bluuebunny/crossref_metadata_2025 huggingface dataset:
The refraction of shock waves at a gaseous interface
Authors: Jahn Robert G.
Published: March 2006

A programme of shock tube experiments has been conducted to study the refraction of plane shock waves at interfaces between two gases. Shocks of strength ζ = 0·85 (weak) and ζ = 0·30 (fairly strong) were allowed to impinge, at various angles of incidence, on interfaces between air/CO2and air/CH4, and the resulting configurations were photographed through a Mach-Zehnder interferometer. From the interferograms, measurements were made of the strengths of the reflected waves, and of the angles of refraction, and the values were compared with the theoretical calculations of Polachek & Seeger (1951). Within the range of parameters for which the refraction model assumed by the theory is applicable–the so-called ‘regular refraction’ region–the observations were in excellent agreement with the theoretical predictions.

When the study was extended to ranges of the parameters for which the theory is clearly inadequate, a succession of rather complex ‘irregular refraction’ patterns was observed. Although these configurations were highly interesting qualitatively, each of them involved curved shocks, non-uniform regions of flow, and other less simple processes which discouraged any formal theoretical analysis. On a less rigorous basis, however, it could be shown that these patterns were internally consistent, and that each represented a distortion of a regular refraction process which was reasonable under the prevailing aerodynamic conditions.

Certain observations in these refraction experiments appear to be of some significance outside the specific problem. (i) The sensitivity of strong shock refractions to the values of the specific heat ratio λ for the two gases suggests a possible technique for the measurement of λ and its temperature dependence. (ii) Two of the irregular refraction patterns display a transition process which would be equally appropriate to the onset of the Mach configuration in the shock reflection problem. (iii) Some irregular refractions can be considered as special cases in the problem of the interaction of a shock and a boundary layer.
Supersonic flow past a bluff body with a detached shock Part II Axisymmetrical body
Authors: Chester W.
Published: March 2006

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
$\delta = (\gamma -1)|(\gamma +1),$
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.
Outline of a theory of turbulent shear flow
Authors: Malkus W. V. R.
Published: March 2006

In this paper the spatial variations and spectral structure of steady-state turbulent shear flow in channels are investigated without the introduction of empirical parameters. This is made possible by the assumption that the non-linear momentum transport has only stabilizing effects on the mean field of flow. Two constraints on the possible momentum transport are drawn from this assumption: first, that the mean flow will be statistically stable if an Orr-Sommerfeld type equation is satisfied by fluctuations of the mean; second, that the smallest scale of motion that can be present in the spectrum of the momentum transport is the scale of the marginally stable fluctuations of the mean. Within these two constraints, and for a given mass transport, an upper limit is sought for the rate of dissipation of potential energy into heat. Solutions of the stability equation depend upon the shape of the mean velocity profile. In turn, the mean velocity profile depends upon the spatial spectrum of the momentum transport. A variational technique is used to determine that momentum transport spectrum which is both marginally stable and produces a maximum dissipation rate. The resulting spectrum determines the velocity profile and its dependence on the boundary conditions. Past experimental work has disclosed laminar, ‘transitional’, logarithmic and parabolic regions of the velocity profile. Several experimental laws and their accompanying constants relate the extent of these regions to the boundary conditions. The theoretical profile contains each feature and law that is observed. First approximations to the constants are found, and give, in particular, a value for the logarithmic slope (von Kármán's constant) which is within the experimental error. However, the theoretical boundary constant is smaller than the observed value. Turbulent channel flow seems to achieve the extreme state found here, but a more decisive quantitative comparison of theory and experiment requires improvement in the solutions of the classical laminar stability problem.
On the motion of small spheroidal particles in a viscous liquid
Authors: Saffman P. G.
Published: March 2006

Small spheroidal particles suspended in a sheared viscous liquid are sometimes observed to take up slowly preferred orientations, relative to the motion of the undisturbed liquid, which are independent of the initial conditions of release. These obsevations cannot be accounted for by the solution, obtained by Jeffery (1922), of the linearized Navier-Stokes equations. It is shown in this paper that the effect of the inertia of the liquid is to alter slowly the orbit of the particle in accordance with Jeffery's hypothesis that the particle ultimately moves in such a way that the dissipation of energy is a minimum, but that this effect is orders of magnitude too small to account for any of the experimental observations.

