The dispersion relations for infinitesimal internal gravity waves (A) and axisymmetric waves in swirling streams (B) are considered. In both cases the mainstream may be sheared and density stratified in the transverse (vertical in case A, radial in case B) direction. The following results are proved for either case: If the maximum speed Wmax (or minimum speed Wmin) (in a meridian plane in case B) of the mainstream occurs at an interior point in the fluid, then the phase speed of any mode takes all values from the Wmax (or Wmin) to +∞ ( —∞) as the overall Richardson number λ2 varies from 0 to ∞. If Wmax(Wmin) is attained at a boundary point with finite rate of strain, there is a positive non-zero critical Richardson number below which one or both branches of the dispersion relation terminate. These results employ variational methods and correct erroneous results concerning problem B stated in Chandrasekhar's treatise on hydrodynamic stability. Furthermore, bounds are given on the group velocity for both branches of the dispersion relation. From these bounds it is shown that in the absence of reversals of the mainstream (Wmin > 0) upstream propagation of wave energy is impossible whenever upstream propagation of constant phase surfaces is impossible.

Oscillations of impinging flows, which date back to the jet-edge phenomenon (Sondhaus 1854), have been observed for a wide variety of impingement configurations. However, alteration of the structure of the shear layer due to insertion of an impingement edge (or surface) and the mechanics of impingement of vortical structures upon an edge have remained largely uninvestigated. In this study, the impingement of a shear layer upon a cavity edge (or corner) is examined in detail. Water is used as a working fluid and laser anemometry and hydrogen bubble flow visualization are used to characterize the flow dynamics. Reynolds numbers (based on momentum thickness at separation) of 106 and 324 are employed. Without the edge, the shear layer produces the same sort of non-stationary (variable) velocity autocorrelations observed by Dimotakis & Brown (1976). When the edge is inserted, the organization of the flow is dramatically enhanced as evidenced by a decrease in variability of autocorrelations and appearance of well-defined peaks in the corresponding spectra. This enhanced organization is not locally confined to the region of the edge but extends along the entire length of the shear layer, thereby reinforcing the concept of disturbance feedback. Comparison of spectra with and without insertion of the edge reveals a remarkable similarity to those of a non-impinging shear layer with and without application of sound at a discrete frequency (Browand 1966; Miksad 1972); with enhanced organization at the fundamental frequency, simultaneous enhancement occurs also at the sub- and higher-harmonics.

This paper contains a study of large amplitude, progressive interfacial waves moving between two infinite fluids of different densities. The highest wave has been calculated using the criterion that it has zero horizontal fluid velocity at the interface in a frame moving at the phase speed of the waves. For free surface waves this criterion is identical to the criterion due to Stokes, namely that there is a stagnation point at the crest of each wave. I t is found that as the density of the upper fluid increases relative to the density of the lower fluid the maximum height of the wave, for fixed wavelength, increases. The maximum height of a Boussinesq wave, which has the density almost the same above and below the interface, is 2·5 times the maximum height of a surface wave of the same wavelength. A wave with air over the top of it can be about 2% higher than the highest free surface wave. The point at which the limiting criterion is first satisfied moves from the crest for free surface waves to the point half-way between the crest and the trough for Boussinesq waves. The phase speed, momentum, energy and other wave properties are calculated for waves up to the highest using Padé approximants. For free surface waves and waves with air above the interface the maximum value of these properties occurs for waves which are lower than the highest. For Boussinesq waves and waves with the density of the upper fluid onetenth of the density of the lower fluid these properties each increase monotonically with the wave height.

Our analysis of 18th and 19th century cannon firings shows that knowledge of sphere drag can be substantially extended into the region of 0·3 ≤ M∞ ≤ 2·0 and Re∞d up to 107. Bashforth’s chronographic measurements (1868) are of a quality comparable to modern measurements. The data of Mayevski (chronograph, 1868), Hutton (ballistic pendulum, 1787-1791), and Didion (ballistic pendulum, 1839-1840) are of lesser accuracy but in agreement with Bashforth’s. These cannon data are combined with modern data to provide the most extensive curves available of CD vs. Re∞d in this region. Interesting features of these curves for M∞ ≤ 1·0 are briefly described.

Local wall shear rates in steady flow in the entry region of a curved tube have been measured by the electrochemical limiting current method. A semi-circular rigid tube of circular cross-section with radius ratio 1/7 has been employed for a range of Dean number between 139 and 2868. The circumferential and axial distributions of the wall shear rates have been measured at 20° circumferential increments at five different sections of the entry region.

