When plane pressure waves in a duct reach an open end, they establish a complicated three-dimensional wave pattern in the vicinity of the exit which tends to readjust the exit pressure to its steady-flow level. This adjustment process is continually modified by further incident waves, so that the effective instantaneous boundary conditions which determine the reflected wave depend on the flow history. In the analysis of a nonsteady-flow problem by means of a wave diagram, it has been customary to assume that the steady-flow boundary conditions are instantaneously established. While this simplifying assumption appears reasonable, the resulting errors have been undetermined. It is the purpose of the present investigation to obtain improved boundary conditions. The results of a previous study of the reflection of shock waves from an open end have now been extended to other waves of finite amplitude. The reflected waves computed by means of the new procedure are in good agreement with experimental data observed in a shock tube for a variety of flow conditions. The pressure variations in a reflected wave lag behind those derived in the conventional manner by the time in which a sound wave travels about one or two duct diameters. Such lags are small, but may occasionally become significant. As a consequence of the lag, certain discontinuities of the incident wave do not reappear in the reflected wave. This improved understanding of the reflection process has made it possible to clarify some previously unexplained experimental observations.

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.

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.

The linearized equations of motion show that in a viscous heat-conducting compressible medium three modes of fluctuations exist, each one of which is a familiar type of disturbance. The vorticity mode occurs in an incompressible turbulent flow, the entropy mode is familiar as temperature fluctuations in low speed turbulent heat transfer problems, and the sound mode is the subject of conventional acoustics. A consistent higher order perturbation theory is presented with the only restrictions being that the Prandtl number is 3/4 and the viscosity and heat conductivity are monotinic functions of the temperature alone. The theory is based on expansion of the disturbance fields in powers of an amplitude parameter α. The non-linearity of the full Navier-Stokes equations can be interpreted as interaction between the three basic modes; in order to help physical insight the interactions are classed as ‘mass-like’, ‘force-like’, and ‘heat-like’ effects.

Besides the amplitude parameter α there is another subsidiary non-dimensional parameter ε which indicates the relative importance of viscosity and heat conduction effects as compared to the inertial effects, ε is proportional to the ratio of the molecular mean free path and the characteristic length of the flow pattern (Knudsen number). The main contribution of the paper is the outline of a consistent successive approximation for an arbitrary order in α and the presentation of explicit formulae for the second order (bilateral) interactions.

A special case of rather general significance is treated in more detail. This is when all three basic modes have intensities and length scales of the same orders of magnitude and in addition to α the parameter ε is also small; the second-order interactions are then relatively few and easily identifiable and are shown in table 1.

The present analysis also sheds some light on the ‘zero order’ approximation which treats the vorticity and entropy disturbances as a ‘frozen pattern’ and the sound field as propagating nondissipative waves. The interpretation of hot-wire measurements relies heavily on these simplified models and the present paper lends some support to these current hot-wire practices.

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.

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.

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.

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.

A simple approximate model is set forth for the flow field between the nose of a blunt body of revolution and its detached shock wave. The model tends to explain the poor convergence of Chester's solution, which is based on an improvement of the Newtonian approximation. It suggests a modification of his series for the body shape which appears to improve its convergence considerably.

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.

Experiments have been performed with a viscous fluid contained in a rotating cylinder heated from below. Previous theoretical explanations of the flow patterns obtained in these experiments assumed that the convective terms in the heat transfer equation were negligible. The present paper gives a treatment which includes one of these convective terms. The results confirm the physical reasoning that, when the laps rate is positive, the stability is increased and that the motion is therefore decreased.

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.

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.

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.

A dispersed shock wave may be defined as one in which finite changes occur over distances large compared to the mean free path in the gas. In contrast a shock wave in air extends over only a few mean free paths. When internal motions in a molecule are excited rather slowly by collisions, as is the case for molecular vibration, the shock wave may be partly dispersed; then, the sharp shock front is followed by a diffuse tail leading to complete thermal equilibrium. Alternatively, it may be fully dispersed, so that adjustments in the energy in all the degrees of freedom proceed slowly and in parallel. The purpose of this note is to point out that within a narrow speed range, from a shock Mach number of 1 to 1.042, shocks in carbon dioxide are fully-dispersed in the above sense. Such waves have been observed experimentally using a shock tube and interferometer. The possible existence of such waves was first pointed out by Bethe & Teller (1941) purely as a matter of academic interest. This note treats the problem in the same spirit.

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.

Differentiation of the Hugoniot function $H (p,v) = E(p,v)-E(p_0,v_0)+\frac {1}{2}(p + p_0)(v_0 - v)$ and use of the first and second laws of thermodynamics leads to the relation dH = T dS+dA, where dA is the element of area in the (p, v) plane swept out (in a counter-clockwise direction) by the line segment (p0, v0) → (p, v) as the point (p, v) is moved from some point (p1, v1) to a neighbouring point (p1+dp1, v1+dv1). This relation, together with rather general assumptions regarding the shape of the isentropic curves dS = 0 for the material behind the shock, makes possible the geometrical derivation of a number of properties of the function H and of the Hugoniot curves dH = 0.

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.

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.