when a supersonic stream impinges on an axisymmetrical body with a rounded nose. The solution is obtained as a double expansion in [formula omitted] where γ is the adiabatic index, and M−2; the expansion is developed to within terms of order (δ+M−2)3. Expressions are obtained for the distance between the body and the shock, the radius of curvature of the shock compared with that of the body, and the pressure distribution on the body. © 1956, Cambridge University Press. All rights reserved.

Description: An inequality is demonstrated involving the rate of production S of mean-square vorticity in isotropic turbulence and a factor γ which may be said to allow for intermittency or for the non-vanishing of fourth-order cumulants. An extreme state, corresponding to equality of this relationship, occurs if S = 0·64 and γ = 2·14. The experimental values are S = 0·4 and γ = 4. Another kinematical relation shows that the mechanism of vorticity production resembles collision between fluid particles rather than the swirling of contracting jets. © 1956, Cambridge University Press. All rights reserved.

Description: The method of characteristics for steady supersonic flow problems in three dimensions, due to Coburn & Dolph (1949), is extended so that flow with shocks and entropy changes may be treated. Equations of motion based on Coburn & Dolph's characteristic coordinate system are derived and a scheme is described for solving these by finite differences. A linearized method of characteristics is developed for calculating perturbations of a given three-dimensional field of flow. This is a generalization of the method evolved by Ferri (1952) for perturbations of plane flow and conical flow. © 1956, Cambridge University Press. All rights reserved.

Description: The ‘Newtonian-plus-centrifugal’ approximate solution (Busemann (1933) and Ivey (1948)) for hypersonic flow past plane and axially symmetric bluff bodies in gases with the ratio of the specific heats λ constant and equal to unity is rederived using ‘boundary layer’ techniques together with the von Mises variables x and ψ. A method of successive approximations then gives a closer approximation to this solution for ε (λ − 1)/(λ + 1) small and the free-strea Mach number infinite. Formulae for the streamlines, shock shape and pressure distribution are determined to this approximation. These formulae are valid for any plane or axially symmetric shape, giving the ‘stand-off’ distance of the shock wave from the body as ½εlog(4|3ε) and ε times the nose radius of curvature for plane and axially-symmetric flows respectively. Particular results are computed for a number of special shapes. For certain shapes, the theory has a singular point where the first approximation to the pressure vanishes (θ = 60° for a sphere). Actually, the theory is not applicable where the pressure becomes too small. The corresponding theory for gases of general thermodynamic properties is deduced, the approximation being valid provided the total energy of the gas is large compared with the energy contained in the translational modes of the gas molecules. © 1956, Cambridge University Press. All rights reserved.

Description: Propagation of spherical shock waves through self-gravitating polytropic gas spheres such as stars, caused by an instantaneous central explosion of finite energy E, is discussed theoretically. The problem is characterized by two lengths R0, L, where [formula omitted] p0 and C0 are the values of pressure, density and velocity of sound at the centre of the equilibrium pre-explosion state, and G is the constant of gravitation. R0 and L are scales connected with the power of the explosion and the dimensions of the star respectively, and their ratio A = R0/L has a fundamental significance. A solution especially suitable in the case of A = O(1) is developed in the form of power series in R/R0 (R is the distance between the shock front and the centre) by a method similar to that used in previous papers by the present author (1953, 1954). An approximation to this solution is carried out up to the term in R3. In particular, the velocity of the shock wave U is found to be [formula omitted] for the case of λ = 1.4, where λ is the ratio of specific heats. © 1956, Cambridge University Press. All rights reserved.

Description: The compression of a perfect gas between a uniformaly moving piston and a rigid wall is discussed in the one-dimensional case. If the piston moves with a finite speed, it will initiate a shock in the gas which will reflect successively from rigid wall and piston and cause the compression process to deviate from a reversible adiabatic process. Expressions are derived for the relative changes in pressure and density at each shock reflection. Then values of density and pressure after any number of shock reflections are computed relative to their initial values, and, in terms of these, the corresponding values of temperature and entropy, as well as shock speeds, are determined. The limiting value of the entropy change, as the number of reflections goes to infinity, is obtained as a function of the ratio of specific heats of the gas and the strength of the initial shock. Hence it is possible to estimate an upper limit to the deviation of the shock compression process from a reversible adiabatic process. Some illustrative numerical examples are given. © 1956, Cambridge University Press. All rights reserved.

Description: An extension of the linearized theory of supersonic flow past quasi-cylindrical bodies of almost circular cross-section has been found which enables a direct calculation to be made of the overall forces on wings mounted on such bodies, subject to certain restrictions on the plan-form. The method is applied to two examples: (i) the effect of an arbitrary body distortion on static stability at supersonic speeds; and (ii) the effect of wing-body interference on rectangular wings mounted on a cylindrical body. The drag calculations in the second example are compared with the results of the supersonic area rule, which is found to be in error for moderate values of the ratio of wing chord to body radius, though the discrepancy is not serious from a practical point of view. © 1958, Cambridge University Press. All rights reserved.

