This paper is the first part of an investigation of the molecular velocity distribution function in non-equilibrium flows. In this part, the general features of the distribution are discussed and illustrated by a detailed study of its asymptotic expansions in different velocity domains for a weak shock, employing a simple relaxation model for the collisions. Using the strength of the shock as a small parameter, the Chapman–Enskog distribution is derived, in a less restrictive way than in previous analyses, as the first two terms in a suitable ‘inner’ asymptotic expansion of the distribution in velocity space, valid only for not very high velocities. It is shown that the consideration of two further limits, called the intermediate and outer, is necessary for a complete description of the distribution in velocity space. The uniformly valid composite expansion demonstrates the slow approach to equilibrium of fast molecules. The outer solution depends on integrals over the flow and is in general ‘global’, in contrast to the inner solution which is essentially local; this introduces certain asymmetries on a fine scale even in a weak shock. It is shown, for example, that fast molecules moving towards the hot side accumulate by collisionless streaming, whereas those moving towards the cold side attenuate like a molecular beam and represent essentially a ‘precursor’ of the hot side. A simple approximation for the distribution in the precursor is derived, and found to contain, in the outer limit, a large perturbation on the local Maxwellian; this results in an approach to equilibrium like $\exp (-|x|^{\frac{2}{3}})$ on the cold side.

A heuristic extension of the argument to the true Boltzmann equation leads to the result that for molecules with an interparticle potential varying as the inverse m-power of the distance, the approach to equilibrium through the precursor is like $\exp (-|x|^l)$, where l = m/(m + 2).

A comparison is made between the effects of spreading the source (thus causing interference) and of viscosity in the Kelvin ship-wave problem. Two simple pressure distributions are considered and in both it is found that spreading, like viscosity, introduces a multiplicative damping factor into the inviscid pressure-point solution. In one case it is this factor, rather than that of viscosity, which dominates the decay of the wave profile, while, in the other, the effects alternate in importance as one travels along any particular wave crest.

An analytic solution is obtained for the head-on collision of strong shock waves with supersonically moving, axisymmetric, slender bodies. A new method of the solution is developed using integral transforms. Pressure distributions on the surface of a cone illustrate the effects of shock strength and body speed. The results are compared with those of Blankenship (1965), which were obtained by numerical methods.

It is also shown that the same analysis can, with certain modifications, be adapted for treating the shock-shock interaction on thin two-dimensional aerofoils of arbitrary shape at small incidence.

In a large range of Reynolds numbers, 6 × 104 < Re < 5 × 106, the flow around single cylinders with smooth surfaces has been investigated. The high values of the Reynolds numbers were obtained in a test channel which could be pressurized up to 40 bar of static pressure. New experiments were performed to measure the local pressure and skin friction distribution around the cylinder. From these results the total drag, the pressure drag and the friction drag were calculated. By means of the skin friction distribution the position of the separation points, separation bubbles or transition points can be localized. These data allow one to define three states of the flow: the subcritical flow, where the boundary layer separates laminarly; the critical flow, in which a separation bubble, followed by a turbulent reattachment, occurs; and the supercritical flow, where an immediate transition from the laminar to the turbulent boundary layer is observed at a critical distance from the stagnation point. According to the total drag coefficient the values found in this paper connect the subcritical region represented by the measurements of Wieselsberger (1923) and Fage & Warsap (1930) with the supercritical range in which Roshko (1961) carried out his experiments.

Temperature and velocity fluctuations have been recorded in the open sea and in a tidal channel, and power spectra have been determined from the records. The one-dimensional spectra of temperature fluctuations are found to have an inertial subrange. At larger wave-numbers the data can be fitted by Batchelor's spectrum function for the viscous-convective range. The spectra are inconsistent with the form proposed by Pao for the viscous-convective range.

