Nonlinear internal gravity waves in an inviscid incompressible fluid are discussed for the case when the properties of the medium vary slowly on a scale determined by the local wave structure. A two-time-scale technique is used to obtain transport equations which describe the slowly varying modulations of the waves. Various solutions of these transport equations are discussed.

Alfvén-gravitational waves are found to propagate in a Boussinesq, inviscid, adiabatic, perfectly conducting fluid in the presence of a uniform transverse magnetic field in which the mean horizontal velocity U is independent of vertical height z. The governing wave equation is a fourth-order ordinary differential equation with constant coefficients and is not singular when the Doppler-shifted frequency Ωd = 0, but is singular when the Alfvén frequency ΩA = 0.If Ω2d < Ω2A the waves are attenuated by a factor exp − [2ΩA(N2−Ω2d)½−Ω2d + Ω2A]z, which tends to zero as z → ∞. This attenuation is similar to the viscous attenuation of waves discussed by Hughes & Young (1966). The interpretation of upward and downward propagation of waves is given.

Alfvén-gravitational waves propagating in a Boussinesq, inviscid, adiabatic, perfectly conducting fluid in the presence of a uniform aligned magnetic field in which the mean horizontal velocity U(z) depends on height z only are considered. The governing wave equation has three singularities, at the Doppler-shifted frequencies Ωd = 0, ± ΩA, where ΩA is the Alfvén frequency. Hence the effect of the Lorentz force is to introduce two more critical levels, called hydromagnetic critical levels, in addition to the hydrodynamic critical level. To study the influence of magnetic field on the attenuation of waves two situations, one concerning waves far away from the critical levels (i.e. Ωd [Gt ] ΩA) and the other waves at moderate distances from the critical levels (i.e. Ωd > ΩA), are investigated. In the former case, if the hydrodynamic Richardson number JH exceeds one quarter the waves are attenuated by a factor exp{−2π(JH −¼)½} as they pass through the hydromagnetic critical levels, at which Ωd = ± ΩA, and momentum is transferred to the mean flow there. Whereas in the case of waves at moderate distances from the critical levels the ratio of momentum fluxes on either side of the hydromagnetic critical levels differ by a factor exp {−2π(J −¼)½}, where J (> ¼) is the algebraic sum of hydrodynamic and hydromagnetic Richardson numbers. Thus the solutions to the hydromagnetic system approach asymptotically those of the hydrodynamic system sufficiently far on either side of the magnetic critical layers, though their behaviour in the vicinity of such levels is quite dissimilar. There is no attenuation and momentum transfer to the mean flow across the hydrodynamic critical level, at which Ωd = 0. The general theory is applied to a particular problem of flow over a sinusoidal corrugation. This is significant in considering the propagation of Alfvén-gravity waves, in the presence of a geomagnetic field, from troposphere to ionosphere.

Results on the behaviour of controlled wave disturbances introduced artificially into turbulent channel flow are reported. Weak plane-wave disturbances are introduced by vibrating ribbons near each wall. The amplitude and relative phase of the streamwise component of the induced wave is educed from a hot-wire signal, allowing the wave speed, the attenuation characteristics and the wave shape to be traced downstream. These results form a basis for evaluation of closure models for the dynamical equations describing wave components in shear-flow turbulence.

The dynamical equations governing small amplitude wave disturbances in turbulent shear flows are derived. These equations require additional equations or assumptions about the wave-induced fluctuations in the turbulence Reynolds stresses before a closed system can be obtained. Some simple closure models are proposed, and the results of calculations using these models are presented. When the predictions are compared with our data for channel flow, we find it essential that these oscillations in the Reynolds stresses be included in the model. A simple eddy-viscosity representation serves surprisingly well in this respect.

The paper presents the outcome of experimental research on turbulence-induced secondary flows in square-sectioned ducts. The main emphasis of the experiments has been on the measurement of the secondary flows in a duct with equally roughened surfaces. Here the secondary flow is a substantially larger proportion of the axial flow than is the case in smooth-walled ducts. With the secondary velocities normalized by the friction velocity, however, the resultant profiles for smooth and rough surfaces are the same, within the precision of the measurements.

