The introduction to Part 1 (Lighthill 1957) proposed a paper in three parts, in the nature of a broad introductory discussion of the dynamics of a pure dissociating gas, which might act as a source of ideas for later, more comprehensive investigations. Part 1 treated ‘equilibrium flow’, Part 2 was to treat transport properties under ‘quasi-equilibrium’ conditions, and Part 3 flows in which large departures from equilibrium occur. When Part 1 was submitted for publication, much of Part 2 had already been written. At this moment, however, the author suddenly became obliged to abandon the project for two years, in order that his contributions to two collective works, Laminar Boundary Layers and Incompressible Aerodynamics, could be produced in time to avoid serious delay in the appearance of the contributions of other authors.

A unified treatment of wave motion in a stratified fluid, with or without density discontinuities, is achieved by reducing the governing differential system to a Sturm-Liouville system. With the aid of Sturm's comparison theorem, it is found (without detailed calculations) that, for any stratification, the phase velocity increases as the wave-number decreases and that, for the same wave-number, the phase velocity increases as the density gradient is increased everywhere and decreases as the density is increased everywhere by a constant amount. Sturm's oscillation theorem provides upper and lower bounds for the phase velocity for a given stratification, a given wave-number, and a given number of zeros of the eigenfunction (or a given number of stationary surfaces in the fluid). The inequalities giving these bounds are used to explain the well-known tendency for surfaces of density discontinuities to behave as rigid boundaries when the stratification in each layer is slight. The rigid-boundary behaviour of interfaces in such cases enables one to obtain the approximate eigenvalue spectrum by superimposing the spectra of the individual layers (with the interfaces treated as rigid) on the spectrum of the interfacial (or free surface) waves, obtained by ignoring the slight continuous stratification in each layer. It is pointed out that the Ritz method can be used for calculating the eigenvalues even when the density is discontinuous, and examples are given to show the accuracy of the Ritz method. The nature of the spectrum when the depth is infinite is also clarified.

In the course of the development of the theory, the effects of compressibility and of three-dimensionality are determined and given explicitly, the rate of growth of unstable stratifications is related to the phase velocity of waves in stable ones, and equipartition of energy is proved. Motion due to a wave-maker is discussed in order to bring out the connexion between the type of the governing partial differential equation and the nature (local or not local) of the disturbances. The effect of surface tension and the stability of a stratified fluid under vertical oscillation are also discussed.

An analytical treatment is given of the problem of the establishment of the flow through a dam or levee with a horizontal underdrain, when the head behind it is raised and then kept at a constant value. The essential idea employed in the analysis is to consider the unsteady flow as a time-dependent perturbation of the final steady flow. The unsteady potential ϕ(x, y, t) is expanded in a power series of e−λt, of the form $\phi (x,y,t) = \phi_0(x,y) + \phi_1(x,y)e^ {\lambda t} + O(e^{-2\lambda t})$ where ϕ0(x, y) is the known steady-state potential, ϕ1(x, y) is a perturbation potential and O(e−2λt) = ϕ2(x, y) e−2δt + ϕ3(x, y) e−3λt + …. Each of the terms ϕn(x, y)e−nλt can be thought of as being a perturbation term of its precursor in the series, and the present approach is limited to the computation of the first perturbation term ϕ1(x, y)e−λt.

It is shown that ϕ1 satisfies Laplace's equation ∇2ϕ1 = 0 in a dimensionless hodograph plane. The free-boundary condition is linear but complicated, containing the eigenvalue λ, which is fixed by a determinantal equation. The amplitude of the displacement of the free surface is left undetermined; only the mode of the motion and the eigenvalue are computed. The results of a numerical example are summarized.

This paper deals theoretically with the problem of the hydrodynamic stability of a stratified flow of a viscous fluid. The primary flow consists of two laminar streams of viscous fluids of different densities flowing in opposite directions between two parallel inclined planes under the action of gravity. The effect of surface tension at the interface of the two fluids is included in the formulation of the problem.

Since instability can be expected to occur at low Reynolds numbers when the inclination is nearly vertical, the solution of the Orr–Sommerfeld equations is developed as a power series in the transverse space co-ordinate. It is shown that for the vertical case, the flow is unstable for all values of the Reynolds number. Surface tension is found to influence both the direction and celerity of the disturbance. Results are also given for inclinations slightly away from the vertical, where small critical Reynolds numbers do exist.

