The sound field generated by the interaction of spatial instabilities on the shear layer shed from a duct with the nozzle lip is studied. It is shown that the intensity varies with direction θ from the exhaust and with the subsonic exhaust speed U according to I ∼ U6(1 − cosθ)2 and I ∼ U6 sin2 θ for the axisymmetric and first azimuthal (sinuous) modes respectively. The first of these results is interpreted in terms of monopole and dipole sources at the exit plane, representing the acoustic effect of fluctuating mass flow and axial thrust across the exit plane, and the second in terms of a transverse dipole at the exit plane, corresponding to fluctuations in cross-stream thrust. A correlated thrust fluctuation of 1% is shown to overwhelm the jet mixing noise in the forward arc, θ > 90° while the acoustic efficiency of the interaction process is never less than 10−6M3 even under the cleanest possible exit conditions. Forward flight of the duct a t Mach number Mα is shown to increase the forward-arc intensity by the factor (1 + Mα cos θ)−4. It is suggested that much of the discrepancy between the noise fields of real engines and the predictions of Lighthill's theory of jet mixing noise – the so-called ‘excess noise’ problem – can be explained in terms of this interaction mechanism.

The structure and propagation rates of premixed flames are determined by singular perturbation in the limit where the activation temperature is large relative to other flow temperatures for several basic flows. Specifically, the simple kinetics of an exothermic first-order monomolecular decomposition under Arrhenius kinetics is studied for one-dimensional laminar flame propagation, spherically symmetric quasi-steady monopropellant droplet burning, and other simple geometries. Results elucidate Lewis-number effects, losses owing to fuel gasification processes, and conditions under which the thin-flame approximation is a limit of finite-rate Arrhenius kinetics.

The behaviour of an initially plane, strong shock wave propagating into a conical convergence is investigated experimentally and theoretically. In the experiment a 10° half-angle cone is mounted on the end of a pressure-driven shock tube. Shock waves with initial Mach numbers varying from 6.0 to 10·2 are generated in argon a t a pressure of 1·5 Torr. During each run local shock velocities a t several positions along the cone axis are measured using a thin multi-crystal piezoelectric probe inserted from the vertex. This technique produces accurate velocity data for both the incident and reflected shock waves. In the corresponding analysis, a simplified characteristics method is used to obtain an approximate solution of the axisymmetric diffraction equations derived by Whitham (1959).

Both the shock velocity measurements and the axisymmetric diffraction solution confirm that the incident shock behaviour is dominated by cyclic diffraction processes which originate at the entrance of the cone. Each diffraction cycle is characterized by Mach reflexion on the cone wall followed by Mach reflexion on the axis, These cycles evidently persist until the shock reaches the cone vertex, where the measured velocity has increased by as much as a factor of three. Real-gas effects, enhanced in the experiment by increasing the initial Mach number and decreasing the pressure, apparently alter the shock wave behaviour only in the region near the vertex. Velocity measurements for the reflected shock within the cone show that the shock velocity is nearly constant throughout most of the convergence length.

A finite-difference technique for the numerical calculation of two-dimensional stratified incompressible fluid flows is presented. Small density variations are not assumed, so that this method is generally applicable to a wide variety of problems. To illustrate this new technique a calculation has been made of the collapse of a uniformly mixed region in a linearly stratified fluid. In addition to giving excellent agreement with experimental data, the calculations also reveal the mechanism for an observed change in scaling behaviour.

It is shown that for low values of the ambient pressure the flow field for a steady spherically symmetric expansion can be divided into three parts termed the inviscied region, the intermediate layer and the shock layer. Analytic solutions are available in the first two regions and a complete integration of the equation is required in the third. Numerical solutions indicate that such a structure is achieved in the limit and the universality of the solutions in the individual regions is confirmed.

This note intends to clarify the way in which the expansion of an initially uniform gas into an ambient atmosphere degenerates into an expansion into a perfect vacuum. The starting-point is a paper by Greenspan & Butler (1962) in which certain tentative remarks are made reprding this question. We show that the flow consists of an infinite-strength contact front coinciding with the gasvacuum interface of the perfect-vacuum expansion, driving an infinite-strength shock into the vacuum ahead of it. The ga,s-vacuum interface and the shock are separated a t all times even though the density of the gas through which the shock travels vanishes in the vacuum limit.