Description: At sufficiently high temperature the oxygen and nitrogen molecules in air dissociate into atoms. Energy is required for this decomposition and, furthermore, if the temperature is not uniform, concentration gradients are formed and energy is transferred by the consequent interdiffusion of atoms and molecules. In this paper, a simple model of a gas is formulated to illustrate the effect of these processes on heat transfer in three situations: (1) in a layer of gas at rest between two walls at different temperature, (2) in Couette flow, and (3) in a laminar boundary layer on a flat plate. In the model, the law for the rate of reaction (dissociation and recombination) is of an especially simple form with the speed of reaction characterized by a single parameter. When the parameter becomes large, the gas approaches chemical equilibrium; at the other extreme, no reaction occurs within the gas. Two chemical conditions of the walls are considered, one being that the walls are catalytic (the surface reaction rate is assumed sufficiently high to hold the gas at the wall in chemical equilibrium), and the other that the walls have no effect on the reaction, i.e. they are non-catalytic. The most important of the simplifications made are: (a) The reaction rate law is put in a form in which the equilibrium concentration of atoms varies linearly with temperature. Thus, there is only one temperature at which the gas is undissociated instead of the actual range of temperature. (b) In problems (1) and (2), μ, κ, and ρD are taken to be constant. (c) In problem (3) the Lewis number is assumed to be unity. (d) Only binary mixtures of atoms and molecules are considered (so that ionization and more complex dissociation processes are not covered). (e) Coupling of irreversible flows, such as that causing thermal diffusion, is neglected. (f) Only problems in which the pressure is constant are considered. © 1958, Cambridge University Press. All rights reserved.

Description: The approximate Tricomi equation relevant to sonic speed of two-dimensional small-distrubance flow is solved by separation of variables, where these are certain stipulated functions of the Cartesian velocity perturbations. The symmetric flow patterns obtained from this solution are shown to correspond to those about half-bodies, whose ordinates vary as xn where 0·4 &les n,≪1, and x is the distance along the plane of symmetry. The surface pressures on such bodies are deduced. In particular, the body whose ordinates vary as has a sonic surface velocity, except at the nose, where an edge-force which causes a drag force is shown to exist. On the assumption that a free-stream breakaway (at sonic velocity) occurs at the shoulder of a body, this solution thereby yields the flow about an aerofoil of the same shape having a flat base. This bluff-nosed section has only a little more than half the drag of a wedge on the same chord and base. © 1958, Cambridge University Press. All rights reserved.

Description: A mathematical model is constructed for cavitating flow past a wedge with sides of equal length but with its axis of symmetry placed at an angle to the incident stream. The model involves a subsidiary cavity with a re-entrant jet at the vertex. Only the case of zero cavitation number is considered. The flow field is worked out in some detail for small angles of incidence, and the lift, drag and moment coefficients are calculated as far as first-order terms in the angle of incidence. It is shown that the effect of the rate of loss of momentum in the re-entrant jet on these force coefficients is negligible to this order. Experimentally, it is shown that the secondary cavity does exist under suitable conditions, and the force coefficients obtained agree with the theory. © 1958, Cambridge University Press. All rights reserved.

Description: An estimate is given of the distribution of skin frictional force per unit length, and of displacement area, on the outside of a semi-infinite cylinder, of arbitrary cross-section, moving steadily in a direction parallel to its generators. A Pohlhausen method is employed with a velocity distribution chosen to yield zero viscous retarding force on the boundary layer approximations. (The smallness of the fluid acceleration far from the leading edge has been pointed out by Batchelor (1954).) Like the Rayleigh method, this method is expected to yield reasonable results at large distances from the leading edge. However, for a large class of cross-sections, which includes all convex cross-sections and locally concave cross-sections with re-entrant angles greater than ½Π, the method yields the expected square root growth of the boundary layer at the leading-edge, with a fairly close approximation to the coefficient, and it is supposed that the skin-frictional force and displacement area are given with reasonable accuracy along the whole length of the cylinder. Results for the elliptic cylinder and the finite flat plate are given in closed form, valid for the whole length of the cylinder, and are expected to be in error by at most 20%. In addition, some estimate is given of the effect of corners on skin frictional force and displacement area. © 1958, Cambridge University Press. All rights reserved.

Description: This paper describes theoretical and experimental studies of the effects on shock tube flows of a monotonic convergence at the diaphragm section. Systematic flow equations are developed for tubes of uniform bore and tubes having either a monotonic convergence or a convergence-divergence in the diaphragm section. Except across the shock front itself, isentropic processes and ideal-gas behaviour have been assumed. Simplified procedures are presented for predicting the ideal-flow parameters over a wide range of operating conditions, as well as for comparing straight and convergent tubes. Such comparisons made by other investigators are found to be incomplete or in error. The experiments described utilize a very simple device for altering the diaphragm section convergence and a multi-station measurement of shock velocity. The expected effect of convergence is verified over a wide range of Mach numbers. Even at Mach numbers where the processes of shock formation can no longer be ignored, it is found that the relative performance between a uniform and convergent tube is preserved. © 1958, Cambridge University Press. All rights reserved.

Description: A large number of small spheres of the same size were injected successively into a horizontal pipe conveying water at constant mean velocity, and their times of transit were measured. The mean velocity of the spheres that were either somewhat heavier or lighter than water was less than that of those of neutral density; for those having a terminal velocity in water within ± 1% of the mean velocity of the water in the pipe, the discrepancy was only about 0.1%. The dispersion of the times of transit of the spheres was almost independent of their density. A theory is developed to show how the mean velocity of the spheres depends upon their relative density and size. © 1958, Cambridge University Press. All rights reserved.