The acceleration of a shock wave in an ideal gas of decreasing density has previously been studied. The problem is reconsidered here with empirical inclusion of real-gas effects for strong shocks in hydrogen. Experimental results suggest that previous shock acceleration models are valid only for a limited range of the Knudsen number in finite geometries and that for large final-state Knudsen numbers a free-expansion model best describes the experimental results.

Messiter's thin shock layer approximation for hypersonic wings is applied to several non-conical shapes. Two calculation methods are considered. One gives the exact solution for a particular three-dimensional geometry which possesses a conical planform and also a conical distribution of thickness superimposed upon a surface cambered in the chordwise direction. Agreement with experiment is good for all cases, including that where the wing is yawed. The other method is a more general approach whereby the solution is expressed as a correction to an already known conical flow. Such a technique is applicable to conical planforms with either attached or detached shocks but only to the non-conical planform for the region in the vicinity of the leading edge when the shock is attached.

The motion of a rigid sphere responding to the passage of an acoustic pulse is considered by means of a simple approximate model which neglects the diffraction of the pulse front. The model is based on a solution of the equivalent inviscid problem and assumes that, initially, the flow field around the sphere corresponds to the steady incompressible flow of an inviscid fluid over a sphere at rest. For t > 0, the motion is studied by means of the unsteady Stokes equations. Results for the sphere's velocity, displacement and drag are obtained in closed form in terms of tabulated functions and compared with results obtained by using the Stokes drag. It is found that, when the ratio of gas density to sphere material density is finite, the initial response of the sphere differs considerably from that predicted by the use of the Stokes drag. However, when the ratio of gas density to sphere material density is infinitesimal, the differences disappear. These results may be of some importance in the study of shock-induced droplet collisions in aerosol clouds.

The average roll diameter in Rayleigh convection for 2000 < R < 31000, where R is the Rayleigh number, has been measured from photographs of three convecting fluids: air, water and a silicone oil with a Prandtl number σ of 450. For air the average dimensionless roll diameter was found to depend uniquely upon R and to increase especially rapidly in the range 2000 < R < 8000. The fluids of larger σ exhibited strong hysteresis but also had average roll diameters tending to increase with R. The increase in average roll diameter with R tended to decrease with σ. Through use of two-dimensional numerical integrations for the case of air it was found that the increase in average roll diameter with R provides an explanation for the usual discrepancy in heat flux observed between experiment and two-dimensional numerical calculations which prescribe a fixed wavelength.

McIntyre (1972) has demonstrated that transient nonlinear self-interactions among the lee waves downstream of an obstacle in an axisymmetric, inviscid, rotating flow yield a columnar disturbance that moves upstream of the obstacle. This disturbance is calculated for a dipole, the moment of which increases slowly from 0 to a3, in a tube of radius R as a function of δ = a/R and K = 2ΩR/U (Ω and U being angular and axial velocities in basic flow). The asymptotic limit δ ↓ 0 with k ≡ K δ fixed, which is relevant for typical laboratory configurations and for the abstraction of unbounded flow, yields: (a) an upstream velocity with an axial peak of u0 = 0·011k5 δ U and a first zero at a radius of 2·8U/Ω; (b) an upstream energy flux of roughly $- \frac{3}{2}\pi\rho U^2 R^2 u_0$; (c) an upstream impulse flux of 0·20D (D = wave drag on dipole). The results (a) and (b), albeit based on the hypothesis of small disturbances, suggest that nonlinear self-interactions among the lee waves could be responsible for upstream blocking. The results (a) and (c) imply that, although upstream influence is absent from an unbounded (δ = 0) flow in the sense that the axial velocity vanishes like δ, it is present in the sense that approximately one-fifth of the impulse flux associated with the wave drag appears in the upstream flow.