bluuebunny/crossref_metadata_2025_split huggingface dataset:
On the stability of plane shock waves
Authors: Freeman N. C. 
Published: March 2006

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.
On almost rigid rotations
Authors: Stewartson K. 
Published: March 2006

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.
On the instability of small gas bubbles moving uniformly in various liquids
Authors: Hartunian R. A. , Sears W. R. 
Published: March 2006

The instability of small gas bubbles moving uniformly in various liquids is investigated experimentally and theoretically.The experiments consist of the measurement of the size and terminal velocity of bubbles at the threshold of instability in various liquids, together with the physical properties of the liquids. The results of the experiments indicate the existence of a universal stability curve. The nature of this curve strongly suggests that there are two separate criteria for predicting the onset of instability, namely, a critical Reynolds number (202) and a critical Weber number (1.26). The former criterion appears to be valid for bubbles moving uniformly in liquids containing impurities and in the somewhat more viscous liquids, whereas the latter criterion is for bubbles moving in pure, relatively inviscid liquids.The theoretical analysis is directed towards an investigation of the possibility of the interaction of surface tension and hydrodynamic pressure leading to unstable motions of the bubble, i.e. the existence of a critical Weber number. Accordingly, the theoretical model assumes the form of a general perturbation in the shape of a deformable sphere moving with uniform velocity in an inviscid, incompressible fluid medium of infinite extent. The calculations lead to divergent solutions above a certain Weber number, indicating, at least qualitatively, that the interaction of surface tension and hydrodynamic pressure can result in instabilities of the bubble motion.A subsequent investigation of the time-independent equations, however, shows the presence of large deformations in shape of the bubble prior to the onset of unstable motion, which is not compatible with the approximation of perturbing an essentially spherical bubble. This deformation and its possible effects on the stability calculation are therefore determined by approximate methods. From this it is concluded that the deformation of the bubble serves to introduce quantitative, but not qualitative, changes in the stability calculation.
Diffusion in free turbulent shear flows
Authors: Batchelor G. K. 
Published: March 2006

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.
The laminar and turbulent mixing of jets of compressible fluid. Part II The mixing of two semi-infinite streams
Authors: Crane L. J. 
Published: March 2006

This paper presents the application of the methods developed in a previous paper (Part I, Crane & Pack 1957) to the mixing of two parallel streams for both laminar and turbulent flows. The effects of both high velocity and large temperature difference are treated together. The method used consists in developing the stream function in a double series of powers of two parameters, the first being the Mach number and the second depending on the temperature difference of the streams. Analytical expressions are found for the terms up to the second order in the series for the stream function when the streams do not differ too greatly in velocity and temperature. However, when one of the streams is at rest the analytical method is no longer sufficiently accurate, and for this case numerical solutions are given.For laminar mixing the most important effect is that of ’change of scale’, as was found in Part I for a laminar jet at large distances from the orifice. For turbulent half-jets the effect of ’change of scale’ and the effect of the perturbation terms due to the Mach number of the flows are approximately equal and opposite, leaving the form of the velocity profile sensibly unchanged from that in incompressible flow. This last result is confirmed by comparison with some experiments of Laurence (1955) on a two-dimensional jet at M = 0·7. Lastly, the effect of temperature differences is shown to be relatively unimportant even when these are fairly considerable.
Two-dimensional subsonic and sonic flow past thin bodies
Authors: Helliwell J. B. , Mackie A. G. 
Published: March 2006

Hodograph methods are applied to determine the flow at high subsonic and sonic velocities past two-dimensional, thin, symmetrical bodies. The boundary value problem for the determination of the stream function Ψ, which in the present theory is a solution of Tricomi's equation, is simplified by the assumption of a free stream breakaway at sonic velocity from the shoulder of the body. A solution is obtained in terms of Bessel functions.In § 2 and 3 the flow past a wedge of small angle is discussed and expressions are obtained for the pressure on the nose, the drag coefficient and the width of the wake. A comparison with the corresponding results in the case of sonic velocity derived by the more complex analysis of Guderley & Yoshihara (1950) shows that the present simpler theory yields very similar values for the pressure over the nose.In § 4 the flow at sonic velocity past a profile which is a first-order perturbation upon a wedge profile is analysed on the basis of the same free streamline theory. The flow pattern is obtained past an arbitrarily specified body by an application of the Hankel inversion theorem and an expression is deduced for the drag.
Some shock tube experiments on the chemical kinetics of air at high temperatures
Authors: Feldman Saul 
Published: March 2006