The effect of increasing the rate of mixing in turbulent boundary layers in a region of adverse pressure gradient has been investigated experimentally. Only the two-dimensional case was considered. The boundary layer was formed on a flat wall in a special wind tunnel in which a variety of adverse pressure gradients could be obtained. Speeds were low enough to justify the neglect of compressibility. The main objective was to compare the effect of increasing the rate of mixing with the effect of reducing the pressure gradient on boundary-layer development and separation. A Variety of mixing schemes was tried, all of them involving fixed devices arranged in a row on the surface in the region of rising pressure. While these differed considerably in effectiveness, they had a generally similar effect on the flow; and, except for effects arising from changes in displacement and momentum thickness introduced at the devices, their effect on the layer was basically equivalent to that of a decrease in pressure gradient. Apart from forced mixing, the shape of the pressure distribution was found to have a significant effect on displacement and momentum thickness, these being minimized and the wall distance decreased for a given pressure rise by a distribution with an initially steep and progressively decreasing gradient.

The utility of piston theory as a means of solving insentropic two-dimensional aerodynamic problems has been aptly demonstrated by many writers in recent years. The present treatment removes the restriction of isentropic flows, extending the applicability of piston theory to flows with strong shocks. Sample calculations for a thin biconvex airfoil are carried out in which the local flow is assumed to be isentropic and non-isentropic. Comparison of the result is made with that of the shock expansion theory of Cole, Gazley & Williams (1956).

Problems of the propagation of shock waves and strong detonation waves through ducts of variable cross-section and ducts with porous walls, and the interaction of a rarefaction wave with a shock and a contact surface in one-dimensional unsteady, or two-dimensional steady, flow are solved using a simple rule. Tables enclosed in this paper permit efficient calculation of several problems of non-uniform shock motion.

This paper attempts to explain theoretically the observed results that (1) the cells in steady convection approach a hexagonal form, and (2) the occurrence of ascent or descent in the middle of the cell depends on how the kinematical viscosity varies with temperature. The theory is based on non-linear equations and, of course, a variable coefficient of viscosity.

It is found that, due to the variation of viscosity with temperature, the non-linear terms contain a second-order term which is destabilizing. This second-order term regulates the development and leads to a final motion composed of regular hexagons with ascent or descent in the middle of the cell according as the viscosity decreases or increases with temperature.

The influence of a variable viscosity on Rayleigh's result concerning the initiation of convection is obtained as a by-product.

This paper considers the problem of the propagation of a shock wave down a non-uniform tube. Linearized solutions to the problem (Chester 1954) do not hold when the velocity behind the shock is near or at the sonic speed. By retaining appropriate non-linear terms of the flow equations, a solution is obtained which holds for all conditions behind the shock, and reduces to the linearized solution for conditions away from sonic.

The behaviour of supersonic or subsonic flow entering regions of expanding or contracting area changes is discussed. It is found that subsidiary shocks may be formed; these can be located and described using the present solution. Explicit solutions are given for the cases of supersonic or subsonic flow entering a region of linearly expanding or contracting area. The point of shock formation as well as the path of the subsidiary shock is obtained for the case in which the area contracts.

The object of this paper is to present a unified account of the distributions of velocity, shear stress and enthalpy in the compressible turbulent boundary layer on a flat plate. As a start, the set of velocity profiles measured over a range of heat-transfer conditions at Mach numbers between 5 and 8 by Lobb, Winkler & Persh (1955) is examined. It is found that by plotting in terms of the Howarth variable $\eta = \int ^y_0 (\rho|\rho_ 0)dy$, the outer parts of the profiles for different Mach numbers are brought together on a single curve of the approximate form u/u∞ = (η/Δ)1/n, Δ being the transformed boundary-layer thickness. By evaluating the reference density ρ0 and kinematic viscosity ν0 at the so-called ‘intermediate’ enthalpy (Eckert 1955) the innerparts of the profiles can also be collapsed, although less completely, to fit a ‘law of the wall’ u/uτ = A log (ηuτ/ν0 - c) + B. Here uτ = (τw/ρ0)½, and A, B and c are the same constants as in incompressible flow.