Description: The boundary layer on a hot flat plate which is fixed at zero incidence in a slow stream carrying a progressive sound wave is investigated. Formulae are obtained for the skin friction and the heat transfer in the extreme cases when the frequency is very low and very high. In addition, different methods of simplifying the boundary layer equations in unsteady compressible flow are briefly compared. © 1958, Cambridge University Press. All rights reserved.

Description: Transition in the boundary layer on a flat plate is examined from the point of view of intermittent production of turbulent spots. On the hypothesis of localized laminar breakdown, for which there is some expermental evidence, Emmons’ probability calculations can be extended to explain the observed statistical similarity of transition regions. Application of these ideas allows detailed calculations of the boundary layer parameters including mean velocity profiles and skin friction during transition. The mean velocity profiles belong to a universal one-parameter family with the intermittency factor as the parameter. From an examination of experimental data the probable existence of a relation between the transition Reynolds number and the rate of production of the turbulent spots is deduced. A simple new technique for the measurement of the intermittency factor by a Pitot tube is reported. © 1958, Cambridge University Press. All rights reserved.

Description: In the case of turbulent flow in a pipe there is a lower experimental number to the Reynolds limit for which fully developed turbulent flow occurs. From the similarity and close agreement of the curves showing the coefficient of skin friction cf as a function of the Reynolds number Rθ (based on the momentum thickness θ) for the circular pipe and flat plate, it is suggested that there should be a lower limit to Rθ for fully developed turbulent flow on a flat plate. Rather limited experimental data confirm this and place the lower limit at Rθ = 320. The choice and size of transition device is examined in relation to this minimum Rθ and an approximate theory leads to a ‘wire’ Reynolds number in fair agreement with experience. © 1958, Cambridge University Press. All rights reserved.

Description: Fluctuations of velocity and temperature which occur in a turbulent flow in a stably-stratified atmosphere far from restraining boundaries are discussed using the equations for the turbulent intensity and for the mean square temperature fluctuation. From these, an equation is derived for the flux Richardson number in terms of the ordinary Richardson number and some non-dimensional ratios connected with the turbulent motion. It is shown that the interaction between the temperature and velocity fields imposes on the flux Richardson number an upper limit of 0·5, and on the ordinary Richardson number a limit of about 0·08. If these values are exceeded, no equilibrium value of the turbulent intensity can exist and a collapse of the turbulent motion would occur. Although the analysis applies strictly only to a homogeneous non-developing flow, it should have approximate validity for effectively homogeneous, developing flows, and the predictions are compared with some recent observations of these flows.

Description: The speed of free floating bodies on the surface of a water stream in a sloping channel has been found to be sensibly the same as the mean speed of the layer of water in which the bodies are floating, contrary to some recorded opinions. © 1956, Cambridge University Press. All rights reserved.

Description: This paper proposes a theory of collisions between small drops in a turbulent fluid which takes into account collisions between equal drops. The drops considered are much smaller than the small eddies of the turbulence and so the collision rates depend only on the dimensions of the drops, the rate of energy dissipation ε and the kinematic viscosity v. Reasons are given for believing that the collision rate due to the spatial variations of turbulent velocity is shown to be [formula omitted], valid for [formula omitted] between one and two. A numerical integration has been performed using this expression to show how an initially uniform distribution will change because of collisions. An approximate calculation is then made to take account also of collisions which occur between drops of different inertia because of the action of gravity and the turbulent accelerations. The results are applied to the case of small drops in atmospheric clouds to test the importance of turbulence in initiating rainfall. Estimates of ε are made for typical conditions and these are used to calculate the initial rates of collision, the change in mean properties and the rate of production of large drops. It is concluded that the effects of turbulence in clouds of the layer type should be small, but that moderate amounts of turbulence in cumulus clouds could be effective in broadening the drop size distribution in nearly uniform clouds where only the spatial variations of velocity are important. In heterogeneous clouds the collision rates are increased, and the effects due to the inertia of the drop soon become predominant. The effect of turbulence in causing collisions between unequal drops becomes comparable with that of gravity when ε is about 2000 cm2 sec−3. © 1956, Cambridge University Press. All rights reserved.

Description: This paper is concerned with a progressive wave of general form in an infinitely long estuary of uniform cross-section when there is a permanent current independent of the wave. The only approximation is the neglect of friction. Explicit formulae in the form of infinite series are found for the surface-elevation and for the current. In the special case in which there is no permanent current and when the oscillation at the mouth of the estuary reduces to a single harmonic constituent, the first five harmonic shallow water constituents at any place up the estuary are evaluated. © 1958, Cambridge University Press. All rights reserved.