This paper is concerned with the rates at which atoms and molecules react in the air that flows over a body flying through the atmosphere at hypersonic speeds. Using air as a working fluid, a series of shock tube experiments were carried out to provide information about these rates. Mach angle measurements were made to determine the state of the gas in three situations of interest.Flow over flat plates was used to determine the state of the gas behind the incident normal shock; temperatures in the gas that passed through the shock varied between 2000 and 6000°:K and densities between standard and 1/80 of standard density.Flow over wedges was employed to decelerate the flow behind the incident shock to a small supersonic Mach number; here temperatures downstream of the oblique shock increased, at most, 2000°:K above the free stream value.A Prandtl-Meyer expansion was used to cool rapidly the dissociated gas, so that the recombination process could be investigated; temperatures dropped at most 2500°:K and the densities varied between standard and 1/200 of the standard value. In some cases, the initial degree of dissociation of air was over 45%.The results (figure 11) indicate that the dissociation and recombination relaxation times of the chemical species found in air are very fast, when compared to the time it takes a particle of gas to flow either around a blunt body in hypersonic flight or past smtill models in shock tubes. Thus the shock tube is shown to be an instrument capable of supplying air at high temperatures in thermodynamic equilibrium (figure 5).In the case of a non-melting blunt body of about 1 ft. diameter flying through the atmosphere at hypersonic speeds, the present results imply that, when the gas behind the detached shock is in thermodynamic equilibrium, the flow will also be in equilibrium as it expands around the body, provided its speed is greater than 10 000 ft./sec at altitudes below 180 000 ft. (figure 12).
The diffraction of a rarefaction wave by a corner
Authors: Powell J. B. L. 
Published: March 2006

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.
The structure of a shock wave in a fully ionized gas
Authors: Jukes J. D. 
Published: March 2006

The structure of a plane shock wave moving through a completely ionized plasma of protons and electrons is calculated. It is assumed that the two species of particles behave as two gases, each separately in a quasi-equilibrium state corresponding generally to two different temperatures. Navier-Stokes type equations with coefficients of viscosity and thermal conductivity appropriate to the two species are solved by numerical iteration.For very strong shocks it is found that both the velocity of electrons and protons and the temperature of the protons change in a distance about twice the mean path for momentum transfer between protons in the hot (shocked) gas. The electron temperature changes in about eight of these mean free paths, causing a relatively wide zone of hot electrons at low density ahead of the usual velocity shock-front. The density and temperature gradients of protons and electrons create an electric field.
The instability of a shear layer between two parallel streams
Authors: Esch Robin E. 
Published: March 2006

An unbounded parallel flow, consisting of a linear shear layer between uniform streams, is investigated for stability. A conventional eigenvalue problem is formulated, and solved by both analytical and numerical methods. The region of instability in the plane of Reynolds number R and disturbance wave number α is determined, and typical growth rates in the unstable region are computed.Unstable disturbances are found at all values of R. Results for αR > 100 are found to agree closely with inviscid theory results. An analytic method useful for αR < 1 is developed.The extent to which the present results can be applied to the laminar boundary layer between free streams is discussed.
The impact of a shock wave on a movable wall
Authors: Meyer R. F. 
Published: March 2006

An approximate solution is devised for the one-dimensional motion following the impact of a shock wave on a wall which is free to move. The approximate solution neglects changes in entropy occurring through the reflected and transmitted shocks, thus reducing the problem to one of a simple wave type. The asymptotic behaviour of the system is considered and it is shown by exact physical argument that the transmitted shock eventually attains the same strength as the incident shock and that the reflected shock ultimately decays to a sound wave.An experimental investigation of the interaction was made, using thin walls of cellulose acetate, in a shock tube at an incident shock Mach number of 1·50. Agreement between the theoretical and experimental results, especially for the path followed by the wall, was found to be good.
A boundary layer theorem, with applications to rotating cylinders
Authors: Glauert M. B. 
Published: March 2006

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.
Two-dimensional flow at high subsonic speeds past wedges in channels with parallel walls
Authors: Helliwell J. B. 
Published: March 2006