These properties provide a physical starting point from which the remaining features of the mean flow can be calculated. By substitution of appropriate stream functions in the equation of motion the distribution of shear stress r in inner and outer regions is found; this approximates to the form $\tau |\tau_w = 1 - (u|u_\infty)^{(n+2)}$ over the whole layer. A relation between the distributions of enthalpy and shear stress is then found from the energy equation, using a turbulent Prandtl number a which is assumed constant across the layer to relate eddy conductivity to eddy viscosity. The final expression is similar in form to Crocco's integral for the laminar boundary layer with α taking the place of the laminar Prandtl number σ, but contains two extra terms proportional respectively to (α − σ)cf and (α − σ)c½j, which represent the effect of the inner viscous regions.

The enthalpy integral is evaluated using the stated velocity-shear relation, and an expression which agrees well with the available experimental data is found for the heat-transfer coefficient as a function of recovery factor and skin-friction coefficient. It is also found that the usual quadratic enthalpy-velocity relation, exact for α = σ = 1, remains an acceptable approximation for Prandtl numbers considerably different from unity.

The mean velocity profile near the surface of turbulent flow in a broad open channel is discussed with dimensional arguments, and a new empirical constant m is introduced which is analogous to von Kármán's constant for flow near a rigid boundary. It is shown that, while the velocity profile depends only rather weakly on m, the dependence of the coefficient of apparent longitudinal diffusion is stronger, and measurements of diffusion could, in principle, provide an accurate determination of its value. The new profiles for various values of m are compared with those in current use, and finally the correction for finite Reynolds number is discussed.

This paper concerns an investigation into the behaviour of a layer of dense salt solution on the floor of a sloping rectangular pipe in which there is a turbulent flow. The various phenomena which are observed are described qualitatively and by the presentation of typical concentration profiles.

Numerical values are obtained experimentally for the rates of spread of the edge of the layer in the case where the salt is moving entirely in the direction of the main stream. The rate of spread is found to depend mainly on the slope α and on the pipe Richardson number, defined by Rip = DΔd cos α/V2, where D is the depth of the pipe, Δd = g(ρd−ρa)/ρa, ρd is the density of the fully mixed discharge and ρa is the density of the ambient flow. In the range of Riρ from 0 to 0·005 the rate of spread of the layer decreases by a factor of about 3 at small slopes.

Some discussion is given of the factors determining the initial rate of spread just after the layer leaves the slit. Finally, it is shown how the depth measurements can be related to the determination of the concentration at the floor.

Two parallel infinite plane disks, between which is contained a viscous fluid, oscillate torsionally about a common axis. Of specific interest are the cases where (i) one disk only is in motion, and (ii) the disks oscillate 180° out of phase, but with the same frequency and amplitude. The basic parameter is found to be the Reynolds number derived from the frequency, the kinematic viscosity and the disk separation. Solutions of the Navier-Stokes equations for both small and large Reynolds numbers are developed and the transverse and radial-axial flows investigated.

The assumption of dynamic similarity is used to determine the velocity potential of an unsteady line source which lies on a plane separating two media of different density and sound velocity. The solution first derived is for a source whose strength varies with time as a step function; the pressure in this case may be identified with Hadamard's elementary solution which in a homogeneous fluid is (c/2π) (c2t2-r2)−½ if r < ct, and 0 if r > ct. We next derive the solution for a source whose strength has a delta-function time dependence; we then describe the results for a supersonic point source moving on the interface, and finally we transfer the results to solve the corresponding electromagnetic problem.

This paper continues an investigation into the mixing of a dense layer of salt solution in a turbulent pipe flow in order to obtain a more detailed understanding of the underlying physical processes. The effect of the density difference on the velocity profile in a sloping pipe is calculated using a simplified model, and the results compared with direct measurements obtained by timing streaks of dye at various levels in the pipe. These velocity profiles are also used in conjunction with density profiles to estimate the dependence of the transfer coefficients for salt and momentum KS and KM, on stability.

It is found that KS is much more greatly affected by the density gradient than KM, and that the ratio KS/KM may be represented, to the accuracy of the experiments, as a function of the local Richardson Number Ri. The results agree with what is known of KS/KM in neutral and very stable conditions, and they confirm an earlier prediction by Ellison that the critical flux Richardson number, at which KS becomes zero, is much less than unity.

Finally, a crude semi-empirical method is outlined which indicates how the new measurements of the transfer coefficients may be related to the overall properties of the flow discussed in the first part of the paper.