Description: The Kelvin-Helmholtz model for the formation of surface waves at the interface between two fluids in relative motion is generalized for parallel shear flows. It is assumed that phase changes across the flow are negligible and hence that the aerodynamic pressure on the wave is in phase with its displacement (rather than its slope). A variational formulation is established and leads to the determination of appropriately weighted means for the velocity profiles. The principal application is to flow of a light inviscid fluid over a viscous liquid; it is shown that the principle of exchange of stabilities is applicable to such a configuration, and a critical wind speed in satisfactory agreement with observation is predicted for an air-oil interface. The results also are applied to an air-water interface and lead to the conclusion that Kelvin-Helmholtz instability of such an interface is unlikely at commonly observed wind speeds. A more general formulation of the Kelvin-Helmholtz boundary-value problem and variational principle, allowing for variations in both velocity and density, is given in two appendices.

Description: In the following it will be shown that a simple argument based on the use of the energy integral equation of the laminar boundary layer permits the derivation of a heat transfer formula valid for non-uniform temperature distribution and non-zero pressure gradients. The formula is then shown to be identical in structure with Lighthill's (1950) well-known results. Lighthill obtained his formula by solving the boundary layer equations in the von Mises form using operational methods. An elegant way to obtain the same results using exact similarity consideration was given by Lagerstrom (not yet published). The derivation given here is probably the most simple-minded one and the method may be useful for other applications as well. Furthermore, it is shown that the approach can be slightly modified to permit application of the formula to flow near separation. The latter result is applied to the Falkner-Skan solution for just separating flows and is found to be in excellent agreement with the exact solutions.

Description: A procedure is given for translating boundary-value problems of gas dynamics from microscopic form into approximately equivalent continuum form. The continuum formulations involve state-variables that are either half-space moments, or complete moments of the molecular distribution functions. Moment equations derived from the kinetic equations are reduced to a determinate set by representing the distribution functions as sums of ‘modified Maxwellian functions based on various characteristic temperatures and velocities’. The particular choice of such a representation depends on the Knudsen number and on the nature of the microscopic boundary conditions.

Description: A previous analysis for the generation of surface waves by a parallel shear flow (Miles 1957 a) is extended by: (a) presenting results based on a more accurate solution of the differential equation; (b) imposing the boundary condition at the surface wave, rather than at the mean surface; and (c) including the dominant viscous term in the complete Orr-Sommerfeld equation. The modification (a) yields an energy transfer somewhat smaller than that predicted previously but of the same order of magnitude as, and in rather better agreement with, observation, while (b) has no effect and (c) only a small effect for gravity waves. The analysis is based on the equations of motion in intrinsic co-ordinates (rather than the usual Orr-Sommerfeld equation) and may be of interest in other problems of hydrodynamic stability.

Description: The effects of density variation in the absence of gravity on the stability of a horizontal shear layer between two streams of uniform velocities is investigated. The density is assumed to decrease exponentially with height and the velocity is represented by U(y) = tanh y.The method of small disturbances is employed to obtain the neutral stability curve. It is demonstrated that disturbances with wave-numbers larger than the width of the transition layer are attenuated.Qualitative agreement with experimental evidence is obtained.

Description: First, from a volumetric formulation of the momentum theorem of linearized theory, a general analytic proof is presented of the invariance of the drag of an arbitrary spatial distribution of horseshoe vortices and sources under reversal of the undisturbed flow. By consideration of the interference drag of two such singularity distributions, a reverse-flow relation for steady subsonic or supersonic flow is then obtained. This relation, a generalization of the Ursell-Ward theorem, may be applied to configurations with bodies whose surfaces are not quasi-cylindrical and whose surface pressures are quadratically related to the perturbation velocity.The relation is used to discuss several interfering two-body arrangements in supersonic flow. It is shown that, in certain cases, the drag and lift may be determined without knowledge of the interference flow field associated with the arbitrarily prescribed body geometry. The simplicity of the results permits the formulation of optimum problems. The invariance of the drag under flow reversal with unchanged geometry is also established.

Description: To determine experimentally the mean value of a randomly fluctuating quantity, it may be necessary to measure the average value over a considerable interval of time. This problem arose in a recent study of the temperature fluctuations over a heated horizontal plate, and a system was used that depended on the counting of electrical pulses generated at a rate proportional to the quantity being measured. The advantage of this technique is that mean values may be measured over time intervals of almost unlimited length with little added difficulty for the experimenter. Circuits are described which measure: (a) the mean square of a fluctuating quantity and of its time-derivative, (b) the statistical distribution of the fluctuations, (c) the mean frequency of the fluctuation assuming a particular value, and (d) the mean product of two fluctuating quantities. Over the range of use, the stability and linearity of the calibrations is better than 1%, more than sufficient for work on natural convection. In its present form, the equipment responds uniformly to all frequencies below 100 c/s, but it would not be difficult to extend this range of response to higher frequencies.

Description: On the assumption that the rotational oscillations of a rigid plane are small, boundary-layer type solutions of the Navier-Stokes equations are attempted by an expansion of the velocity components in power series of the amplitude. First-and third-order approximations to the transverse velocity are obtained, from which a correction to the moment on a disk of finite radius is found.The first non-vanishing approximation to the radial-axial flow (a second-order term) is seen to have a steady component and a component with frequency twice that of the plate. The former component appears to persist outside the boundary layer, and at large distances from the plate to have the character of an irrotational stagnation flow. A re-examination not involving the series approximation reveals that although steady radial flow does exist outside the boundary layer, it is a viscous flow and is confined within a secondary layer. The ratio of the thicknesses of the two layers is found to be inversely proportional to the amplitude of the oscillations. These results indicate that a second-order flow in a region where the first-order flow field vanishes should not be accepted without further discussion.

