The structure of a strong blast wave under the influence of an expanding inner contact surface is studied asymptotically in the Newtonian limit: $\epsilon \equiv (\gamma - 1)/2\gamma \ll 1, \epsilon \dot{y}^2_s \gg a^2_{\infty}$. The theory treats the interaction of a shock layer and an inner flow region (the entropy wake) and reduces the problem to an ordinary differential equation for the shock radius. The pressure–volume relation of Cheng et al. (1961) is recovered and extended to a higher order of ε.

It is shown that, depending on the rate of growth of the contact surface, the shock layer may ‘reattach’ to the surface at large time. In a number of cases, the reattachment is approached in an oscillatory manner which leads to a period of non-uniformity. The associated problem of multiple time scales (treated in sequels to this paper) is identified.

The modifying effect of base bleed on the steady separated flow past a two-dimensional bluff body is considered. Detailed experimental results are presented for Reynolds numbers R between 50 and 250 and for bleed coefficients b in the range 0 to 0·15. The streamline pattern near the object is found to be strongly affected by small changes in the rate of bleed, with the recirculating closed wake disappearing altogether for b > 0·15. Nevertheless, the qualitative dependence on R of the physical dimensions of the near-wake region and the associated streamwise pressure profile appear to be unaffected by base bleed.

A study is made of the wind-driven circulation of a two-layer ocean within a square basin, with a view to describing the observed separation of western boundary currents. The lower layer is allowed to surface and the line along which the upper-layer depth vanishes is interpreted as the region of the surfacing thermocline. For a representative wind stress the theory predicts the gross features of the Gulf Stream flow, the region adjacent to the surfacing line containing the separated boundary current. By assuming that the effects of friction and inertia are confined to regions of a boundary-layer character, the position of a separated current is shown to depend only on the degree of stratification and certain integral properties of the applied wind stress.

The method of matched asymptotic expansions is used to determine the lateral flow of an ideal fluid past a slender body, when the flow is constrained by a pair of closely spaced walls parallel to the long axis of the body. In the absence of walls, the flow field would be nearly two-dimensional in the cross-flow plane normal to the body axis, but the walls introduce an effective blockage in the cross-flow plane, which causes the flow field to become three-dimensional. Part of the flow is diverted around the body ends, and part flows past the body in the inner cross-flow plane with a reduced ‘inner stream velocity’. An integro-differential equation of identical form to Prandtl's lifting-line equation is derived for the determination of this unknown inner stream velocity in the cross-flow plane. Approximate solutions are applied to determine the added mass and moment of inertia for accelerated body motions and the lift force and moment acting on a wing of low aspect ratio. It is found that the walls generally increase these forces and moments, but that the effect is significant only when the clearance between the body and the walls is very small.

An electrically conducting fluid is contained above a horizontal plane. A uniform vertical magnetic field is applied externally and the plane is maintained at a uniform temperature except for a point or a line heat source. Density variations are ignored except where they give rise to buoyancy forces.

(i) The point heat source. Non-linear effects are small sufficiently far from the source. The resulting buoyancy forces interact with the magnetic forces to maintain a radial inflow towards the heat source. This fluid then escapes vertically as a jet, its structure now depending on the additional influence of viscosity. The perturbations of the temperature distribution and the magnetic field due to the motion are obtained. Finally, the effects of these perturbations back on to the fluid velocity are considered. The most striking features of the perturbations are (a) the action of the jet as a line source of heat for the fluid in the outer regions, (b) the large (compared to other perturbations) eddy in the jet.

(ii) The line heat source. The temperature distribution and magnetic field are weakly perturbed only if the thermal and electrical conductivities are sufficiently small. Similar results are obtained, as in (i) above, provided ε (a dimensionless number characterising the strength of thermal convection: see (1.32), (3.24)) is less than ¼. However, even for small ε, the effects of thermal convection cannot be ignored. Hence, superimposed on the jet is an eddy (driven by buoyancy forces) whose flux of fluid increases indefinitely with its height above the plane. When ε > ¼, the results suggest that numerous eddies will be formed.

The reattachment of a supersonic jet with a turbulent boundary layer abruptly expanding into an axisymmetric parallel diffuser has been experimentally investigated using a surface-flow technique. Measurements were made in the started condition, where the blowing pressure is sufficiently high to establish an oblique shock system in the diffuser. The proposed reattachment criterion correlates present measurements in terms of the diffuser area ratio, and also those of other workers for unconfined flow in terms of the free stream Mach number after separation. As already reported for unconfined flow, it is found that disturbances downstream of reattachment do not affect the upstream region.