It is suggested that non-Newtonian properties of the liquid account for the observations. It is shown that the rate of orientation of a particle would then be independent of its size, and this prediction is verified experimentally. Other experimental evidence in support of this suggestion is also described.

Some remarks are also made about the possible effect of collisions between the particles when more than one particle is present.
An experimental investigation of heat transfer effects on boundary layer separation in supersonic flow
Authors: Gadd G. E.
Published: March 2006

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.
A new approach to problems of shock dynamics Part I Two-dimensional problems
Authors: Whitham G. B.
Published: March 2006

In this paper, two-dimensional problems of the diffraction and stability of shock waves are investigated using an approximate theory in which disturbances to the flow are treated as a wave propagation on the shocks. These waves carry changes in the slope and the Mach number of the shock. The equations governing the wave propagation are analogous in every way to the non-linear equations for plane waves in gas dynamics, and their solutions can be deduced by the same mathematical techniques. Since the propagation speed of the waves is found to be an increasing function of Mach number, waves carrying an increase in Mach number will eventually break and form what we may call a ‘shock’, corresponding to the breaking of a compression wave into a shock in the ordinary plane wave case. Such a ‘shock’ moving on the shock is called ashock-shock.The shock-shock is a discontinuity in Mach number and shock slope, and it must be fitted in to satisfy the appropriate relations between these are interpreted as the trace of cylindrical sound waves in the flow behind the shock. In particular a shock-shock is the trace of a genuine shock in the flow behind, and thus corresponds to Mach reflection.

The general theory of the wave propagation is set out in § 2. The subsequent sections contain applications of the theory to specific problems, including the motion of a shock along a curved wall, diffraction by a wedge, stability of plane shocks and the instability of a converging cylindrical shock.
A numerical method for a converging cylindrical shock
Authors: Payne R. B.
Published: March 2006

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.
Upward ‘falling’ jets and surface tension
Authors: Keller Joseph B. , Weitz Mortimer L.
Published: March 2006

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.
Interferometric test for N-wave flow
Authors: Steininger D. H. , Bennett F. D.
Published: March 2006

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.
Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder
Authors: Proudman Ian , Pearson J. R. A.
Published: March 2006

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
$\Sigma \;f_n(R) \psi_n(r, \theta)$ and $\Sigma \; F_n(R) \Psi_n(R_r, \theta)$
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.
The inhomogeneity of grid turbulence
Authors: Grant H. L. , Nisbet I. C. T.
Published: March 2006

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.
Steady state free thermal convection of liquid in a saturated permeable medium
Authors: Wooding R. A.
Published: March 2006

The partial differential equations which describe steady flow of fluid in saturated homogeneous permeable solid material under non-isothermal conditions are stated. From these are derived the equations for flow of liquid (in particular, water) using suitable approximations and making use of empirical laws when necessary.

It is then postulated that the only ‘ponderomotive’ (i.e. massmoving) forces present are those due to thermal expansion effects. Free convection results. An approximate solution of the equations is attempted for plane flow by means of classical perturbation methods, the temperature and stream-function variables being represented by power series in a convection parameter proportional to the Rayleigh number.

A numberical example of the method, with boundary conditions based on a geothermal area at Wairakei, New Zealand, is given. The results show features which are in fair agreement with temperature measurements made in the area, and it appears that the convection parameter η is of the order of 10.
The motion of a shock wave in a channel, with applications to cylindrical and spherical shock waves
Authors: Chisnell R. F.
Published: March 2006

A first-order relationship between changes in area and shock strength is derived for the case of a shock moving through a small area change in a channel. By integration of this relationship the area of the channel is obtained as a function of the shock strength in closed form. This result is interpreted as giving the average strength of a shock at a given time as it moves along a channel of arbitrary shape.

