The angular momentum A per unit horizontal distance of a train of periodic, progressive surface waves is a well-defined quantity, independent of the horizontal position of the origin of moment.

The Lagrangian-mean angular momentum $\overline{A}_L$ consists of two parts, arising from the orbital motion and from the Stokes drift respectively. Together these contribute a positive sum, nearly proportional to the energy density (when the origin is taken in the mean surface level). If moments are taken about some point P not at the mean surface level, the angular momentum will differ by an amount proportional to the elevation of P. There is just one elevation for which the Lagrangian-mean angular momentum about P vanishes. This elevation is called the level of action. For infinitesimal waves in deep water the level of action is at a height above the mean surface equal to ½k, that is ¼π times the wavelength.

Just as for ordinary fluid velocities, the Lagrangian-mean angular momentum $\overline{A}_L$ differs from the Eulerian-mean $\overline{A}_L$, the latter being zero to second order. The difference between $\overline{A}_L$ and $\overline{A}_E$ is associated with the displacement of the lateral boundaries of any given mass of fluid.

For waves of finite amplitude, an initially rectangular mass of fluid becomes ultimately quite distorted by the Stokes drift. Nevertheless it is possible to define a long-time average l.t.$\overline{A}$ and to calculate its numerical value accurately in waves of finite amplitude. In low waves, l.t.$\overline{A}$ is equal to $\overline{A}_L$. Defining the level of action ya in the general case as l.t.$\overline{A}/I$, where I is the linear momentum, we find that ya rises from 0·5k−1 for infinitesimal waves to about 0·6k−1 for steep waves. Thus ya is about the same as the height yx of the wave crests above the mean level in limiting waves, a fact which may account for why steep irrotational waves can support whitecaps in a quasi-steady state. The same argument suggests that Gerstner waves (in which the particle orbits are theoretically circular) could not support whitecaps.

Large-Reynolds-number analysis is given for separated flows that have the characteristic feature of being confined in the transverse direction. Two principal limits of the governing equation are obtained depending on whether the streamwise length scale is of order one or of the order of the Reynolds number. The corresponding two types of separated flows are discussed. A method of calculation is given for the second type of flow. It employs an expansion in the eigenfunctions of the Poiseuille flow development and the problem is reduced to solving nonlinear first-order ordinary differential equations that have a tendency to decouple rapidly. The method is tested by a detailed comparison of the results of the present calculation with finite-difference solutions of the Navier–Stokes equations for a channel with sudden expansion. Applicability to other configurations is illustrated by solving the problem of the flow in a channel with a base.

A theoretical account is given of experiments performed by Saffman & Taylor (1958) in which a fluid drives a liquid out of a long straight channel of very small thickness formed between two parallel sheets sealed at the edges. The penetrating fluid forms a long finger, whose sides are parallel to the edges of the channel, and which has a rounded tip, which advances with unaltered shape at a constant speed U. The theory correctly predicts the shape of the finger as a function of the ratio λ = (asymptotic width of finger)/(width of channel) and gives the relation between λ and U, which is in good agreement with experiment. In particular it shows that, as U increases from zero to infinity, λ steadily decreases from 1 to 0·5.

Measurements of turbulent flow through a constricted tube with a contoured 75% constriction have been performed with a laser-Doppler anemometer. One motivation for these studies was the problem of flow through arterial stenoses and consequently a range of relatively low Reynolds numbers (5000–15000) was employed. Velocity profiles, r.m.s. turbulence velocities, and energy spectra were recorded along with determinations of the wall pressure variation and length of recirculation region. Results show extremely high levels of turbulence and considerable turmoil within the so-called recirculation region. A method for determining the reattachment point in an intensely turbulent flow is given which should offer improvement over flow visualization techniques. Similarity in turbulence energy spectra measured at various radial positions is found at several axial locations, but does not occur in the region immediately downstream of separation. The downstream recovery of the turbulence velocity to upstream values takes place slowly. A secondary motivation of providing experimental data useful in evaluating turbulence models for computational fluid dynamics led to extensive measurements of radial and axial r.m.s. turbulence velocity components at a Reynolds number of 15000.