Investigations are made of the plane flow of an inviscid isentropic gas at high subsonic and sonic speeds past a finite wedge of small angle set at zero incidence in a channel with parallel walls. Hodograph methods are applied to the determination of the stream function, which, under the usual transonic approximation, is a solution of Tricomi's equation. In § 2 a brief summary is given of the derivation of the fundamental solution of this equation in terms of Bessel functions.Two models of the flow pattern are discussed. The model of Cole is examined in § 3 with the sonic line from the shoulder extending completely across the channel at right angles to the wall. In § 4 the Helliwell-Mackie model is taken in which there is a free stream breakaway at sonic velocity from the shoulder of the wedge and the velocity far downsteam may be either uniformly subsonic or sonic. Values are obtained for the drag coefficient in both cases and a high degree of both qualitative and quantitative agreement is found. On the basis of the free streamline theory explicit formulae are given relating, in terms of the channel width and upstream Mach number, the drag of the wedge in the channel to that of the same wedge in the free stream.Attention is drawn in § 5 to certain properties of plane jet flows that may be deduced from the investigation of the flow with free stream breakaway past a wedge in a channel.
On the series that represent tides and surges in an estuary
Authors: Proudman J. 
Published: March 2006

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.
Mixing and chemical reaction in the laminar wake of a flat plate
Authors: Cheng S. I. , Kovitz A. A. 
Published: March 2006

The initial value problem presented by mixing and chemical reaction in the wake of a flat plate is solved using the boundarylayer approximation. When a cool combustible mixture and its hot combustion products are separated by a finite, perfectly insulating flat plate, the velocity, temperature, and combustible concentration are determined in the vicinity of the trailing edge.The mixing problem without chemical reaction is solved in terms of a ‘universal solution’ for a given initial temperature ratio and Prandtl number from which the solution for arbitrary temperature ratios can be obtained.The mixing problem with chemical reaction is solved in terms of a ‘universal solution’ for the first two terms of an assumed series solution for the temperature. In this case the ‘universality’ is with respect to a parameter B characterizing the chemical and hydrodynamic initial conditions.The axial distance from the trailing edge to the first local temperature maximum is given in terms of the initial conditions and is shown to be greatly shortened by the presence of the viscous wake as compared with non-viscous mixing.
Calculations of unsteady viscous flow past a circular cylinder
Authors: Payne R. B. 
Published: March 2006

A numerical solution has been obtained for the starting flow of a viscous fluid past a circular cylinder at Reynolds numbers 40 and 100. The method used is the step-by-step forward integration in time of Helmholtz's vorticity equation. The advantage of working with the vorticity is that calculations can be confined to the region of non-zero vorticity near the cylinder.The general features of the flow, including the formation of the eddies attached to the rear of the cylinder, have been determined, and the drag has been calculated. At R = 40 the drag on the cylinder decreases with time to a value very near that for the steady flow.
The propagation of long waves into a semi-infinite channel in a rotating system
Authors: Crease J. 
Published: March 2006

In this paper the propagation of long waves (tides) into a canal is studied. The canal is assumed to be rotating at constant angular velocity and the depth of the fluid is uniform.The rate of rotation can have a considerable effect on the amplitude of the unattenuated modes in the channel. This is due partly to the modification of the known solution when the rotation is zero, and partly to the fact that even if there is only a single semi-infinite barrier unattenuated waves of a special type (Kelvin waves) may be propagated in the rotating system into what is normally called the ‘shadow’ region behind the barrier (Crease 1956).The purpose of this theoretical investigation is to seek a partial explanation of the behaviour of tides and storm surges in the North Sea. For this reason two models of this region are discussed. In Example I the model is of two parallel semi-infinite barriers in the path of a plane progressive wave, and in Example II there are two parallel barriers, one semi-infinite and the other infinite.The ratio of the observed height of the semi-diurnal M2 tide in the North Sea to the height of the incident tide from the Atlantic lies between the results predicted by the two models and is in closer agreement with the model of Example I.
Energy transfer through a dissociated diatomic gas in Couette flow
Authors: Clarke John F. 
Published: March 2006

The transfer of energy through a dissociated diatomic gas in Couette flow is considered, taking oxygen as a numerical example. The two extremes of chemical equilibrium flow and chemically frozen flow are dealt with in detail, and it is shown that the surface reaction rate is of prime importance in the latter case. The chemical rate equations in the gas phase are used to estimate the probable chemical state of the gas mixture, this being deduced from the ratio of a characteristic chemical reaction time to a characteristic time for atom diffusion across the layer. The influence of the surface reaction appears to spread outwards through the flow from the wall as gas-phase chemical reaction times decrease. For practical values of the surface reaction rate on a metallic wall, the energy transfer rate may be significantly lower in chemically frozen flow than in chemical equilibrium flow under otherwise similar circumstances.Similar phenomena to those discussed will arise in the more complicated case of boundary layer flows, so that a treatment of the simpler type of shear layer represented by Couette flow may be of some value in assessing the relative importance of the various parameters.