Description: Some ablating bodies entering the atmosphere will melt or soften. Under deceleration, the soft or melted surface will tend to develop instabilities of the Lamb-Taylor type. Two situations involving viscous incompressible fluids are investigated here: one where the liquid layer has constant viscosity and finite thickness, and the other, where the viscosity increases exponentially with distance away from the interface, and the layer is semi-infinite in extent.It can be demonstrated, that if one neglects gradients in the flow direction, the rate of growth of interface disturbances in a plane normal to the axis of an axially symmetric blunt body is independent of the flow velocity. The fact that the deceleration and liquid thickness vary with time along a trajectory is also included in the analysis. Results of calculations for the amplification factor and the most amplified wavelength are given.A mechanism due to the deceleration is postulated, which would cause the formation of longitudinal grooves on the surface of an axially symmetric blunt body while entering the atmosphere.

Description: A hinged vane and a sensitive electrical system for recording the motion of the vane have been developed for the observation of fluctuating y-components of velocity in boundary layers. An approximate theory of the natural oscillations of such vanes is presented and experimentally verified. Using vanes as resonant detectors, meassurements have been made of oscillations injected into the laminar boundary layer on a flat plate in a wind tunnel with 0·3% free-stream turbulence. Points on the neutral Tollmien-Schlichting curve have thereby been obtained which lie close to the theoretical neutral curve.

Description: The uniform steady flow of an incompressible, viscous, electrically conducting fluid is distorted by the presence of a symmetrically oriented semi-infinite flat plate. The ambient magnetic field is coincident with the ambient velocity field. The description of the resulting fields depends on the physical co-ordinates measured in units of Reynolds number and on the two parameters ε = ωμν and β = μH2/ρv2. This description of the fields is approximated in three different ways and essentially covers the full range of ε and β. In particular, when β [Gt ] 1, no steady flow which is uniform at large distances from the plate exists.

Description: Low Reynolds number flow of a conducting fluid past a sphere is considered. The classical Stokes solution is modified by a magnetic field which, at infinity, is uniform and in the direction of flow of the fluid. The formula for the drag is found to be [Formula omitted] Where DS is the Stokes drag and M is the Hartmann number. © 1957, Cambridge University Press. All rights reserved.

Description: Exact transfer calculations for boundary layers with longitudinal pressure gradients are very complicated, but in the literature several approximate methods are known for the rapid calculation of both the wall friction and the heat transfer. A ‘wedge method’ propounded by Meksyn turns out to be one of the most rapid methods, being no less accurate than other approximate methods. A way of refining this method is proposed.This paper also shows that asymptotic expansions provide convenient relations which are capable of expressing the Nusselt number explicitly in terms of the Prandtl number.It is shown that, together with the asymptotic expansions, Meksyn's method permits rapid calculation of local heat transfer numbers. Some examples of application are given for elliptical cylinders and spheres for several values of the Prandtl number.

Description: A physical theory is proposed for the skin friction on a flat plate at zero incidence in the transition flow regime, i.e. in the flow of a moderately rarefied gas. The ratio of the molecular mean free path to the characteristic size of the plate is assumed of order unity or larger. A general formula for the perturbation to the well-known friction of the free-molecule theory is given. This perturbation is attributed to the intermolecular collisions which are neglected on the basis of the free-molecule hypothesis. The expected rate of collisions are calculated for rigid spheres, using the classical kinetic theory.Although this is intended as an approximate theory, the theoretical results check surprisingly well with the limited experimental data that are available. The present theory shows that the ratio of the Reynolds number to the Mach number squared is the governing parameter for determining the intermolecular collision effect on skin friction in the transition flow regime.

Description: A mechanism for the generation of surface waves by a parallel shear flow U(y) is developed on the basis of the inviscid Orr-Sommerfeld equation. It is found that the rate at which energy is transferred to a wave of speed c is proportional to the profile curvature -U“(y) at that elevation where U = c. The result is applied to the generation of deep-water gravity waves by wind. An approximate solution to the boundary value problem is developed for a logarithmic profile and the corresponding spectral distribution of the energy transfer coefficient calculated as a function of wave speed. The minimum wind speed for the initiation of gravity waves against laminar dissipation in water having negligible mean motion is found to be roughly 100cm/sec. A spectral mean value of the sheltering coefficient, as defined by Munk, is found to be in order-of-magnitude agreement with total wave drag measurements of Van Dorn. It is concluded that the model yields results in qualitative agreement with observation, but truly quantitative comparisons would require a more accurate solution of the boundary value problem and more precise data on wind profiles than are presently available. The results also may have application to the flutter of membranes and panels. © 1957, Cambridge University Press. All rights reserved.

