The peeling of a flexible strip from a rigid surface to which it is attached by a thin layer of adhesive is discussed, treating the adhesive as a Newtonian viscous fluid. This makes it possible to examine the flow and stress distributions ahead of the point where separation occurs. The conditions at this point are taken to be the same as those observed in other cases where a stream of viscous fluid separates into two. In particular, the effect of surface tension at the separating meniscus on the speed of peeling is predicted.

Experiments are described in which a sheet of ‘Melanex’ 4μm thick was laid on a sheet of fluid covering a piece of plate glass. The apparatus was designed to ensure that this was peeled off at a constant angle, and the speed of the separation meniscus, as well as the load on the sheet, was measured. The experimental results are analysed in the light of the theory and shown to be consistent with it.

An interesting feature is the prediction that at low peeling speeds there is a great reduction in the thickness of the adhesive layer immediately ahead of the line of separation. Although the initial thickness of the layer dictates the scale of the shape adopted by the strip ahead of this line, it exerts no effect upon the relation between the external variables.

It is noted that, when the adhesive layer remains intact ahead of separation, the physical appearance of commercially available tapes in slow peeling can resemble that of simple viscous adhesives.

The distance between the separation point and the final approach to a fully developed turbulent mixing layer is found to be of the order of a thousand times the momentum-deficit thickness of the initial boundary layer, whether the latter be laminar or turbulent. There are correspondingly large shifts in the virtual origin of the mixing layer, resulting in spurious Reynolds-number effects which cause considerable difficulties in tests of model jets or blunt-based bodies, and which are probably responsible for the disagreements over the influence of Mach number on the development of free shear layers. These effects are explained.

The paper deals with an extension of previous work on the radiation properties of turbulent flow formed on compliant surfaces. The effect of simple supports is shown to be acoustically equivalent to an extended dipole system of strength equal to the support stress. The dipole radiation is reduced by a transmission factor below that radiated into a uniform environment. A particular example is worked out in detail. That example deals with the case of a single point support on an otherwise homogeneous surface excited by boundary-layer turbulence.

Expressions are obtained for the jump in vorticity across a discontinuity surface which is not a contact surface. Admitting arbitrary motion of the fluid and the discontinuity, the most general of these is established on a purely kinematical basis. More specific expressions are obtained by successively enforcing the momentum equation of the flow and the conditions of conservation of mass and momentum across the discontinuity. Eventually Hayes's (1957) result for a gasdynamic discontinuity is recovered.

The development of waves and currents resulting from the action of a steady wind on initially standing water has been investigated in a wind–water tunnel. The mean air flow near the water surface, the properties of wind waves, and the drift currents were measured as they evolved with increasing fetch, depth and mean wind speed. The results suggest how the stress on the water surface changes with an increasingly wavy surface, and, from a different viewpoint, how the drift current and the waves develop in relation to the friction velocity of the air. The amplitude spectra calculated for the wavy surface reflected certain features characteristic of an equilibrium configuration, especially in the higher frequencies. The observed equilibrium range in the high frequencies of the spectra fits the f−5 rule satisfactorily up to frequencies f of about 15 c/s. The wave spectra also revealed how the waves grow in the channel, both with time at a fixed point, and with distance from the leading edge of the water. These results are discussed in the light of recent theories for wave generation resulting from the action of pressure fluctuations in the air, and from shearing flow instabilities near the wavy surface. The experimental observations agree reasonably well with the predictions of the recent theory proposed by Miles, using growth rates calculated for the mechanism suggesting energy transfer to the water through the viscous layer in the air near the water surface.

Experimental and theoretical investigations were made of the instability and transition of the wake behind an axisymmetric slender body at high Reynolds number. The sound from a loudspeaker was used as an artificial disturbance. The velocity fluctuations induced by the sound are selectively amplified depending on the frequency. The wavelength, the phase velocity and the amplification rate of the velocity fluctuations were measured. A linearized stability equation of the disturbance superposed on the axisymmetric wake was solved numerically for both neutral and amplified disturbances. Theoretical results on eigenvalues and eigenfunctions of the stability equation show good agreement with experimental results.

