This paper describes a theoretical investigation into (i) the response of a spherical particle to a one-dimensional fluid flow, (ii) the motion of a spherical particle in a uniform two-dimensional fluid flow about a circular cylinder and (iii) the motion of a particle about a lifting aerofoil section. In all three cases the drag of the particle is allowed to vary with (instantaneous) Reynolds number by using an analytical approximation to the standard experimental drag-Reynolds-number relationship for spherical particles.

Measurements of velocity and temperature fluctuations have been made in flows of air over a strongly heated horizontal surface in conditions of large negative Richardson number and approximating to horizontally homogeneous turbulent flow with constant shear stress and upwards flux of total heat. Within a layer that includes most of the total variation of mean temperature and velocity, the effect of the mean flow on the thermal structure is nearly confined to a change in the length scale which measures the thickness of the viscous-convective layer at the surface but which is also the length scale for the fully turbulent region. Measurements of the mean velocity and the mean-square fluctuation in the stream direction were made for a flow velocity of about 0·70 ms−1 and showed that the profiles of mean temperature and mean velocity are closely similar, implying proportionality at each height of the transfer coefficients for heat and for momentum. The ratio is estimated to be 1·4. The measurements were carried out in conditions such that the Monin-Obukhov length was in the range 8–60 mm and the maximum height of observation was 80 mm.

An instability mechanism, leading to the generation of cross-waves in a closed channel, was examined recently by Garrett (1970). His theory is not applicable to long channels where the wavemaker produces a primary field which is a progressive wave train. In such cases, the heaving of the mean surface, of considerable significance in the instability mechanism, is confined to the nonpropagating field near the wavemaker. Here the theory of resonant interactions is extended to describe the energy transfer from this forced localized field to the cross-wave field. There are close analogies between the present results and Garrett's, although the resonant bandwidth estimated here is an order of magnitude smaller. The theory indicates that nonlinear effects may control the decay of cross-waves down the channel.

In this paper experiments in which cross-waves were generated in front of a wave-maker at one end of a long channel are described. The primary field produced was a progressive wave train, but, at certain frequencies, a standing wave developed in front of the wavemaker. This wave, whose crests were at right angles to the wavemaker and which had frequency half that of the wavemaker, is known as a cross-wave.

An instability mechanism for the formation of cross-waves in a long channel has been presented in Part 1 (Mahony 1972). In the present paper we describe some measurements concerning the formation of the waves. The initial growth rate of the waves has been measured and the curves of neutral stability determined for two of the possible modes. The results are in good agreement with Mahony's theory. After the early stages in the development of the waves, there is an increase in their growth rate. This new rate was found to be about twice the initial growth rate. Also reported are some measurements of the amplitude of the cross-wave field along the channel.

The paper explores the whole class of self-similar solutions for blast waves bounded by Chapman-Jouguet detonations that propagate into a uniform, quiescent, zero counter-pressure atmosphere of a perfect gas with constant specific heats. Since such conditions can be approached quite closely by some actual chemical systems at N.T.P., this raises the interesting possibility of the existence of Chapman-Jouguet detonations of variable velocity. The principal virtue of the results presented here is, however, more of theoretical significance. They represent the limiting case for all the self-similar blast waves headed by gasdynamic discontinuities associated with a deposition of finite amounts of energy, and they exhibit some unique features owing to the singular nature of the Chapman-Jouguet condition.

Energy transport at Rayleigh numbers up to 675 times the critical (linear stability theory) value is measured in a layer of dilute electrolyte bounded horizontally by two rigid planes of constant and equal temperature; Joule heating by an alternating current passing horizontally through the layer provides the volumetric energy source. Horizontally averaged temperature profiles are determined optically. Mean temperature distributions are asymmetric at elevated Rayleigh numbers, the energy transport at the upper boundary being more than twice that at the lower boundary. Three regimes of flow are identified and discrete transitions in the energy transport appear to exist when the flow is turbulent. Extrapolation of the data to the conduction value of the Nusselt number yields a critical Rayleigh number which is within + 10·7% of linear theory values. No subcritical convection is observed when finite amplitude disturbances are introduced into the fluid at a Rayleigh number between the critical values predicted by the linear stability theory and energy theory respectively.

The mean-square angular displacement of a fluid material line element is expressed as an integral of the corresponding angular velocity in material coordinates, with forms like those in Taylor's (1921) linear displacement analysis. Measurements using a hydrogen-bubble tracer in isotropic turbulence show that the mean-square angular velocity of a line is of the same order of magnitude as the mean-square vorticity, and that its ‘Lagrangian’ integral time scale is of the order of the inverse of the r.m.s. vorticity. The angular velocity of a line element is also formulated in spatial co-ordinates. Finally, the connexion between angular dispersion and the approach toward isotropy is pointed out.