Description: The motion of a viscous fluid contained between two rotating, circular cylinders whose axes are set slightly apart is considered. The equations of viscous motion are linearized by expanding the stream function in the form [formula omitted], where γ is a parameter which depends on the distance between the cylinder axes. The ensuing analysis appears to hold for all values of the fluid viscosity v, and in particular for small values of v. The asymptotic behaviour of the solutions for small v is examined, attention being mainly confined to the first order stream function ΰ1 and the corresponding component of vorticity ζ1. Outside the boundary layers, where, for small v, ψ1 may be expanded asymptotically as [formula omitted], the terms ζ(n)1 of the corresponding expansions for the vorticity are shown to be uniform throughout the fluid. It is noted that the asymptotic expansions of ψ1 for the region of the boundary layers and for the region outside the boundary layers may be combined in a single expansion which holds in both regions. The leading terms of this expansion are calculated by boundary layer methods. © 1957, Cambridge University Press. All rights reserved.

Description: This paper considers the two-dimensional laminar boundary layer on an infinite flat plate normal to an oncoming stream, the plate making transverse oscillations in its own plane. The exact solution is shown to depend on a single ordinary differential equation, containing the frequency of oscillation as a parameter. Series methods are employed to evaluate the solution for small and large values of the frequency, and enough terms are calculated to give the solution with satisfactory accuracy over the whole frequency range. It is observed that the solution satisfies the full Navier-Stokes equations. For a cylinder of arbitrary section making transverse or rotational oscillations, it is shown how the results for a flat plate can be used to describe the boundary layer in the neighbourhood of the front stagnation point. Difficulties which arise in extending the solution to cover the remainder of the cylinder are discussed, and an estimate is made of the fluctuating torque on a circular cylinder making transverse oscillations. © 1956, Cambridge University Press. All rights reserved.

Description: The turbulence produced by a multiplicity of small air jets has been investigated and comparisons are made with other turbulent flows. The eddy Reynolds number is large (Rλ = 250). The energy spectrum was measured, as well as the skewness and kurtosis of ∂u/∂x. The effect of the finite length of the hot wire is considered and the corrected results indicate that the spectrum follows the law of Kolmogoroff in an intermediate range and appers to fall off with the (−6)-power of the frequency in the viscous range. This range is limited at the upper end of the frequency range by electrical noise. Special precautions reduced this noise to the level of thermal agitation in the hot wire. The ultimate limit to spectral analysis is imposed by the molecular agitation of the gas. This limit is evaluated and compared with the spectrum of turbulence. It appears that the spectrum of ∂3u/∂x3 merges with the spectrum of molecular agitation without a distinctive separation. © 1957, Cambridge University Press. All rights reserved.

Description: An investigation is made of the effect of a small disturbance on the flow in a complete rarefaction wave, for example, the flow produced by the rupture of a membrane originally separating a compressible gas from a vacuum. The perturbation arises from a rigid boundary slightly inclined to the direction of flow. The growth of the perturbed region is studied and the pressure field is calculated for diatomic gases. The nature of the expanding boundary of the perturbed region is investigated. Arguments are put forward which suggest that this boundary can be a weak shock in certain circumstances. A second shock may also appear in some cases, following the first and of greater strength. In an appendix the solutions are extended to monatomic gases and to fluids with an adiabatic index of 2. The latter results are suitable for a comparison with hydraulic experiments. © 1957, Cambridge University Press. All rights reserved.

Description: A flat strip, of infinitesimal thickness and infinite lenght, is located in a viscous incompressible fluid, and both the strip and fluid are at rest initially. The strip is abruptly set into steady motion parallel to its length. An unsteady uni-directional flow of the fluid results, and there is a variable skin friction on the strip which must be overcome to maintain its velocity. In the early stages of motion, the skin friction is large, with a local behaviour which resembles that for a flat strip of infinite width. The skin friction near each edge of the strip can be more accurately represented by referring to the semi-infinite configuration that is realized on displacement of the other edge to infinity. At this stage, the results depend on separate consideration of the two limiting configurations, and the way to further improvements is not clearly delineated. The object of this paper is to provide a formulation which contains the preceding information and allows a systematic evaluation of all additional refinements. Thus, it is shown that the total skin friction on a strip of width 2a, moving with velocity V in a fluid whose coefficients of viscosity and kinematic viscosity are µ, ν, takes the form [formula omitted] Here the first two terms stem from the infinite and semi-infinite strip distributions, and the integral, containing a complementary error function, furnishes all corrections of the lowest exponential order, in the initial stages of motion, when [formula omitted] 1. The calculation is based on a new integral equation which makes explicit the effects of an isolated edge, and by iteration provides the interaction effects between edges at a finite separation. © 1957, Cambridge University Press. All rights reserved.

Description: This paper is concerned with some statistical properties of the displacement of a marked fluid particle released from a given position in a turbulent shear flow, and in particular with the dispersion about the mean position after a long time. It is known that the dispersion takes a simple asymptotic form when the particle velocity is a stationary random function of time, and that analogous results are obtainable when the particle velocity can be transformed to a stationary random function by suitable stretching of the velocity and time scales. The basic hypothesis of the paper is that, in steady free turbulent shear flows which are generated at a point and have a similar structure at different stations downstream, the velocity of a fluid particle exhibits a corresponding Lagrangian similarity and can be so transformed to a stationary random function.The velocity and time scales characterizing the motion of a fluid particle at time t after release at the origin are determined in terms of the powers with which the Eulerian length and velocity scales of the turbulence vary with distance x from the origin. The time scale has the same dependence on t for all jets, wakes and mixing layers (and also for decaying homogeneous turbulence) possessing the usual kind of Eulerian similarity. The dispersion of a particle in the longitudinal or mean-flow direction (and likewise that in the lateral direction in cases of two-dimensional mean flow) is found to vary with t in such a way as to be proportional to the thickness of the shear layer at the mean position of the particle. The way in which the maximum value of the mean concentration of marked fluid falls off with t (for release of a single particle) or with x (for continuous release) is also found.