Measurements on phase distributions in streamwise and azimuthal directions indicate that the line of same phase forms a helix and not a series of discrete closed loops.

It has been found that ventilated cavities extending behind hydrofoils, plates, and other two-dimensional bodies, oscillate when the air supply rate is sufficient to reduce the cavitation number to about one-fifth of its natural value. As the rate increases further, higher modes of oscillation occur in which the cavity–water interface supports several waves that are convected downstream towards the wake, which, owing to a pinching-off action replacing the usual entrainment sink, consists of a sequence of large bubbles drifting downstream. A theory of such flows that allows both for the convected velocity fluctuations in the cavity, and for the transport of bubble volume down the wake, is given in this paper. Coupled with a rather simple phenomenological relation between the pressure fluctuations within the cavity and the departure of the pinched-off rear portion of the cavity—explained in terms of the action of the re-entrant jet—this theory successfully predicts the resonance frequencies obtained in experiments by Silberman & Song.

The theory also provides a solution of the more general problem of determining the fluctuations in the pressure distribution over the whole surface of the body, when it is in a prescribed unsteady motion along its axis of symmetry (the theory is confined to symmetrical bodies and flows). Thus the growth in drag due to a sudden increment in the upstream velocity can be predicted, and also the damping forces acting on the body when it is forced to oscillate at a given frequency. It is shown that in all cases the body is unstable.

One important feature of the mathematical model chosen is that it completely avoids the presence of a time-dependent sink at infinity—with its associated infinite pressures—by conserving total volume of wake and cavity in just the same way as vorticity is conserved in unsteady aerofoil theory.

In this paper systems of non-linear integral equations are formulated which are equivalent to the Krook kinetic equation for steady problems in two and three space-dimensions. The boundary conditions used are discussed and some of the properties of the equations which are significant in their numerical solution pointed out. This work will then serve as a basis for the consideration of a sequence of particular problems to be presented in subsequent papers.

The changes of surface stress in a deep boundary layer passing from a surface of one roughness to another of different roughness are described fairly accurately by theories that assume self-preserving development of the flow modifications. It has been shown that the dynamical conditions for self-preserving flow can be satisfied if the change in friction velocity is small and if log l0/z0 is large (l0 is the depth of the modified flow and z0 is the roughness length of the surface). In this paper it is shown that, if the change of friction velocity is not small, the dynamical conditions can be satisfied to a good approximation over considerable fetches if log l0/z0 is large. The flow modification is then locally self-preserving, that is, the fields of mean velocity and turbulence are in a moving equilibrium but one which changes very slowly with fetch and depends on the ratio of the initial to the current friction velocity. In the limit of a very large increase in friction velocity, the moving equilibrium is essentially that of a boundary layer developing in a non-turbulent free stream. Equations describing the flow development are derived for all changes of friction velocity, and the form of the velocity changes is discussed. For large increases of friction velocity, the depth of the modified layer is substantially less than would be expected from the theories of Elliott and of Panofsky & Townsend.

The physical mechanisms underlying the relaxation process leading to thermal equilibrium behind ionizing shock waves in argon have been studied through use of optical techniques. The non-equilibrium condition in the relaxation region was investigated experimentally by measuring the shift in the fringes due to a change in the refractive index of the medium with a Mach–Zehnder interferometer. Both electron- and mass-density profiles from the shock front to the equilibrium region were determined. The experimental work has been supplemented by a theoretical analysis of the ionization mechanism to explain the measured profiles and relaxation times.