In this paper time dependent expansions of monatomic gases with spherical symmetry are discussed. For the particular case of Maxwellian molecules closed expressions for the moments up to second order are obtained in regions of the flow where the inviscid solution is no longer valid. These solutions are derived in a general form using the particle path function as a parameter. The structure of the inviscid solution is such that this simplification can be made. The novelty of the present approach is that solutions already derived in previous papers can be obtained from the general solution in various limits; both the results for steady flow and the expansion of a fixed mass of gas can be derived in this manner. Finally, a particular example is constructed in order to illustrate the general theory.

The structure of fully developed turbulence in smooth circular tubes has been studied in detail in the Reynolds number range between 10,700 and 96,500 (R based on centre velocity and radius). The data was taken as longitudinal and transverse correlations of the longitudinal component of turbulence in narrow frequency bands. By taking Fourier transforms of the correlations, crosspower spectral densities are formed with frequency, ω, and longitudinal or transverse wave-number, kx or kz, as the independent variables. In this form the data shows the distribution of turbulence intensity among waves of different size and inclination, and permits an estimate of the phase velocity of the individual waves.

Data taken at radii where the mean velocity profile is logarithmic show that the waves of smaller size (higher (k2x + k2z)½) decrease in intensity more rapidly with distance from the wall than the larger waves, and also possess lower phase velocity. This suggests that the waves might constitute a geometrically similar family such that the variation of intensity with wall distance is a unique function with a scale established by (k2x + k2z)−½). The hypothesis fits the data very well for waves of small inclination, α = tan−1(kx/kz), and permits a collapse of the intensity data at the several radii into a single ‘wave-strength’ distribution. The function of intensity with wall distance which effects this collapse has a peak at a wall distance roughly equal to 0·6(k2x + k2z)−½). For waves whose inclination is not small, it would not be expected that the intensity data could collapse in this way since the measured longitudinal component of turbulence represents a combination of two turbulence components when resolved in the wave co-ordinate system.

Although the similarity hypothesis is strictly true only for data taken where the mean velocity profile is logarithmic, a simple correction procedure has been discovered which permits the extension of the similarity concept to the sublayer region as well. This procedure requires only that the observed total turbulence intensity at any station in the sublayer be reduced by a factor which depends solely on the y+ distance from the wall (i.e. on the distance from the wall, scaled by the viscid parameters of the sublayer). The correction factor is independent of Reynolds number and applies equally to waves of all sizes. In this way, all of the turbulence waves down to the very smallest of any significance, are found to satisfy slightly modified similarity conditions.

From the data taken a t Reynolds numbers between 96,500 and 46,000 wave ‘strength’ is seen to be distributed more or less uniformly over a range bounded at one extreme by the largest waves which the tube can contain (k2x + k2z ≅ (2/a)2, where a is the tube radius) and at the other extreme by the smallest waves which can be sustained against the dissipative action of viscosity (k2x + k2z ≅ (0·04v/Uτ)2, where Uτ is the shear velocity). As the Reynolds number of the flow is lowered, the spread between the bounds becomes smaller. If the data is projected to a Reynolds number of order lo3 the bounds coalesce and turbulence should no longer be sustainable.

Based on a set of approximate equations for long waves over an uneven bottom, numerical results show that as a solitary wave climbs a slope the rate of amplitude increase depends on the initial amplitude as well as on the slope. Results are also obtained for a solitary wave progressing over a slope onto a shelf. On the shelf a disintegration of the initial wave into a train of solitary waves of decreasing amplitude is found. Experimental evidence is also presented.

A NATO Advanced Study Institute on the topic of transition from laminar flow to turbulence was held at Imperial College, London, from 1 to 6 July 1968. Each morning's session was started with a one-hour general lecture, and was followed by five or six half-hour lectures interspaced with discussion periods. The main lecturers were C. C. Lin (general survey), S. Rosenblat (stability of time-dependent flows), L. S. G. Kovasznay (turbulent, non-turbulent interfaces), L. E. Scriven (free surface effects) and A. A. Townsend (shear turbulence). The idea of the meeting was to bring forth and to discuss current ideas in the subject, both from the point of view of developments out of laminar flow and from that of developments into real turbulence. To this end speakers were chosen to introduce a variety of topics ranging from laminar-flow instabilities (with emphasis on aspects at present imperfectly understood), through non-linear effects to the processes affecting turbulence itself.

Many ideas recurred throughout the meeting, both at lectures and in discussion periods. This is true, for example, of several relevant points forcefully made by C. C. Lin. For this reason the present account does not attempt to describe the proceedings of the meeting in chronological order, but rather takes an overall view of the subject matter and points to the areas of agreement and of controversy in relation to various problems.