An experimental study is made of the instability and transition of a laminar free shear layer by sound excitation. Primary emphasis is placed on the nonlinear stages of transition. Transition from laminar instability to turbulent breakdown covers approximately five wavelengths of downstream distance. The instability has six distinct regions of behaviour : a region of exponential growth described by linear theory; a nonlinear region where critical-layer effects are important, and harmonics and subharmonics are generated; a region of finite amplitude equilibration of the fundamental mode; a region of finite amplitude triggered sub-harmonic instabilities; a region of three-dimensional longitudinal vortex formation; and a final region of weak secondary instabilities and turbulent breakdown.

The axial and transverse wave motions of an inviscid perfect gas in isothermal solid-body rotation in a cylinder are investigated. Solutions of the resulting eigenvalue problem are shown to correspond to two types of waves. The acoustic waves are the rotational counterparts of the well-known Rayleigh solutions for a gas at rest in a cylinder. The rotational waves, whose amplitudes and frequencies go to zero in the non-rotating limit, exhibit phase speeds both larger and smaller than the speed of sound. The effect of rotation on the frequency and structure of these waves is discussed.

The radiation and scattering of time-periodic surface waves by partially immersed objects is investigated in the short-wave asymptotic limit ε → 0, where ε is a non-dimensional wavelength. Details are given for the prototype radiation and scattering problems in which the fluid has infinite depth and the body is a two-dimensional dock of finite width and zero thickness. The solutions are then generalized to deal with other two-dimensional geometries, with the restriction that the ends of the obstacle are horizontal for a distance of many wavelengths. The method of matched expansions is used. A first approximation ϕ0, presumed to be a good estimate for the potential throughout most of the fluid region, is obtained by replacing the free-surface condition by its formal limit ϕ0 = 0. In the vicinity of the ends of the obstacle, the correct surface condition is used but the geometry of the problem is simplified. The remaining surface layers are dealt with by superimposing on the function ϕ0 regular wave trains of the appropriate amplitude.

Experiments on fully developed turbulent flow in a channel which is rotating at a steady rate about a spanwise axis are described. The Coriolis force components in the region of two-dimensional mean flow affect both local and global stability. Three stability-related phenomena were observed or inferred: (i) the reduction (increase) of the rate of wall-layer streak bursting in locally stabilized (destabilized) wall layers; (ii) the total suppression of transition to turbulence in a stabilized layer; (iii) the development of large-scale roll cells on the de-stabilized side of the channel by growth of a Taylor-Gortler vortex instability.

An appropriate local stability parameter is the Richardson number formulated by Bradshaw (1969) for this case and the analogous cases of flow over curved walls and of shear-layer flow with density stratification. Local effects of rotational stabilization, such as reduction of the turbulent stress in wall layers, can be related to the local Richardson number in a simple way. This paper not only investigates this effect, but also, by methods of flow visualization, exposes some of the underlying structure changes caused by rotation.

The test-field model for isotropic turbulence is restated in a form which is independent of the choice of orthogonal basis functions for representing the velocity field. The model is then extended to non-stationary inhomogeneous turbulence with a mean shearing velocity, contained by boundaries of arbitrary shape. A modification of the model is introduced which makes negligible changes in the numerical predictions but which greatly simplifies computations when the co-variance matrix and related statistical matrices are non-diagonal. The altered model may be regarded as a kind of generalization of Orszag's eddy-damped Markovian model, with the damping factors determined systematically, in representation-independent form, from dynamical equations. The final equations of the test-field model are presented in a sufficiently explicit form to serve as a starting point for numerical work. To facilitate comparison, the corresponding direct-interaction equations for inhomogeneous turbulence with mean shear are presented also, in a uniform notation. The test-field model is much faster to compute than the direct-interaction approximation because, in the former, only single-time statistical functions need be computed. This advantage is at the cost of a less rich and less faithful representation of the dynamics.

