This paper presents exact solutions using toroidal co-ordinates to the equations of creeping fluid motion with the no-slip boundary conditions for a toroidal particle translating in a direction normal to the axis of symmetry or rotating about an axis normal to the axis of symmetry through an otherwise infinite expanse of quiescent fluid. The associated resisting force and resisting torque are computed for toroids of various geometrical ratios b/a, b being the smallest radius of the open hole and (b + 2a) being the radius to the outermost rim of the torus. These results are compared with approximate calculations based on slender-body theory and on the theory for interacting beads. The exact and approximate calculations become asymptotically equal as b/a becomes very large, but departures from the exact calculations are apparent for b/a less than 10−100 depending on the mode of motion and the method of approximation and the approximations are unreliable for b/a less than 2·0.

The propagation of finite-amplitude internal waves in a shear flow is considered for wavelengths that are long compared to the shear-layer thickness. Both singular and regular modes are investigated, and the equation governing the amplitude evolution is derived. The theory is generalized to allow for a radiation condition when the region outside the stratified shear layer is unbounded and weakly stratified. In this case, the evolution equation contains a damping term describing energy loss by radiation which can be used to estimate the persistence of solitary waves or nonlinear wave packets in realistic environments. A continuous three-layer model is studied in detail and closed-form expressions are obtained for the phase speed and the coefficients of the nonlinear and dispersive terms in the amplitude equation as a function of Richardson number.

The convective instability of a layer of fluid heated from below is studied on the assumption that the flux of heat through the boundaries is unaffected by the motion in the layer. It is shown that when the heat flux is above the critical value for the onset of convection, motion takes place on a horizontal scale much greater than the layer depth. Following Childress & Spiegel (1980) the disparity of scales is exploited in an expansion scheme that results in a nonlinear evolution equation for the leading-order temperature perturbation. This equation which does not depend on the vertical co-ordinate, is solved analytically where possible and numerically where necessary; most attention is concentrated on solutions representing two-dimensional rolls. It is found that for any given heat flux a continuum of steady solutions is possible for all wave numbers smaller than a given cut off. Stability analysis reveals, however, that each mode is unstable to one of longer wavelength than itself, so that any long box will eventually contain a single roll, even though the most rapidly growing mode on linear theory has much shorter wavelength.

The time development of the symmetrical standing zone of recirculation, which is formed in the early stage of the flow due to a circular cylinder impulsively set in motion perpendicular to its generators, has been studied using a flow visualization technique. The Reynolds numbers (based upon the diameter) range from 40 to 104. Some new phenomena indicated in the flow patterns are revealed, and several different regimes are differentiated by a detailed analysis of the evolution of the main flow characteristics. A correlation with some theoretical results is established.

To resolve possible fluctuations about the mean motion of the Desnyansky-Novikov model for Kolmogorov's inertial flow, we have investigated two dynamical systems of the cascade process which are formally derivable from Burgers’ equation. The first cascade model produced no fluctuations, for its trajectory was identical with the Desnyansky–Novikov model's. Disappointingly, the second cascade system, which is similar to the Kerr–Siggia model, has also proved unable to engender fluctuations. This is because the second model when truncated consistently maps an arbitrary initial point into the attainable phase space of the first cascade model. However, when truncated inconsistently the trajectory of second model can exhibit a quite erratic and somewhat sporadic motion, thereby reflecting the apparently random motion of inviscid equilibrium solutions. Therefore, the observation of temporally intermittent fluctuations by a stationary Kerr–Siggia model is due to the inconsistent truncation produced by restricting energy dissipation for all but the upper truncation mode in their model.

This note provides some explanation of the fact that, contrary to the requirements of local isotropy, the skewness S of the streamwise temperature derivative ∂θ/∂x1 has been observed to be a non-zero constant of magnitude of about unity in high-Reynolds-number and high-Péclet-number turbulent shear flows. Measurements in slightly heated homogeneous shear flows and in unsheared grid turbulence suggest that S is non-zero only when the mean shear dU1/dx2 and the mean temperature gradient dT/dx2 are both non-zero. The sign of S is given by –sgn (dU1/dx2).sgn (dT/dx2). For fixed dU1/dx2, S is of the form tanh (αdT/dx2), α being a constant, while for fixed dT/dx2, it is of the form S/S* = 1 − β1 exp (− β2τ), where S* is a characteristic value of S, β1 and β2 are positive constants, and τ can be interpreted as a ‘total strain’. The derivative skewness data in other (inhomogeneous) shear flows are also compatible with the latter relation. Predictions from a simplified transport equation for $\overline{(\partial\theta/\partial x_1)^3}$, derived in the light of the present experimental observations, are in reasonable agreement with the measured values of S. A possible physical mechanism maintaining S is discussed.