Measurements of grid turbulence have been obtained for grid Reynolds numbers ranging from 2·4 × 106 to 1·2 × 105. The decay law and the effect of Reynolds number on the turbulence level are established. The measured power spectra of the turbulence are consistent with Kolmogoroff scaling for kη > 0·1 (k is the wave-number and η is the Kolmogoroff length); but for kη < 0·1, the spectra of the stream-wise turbulence velocity component and of the cross-stream component do not appear to be isotropically related. However, the stream-wise spectrum does display a $-\frac{5}{3}$ region, which increases in extent with increasing Reynolds number.

A considerable quantity of observations and measurements exists concerning the phenomenon of intermittency which is connected closely with the entrainment process in free turbulent flows. A number of these are described in the first part of the paper and conclusions are drawn about the shape and motion of the bounding surface that separates turbulent and non-turbulent fluid. The salient features are that indentations of the surface grow and decay cyclically, that each cycle leads to substantial entrainment of ambient fluid into the turbulent region, that the indentations move at a considerable speed relative to the free stream, and that the surface has a comparatively simple form. The growth–decay cycle of the indentations suggests that a critical condition for growth exists, but the pressure field consequent on the convection velocity of the indentations makes for a Helmholtz type of instability that is unlikely to be stabilized by purely viscous behaviour of the turbulent fluid. It is known that the initial response of turbulent fluid to distortion is elastic in character, with incremental Reynolds stress proportional to increment of total strain, and sufficient rigidity could stabilize the bounding surface. A simple flow model–an inviscid stream flowing over an elastic jelly—is examined and the condition for marginal stability is compared with the observed properties of the flow. The model leads to the conclusion that indentations of more than a critical wave-number are stable, and provides reasons for the comparatively simple form of the surface and for the occurrence of indentations in groups of about three. The relative values of entrainment constants in different flows of uniform density do not depend critically on the nature of the entrainment process provided that the main turbulent motion remains geometrically similar, but the correlation between entrainment constant and relative depth of the indentations found by Gartshore (1966) appears as a consequence of the ‘elastic’ control of the growth–decay cycle. Lastly, the properties of the engulfment mechanism are used to show that the entrainment constant for a jet is proportional to the square root of the ratio of ambient density to the average density inside the jet. In contrast, the corresponding result for engulfment controlled by an eddy viscosity is variation as the ratio of the mean of the ambient and inside density to the inside density. Observations of high-speed jets of water in air and air in water give some support to the ‘elastic’ hypothesis.

The results of a detailed mean velocity survey of a smooth-wall turbulent boundary layer in an adverse pressure gradient are described. Close to the wall, a variety of profiles shapes were observed. Progressing in the streamwise direction, logarithmic, ½-power, linear and $\frac{3}{2}$-power distributions seemed to form, and generally each predominated at a different stage of the boundary-layer development. It is believed that the phenomenon occurred because of the nature of the pressure gradient imposed (an initially high gradient which fell to low values as the boundary layer developed) and attempts are made to describe the flow by an extension of the regional similarity hypothesis proposed by Perry, Bell & Joubert (1966). Data from other sources is limited but comparisons with the author's results are encouraging.

A general theory is presented to account for the small, free, lateral motions of a flexible, slender, cylindrical body immersed in fluid flowing parallel to the position of rest of its axis. The cylinder is either clamped or pinned at both ends, or clamped at the upstream end and free at the other; it lies in a horizontal plane wherein all motion is considered to be confined. It is shown that for sufficiently large flow velocities the cylinder may be subject to buckling and oscillatory instabilities in its first and higher flexural modes, respectively. It is shown that for cylinders with both ends supported the oscillatory instabilities are specifically caused by lateral frictional forces, and that in the absence of hydrodynamic-drag effects only buckling is possible. The same applies for cylinders supported at the upstream end and with a very long, gradually tapering free end. The critical conditions of stability, expressed in dimensionless form, are evaluated extensively for clamped-free and pinned-pinned cylinders, illustrating the effect of the various system parameters on stability.