By suitable choices of the shape of the channel, descriptions of converging cylindrical and spherical shocks are obtained. These descriptions are found to be in close agreement with the similarity solutions valid near the points of collapse of the shocks. The reason for such good agreement is examined.
Corrigenda to “Drift”
Authors: Lighthill M. J.
Published: March 2006

In the author's paper “Drift” (Lighthill 1956) there are mistakes which need correction. Equation (28) should read
$\phi \sim -\frac {m}{4\pi U_r}\;\; as \;\; r \rightarrow \infty$ (28)
since the definition of the disturbance potential ϕ (p. 35) states that the full velocity potential is U(x + ϕ). Hence by (18) equation (29) for the asymptotic form of the secondary flow due to a source should read
$v_x = -\frac {Amy} {4 \pi U (y^2 + z^2)} \left(1 + \frac{\chi}{r}\right),\;\;\;\; v_y = -\frac {Am}{4\pi Ur}, \;\;\;\; v_z=0$ (29)
where as well as the U's a 4π was accidentally missing from the version printed and a y misprinted as x.
Space-time double correlations and spectra in a turbulent boundary layer
Authors: Favre A. J. , Gaviglio J. J. , Dumas R.
Published: March 2006

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.
On the hydrodynamic stability of two viscous incompressible fluids in parallel uniform shearing motion
Authors: Feldman Saul
Published: March 2006

A new problem in hydrodynamic stability is investigated. Given two contiguous viscous incompressible fluids, the fluid on one side of the plane interface being bounded by a solid wall and that on the other side being unbounded, the problem is to determine the hydrodynamic stability when the fluids are in steady unidirectional motion, parallel to the interface, with uniform rate of shear in each fluid. The mathematical analysis, based on small disturbance theory, leads to a characteristic value problem in a system of two linear ordinary differential equations. The essential dimensionless parameters that appear in the present problem are the viscosity ratiom, the density ratior, the Froude numberF, and the Weber numberW, as well as the parameters α,R(which is proportional here to the flow rate of the inner fluid) andc, that occur in the study of hydrodynamic stability of a single fluid. The results obtained are presented graphically for most fluid combinations of possible interest. The neutral stability curve in the (α,R)-plane is single-looped, as in the boundary layer case. The calculated critical Reynolds numbers are higher than the values observed in liquid film cooling experiments. (In these experiments, the outer fluid is usually a turbulent gas, in which the thickness of the laminar sublayer is of the same order of magnitude as the liquid film thickness.) General agreement between the theoretical and experimental values exists for all critical quantities except the Reynolds number. Gravity and surface tension are found here to have a destabilizing effect on the flow, in agreement with experimental evidence. Semi-infinite plane Couette flow is a special case of the present problem and the known stability of this flow is recovered. The linear velocity profile of two adjacent fluids with the same viscosity, but different densities, is shown to be unstable for high enough Reynolds numbers. The Reynolds stress distribution for a neutral oscillation in the general case is discussed qualitatively.
Oscillations of tide and surge in an estuary of finite length
Authors: Proudman J.
Published: March 2006

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.
An electronic analogue for supersonic flow
Authors: Kovásznay Leslie S. G.
Published: March 2006

An analogue method for solving certain quasi-linear hyperbolic partial differential equations is presented. The characteristic lines are formed by scanning electronically the screen of a cathoderay tube. The boundary conditions are introduced in the form of an opaque mask. The solution appears as a picture on the screen of a second cathode-ray tube. The experiments demonstrate the feasibility of the approach, but the development of the machine has not been carried beyond this stage.
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.
The diffusion of smoke from a continuous elevated point-source into a turbulent atmosphere
Authors: Smith F. B.
Published: March 2006

The problem is to determine the downwind concentration of non-buoyant smoke emitted from a continuous elevated pointsource in a turbulent airstream. The velocity and eddy diffusivity coefficients are represented by related powers of the height above ground, and are independent of the position of the source. Exact solutions are obtained for the zero and second moments of the concentration distribution along lines lying in the cross-wind direction at ground level. In special cases, these moments may also be determined along lines at general height. In one such case the concentration is determined exactly (rather than just the two moments) and it is found that the cross-wind distribution always has a Gaussian form. If it is assumed that in all cases the cross-wind profile is Gaussian, a formulation for the concentration can be given purely in terms of the known zero and second moments. When the source is moving with constant velocity across the wind, the first moment as well as the zero and second moments is exactly determined, and under a similar assumption a formula for the concentration is found.
Boundary layers whose streamlines are closed
Authors: Wood W. W.
Published: March 2006