The linear Rayleigh–Taylor stability of superposed viscous fluids with interfacial transfer of mass and heat is first considered for layers of finite thickness. A dispersion relation is obtained. It is then employed to derive stability and instability criteria for the case of two semi-infinite layers as well as the case where one of the layers is finite. From these criteria one arrives at a critical dispersion relation and a new critical wavenumber. This new critical wavenumber is distinct from the classical value owing to the presence of a parameter which depends, in a very simple manner, upon the kinematic viscosity of the fluids, the surface tension and the rate of interfacial transfer of mass and energy. Also it is found that the stabilizing effect of the surface tension is neither affected by the arrangement of the system nor the direction of the temperature gradient. However, the effects of the viscosity and the gravity will depend upon the relative positions of the superposed fluids and the direction of the temperature gradient at the interface.

Previous studies of the distortion of the electric double layer around a charged sphere have assumed that the electric stresses are small compared with the viscous stresses. The flow around the particle is therefore changed only slightly by the presence of the charge cloud. This change is measured by the Hartmann number, and in § 6 we remove the restriction that it should be small. It is found that the previous linearized theory is sufficiently accurate for typical experimental values of the Hartmann number. Previous studies have also assumed that the potential at the surface of the particle is small. This assumption is removed in § 7 of this paper. For values of the non-dimensional surface potential smaller than 2 the predictions are altered by less than 10 %. For higher values the differences between linear and nonlinear theory are not negligible, especially when the charge cloud is thin compared with the radius of the charged sphere.

A new class of three-dimensional, deep-water gravity waves of permanent form has been found using an equation valid for weakly nonlinear waves due to Zakharov (1968). These solutions appear as bifurcations from the uniform two-dimensional wave train. The critical wave heights are given as functions of the modulation wave vector. The three-dimensional patterns may be skewed or symmetrical. An example of the skewed wave pattern is given and shown to be stable. The results become exact in the limit of very oblique modulations.

The boundary-layer equations of mixed convection are examined in the vicinity of separation. The correlation between the uniform wall temperature case and that of compressible boundary layer flow is outlined. Goldstein–Stewartson–Buckmaster theory is thus appropriate and associated indeterminacies in the theory are evaluated from a numerical integration. The case of uniform heat flux at the wall is then examined theoretically. Significantly it is concluded that the original Goldstein–Stewartson theory is sufficient to describe the structure of the singularity at separation in this case. Indeterminacies associated with the theory are determined via a reconciliation between analytical and numerical representation of skin friction and heat transfer coefficients near separation.

The steady movement of a fluid meniscus in a circular capillary tube is analysed by means of finite-element numerical simulation for a range of values of contact angles and contact-line velocities with minute slippage of the fluid on the tube wall, thus relaxing the conventional no-slip boundary condition. The resulting flow field does not produce unbounded forces at contact line, contrary to that with the no-slip condition. The unknown meniscus shape is determined by an iterative scheme in which the imbalance in the normal-stress boundary condition is the basis for improving the shape. Comparison of the numerical results found here and the experimental results of a number of investigators suggests the possibility that the contact angle does not vary with contact-line velocity.

Hydrodynamic measurements were made with a triaxial hot wire in the full-coverage region and the recovery region following an array of injection holes inclined downstream, at 30° to the surface. The data were taken under isothermal conditions at ambient temperature and pressure for two blowing ratios: M = 0·9 and M = 0·4. (The ratio M = ρjetUjet/ρ∞U∞, where U is the mean velocity and ρ is the density. Subscripts jet and ∞ stand for injectant and free stream, respectively.) Profiles of the three mean-velocity components and the six Reynolds stresses were obtained at several spanwise positions at each of five locations down the test plate.

In the full-coverage region, high levels of turbulence kinetic energy (TKE) were found for low blowing and low TKE levels for high blowing. This observation is especially significant when coupled with the fact that the heat transfer coefficient is high for high blowing, and low for low blowing. This apparent paradox can be resolved by the hypothesis that entrainment of the mainstream fluid must be more important than turbulent mixing in determining the heat transfer behaviour at high blowing ratios (close to unity).

In the recovery region, the flow can be described in terms of a two-layer model: an outer boundary layer and a two-dimensional inner boundary layer. The inner layer governs the heat transfer.

Hydrodynamic data are reported in the companion paper (Yavuzkurt, Moffat & Kays 1980) for a full-coverage film-cooling situation, both for the blown and the recovery regions. Values of the mean velocity, the turbulent shear stress, and the turbulence kinetic energy were measured at various locations, both within the blown region and in the recovery region. The present paper is concerned with an analysis of the recovery region only. Examination of the data suggested that the recovery-region hydrodynamics could be modelled by considering that a new boundary layer began to grow immediately after the cessation of blowing. Distributions of the Prandtl mixing length were calculated from the data using the measured values of mean velocity and turbulent shear stresses. The mixing-length distributions were consistent with the notion of a dual boundary-layer structure in the recovery region. The measured distributions of mixing length were described by using a piecewise continuous but heuristic fit, consistent with the concept of two quasi-independent layers suggested by the general appearance of the data. This distribution of mixing length, together with a set of otherwise normal constants for a two-dimensional boundary layer, successfully predicted all of the observed features of the flow. The program used in these predictions contains a one-equation model of turbulence, using turbulence kinetic energy with an algebraic mixing length. The program is a two-dimensional, finite-difference program capable of predicting the mean velocity and turbulence kinetic energy profiles based upon initial values, boundary conditions, and a closure condition.