The fluid is incompressible, inviscid and non-diffusive. It has a uniform Brunt–Väisälä frequency, N, and is of constant depth, D. A body or wing moves horizontally through the fluid at velocity U in a straight line, exerting a vertical force during a given time interval. The force is constant, or oscillatory with frequency σ. The vertical average of the strain rate in a thin surface layer is calculated for a network of points behind the body.

The linearized analysis is first applied with tank walls, then modified for remote walls and a vertical force of long duration.

For moderately high velocity and forcing frequency (U/ND = 5, σ/N≅ 4−16) the recurring internal wave pattern just behind the body is well established in one cycle of the oscillatory force. A tank width one or two times the depth gives good agreement between tank and no-wall calculations for the chosen examples.

For a stationary wing (U/ND = 0) in a cubic tank with forcing frequency one-half the natural frequency (σ/N = ½) the strain rates after one cycle are 103 times greater than for the moving wing case. After five cycles the magnitudes are twenty times larger than after one cycle. Presumably these large increases are due to the continuous and efficient feeding of energy into a small fluid volume which occurs for the stationary wing. No-wall calculations for many cycles give amplitudes roughly one-half those for five cycles in the tank, showing the effect of escaping energy.

The relation of these developments to stationary phase analysis and preferred directions is discussed.

Experiments were conducted in a long channel in which a mixed region was allowed to collapse in the thermocline region of a stratified fluid. Two solitary wave-like disturbances were generated travelling to the right and left of the mixed region. The mixed region fluid was partly entrained in these waves. The waves were allowed to reflect from the end walls and to collide after the reflexions. The velocity structure of the wave was studied before, during and after a collision by means of hot-film anemometry and streak pictures. Wave speeds were accurately determined by two hot-film probes. Permanence of form and amplitude decay of the waves were observed over long distances and through successive collisions and reflexions. An analytical result for the structure of the solitary wave in an ambient stratification of the hyperbolic tangent type, but of finite total water depth, was obtained using Benney's method. Excellent agreement between the experimental and theoretical results was obtained. The results showed that the generated waves were indeed solitary waves.

A possible model of the flow in the neighbourhood of a point of reattachment of a supersonic laminar boundary layer consists of a three-tiered ‘triple-deck’ structure in which the basic problem reduces to that of solving the incompressible boundary-layer equations in the lower deck, a region of lateral and streamwise extent $O(R^{-\frac{5}{8}})$ and $O(R^{-\frac{3}{8}})$, where R is a representative Reynolds number of the flow (Daniels 1979). The present paper describes a scheme for the numerical solution of this problem which provides evidence in support of the proposed model and quantitative values for the O(R−¼) correction to the base pressure in the flow upstream and the O(R−⅝) correction to the position of reattachment relative to the point of intersection of the incoming shear layer and the wall. An important feature of the scheme is that it copes successfully with a flow which contains substantial reverse velocities.

To account for our previous experimental results, the flow past a flat plate set parallel to a uniform stream is investigated on the basis of Oseen's equations of motion. We are mainly concerned with the wake region and calculate the velocity field in detail for the range of Reynolds numbers 20–3000, corresponding to the experiments. The calculation shows that there is a velocity overshoot in the velocity distribution near the trailing edge, with the same magnitude as in the experiments. Furthermore, we confirm the experimental fact that the velocity on the centre-line of wake recovers in proportion to the square root of the distance from the trailing edge in the near wake. The pressure field is also examined to see the streamwise pressure gradient in the near wake.