Certain two-dimensional, laminar bounday layers are considered whose streamlines are closed. The speed at the solid boundary is supposed uniform, the bounday outline being stationary, and the speed in the boundary layer is supposed to differ only slightly from that of the boundary. A formal solution is then obtained for the motion in the boundary layer. The analysis confirms that a closed boundary layer may exist and yields a condition needed to determine the inviscid motion. The condition is extracted in a simplified but approximate form and two examples of its use are given. A further class of closed boundary layers, namely those for which the pressure is uniform, is also considered. For this class the condition needed to determine the inviscid motion may be derived in a form both simple and exact.
A general solution of the equations of hydrodynamics
Authors: Milne-Thomson L. M.
Published: March 2006

Consider a fluid of constant viscosity μ. If the fluid is incompressible, we suppose it to move in a conservative field in which the force per unit mass is - ∇Ω. If the fluid is compressible, we suppose the field of force to be absent. Both cases are included if we write V = ερΩ, where ε = 1 for an incompressible fluid, and ε = 0 for a compressible fluid. The equation of continuity
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.
Flow past a flat plate at low Reynolds numbers
Authors: Janssen E.
Published: March 2006

The flow past a flat plate at Reynolds numbers in the range 0·1 to 10·0 is investigated by an analogue method. The solution gives the stream function and the vorticity in the flow field surrounding the plate. From these are obtained the local coefficient of friction, the pressure distribution along the plate, and the total drag coefficient. The drag coefficient approaches the analytical values of Haaser (1950) and of Tomotika & Aoi (1953) as the Reynolds number decreases toward 0·1. The drag coefficient approaches the Blasius solution as the Reynolds number increases. At Reynolds number 10·0 the drag coefficient is still above the Blasius value, but is below the value obtained experimentally by Janour (1951). The difference from the experimental result is attributed for the most part to truncation error.
Further space-time correlations of velocity in a turbulent boundary layer
Authors: Favre A. J. , Gaviglio J. J. , Dumas R. J.
Published: March 2006

This paper describes the results of further experimental investigation of the turbulent boundary layer with zero pressure gradient. Measurements of autocorrelation and of space-time double correlation have been made respectively with single hot-wires and with two hot-wires with the separation vector in any direction. Space-time correlations reach a maximum for some optimum delay. In the case of two points set on a line orthogonal to the plate, the optimum delay Ti is not zero. In the general case it is equal to the corresponding delay Ti, increased by compensating delay for translation with the mean flow. Taylor's hypothesis may be applied to the boundary layer at distances from the wall greater than 3% of the layer thickness. Space-time isocorrelation surfaces obtained with optimum delay have a large aspect ratio in the mean flow direction, even if they are relative to a point close to the wall (0·03δ); the correlations along the mean flow then retain high values on account of the large scale of the turbulence.
Turbulent flow in a stably stratified atmosphere
Authors: Townsend A. A.
Published: March 2006

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.
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.
Some properties of boundary layer flow during the transition from laminar to turbulent motion
Authors: Dhawan S. , Narasimha R.
Published: March 2006

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.
Heat transfer from surfaces of non-uniform temperature
Authors: Spalding D. B.
Published: March 2006

The paper deals with the calculation of steady heat transfer from a surface of arbitrary temperature distribution to a laminar semi-infinite stream of arbitrary velocity distribution. Lighthill's method is improved by a correction which accounts for the departures from linearity of the velocity profile within the thermal boundary layer, and which comprehends the influences of Prandtl number, pressure gradient, body forces, and non-coincident start of velocity and thermal layers. Methods are given for evaluating the total heat flux directly, and for integrating the differential equations for the growth of the boundary layer thickness by means of quadratures.