A rational asymptotic theory describing the perturbed flow in a turbulent boundary layer encountering a small two-dimensional hump is presented. The theory is valid in the limit of very high Reynolds number in the case of an aerodynamically smooth surface, or in the limit of small drag coefficient in the case of a rough surface. The method of matched asymptotic expansions is used to obtain a multiple-structured flow, along the general lines of earlier laminar studies. The leading-order velocity perturbations are shown to be precisely the inviscid, irrotational, potential flow solutions over most of the domain. The Reynolds stresses are found to vary across a thin layer adjacent to the surface, and display a singular behaviour near the surface which needs to be resolved by an even thinner wall layer. The Reynolds stress perturbations are calculated by means of a second-order closure model, which is shown to be the minimum level of sophistication capable of describing these variations. The perturbation force on the hump is also calculated, and its order of magnitude is shown to depend on the level of turbulence closure; a cruder turbulence model gives rise to spuriously large forces.

The usual way of posing the problem for the reflexion of wave trains from beaches seems inevitably to imply perfect reflexion. Energy considerations show that wave absorption must be associated with the degradation of mechanical energy either through wave breaking or viscous effects. Some experiments reported here showed substantial wave absorption in the absence of any breaking.

We describe some theoretical and experimental work aimed at assessing the role played by friction at the bottom in determining the reflexion coefficient of a beach. The results suggest that, if the parameter (νω3)½½gα2 is not too small, bottom friction can be a significant factor in the absorption process for waves on beaches. Here ν represents the kinematic viscosity (or perhaps an ‘eddy’ viscosity) of the fluid, ω is the frequency of the motions, α is the slope of the beach and g is the acceleration due to gravity.

Previous work on irrotational incompressible inviscid flow (Guiraud & Zeytounian 1977) is extended to rotational and compressible flow. A formal proof is given that, within the core, one may avoid computing with the sheet by defining an equivalent continuous flow. One shows how the vorticity and the entropy gradient between the turns of the sheet are transported along trajectories of the equivalent continuous flow.

A generalized eddy-viscosity function νT, is introduced in order to express the Reynolds stress in an incompressible dusty gas as a linear combination of the Kronecker and rate-of-strain tensors. On the basis of Saffman's dusty-gas model a transport equation for the eddy viscosity is derived from the general turbulence energy equations, thereby introducing two additional functions, the specific turbulence kinetic energy E1, and a scale variable s. In order to determine the three variables modified Prandtl–Wieghardt relation among them is accepted and a transport equation for s is postulated in the same manner as in the clean-gas turbulence transport model (firstly proposed by Harlow & Nakayama 1967) but with the inclusion of an additional term accounting for the dust particles stabilizing action. We are considering values of loading (mass ratio of particles) of order of unity, with particle/gas density ratios of order of 103 and volume concentrations of the order of 10−3, so that particle–particle interactions are neglected. Supposing that the particles nearly follow the gas motion, following well at large scales and poorly at small, an application of the theory to problem of numerical calculations of the dusty-gas parameters such as mean velocity profile of turbulent pipe flow is given.

Using the method of averaged equations, we examine the difference in temperature between the bulk and fixed heated spherical particles under conditions in which ϕ the volume fraction of the particles and ε the Peclet number of the flow past the particles are both small. If ϕ [Lt ] ε2 the particles are effectively isolated, and so their excess temperature has an O(ε) correction to the pure conduction estimate. On the other hand if ϕ [Gt ] ε2, the bulk heating is of sufficient magnitude to produce a significant temperature gradient throughout the fixed bed. This temperature gradient leads to an O(ϕ½) correction to the pure conduction estimate of the excess temperature of the particles, and the correction depends on the details of the flow even though its magnitude is independent of ε. A study of the leading-order terms when ϕ and ε2 are of the same magnitude finds that the two small effects are not simply additive.

Laboratory experiments were conducted to measure the surface elevation probability density function and associated statistical properties for a wind-generated wave field. The laboratory data together with some limited field data were compared. It is found that the skewness of the surface elevation distribution is proportional to the significant slope of the wave field, §, and all the laboratory and field data are best fitted by \[ K_3 = 8\pi\S, \] with § defined as ($(\overline{\zeta^2})^{\frac{1}{2}}/\lambda_0 $, where ζ is the surface elevation, and λ0 is the wavelength of the energy-containing waves. The value of K3 under strong wind could reach unity. Even under these highly non-Gaussian conditions, the distribution can be approximated by a four-term Gram-Charlier expansion. The approximation does not converge uniformly, however. More terms will make the approximation worse.