A stability analysis for high wavenumber perturbations of a Stokes wave of wavenumber k1 and slope ϵ is presented. Except for a correction term the governing equation is shown to be of Hill's type. The analysis predicts instability at wavenumbers k2 = ¼(m + 1)2k1. The two lowest and strongest instabilities are the Benjamin- Feir instability at m = 1, and the quartet resonance at m = 2. Both are incorrectly treated by the present method. For m ≥ 3 the analysis should be asymptotically (ϵ → 0) correct, yielding instability O(ϵm) due to m-fold Bragg-scattering. The non-resonant perturbations behave as predicted by WKBJ theory. The instability is too weak for experimental detection; numerical tests should be possible, but are not available at present.

A grid-generated turbulence is subjected to a pure plane strain and the principal axes of the Reynolds stress tensor become those of the strain. This ‘oriented’ homogeneous turbulence is then submitted to a new strain the principal axes of which have a different orientation. We show that in such a situation it is possible to observe a transfer of energy from the fluctuating motion to the mean one. Such transfer is necessarily associated with a forced decay of the anisotropy of the motion. A detailed analysis of the reorientation of the principal axes of the Reynolds stress tensor in the frame of those of the second strain gives an explanation of the evolution of the principal axes of the Reynolds stress tensor in a shear flow.

The wavelength of turbulent Taylor vortices at very high Taylor numbers up to 40000Tc, has been measured in long fluid columns with radius ratios η = 0·896 and η = 0·727. Following slow acceleration procedures the wavelength (in units of the gap width) of turbulent axisymmetric vortices was found to be λ = 3·4 ± 0·1 with the small gap and about λ = 2·4 ± 0·1 with the larger gap, and thus in both cases substantially larger than the critical wavelength of laminar Taylor vortices. In the narrow and wide gap the wavelength was, within experimental error, independent of the Taylor number for T > 100Tc. In the experiments with the narrow gap a clear dependence of the value of the wavelength of the turbulent vortices on initial conditions was found. After sudden starts to Taylor numbers > 700Tc the wavelength of steady axisymmetric turbulent vortices was only 2·4 ± 0·05, being then the same as the wavelength of the vortices after sudden starts in the wide gap, and being, within the experimental error, independent of the Taylor number. In the narrow gap all values of the wavelength between λmax = 3·4 and λmin = 2·4 can be realized as steady states through different acceleration procedures. In the wide gap the dependence of the wavelength on initial conditions is just within the then larger experimental uncertainty of the measurements.

Steady flow in a cylinder of finite length rotating rapidly at right angles to the earth's gravity and containing a rigid cylindrical float is found under the assumption of small viscosity. The float is modelled by a ‘rigid free surface'. The major new result is a prediction of the rotation rate of the interface, which differs by a factor of O(E1/4) from those given by Greenspan (1976) and Wood (1977) for a rigid interface in an infinitely long cylinder. The difference is accounted for by the appearance of Stewartson layers on the inner boundary in the finite case. The result is supported by an experiment of my own, and is not in conflict with the experimental results given by Greenspan.

Some additional experimental observations are reported: a comparison of free and rigid surface rotation rates; and a brief description of the spin-up from rest.

Further experiments on the detailed study of annular jets are described in which the mean and fluctuating properties in the inner region have been measured. The experiments in the conical jetf have shown, besides the jet† vortices in the outer mixing region, another train of vortices in the inner region. This train of vortices is due to the wake formed by the boundary layer on the surface of the conical bullet.

The experiments in the inner region of the basic annular jet have similar mean velocity profiles to those in the internal recirculating region. Good correlation is found for the location of the vortex centre, the location of reattachment, the minimum and maximum mean static pressure and their locations with the non-dimensional available pressure M°/AiPatm for entrainment behind the interface. A train of wake vortices is generated in the internal recirculating region and a train of jet vortices is found in the inner mixing region. Both types of vortices in the inner region seem to decay fairly rapidly within a distance of one outer diameter D° downstream.

The disturbances associated with the wake vortices in the inner region seem to excite the outer mixing region. This results in another wave or train of vortices observed in the outer shear layer in addition to the jet vortices which are already in existence.

The mean-free-path approach to kinetic theory, initiated by Maxwell, and largely abandoned after the Chapman-Enskog success with Boltzmann's equation, is revised and considerably extended in order to find expressions for the heat flux vector q and pressure tensor p, valid (it is hoped) for all Knudsen numbers, K. These expressions (equations (2.24) and (2.26)) are integrals taken over the whole volume of the fluid plus surface integrals taken over the solid boundaries. The one phenomenological element is the mean free path λ, which takes different values according to whether it is mass, momentum or energy that is transported by the molecules. The need for such an approach is evidenced by the existence of critical values of K, above which the Chapman-Enskog expansion in powers of K, truncated after a finite number of terms, fails to yield a solution. For example with the Burnett equations, which are correct to O(K2), the critical K in a shock wave is only 0·2 based upon the upstream λ.