The equilibrium burning of a spherical drop of pure non-gaseous fuel in a slow convective flow of hot oxidant is examined for Lewis number unity. A Stokes or Oseen flow with modified boundary conditions to permit mass transfer at the drop surface describes the velocity field. The method of inner and outer expansions is then adopted to describe the thermal and mass-fraction profiles under the model of a direct one-step irreversible indefinitely fast chemical reaction. The thin-flame position and surface mass-transfer rate, both functions of polar angle as well as radial position when convection is added to the conventional diffusive transport, are furnished in terms of the Peclet number. It is found that the interaction of the perturbational free-streaming with the asymmetric vaporization it induces can lead to drag coefficients in excess of the Stokes value.

Calculations are presented for two-dimensional Bénard convection between free bounding surfaces for ranges of Rayleigh and Prandtl numbers. The variables are expanded in a series consisting of the eigenfunctions of the stability problem and the system is truncated to take into account only a limited number of terms. The amplitudes of the eigenfunctions are evaluated by numerical integration of the resulting non-linear equations. In all cases considered, the system achieves a steady state with the motion consisting of a single large cell. Results for Nusselt number vs. Rayleigh number are given for a range of Prandtl number varying between 0·01 and 100 and show that heat flux increases slightly with decreasing Prandtl number. The calculations agree with those of Kuo where the ranges of Rayleigh number overlap. A simple heuristic argument based on the assumption that turbulent boundary layers exist is also given and the conclusions of the latter indicate that heat flux should decrease with decreasing Prandtl number. Thus the behaviour is qualitatively different from that of the calculations. The reason appears to be associated with the fact that the single large cell in the computed cases enables the fluid to accelerate through repeated cycles until it achieves a steady state with the amplitude of the motion much larger than could be acquired by a single turbulent blob free-falling in the gravitational field.

The experiments described here were designed to illustrate the dynamical behaviour of flexible cylinders in axial flow theoretically examined in Part 1. Rubber cylinders, either clamped or pinned at the upstream end and free at the other, or pinned at both ends, displayed both buckling and oscillatory instabilities when immersed in flowing water of sufficiently high velocity. The conditions of neutral stability were determined in a number of cases and compared with theory. The observations were in substantial agreement with the theory.

A solution is offered for the relative phase relationship of bed, depth and surface waves observed in alluvial channels, and is found to be in good agreement with observation in laboratory flumes. The solution does not depend on an assumption of a phase lag between the velocity and sediment movement. The emphasis of the paper is on the mechanics of bed-wave formation.

The problem of flow through rotating channels of almost arbitrary cross-section is considered. It is shown that when the ratio of the Rossby number and the Reynolds number is small (ε = Ro/Re [Lt ] 1) and when the Reynolds number is not too large (Re [Lt ] ε−1): (1) the viscous effects are important only in thin boundary layers along the channel walls; (2) the flow in the interior is geostrophic; and (3) the inertia effects may be neglected everywhere. Solutions for the geostrophic region and the boundary layers are obtained and are combined to give the complete velocity field. Experimental results for a circular conduit are presented which are in good agreement with the theory.

The investigation of a separated shear layer was undertaken to clarify the non-linear mechanisms associated with instability and transition to turbulence. Such an investigation is of practical importance since profiles which resemble the separated shear layer are a common occurrence.

A two-dimensional free shear layer was formed by separation of a laminar boundary layer from a rearward-facing step. The free-stream speed was approximately 16 ft./sec. Hot-wire measurements were made in the region directly downstream of the plate trailing edge. The measurements included mean velocity profiles, frequency spectra of the longitudinal fluctuation, and root-mean-square amplitude and phase distributions of various spectral components of the longitudinal fluctuation. Several measurements were designed to detect the presence of periodic spanwise structure.

The most important findings were:

1. Significant non-linear distortion of the initial unstable wave occurred without periodic spanwise structure.

2. Non-linear distortion was first manifest by the growth of a subharmonic oscillation, which was strongly intermittent. Numerous harmonics of the sub-harmonic oscillation were also present.

3. Strong evidence suggests that secondary instabilities were present, which created still higher frequencies.