The problem of the reflexion of tides in an enclosed sea such as the North Sea at a point at which it either enters the ocean or its width suddenly increases is considered by investigating the reflexion of a Kelvin wave at the open end of a rotating uniform semi-infinite channel.

It is shown that for a given channel, if the wave period is less than a pendulum day, then, according to the linearized theory of long waves in a rotating system, the reflexion coefficient increases with the angular velocity of rotation. It is also shown that there is a resonance effect for certain critical channel widths, namely, those at which extra modes within the channel become possible.

This study concerns the hypersonic flow over blunt bodies in two specific cases. The first is the case when the Mach number is infinite and the ratio of the specific heats approaches one. This is sometimes referred to as the ‘Newtonian limit’. The second is the case of infinite Mach number and very large streamwise distance from the blunt nose with a strong shock wave, or the ‘blast wave limit’. In both cases attention is restricted to power law bodies. Experiments are described of such flows at M∞ = 7·55 in air.

The Newtonian flow over bodies of the shape y ∞ xm at zero incidence is shown to be divisible into three regions: the attached layer at small x, the free layer and the blast wave region. As m increases from zero, the free-layer region reduces in extent until it disappears at m = 1/(2+j) (j = 1 and 0 for axisymmetric and plane flow respectively). A difficulty arises in a transition solution of the type given by Freeman (1962b) connecting the free layer with the blast wave result. At m > 2/(3+j) the attached layer merges smoothly into the Lees-Kubota solution which replaces the blast-wave result in this range.

In the blast wave limit, solutions were obtained for flow over axisymmetric power law shapes in the range ½γ < m < ½. Second-order results taking account of the body shape are given. These solutions are compared with experimental results obtained in air at a free stream Mach number of 7·55 and stagnation temperature of 630 °K, as well as with numerical solutions at Mach number of 100. The numerical method is tested by comparing solutions corresponding to the experimental conditions with experiment.

The prototype spin-up problem between infinite flat plates treated by Greenspan & Howard (1963) is extended to include the presence of an imposed axial magnetic field. The fluid is homogeneous, viscous, and electrically conducting. The resulting boundary initial-value problem is solved to first order in Rossby number by Laplace transform techniques. In spite of the linearization the complete hydromagnetic interaction is preserved: currents affect the flow and the flow simultaneously distorts the field. In part 1, we analyze the impulsively started time dependent approach to a final steady Ekman–Hartmann boundary layer on a single insulating flat plate. The transient is found to consist of two diffusively growing boundary layers, inertial oscillations, and a weak Alfvén wave front. In part 2, these one plate results are utilized in discussing spin-up between two infinite flat insulating plates. Two distinct and important hydromagnetic spin-up mechanisms are elucidated. In all cases, the spin-up time is found to be shorter than in the corresponding non-magnetic problem.

As they propagate through a gas, fluctuating pressure signals of moderate amplitude and of ultrasonic frequency are affected by amplitude dispersion, by relaxation damping and, particularly in ‘shock layers’, by diffusive damping. We derive a ‘high frequency’ theory including all these effects, for disturbances of arbitrary wave form excited by a wide variety of boundary conditions. By introducing a phase variable α, and taking account of non-linearity, we show how the signal propagates along the rays of linear acoustics theory, with constantly changing wave profile.

Relaxation dampens the signal, as for linear acoustics, and also diminishes amplitude dispersion. A criterion for shock formation is given, and the importance of non-linearity for signal attenuation exhibited. As shocks form, α surfaces coalesce and diffusive mechanisms are accentuated. Whitham's area rule is shown to be relevant for unsteady three-dimensional flows in relaxing gases, and is used to compute the attentuation of an ultrasonic beam. Supersonic relaxing flow over a wavy wall is also analyzed, and focusing effects are discussed.

The flow fields in two-dimensional, isoenergetic, viscous free mixing with constant β and with initial velocity profiles deviating slightly from those given by wakelike solutions of the Falkner-Skan equation for that β are considered. The similar solutions of the Falkner-Skan equation are investigated in more detail than in the past, e.g. we show that as β → −1 the flows approach the pure jet with the surrounding fluid at rest, and that there are new branch solutions for β < −1. We have investigated the spatial stability of these flows; it is found that for β > − 0·5 the only spatially stable solutions are the trivial ones f′(η) ≡ 1, but for −1 < β < − 0·5 there are non-trivial, jet-like solutions which are spatially stable. As to the new branch solutions for β < − 1, all are spatially unstable.