Description: In this work a method is given for finding the position of ring singularities in a three-dimensional potential field having axial symmetry, by consideration of the very much easier case of a similar two-dimensional potential function. It is seen that the traces of these ring singularities on a plane through the axis of symmetry occur at points corresponding to those of the singularities existing in the two-dimensional plane when the axial velocity potential functions are the same. It is thought that this might be of value in the plotting of stream surfaces. © 1957, Cambridge University Press. All rights reserved.

Description: Measurements of mean velocity, turbulent stress and static pressure were made in the mixing region of a jet of air issuing from a slot nozzle into still air. The velocity was low and the two-dimensional flow was effectively incompressible. The results are examined in terms of the unsimplified equations of fluid motion, and comparisons are drawn with the common assumptions and simplifications of free jet theory. Appreciable deviations from isobaric conditions exist and the deviations are closely related to the local turbulent stresses. Negative static pressures were encountered everywhere in the mixing field except in the potential wedge region immediately adjacent to the nozzle. Lateral profiles of mean longitudinal velocity conformed closely to an error curve at all stations further than 7 slot widths from the nozzle mouth. An asymptotic approach to complete self-preservation of the flow was observed.

Description: In order to answer some of Proudman's questions (1956) concerning shear layers in rotating fluids, a study is made of the flow between two coaxial rotating discs, each having an arbitrary small angular velocity superposed on a finite constant angular velocity. It is found that, if the perturbation velocity is a smooth function of r, the distance from the axis, then the angular velocity of the main body of fluid is determined by balancing the outflow from the boundary layer on one disc with the inflow to the boundary layer on the other at the same value of r. At a discontinuity in the angular velocity of either disc a shear layer parallel to the axis occurs. If the angular velocity of the main body of the fluid is continuous, according to the theory given below the purpose of this shear layer is solely to transfer fluid from the boundary layer on one disc to the boundary layer of the other. It has a thickness O(v1/3), where v is the kinematic viscosity, and in it the induced angular velocity is O(v1/6) of the perturbation angular velocity of the discs. On the other hand, if the angular velocity of the main body of fluid is discontinuous, according to the theory given below the thickness of the shear layer is O(v1/4). A secondary circulation is also set up in which fluid drifts parallel to the axis in this shear layer and is returned in an inner shear layer of thickness O(v1/3).The theory is also applied to the motion of fluid inside a closed circular cylinder of finite length rotating about its axis almost as if solid.

Description: The intensity of turbulent motion in the wake of square mesh grids of bars has been examined experimentally. The measurements show that there are considerable departures from homogeneity over surfaces normal to the mean stream. Large variations of intensity have been observed at a distance of 80 meshlengths from the grid, where they are diminishing slowly, if at all. This inhomogeneity is believed to be the cause of the discrepancies between various sets of measurements of energy decay in the ‘initial period’. There is also a possibility of error in other measurements made in grid turbulence and analysed on the assumption of homogeneity. Observations with a grid of 2 in. mesh show that v2 is significantly less than u2, which is consistent with results from other laboratories.

Description: The purpose of this note is to comment on the calculation of equilibrium gas dynamic parameters behind strong shock waves. The writer has been aroused by the appearance of a paper by Guman (1956) presenting a generalized computing scheme for ionizing shock waves in monatomic gases. In that paper the reader is not cautioned about including excited states in the Saha equation for the computation of the degree of ionization behind shock fronts at appropriate temperatures and densities. The same paper treats the adiabatic exponent γ = cp/cv as constant across strong shocks when at the same time it is implied that the computing scheme is of general validity. Hence, the unwary reader might attempt to apply the scheme in a regime where γ is not only no longer constant but is no longer a useful quantity for characterizing the shock conditions. Other authors (see, for example, Glass, Martin & Patterson (1953)) characterize flows in which a shock has excited internal degrees of freedom in terms of a variable specific heat ratio when in fact one cannot use this quantity in calculating shock front conditions.

Description: Atmospheric dissociation will be appreciable in the neighbourhood of projectiles travelling at speeds greater than 2 km/sec. This introductory paper on possible effects of dissociation on the airflow, and hence on the drag, stability and aerodynamic heating of such projectiles, is intended mainly as a source of ideas for later, more comprehensive investigations.

Description: The effects of compressibility on a radial laminar wall jet are investigated. On the assumption that the coefficient of viscosity is proportional to the temperature, it is shown that a similarity solution for the velocity distribution exists, which is expressible directly in terms of the corresponding solution for an incompressible wall jet. For arbitrary Prandtl number the energy equation is studied in detail and solutions are obtained for a variety of temperature conditions. © 1958, Cambridge University Press. All rights reserved.