Estimates are given for the constants in Batchelor's spectrum function, but these depend upon knowledge of the rate of dissipation of kinetic energy, which is determined from the velocity spectra. There is doubt about the validity of some of the velocity spectra, and in other cases there is reason to suspect that the turbulence is not locally isotropic.

Detailed experimental results are presented for the steady separated flow past a variety of bluff objects. Included in this work are measurements of: rear stagnation point pressure coefficients; the pressure distribution along the base of a backward-facing step; velocity profiles within the circulating wake-bubble as well as in the outer flow; lengths, maximum widths and vortex centre locations of the corresponding wake-bubbles; and the local shear stress along the surface of a circular cylinder. In every case, the experimental data were found to be consistent with the theoretical model proposed by Acrivos et al. (1965) for the steady separated flow past bluff objects in the limit of large Reynolds numbers R.

A numerical solution is presented for predicting the flow and heat transfer due to free convection in the entry region of a cooled vertical pipe, open at both ends. Following Lighthill (1953) the Kármán-Pohlhausen method is used in this analysis. Velocity and temperature profiles are assumed satisfying the physical boundary conditions, and the integrated forms of the equations of motion and the equations themselves at the axis and the walls of the pipe are used to calculate the various parameters involved in the profiles assumed. Two cases of constant wall temperature and linearly decreasing temperature with height were considered. Cooling decelerates the fluid in the boundary layer that forms at the wall and the fluid in the core is accelerated by virtue of continuity. As cooling progresses, at Rayleigh number O(103) the fluid in the boundary layer starts reversing at first and then assumes progressively increasing velocities. Graphs are presented for the development of the predicted velocity and temperature profiles and the other parameters involved, for a situation when the boundary layer fills the whole pipe.

The drag experienced by a sphere moving with velocity dependent on a single time scale t0 in an unbounded viscous liquid is considered under the assumption that the Reynolds number is small. It is shown that, unless t0 is sufficiently large, an asymptotic expansion in the Reynolds number becomes invalid for large times. Moreover, when the expansion is valid for large times, the drag can differ considerably from that predicted by the unsteady Stokes equations.

The surge studied in this paper is due to the sudden closure of the sink of a bathtub vortex. Experimental evidence established that the surge is not initially a wave but is in fact a slug of fluid which has, for a very short period of time, a distinctive motion of its own. The purpose of this paper is to present the salient features of an experimental investigation of the effect of circulation, geometry, and viscosity on the speed of the surge and to report some rather unexpected observations.

The possibility of long waves in a rotating ocean being trapped along a straight discontinuity in depth was demonstrated in a recent paper (Longuet-Higgins 1968). The analysis is now extended to the situation where the depth varies continuously, in a zone separating two regions of different depths. The trapping of waves in the transition zone is investigated, taking full account of the horizontal divergence of the motion.

If the profile of the depth is assumed to be monotonic, then it is shown that the trapped waves always travel along the transition zone with the shallower water to their right in the northern hemisphere and to their left in the southern hemisphere. The wave period must always exceed a pendulum-day. The period is also bounded below by a quantity depending inversely on the maximum bottom gradient.

By allowing the width W of the transition zone to vary, asymptotic forms for the trapped modes are obtained, both as W → 0 and as W → ∞. In the limit as W → 0 the depth becomes discontinuous, and it is shown that the lowest mode then becomes a double Kelvin wave (Longuet-Higgins 1968) propagated along the discontinuity. The periods of the higher modes, on the other hand, all tend to infinity; these modes become steady currents.

Numerical calculations of the trapped modes are presented for two different laws of depth in the transition zone. It is found that as W → 0 the lowest mode is insensitive to the form of the depth profile. Higher modes depend on the details of the profile. Hence the lowest mode is the most likely to be observed in the real ocean.

The dispersion relation is also investigated. It is shown that the group-velocity of all modes must change sign at some point in the range of wave-numbers, if the divergence is taken into account. When the divergence was neglected the lowest mode appeared to be exceptional, in that the group-velocity was always in the same direction. This anomaly is now removed.