The paper describes a simple but general method for solving 'steady-state’ problems involving internal gravity waves in a stably stratified liquid. Under the assumption that the motion is two-dimensional and that the Brunt-Väisälä frequency is constant, the method is used to re-derive in a very simple way the solutions to problems where the boundary of the liquid is either a wedge or a circular cylinder. The method is then used to investigate the effect that a model of the continental shelf has on an incident train of internal gravity waves. The method involves analytic continuation in the frequency of the disturbance, and may well prove to be effective for other types of wave problem.

A hypersonic gun tunnel has been used to measure the heat-transfer-rate distribution over a compression corner under turbulent boundary-layer conditions. Attached, incipient and separated flows are considered. The results are compared with other experimental data and with the predictions of a simple theory.

The objectives of this study were to determine whether the addition of drag-reducing macromolecules alters the structure of the viscous sublayer and thereby modifies the production of kinetic energy in turbulent wall flows. This was accomplished by visualizing the near-wall region of a fully developed two-dimensional channel flow. Motion pictures were taken of dye injected into the near-wall region. Both water and a dilute drag-reducing polyethylene oxide-FRA solution were used as working fluids. The motion pictures were analysed to determine the spanwise spacing and the bursting rate of low-speed streaks that are characteristic of the viscous sublayer. The amount of drag reduction was established from pressure-drop measurements in pipe flows and a correlation that is independent of hydraulic diameter.

The data show that the time between bursts for an individual streak in a drag-reducing flow has the value for a water flow at the reduced wall shear. However, both the physical and the non-dimensional streak spacing is significantly increased in the drag-reducing flows and thus the spatially averaged bursting rate is decreased. This evidence strongly suggests that the dilute polymer solution decreases the production of turbulent kinetic energy by inhibiting the formation of low-speed streaks. A tentative explanation for this behaviour which is based upon the solution's high resistance to elongational strains and vortex stretching is offered.

The general analysis developed in Parts 1 and 2 of three-dimensional duct flows subject to a strong transverse magnetic field is used to examine the flow in diverging ducts of rectangular cross-section, the walls of which are electrically non-conducting. A dramatically different flow is found in this case from that studied in Part 2, where the side walls parallel to the magnetic field were highly conducting. Now it is found that the core velocity normalized with respect to the mean velocity is of O(M−½) while the velocity in the side-wall boundary layers is of O(M½), so that these boundary layers carry most of the flow. The problem of entry is solved by analysing the change from fully developed Hartmann flow in a rectangular duct to the flow in the diverging duct. It is found that the disturbance in the upstream duct decays exponentially. The analysis of the side-wall boundary layers is more difficult than that in Part 1 on account of the different boundary conditions and requires the solution of two coupled integro-differential equations. Numerical solutions are obtained for a duct whose width increases linearly in the flow direction.

It is shown that the equilibrium shape of an incompressible dielectric fluid drop rotating with constant angular velocity in the presence of a uniform external electric field of appropriate magnitude along the axis of rotation is spherical. For an inviscid fluid drop, the stability of this spherical configuration to small deformations is investigated by means of Chandrasekhar's virial method. We find that a rotating drop in the presence of an electric field parallel to the axis of rotation is, in some respects, more stable than when either only the electric field or only rotation is present. This is due to the fact that the application of an electric field parallel to the axis of a rotating drop, or of rotation parallel to an electric field in which a drop is immersed, shifts the instability mechanism to another normal mode.

The viscous compressible flow in the vicinity of a right-angle corner, formed by the intersection of two perpendicular flat plates and aligned with the free stream, is investigated. In the absence of viscous-inviscid interactions and imbedded shock waves, a theory is developed that is valid throughout the subsonic and supersonic Mach number range. Within this limitation and the additional assumptions of unit Prandtl number and a linear viscosity-temperature law, a consistent set of governing equations and boundary conditions is derived. The method of matched asymptotic expansions is applied in order to distinguish the relevant regions in the flow field.