Description: The results of calculations of the compression of air by repeated shock waves are compared with the perfect-gas values given by Evans & Evans (1956). The comparison emphasizes the increasing divergence of real from perfect-gas results as shock strengths are raised. The equations relating conditions across a shock wave are obtained in a convenient form for solution using a Mollier diagram. © 1958, Cambridge University Press. All rights reserved.

Description: If, in a given solution of the boundary layer equations, the position of the wall is varied, then additional solutions of the boundary layer equations may be deduced. The theorem considers the nature of such solution, for the general case of time-dependent three-dimensional compressible flow.Applications of the theorem arise in several different fields, and it is shown that useful quantitative results can often be obtained with the minimum of calculation. In this paper, chief attention is focused on the case of a rotating circular cylinder, and explicit formulae are developed for the skin friction, valid for sufficiently low rotational speeds. The important results which the theorem gives for slip flow have been noted by previous extenions to these previous treatments are made. Other applications of the theorem are briefly mentioned.

Description: The plane steady decelerated flow of a dust-gas mixture is analysed in an approximate manner. The problem, which has a five-parameter family of solutions, is reduced to a form such that the analysis can be completed by the integration of a first-order non-linear differential equation and a quadrature. A few integral curves are given and the characterizing features of the flow field are discussed. © 1958, Cambridge University Press. All rights reserved.

Description: A new water tunnel, incorporating a slotted wall working section, was found to suffer from severe vibration. A theoretical explanation is given for this, together with experimental evidence gleaned from this water tunnel and a small wind tunnel. It is shown that the oscillations are hydrodynamic in origin and are associated with the slotted wall design. Consideration is given to methods of elimination or reduction of the oscillations. © 1958, Cambridge University Press. All rights reserved.

Description: A study is made of the propagation of sound in both a constant gradient shear flow and a turbulent shear flow above a flat surface. Curves are presented showing how, in the case of downstream propagation, the flow gradient tends to channel the sound energy into a narrow layer next to the wall. These results are used in estimating the effect of a flow on the attenuation of sound in a duct with absorbing side walls. © 1958, Cambridge University Press. All rights reserved.

Description: In a recent paper Carrier & Greenspan (1958) showed that, within the framework of the non-linear shallow-water theory, there exist waves which do not break as they climb a sloping beach. The formation of a shock or bore is dependent on a variety of factors (wave shape, particle velocity, etc.) and, as yet, no general criteria for breaking have been found. In this paper, we consider waves which propagate shoreward into quiescent water; it is shown that any compressive wave (a wave of positive amplitude) which has a non-zero slope at the wave-front eventually breaks before reaching the coastline. In fact, an explicit relation is obtained between the initial conditions and the position where breaking occurs. © 1958, Cambridge University Press. All rights reserved.

Description: This paper considers the stabilizing effect of density stratification on the horizontal shear layer between two parallel streams of uniform velocities. A simple continuous velocity distribution, [formula omitted], is used to represent the laminar shear layer. The density of the fluid is assumed to vary as exp (-βy), y being the vertical coordinate, with a small total change in density across the shear layer. The fluid is unbounded, and is assumed to be inviscid, incompressible and under the action of gravity. By the methods of hydrodynamic stability theory, it is shown that a disturbance of small amplitude and wave-number α is neutrally stable if the Richardson number, defined as [formula omitted], has the value [formula omitted], and the form of the neutral disturbance is obtained. It follows that the critical Richardson number is [formula omitted], so that the flow is stable if [formula omitted]. The relation between these results and Goldstein's derivation of the same critical Richardson number for a flow with discontinuous velocity and density gradients is discussed. © 1958, Cambridge University Press. All rights reserved.

Description: In a stationary weak sound wave the four gas dynamical quantities, entropy, stagnation enthaply, mass flow per unit area and impulse per unit area are constant. The six processes in which pairs of these four variables are kept constant are studied for the case of any single phase fluid or mixture of fluids in equilibrium, and it is shown that the remaining variables are stationary at sonic points. Such points are shown to occur once only in each process and are identified as maxima or minima, on the assumption that the fluid is a normal one which expands on heating at constant pressure and for which (∂2p/∂v2)s is positive. Newton's theory of sound assumed that the fluid temperature was invariant. Across a stationary Newtonian sound wave, the four quantities, temperature, mass flow and impulse per unit area, and a dynamical variant of the Gibbs function are constant. The six processes in which pairs of these four variables are kept constant are studied, and it is shown that the remaining variables are stationary at points where the fluid speed equals Newton's sound velocity. These points are shown to occur once only in each process and are identified as maxima or minima with the further proviso that (∂2p/∂v2)T is positive. The two sets of processes have one member in common, that usually referred to as the Rayleigh line. The Fanno line and the usual isentropic ‘nozzle’ process also belong to the first set. Finally, the variation of stagnation temperature and pressure in some of the processes and in stationary shocks is investigated. © 1958, Cambridge University Press. All rights reserved.

Description: A comparison is made between various methods for the calculation of the pressure distribution on an elliptic cone in supersonic flow. In particular, a criterion first put forward by Van Dyke (1956) is reconsidered. The paper is presented in two parts under separate authorship, taking the form of a discussion of some controversial aspects of this subject. © 1958, Cambridge University Press. All rights reserved.