The question of whether or not waves exist upstream of an obstacle that moves uniformly through an unbounded, incompressible, inviscid, unseparated, rotating flow is addressed by considering the development of the disturbed flow induced by a weak, moving dipole that is introduced into an axisymmetric, rotating flow that is initially undisturbed. Starting from the linearized equations of motion, it is shown that the flow tends asymptotically to the steady flow determined on the hypothesis of no upstream waves and that the transient at a fixed point is O(1/t). It also is shown that the axial velocity upstream (x < 0) of the dipole as x → − ∞ with t fixed is O(|x|−3), as in potential flow, but is O(|x|−1) as t → ∞ with |x| fixed. The results extend directly to closed obstacles of sufficiently small transverse dimensions and suggest the existence of a finite, parametric domain of no upstream waves for smooth, slender obstacles. The axial velocity in front of a small, moving sphere at a given instant in the transient régime is calculated and compared with Pritchard's laboratory measurements. The agreement is within the experimental scatter for Rossby numbers greater than about 0·3 even though the equivalence between sphere and dipole is exact only for infinite Rossby number.

Experiments have been made to investigate the motion generated by a body moving along the axis of a uniformly rotating fluid.

Part of the investigation concerns the motion generated in a cylinder whose radial dimensions are much greater than those of the body. Measurements have been made of the velocities of particles on the axis of rotation both ahead of and behind the body, and the results indicate that there is a significant axial motion generated by the body over a wide range of Rossby numbers. A measurement of the instantaneous velocity profile ahead of the body, determined as a function of the radius, agrees fairly well with a low Rossby number calculation of the flow due to a circular disk (Morgan 1951). In addition, the forward influence of the body has been measured as a function of the Rossby number and from these results it is suggested that the body has a finite influence far upstream at all Rossby numbers and that the blocking phenomenon first reported by Taylor (1922) probably occurs for all values of the Rossby number (UΩa) less than a critical value which is about 0·7.

Experiments have also been made in a long cylindrical tube which acts as a wave guide. At large distances from the body the separate effects of the various modes can be observed and hence it is possible to measure the flow corresponding to an individual wave-number: these measurements show that, as a result of the Doppler effect, the motion a large distance ahead of the body is different from that far behind (see Lighthill 1967). Moreover, the experiments indicate that no disturbances propagate ahead of the body when its velocity exceeds the maximum group velocity in the fluid, but that disturbances trail behind the body when its velocity is far in excess of the maximum group velocity. Measurements of the maximum group velocity are in good agreement with the theoretical value.

The low Reynolds number flow of a variable property gas past an infinite heated circular cylinder is studied when the temperature difference between the cylinder and the free stream is appreciable. The velocity field (and hence the drag on the cylinder) is calculated by the method of matched asymptotic expansions. It is found that the zero-order velocity field calculated on the Stokes approximation satisfies both the no slip condition at the cylinder and the uniform stream condition at infinity which is in strong contrast with the corresponding velocity field for incompressible slow flow past an unheated cylinder where the uniform stream condition at infinity cannot be satisfied. When the temperature of the cylinder is twice the temperature at infinity it is found that the drag on the cylinder is almost twice the drag on a similar unheated cylinder.

Waves at an unstable horizontal interface between two fluids moving vertically through a saturated porous medium are observed to grow rapidly to become fingers (i.e. the amplitude greatly exceeds the wavelength). For a diffusing interface, in experiments using a Hele-Shaw cell, the mean amplitude taken over many fingers grows approximately as (time)2, followed by a transition to a growth proportional to time. Correspondingly, the mean wave-number decreases approximately as (time)−½. Because of the rapid increase in amplitude, longitudinal dispersion ultimately becomes negligible relative to wave growth. To represent the observed quantities at large time, the transport equation is suitably weighted and averaged over the horizontal plane. Hyperbolic equations result, and the ascending and descending zones containing the fronts of the fingers are replaced by discontinuities. These averaged equations form an unclosed set, but closure is achieved by assuming a law for the mean wave-number based on similarity. It is found that the mean amplitude is fairly insensitive to changes in wave-number. Numerical solutions of the averaged equations give more detailed information about the growth behaviour, in excellent agreement with the similarity results and with the Hele-Shaw experiments.

A non-linear theory of internal gravity waves of finite amplitude is developed in terms of conservation equations averaged with respect to the phase. The theory overcomes the failure of linear ray theory in regions in which waves are trapped and establishes the conditions under which finite amplitude waves may propagate. It gives a geometrical representation of the degeneration of waves into quasi-turbulence and predicts the dependence of the energy density on its parameters.

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.

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.

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 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.

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 vibrational relaxation frequency measurements were made on mixtures of carbon dioxide and the light gases using a shock tube and a Mach-Zehnder interferometer. The temperature range covered was 350–1200 °K. It was discovered that in the case of helium the effectiveness of the carbon dioxide-helium collision increases with increasing temperature while in the case of hydrogen and deuterium the collision number displays an anomalous temperature behaviour. At about 1000 °K all the three light gases are almost equally efficient in exciting the vibrational modes of carbon dioxide.

A disturbance of finite amplitude λ, which is periodic in the direction of the axis of the channel, is superimposed on plane Poiseuille flow, and the subsequent development of the disturbance is studied. The disturbance is represented by an expansion in the eigenfunctions of the Orr-Sommerfeld equation with coefficients which are functions of the time, and an accurate numerical solution of the truncated system of non-linear ordinary differential equations for the coefficients is obtained.

It is found that even for Reynolds numbers R less than the critical value Rc, the flow breaks down when λ exceeds a critical value λc(R). This is shown in figure 11 for the case when the initial disturbance is represented by the first mode of the Orr-Sommerfeld equation. The development of this type of disturbance is illustrated in figures 1, 3 and 13 and, for the case of a higher-order mode initial disturbance, in figure 14. Near the time of breakdown, the curvature of the modified mean flow changes sign (figure 15), but a disturbance may die down even after a reversal in the sign of the curvature has taken place (see figure 2).

The stability of plane Poiseuille flow to disturbances of finite amplitude is affected by the characteristics of the higher-order modes of the Orr-Sommerfeld equation. As shown in figures 4, 10, and 12, and in figures 5, 6, and 7, these modes are either of a ‘boundary type’, characteristic of the region near the wall, or of an ‘interior type’, characteristic of the centre of the channel. The modes in the transition zone, where the two types merge, are easily amplified through mutual constructive interference, even though individually they have high damping coefficients. It is these transition modes which are mainly responsible for the breakdown through finite amplitude effects.

It is shown that there exist undamped solutions for perturbations of finite amplitude of plane Poiseuille flow, which are periodic in the direction of the axis of the channel. The shift in the ‘neutral curve’ as a function of the amplitude λ* of the disturbance is shown in figure 2. The solution is obtained by a perturbation method in which the eigenfunctions and the eigenvalue c are expanded in power series of the amplitude λ, as shown in (14), (15), (16) and (17). Near the neutral curve for a finite amplitude disturbance, the curvature of the mean flow shows a tendency to become negative (figure 5).