Reverse-flow relations are used to provide convenient expressions for the aerodynamic forces which act on a general wing-fuselage combination in supersonic flow. The drag, lift, spanwise and chordwise lift distributions, and wing moments are treated. Consideration is given to available methods and results, including tables, which apply to the wing-fuselage interference problem, and reverse-flow relations are then employed to effect desirable changes in the detining force expressions by introducing wings or fuselages in the reverse flow. It is shown that the aerodynamic forces can be determined from a solution for the pressure on only the fuselage surface within the domain of dependence of the wing, this region being selected on the basis of relative ease of computation. In some cases the simplification achieved is quite substantial. Certain results in the literature arise naturally as special cases. The reverse-flow theorem is re-examined in the light of the procedure considered, and it is found to be inapplicable in one respect. The difficulty is circumvented by constructing an acceptable configuration which is equipollent to the prescribed one. It appears that the method given is applicable in other linear hyperbolic problems in mathematical physics when bulk or gross information is required and a suitable reciprocity relation can be constructed.

An experimental investigation has been made of the effect of ejecting nitrogen and helium coolant gases at the nose of a bluff body at a nominal Mach number of 5·8. The gases were ejected with ‘swirl’, to encourage them to flow tangentially to the model surface at ejection, and also straight out. Measurements were made of the pressure and temperature on the surface of the model at incidences of 0, 4, 8°, and for a range of coolant gas flows. From these measurements the flow patterns and distributions of heat flux were deduced.

It was found that ejection with swirl did not in fact lead to an easing of the heating problem because the high tangential velocity with which the coolant was injected into the boundary layer increased the wall shear stress, and hence, by the Reynolds analogy, the heat flux, so that it predominated over the reduced driving temperature difference associated with the cooled boundary layer.

With straight-out ejection it was found that the heat alleviation capabilities of the ejected coolant were reduced considerably if the flow rate was sufficiently large that the bow shock wave was bulged out. However, provided that the external flow is not disturbed, straight-out ejection provides an effective way of reducing the heat flux.

Measurements of spectra of sound emission to the far field from jets at high subsonic velocities are presented. The similarity relations found in the experiments suggest a mechanism of sound generation and scattering where the latter is of dominant importance. A possible mechanism is described.

An exact solution is presented in this paper for the problem of laminar convective flow under a pressure gradient along a vertical pipe, the walls of which are heated or cooled uniformly; the solution is based on the assumption that velocity and buoyancy profiles far from the pipe entrance do not change with height, and entry-lengt effects are ignored. Two different types of behaviour are found accordingly as the pressure gradient and buoyancy forces act together or in opposition near the centre of the pipe.

When an upflow is heated (or a downflow cooled) the velocity near the walls is increased relatively and that near the axis decreased until, for sufficiently large Rayleigh numbers, definite velocity and thermal boundary layers are formed.

In the case of cooled upflow (or heated downflow) there is an increase in the velocity across the whole profile for small Rayleigh numbers. As the Rayleigh number is increased the velocity and buoyancy increase, slowly at first and then rapidly, and the solution ‘runs away’ at a Rayleigh number of about 33. For higher Rayleigh numbers, laminar Poiseuille flow of an increasingly complicated profile is theoretically possible, but is unlikely to be found in practise.

Dungey (1956) has shown that the number densities of electrons and positive ions, under the action of turbulence and the magnetic field in the ionosphere, are closely equal—in other words the only fluctuations in electron density are those controlled electrostatically by the ions—and he has estimated the magnitude of the fluctuations. The present paper uses Dungey's model and results, reduces his equations to a single equation for electron density, in each of several cases, and then investigates the possible spectra of fluctuations. It is concluded that, in the circumstances that commonly arise, the spectrum on this model should be nearly isotropic, though exceptionally there could be a strong elongation of irregularities at right angles to the magnetic field. Thus some other mechanism is required to account for the elongation that is observed, parallel to the field.

Below 110 km, where the magnetic effect on the ions’ motion is small, and at wave-numbers in the inertial subrange of the turbulence, dimensional argument shows that the spectrum function (integrated over all directions) is proportional partly to κ−1 and partly to $\kappa^{-\frac{5}{3}$. Above 120 km the magnetic effect is large; a more detailed study shows that when turbulence is present, which probably is not often, the spectrum function in the inertial subrange is proportional to $\kappa {\frac{2}{3}$, with considerable anisotropy.