This paper presents an approximate method for solving Oseen's linearized equations for a two-dimensional steady flow of incompressible viscous fluid past arbitrary cylindrical bodies at low Reynolds numbers. The formulation is based on a discrete singularity method with a least squares criterion for satisfying the no-slip boundary condition. That is, sets of Oseenlets, sinks, sources and vortices are discretely distributed in the interior of the body, and then the least squares criterion attempts to minimize the integrated squares of velocities along the body contour, thus leading to a system of simultaneous algebraic equations. Complex-variable arithmetic, usually available on modern computers, makes the computation algorithm very simple. Furthermore, the method is applicable to cases that cannot be solved by classical analytical approaches. As examples of application, we computed the forces acting on a single circular cylinder, two circular cylinders of equal radius separated by a distance, an inclined elliptic cylinder and an inclined square cylinder all of which are immersed in uniform flow fields. The computed results agree very well with those of classical analytical methods.

In a recent paper, Tatsumi, Kida & Mizushima (1978) have made a numerical study of the quasi-normal Markovian (QNM) equation for homogeneous isotropic incompressible turbulence at Reynolds numbers R up to 800.

Analytical investigations of the QNM equation support the contention of Tatsumi et al. that, at R = ∞, the decay of an initial energy spectrum of the form ka exp (− k2) leads to an initial energy-conserving regularity phase followed by a self-similar decay phase. During the former we give explicit expressions for the enstrophy and skewness. During the latter we show that for 1 < a < 4 the energy follows, for t → ∞, a t−b law with the usual value b = 2(a + 1)/(a + 3); when a [ges ] 4 deviations from Kolmogorov's (1941) $t^{\frac{10}{7}}$ law originate from non-local ‘beating’ interactions between eddies with sizes of the order of the integral scale.

We also show, analytically, that the QNM equation has a k−2, not a $k^{-\frac{5}{3}}$, inertial range and that its dissipation range is of the form k3e−k/kD, rather than e−σk1.5.

Our results are illustrated by numerical integration of the QNM equation for R up to 106 and by comparison with results from the eddy-damped quasi-normal Markovian equation which is known to produce a k−5/3 spectrum.

One-dimensional bispectra are computed from the statistical theory of turbulence (using the Test Field Model) and are compared with experiments. For an inertial range, we obtain B(k1, p1) = εk−3F(θ), where B is the two-dimensional Fourier transform of $\langle u({\bf x})u({\bf x}+\hat{\imath}\xi_1)u({\bf x}+\hat{\imath}\xi_2)\rangle $ with respect to (ξ1, ξ2), ε is the energy dissipation and F(θ) (θ = tan−1(k1/p1) is an angular distribution of order unity, which is compared to measurements of planetary boundary-layer turbulence. We also compare theory to wind tunnel data, as reported by Helland et al. (1978). Finally, we discuss to what extent the bispectra give insight into the dynamics of the flow.

This note summarizes a detailed numerical computation of bispectra arising from weak nonlinear resonant interactions of internal waves whose energies are represented by the Garrett & Munk (1975) model spectrum. Two spectra are computed – the bispectrum of power and the auto-bispectrum of vertical displacement. These are chosen because the first is the most informative and the second is easy to observe. The numerical computations indicate that the level of the bispectral signal is much too low to be detected by any reasonable observational programme. Even more disturbing, bispectra of Eulerian variables are subject to a kinematic contamination causing a significant bispectral level which can easily be misinterpreted as a nonlinear interaction.

The energy transfer in a finite-depth gravity-wave spectrum is investigated in the approximation of a narrow spectrum. It is shown that for ocean depths larger than approximately one tenth of the wavelength (kh [ges ] 0·7) the finite-depth case can be reduced to Longuet-Higgins’ (1976) result for an infinitely deep ocean by a similarity transformation involving changes in scale of the angular spreading function and the transfer rate. For shallower water (kh < 0·7) Longuet-Higgins’ expansion technique is no longer applicable without modification, as the nonlinear coupling coefficient develops a discontinuity at the origin of the expansion. In the range kh [ges ] 0·7 both the magnitude and the two-dimensional frequency-directional distribution of the energy transfer are found not to differ significantly (to within variations by a factor of 2) from the case of an infinitely deep ocean. The transformation rules relating the infinite-depth and finite-depth cases may provide a useful guide for constructing parametrizations of the nonlinear transfer for finite-depth wave prediction models.