In the corner region the Crocco integral is shown to apply, even for a three-dimensional flow field. The equations governing the flow in the corner layer consist of four coupled nonlinear elliptic partial differential equations of the Poisson variety. Since they do not lend themselves to analytic solution, numerical methods are employed. Two such methods used here are the Gauss-Seidel explicit technique and the alternating direction implicit method. The merits of both techniques are discussed with regard to convergence rate, accuracy and stability. The calculations show that in cases where the Gauss-Seidel method fails to give converged solutions, owing to instability, the alternating direction implicit method does provide converged solutions. However, in cases where both methods are convergent, there is no appreciable difference in convergence rates. The numerical calculations were done on a CDC 6600 computer.

Results of calculations are presented for representative compressible-flow conditions. The extent of the corner disturbance is controlled by the Mach number and wall temperature ratio in a manner analogous to the two-dimensional boundary layer. A swirling motion is noted in the corner layer which is influenced to a great extent by the asymptotic cross-flow profiles. The skin-friction coefficient is shown to increase monotonically from zero at the corner point to its asymptotic two-dimensional value. For cold wall cases, this value is approached more rapidly. The asymptotic analysis indicates that for even colder wall cases, not considered here, an overshoot is possible.

According to classical boundary-layer theory, uniform flow past a semi-infinite wedge, inclined at a negative angle ½πβ to the direction of the free stream, does not separate unless β ≤−0·1988. It has been assumed, therefore, that, in the range −0·1988 < β < 0, the flow within the boundary layer is represented by the Falkner-Skan equation, which, as was shown by Stewartson (1954), has two admissible solutions. All such solutions for p < 0 appear to be somewhat unsatisfactory, however, because they require an adverse pressure gradient, which, by becoming infinite as the corner of the wedge is approached, could lead to separation even if, β > −0·1988. In addition, the structure of the high Reynolds number flow for β < −0·1988 has remained, to date, unresolved.

We present here a fundamentally different solution to this classical problem which eliminates the singularity in the potential region by allowing the flow to separate at the leading edge of the inclined surface. The associated flow field is then characterized by an essentially uniform free stream flowing over an inviscid and, to a high approximation, irrotational region of reverse flow in which the velocity is of O(R−½) in magnitude, R being the Reynolds number. Mixing of these two streams is confined to a free shear boundary layer, of O(R−½) in thickness, extending downstream from the leading edge and parallel to the direction of the undisturbed main flow. Finally, an additional boundary layer, of O(R−½) in thickness, is shown to exist between the separated region and the surface of the wedge. Owing to the absence of a characteristic length in the problem, similar solutions to the appropriate equations describing the flow in each region are obtained and are valid for all β < 0, provided that the Reynolds number is sufficiently large. The analysis is then extended to higher order in R to increase its range of validity and to demonstrate that the proposed structure of the flow field remains self-consistent. Although the solution is developed only for a semi-infinite wedge with β < 0, it is believed that certain of its features may be of value in the analysis of other problems involving high Reynolds number separated flows.

The theory of concentrated two-phase mixtures developed in the previous parts of this paper is applied to analysis of the structure of the local random motion (pseudo-turbulence) occurring in flows of suspensions of small solid spheres. Suspensions under study are assumed to be locally homogeneous in the sense that large-scale agglomerates of many particles or voids filled with the pure liquid do not arise in their flows and particles can be approximately regarded as statistically independent units.

Coefficients of the particle diffusion caused by pseudo-turbulence are calculated without restrictions imposed on the value of the Reynolds number Re characterizing the fluid flow around one particle. Other pseudo-turbulent quantities (the r.m.s. pseudo-turbulent velocities of both phases, their effective pseudo-turbulent viscosities in a shear flow, etc.) are considered for small Re. In particular, a natural explanation is given to the known effect of the reduced hydraulic resistance of a fluidized bed as compared with that of a stationary particulate bed of the same porosity.

Additionally, some properties of the mean motion of a suspension influenced by pseudo-turbulence are discussed brifly. By way of example, two problems are considered: stability of the upward flow of a homogeneous suspension with respect to small perturbations depending upon the vertical co-ordinate and time, and the spatial distribution of particles suspended by the upward flow of a fluid under gravity.

The 36th European Mechanics Colloquium (EUROMECH 36) which was held at Imperial College from 17 to 19 April 1972 was the first to be devoted to laser anemometry. It brought together European researchers whose primary interests included physics, electronics and fluid mechanics with the following purposes.