A hot film flowmeter, a thin film resistance thermometer and several thermistors were mounted on a submarine and used to study turbulence and temperature microstructure in the open sea. The surface layer is found to be continuously turbulent and patches of turbulence have been found in and below the main ocean thermocline.

The conservation equations describing steady, incompressible flow in a variable-area duct with mass transfer occurring at its walls are simplified by linearizing the inertial and convective terms. Solutions to a large class of problems can be obtained by means of the general method which is presented. Particular examples considered are entrance flow with heat transfer to an isothermal wall in the presence of mass addition, constant rate of injection of a foreign gas through the wall, vaporization and sublimation of a volatile wall material, and gas-phase combustion of a fuel which enters the duct from its wall. Comparison of the present results with previous work and with new experimental results is discussed for the first of these applications. It is concluded that the present results for velocity and pressure fields are likely to be accurate for small values of the Reynolds number based on wall injection velocity, that the present results for temperature and composition fields are likely to be accurate in the absence of wall mass transfer, and that in the absence of wall mass transfer the linearization technique is likely to exhibit its highest accuracy for flows with uniform entrance conditions.

Small-scale temperature fields in water were used to test the dynamic response of towed thermometers of the platinum film resistance type. Laminar buoyant plumes rising from submerged heaters below the line of motion were the test temperature fields. The analysis of results was based on an approximate ‘diffusion-layer’ model of the dynamic heat-transfer process occurring near the platinum film on the probe tip. The model represents the linear heat transfer into a two-layer semi-infinite medium, with the platinum thin film located at the interface between a water layer of thickness Δ and a semi-infinite substrate of glass. The differences of the thermal properties of water and glass were found to be negligible. The characteristic time Δ2/D, where D is the thermal diffusivity of water, was determined by the ratio of actual to film-indicated plume-peak temperature, assuming a sinusoid approximation to the plume profile. The frequency response for the same operating conditions as the plume tests could then be obtained from the diffusion-layer model.

This paper concerns the stability characteristics of laminar natural convection in external flows. Until recently, very little was known about such stability because of the inherent complexity of temperature-coupled flows and because of the complicated mechanisms of disturbance propagation. In this work the stability of the laminar natural convection boundary layer is examined more closely in an attempt to predict the experimental results recently obtained. In particular, it is shown that an important thermal capacity coupling exists between the fluid and the wall which generates the flow. This thermal capacity coupling is shown to have a first-order effect for particular Grashof-number wave-number products. Solutions are obtained for a Prandtl number of 0·733 and several values of relative wall thermal capacity. These solutions indicate the important role of this wall coupling. In particular, the results predict the experimental data previously obtained.

In addition, solutions with ‘zero wall storage’ are obtained for a range of Prandtl numbers from 0·733 to 6·9. The relative disturbance u-velocity and temperature amplitudes and their phases are shown for Pr = 0·733 and several wall-storage parameters, and for Pr = 6·9 with zero wall storage. A comparison between the disturbance temperature distribution and the data obtained from a recent experimental investigation shows close agreement when the thermal capacity of the wall is taken into account.

In the appendix, it is shown that for temperature-coupled flows and wall-coupled boundary conditions the flow is unstable at a lower Grashof number for two-dimensional disturbances than it is for three-dimensional disturbances. This result has been supported by the recent experimental observations.

A thin disk, immersed in a fluid consisting of two superposed homogeneous inviscid liquids, moves in its own vertical plane. A Green's function is found for the potential and a formula for the gravity wave energy is found and evaluated for a specific case.