We consider the propagation of waves of small finite amplitude ε in a gas whose internal energy is characterized by two temperatures T (translational) and Ti (internal) in the form e = CvfT + CvfTi, and Ti is governed by a rate equation dTi/dt = (T − Ti)/τ. By means of approximations appropriate for a wave advancing into an undisturbed region x > 0, we show that to order εδ, the equation satisfied by velocity takes the non-linear form \[ \bigg(\tau\frac{\partial}{\partial t}+1\bigg)\bigg\{\frac{\partial u}{\partial t}+\bigg(a_1+\frac{\gamma + 1}{2}u\bigg)\frac{\partial u}{\partial x}-{\textstyle\frac{1}{2}}\lambda\frac{\partial^2u}{\partial x^2}\bigg\}=(a_1-a_0)\frac{\partial u}{\partial x}, \] where a1, a0 are the frozen and equilibrium speeds of sound in the undisturbed region, δ = ½(1 − (a20/a21)), and λ is the diffusivity of sound due to viscosity and heat conduction (λ may be neglected except when discussing the fine structure of a discontinuity). Some numerical solutions of this model equation are given.

When ε is small compared with δ, it is also possible to construct a solution for the flow produced by a piston moving with a constant velocity by means of a sequence of matched asymptotic expansions. The limit reached for large times for either compressive or expansive pistons is the expected non-linear solution of the exact equations. For a certain range of advancing piston speeds, this is a fully dispersed wave with velocity U in the range a0 < U < a1. If U > a1 the solution is discontinuous, and indeterminate in the absence of viscosity; a singular perturbation technique based on λ is then used to determine the structure of the wave head.

The energy theory, giving a sufficient condition for stability, is developed for the motions in a horizontal, heated layer subject to buoyancy and surface tension effects. The free surface is assumed to be non-deformable (Pearson's 1958 model).

It is shown that the equations governing the energy theory are the symmetric part of the time-independent linear theory problem, and that the surface tension terms behave like a bounded perturbation to the Bénard problem. The qualitative behaviour of the optimal stability boundary as a function of its parameters is given. The optimal stability boundary is computed, and compared with previous linear and non-linear stability theories in terms of allowable subcritical instabilities.

This paper is concerned with the propagation of small amplitude gravity waves over a flow with non-uniform velocity distribution. For such a flow Burns derived a relation for the velocity of propagation in terms of the velocity distribution of the mean flow. This result is derived here in another way and some of its implications are discussed. It is shown that one of these is hardly acceptable physically. Burns's result holds only when a real value of the propagation velocity is assumed; the mentioned difficulties vanish if complex values are allowed for, implying damping or growth of the waves. Viscous effects which are the cause of damping or growth are important in the wall layer near the bottom and also at the critical depth, which is present when the wave speed is between zero and the fluid velocity at the free surface.

In § 2 the basic equations for the present problem are given. In § 3 exchange of momentum and energy between wave and primary flow is discussed. This is analogous to what happens at the critical height in a wind flow over wind-driven gravity waves. In § 4 the viscous effects at the bottom are included in the analysis and the complex equation for the propagation velocity is derived. Finally in § 5 illustrations of the theory are given for long waves over running flow and for the flow along a ship advancing in a wavy sea. In these examples a negative curvature of the mean velocity profile is shown to have a stabilizing effect.

In this paper, we study the propagation of a shock wave in water, produced by the expansion of a spherical piston with a finite initial radius. The piston path in the x, t plane is a hyperbola. We have considered the following two cases: (i) the piston accelerates from a zero initial velocity and attains a finite velocity asymptotically as t tends to infinity, and (ii) the piston decelerates, starting from a finite initial velocity. Since an analytic approach to this problem is extremely difficult, we have employed the artificial viscosity method of von Neumann & Richtmyer after examining its applicability in water. For the accelerating piston case, we have studied the effect of different initial radii of the piston, different initial curvatures of the piston path in the x, t plane and the different asymptotic speeds of the piston. The decelerating case exhibits the interesting phenomenon of the formation of a cavity in water when the deceleration of the piston is sufficiently high. We have also studied the motion of the cavity boundary up to 550 cycles.

The paper discusses free convective flows above a horizontal plate, both theoretically and on the basis of experiments which yield quantitative data. The theory is applicable to the semi-infinite plate and is extended to cover the complete range of Prandtl number values including Pr → 0 and Pr → ∞. Experiments were carried out to demonstrate the existence of a laminar boundary layer above a horizontal plate at intermediate Grashof (respectively Rayleigh) numbers, and its extent along the plate. This layer breaks down into large-eddy instability some distance from the leading edge. The value of the critical Rayleigh number for this to occur, obtained experimentally using semi-focusing colour-Schlieren photography is in reasonable qualitative agreement with previously known data (Tritton 1963a,b).