Stimulated by considerations of the shear structure of the tropospheric jet stream, we have performed a series of experiments in a rotating vessel (using water, barotropic model) to study the angular momentum exchange in a rotating fluid. In the experiments, a cylinder was fitted in the centre of a cylindrical vessel (of large diameter) rotating around the vertical axis and could rotate independently around the same axis in both directions. The experimental results are as follows. (i) For constant relative rotation with the inner cylinder acting as a source (sink) of angular momentum, the momentum exchange becomes stationary, the bottom of the vessel being a sink (source) of momentum. The zone for this exchange is confined to a relatively small area around the inner cylinder. Beyond this zone, the ‘friction’ zone, no relative motion can be observed. Essentially the radial extent b of the ‘friction’ zone depends on the dimensionless parameter ε = Δω/ω (Δω = relative rotation rate of inner cylinder, ω = rotation rate of vessel): b(|ε|) > b(–|ε|). (ii) The velocity field shows a strong tendency to be two-dimensional, the radial and vertical components being about one to two orders of magnitude smaller than the zonal component, whose absolute value decreases monotonically from the inner cylinder to the outer limit of the ‘friction’ zone. Elementary analytical considerations indicate that inertial stability seems to be the key for the understanding of these results.

Nonlinear waves on the interface of two incompressible in viscid fluids of different densities and arbitrary surface tension are analysed using the method of multiple scales. Third-order equations are presented for the space and time variation of the wavenumber, frequency, amplitude and phase of stable waves. A third-order expansion is also given for wavenumbers near the linear neutrally stable wave-numbers. A second-order expansion is presented for wavenumbers near the second harmonic resonant wavenumber, for which the fundamental and its second harmonic have the same phase velocity. This expansion shows that this resonance does not lead to instabilities.

An incompressible fluid of constant thermal diffusivity D flows with velocity u = Sβ(ωt) y in the x direction, where S is a scaling factor for the velocity gradient at the wall y = 0, and β(ωt) is a positive function of time t, with characteristic frequency ω. The region 0 [les ] x [les ] l of the wall is occupied by a heated film of temperature T1, the rest of the wall being insulating. Far from the film the fluid temperature is T0 < T1. Using boundary-layer theory, we calculate the heat transfer from the film by means of two asymptotic expansions, a regular one for small values of the frequency parameter $\epsilon (x) = \omega(9x)^{\frac{2}{3}} D^{-\frac{1}{3}} S^{-\frac{2}{3}}$ and a singular one (requiring the use of matched asymptotic expansions) for large values of ε. We notice the appearance of eigenfunctions in the large-ε expansion, where they are to be expected on physical grounds in order to take account of upstream conditions. Numerical computations are made for the case of sinusoidal oscillations, where β(ωt) = 1 + α sin ωt, α < 1 (three values of α, = 0·2, 0·5, 0·8, were chosen); there is seen to be no satisfactory overlap between the two expansions–the small-ε expansion is quite accurate for ε < 5·0 (especially for the smaller values of α) and the large-ε expansion is quite accurate for ε > 10·0. Approximate overlap is declared to occur at ε = 8·0.

The theory is used to calculate the response in oscillatory flow of the hot-film anemometer developed by Seed & Wood (1970a, b) to measure blood velocities in large arteries. The velocity gradient over the film (embedded in the surface of a larger probe) is obtained from the theory of the companion paper (Pedley 1972) on the assumption that the probe resembles a semi-infinite flat plate. The deviations observed in unsteady calibration experiments between the unsteady response of the anemometer and its steady response are predicted qualitatively by the theory, but quantitative agreement is in general unsatisfactory. The probable sources of this error, and the possibility of removing them, are discussed. The quasi-steady calibration curve used by Seed & Wood (1971) is suspect at low instantaneous velocities, but is shown to be adequate for the turbulence measurements of Nerem & Seed (1972). The theory is also applied to the experiments of Caro & Nerem (1972) on mass transfer to segments of arterial wall, and it is shown that oscillations characteristic of the cardiovascular system will have a negligible effect on the mean mass transfer.

The laminar boundary layer over the semi-infinite rigid plane y = 0, x > 0 is examined for the case when the free-stream velocity takes the form \[ U(x, t) = U_0(x) (1+\alpha_1 \sin \omega t), \] where 0 [les ] α1 < 1 and U0(x) ∝ xn, 0 [les ] n [les ] 1. The corresponding steady solution is the Falkner-Skan boundary layer with zero (n = 0) or favourable pressure gradient (n = 1 corresponds to the stagnation-point boundary layer). The skin friction, and the heat transfer from the wall when that is maintained at a uniform temperature T1 greater than the temperature T0 of the oncoming fluid, are calculated by means of two asymptotic expansions: a regular one for small values of the frequency parameter ε1(x) = ωx/U0(x) and a singular one (requiring the use of matched asymptotic expansions) for large values of ε1. The principal difference between this work and that of earlier authors is that here α1 is not required to be small. Numerical computations are made for three values of n (0, $\frac{1}{3}$, 1), three values of α1 (0·2, 0·5 and 0·8) and (in the case of the heat transfer) two values of the Prandtl number σ (0·72 and 7·1). It is demonstrated that for each n there is a value of ε1 at which the small-and the large-ε1 expansions for the skin friction overlap quite well, and that, near the overlap region, two terms of the small-ε1 expansion provide a more accurate asymptotic representation of the solution than three. It is also shown that there is no region of overlap between the small- and the large-ε1 heat-transfer expansions, except in one case (n = 1/3, σ = 0·72) where the overlap value of ε1 (= 2·0) is the same as for the skin-friction expansions. The question of the existence of eigensolutions in the large-ε1 expansion, where on physical grounds they can be expected to appear, is discussed in the appendix.