A numerical solution is presented for the development of free convection from a semi-infinite vertical flat plate which is uniformly heated up to a length l from the base and insulated for the rest of its length. At great heights above the heated part of the plate, the velocity and temperature distributions behave as if the heat were put in as a line source of heat at the base of the plate. Matching of the solutions for the heated and the insulated parts of the plate, by keeping the fluxes of heat and momentum continuous, determines the position of the effective origin of the similarity solution for the insulated plate in terms of the length, l, of the heated part of the plate. Graphs of the dimensionless velocity, temperature, heat flux and axial length parameters are given for different values of the Prandtl number.

Intermittency measurements have been performed within the first 100 virtual diameters of an axisymmetric compressible wake. The primary measurements, performed with the hot-wire anemometer, concerned the intermittency factor and the frequency of ‘zero crossings’. The intermittent flow covered most of the wake profile but for a narrow region about the axis, and was found to be normally distributed around the average front position. The latter, along with the extent of the front standard deviation, was found to agree numerically with expectations based on low speed wakes and to grow as the 1/3 power of axial distance. A weak periodicity of the front was detected at a wavelength about nine times greater than the longitudinal scale of turbulent velocity fluctuations. This periodicity affects the turbulent spectra.

Stewartson's theory for axisymmetric hypersonic flow of a model gas over slender bodies with strong viscous interaction and strong shock wave is extended to power-law viscosity variation and Prandtl numbers other than one. Flow properties at the body surface and shock are obtained without recourse to numerical integration. Numerical computations are presented for axisymmetric flow over a three-quarter power-law body with strong shock wave and viscous interactions that range from weak to strong.

The present study consists of wind profile surveys, drift current measurements and water surface observations for a wide range of wind velocities in a wind–wave tank. It is confirmed that the velocity distribution essentially follows the logarithmic law near the water surface and the velocity-defect law toward the outer edge of the boundary layer. The wind stresses and surface roughnesses calculated from these distributions are divided into two groups separated by the occurrence of the wave-breaking phenomenon. For low wind velocities the surface roughness is dictated by ripples, and the wind-stress coefficient varies with U0−½, where U0 is the free-stream wind velocity. The surface roughness is proportional to the average height of the basic gravity wave at higher wind velocities; the stress coefficient is then proportional to U0. In addition, it is found that Charnock's expression (k ∝ u*2/g) holds only at high wind velocities, and that the constant of proportionality determined from the present experiment correlates very well with field observations. A new technique, involving the use of various-sized surface floats to determine the drift current gradient and the surface drift current, has been developed. A good agreement is shown between the gradients obtained from the measured currents and those determined from the wind stresses. Finally, the wind-stress coefficient is shown to be larger than the friction coefficient for turbulent flow along a solid rough surface; the difference is shown to be the wave drag of the wind over the water surface.

A general theory is presented to account for the small, free, lateral motions of a flexible, slender, cylindrical body with tapered ends, totally submerged in liquid and towed at steady speed U. For particular shapes of the ends and length of tow-rope, it is shown that the body may be subject to oscillatory and non-oscillatory instabilities for U > 0; at small U, these instabilities correspond to those of a rigid body. At higher U, the system generally regains stability in the above modes, but may be subject to higher-mode, flexural oscillatory instabilities. The critical conditions of stability are calculated extensively and the effect on stability of a number of dimensionless parameters is discussed. It is shown that optimum stability is achieved with a streamlined nose, a blunt tail and a short tow-rope.

Some experiments are described which were designed to test the theory. Rubber cylinders of neutral buoyancy were held in vertical water flow by a nylon ‘tow-rope’. Provided the tail was streamlined and the tow-rope not too short, ‘criss-crossing’, non-flexural oscillations developed at very low flow. Increasing the flow, these oscillations ceased and the cylinder buckled like a column; subsequently higher-mode flexural oscillations developed. However, for a sufficiently blunt tail and short tow-rope, the system was completely stable.

The experimental observations are generally in qualitative agreement with theory. Quantitative comparison of the various instability thresholds and stable zones between experiment and theory, based on estimated values of some of the theoretical dimensionless parameters, is also fairly good.