We consider the flow of a slightly viscous homogeneous fluid over a small two-dimensional obstacle perpendicular to the vertical side walls in a channel rotating about a vertical axis. The flow in the channel is obtained from the solution of the quasi-geostrophic potential vorticity equation in the limit ε = D/L→ 0, where D is the obstacle width and L the channel width. The lowest order, interior flow is shown to be a combination of three effects: a rotational flow caused by vortex stretching and Ekman boundary-layer pumping; a significant irrotational flow induced by the magnitude of the former flow at the vertical boundaries; and the interior Ekman drift due to the basic current. The maximum streamline displacement is calculated and compares very well with recent experiments in the identical parameter range by Boyer (1971a, b). This theory explains how the side walls are responsible for the dependence of the maximum streamline displacement on Rossby number.

In this paper we study the development of large-scale wavelike eddies, or instability waves, in a turbulent free shear flow. The model is based on splitting the flow into three components: the mean flow, the instability wave and the fine-scale turbulence. The wave is considered to be sufficiently weak so that it is developing in a pre-existing, known turbulent mean shear flow. The basis for the wave development is its time-averaged kinetic energy flux equation in integral form and the wave description is obtained through a shape assumption: the amplitude is determined by the energy equation; the shape function and local characteristics are determined by the local linear stability theory. The wave energy changes as it is convected into a different streamwise position where its instability properties change. The energy balancing mechanisms are production, work done by the wave pressure gradients and the energy transfer between the wave and the fine-scale turbulence via the wave-induced Reynolds stresses. The latter is taken to be dissipative via an eddy-viscosity model, inertia—elastic effects not being considered. According to forceful evidence from observations in turbulent free shear flows, the wave development is taken as being upstream controlled and begins from a distinct origin rather than being the result of local forcing by variations of the fine-scale turbulent Reynolds stresses. The wave energy flux initially grows via energy supplied by the inflexional mean flow when the shear layer is relatively thin but eventually decays through action of the finescale turbulence, directly via the dissipative energy transfer and indirectly via the turbulence-diffused, rapidly thickened mean shear flow, which renders the production mechanism less available. Numerical calculations are carried out for a turbulent mean shear flow, with speed Tie on one side and zero in the ambient region, its distribution being approximated by a sine profile in the Howarth-Dorodnitsyn co-ordinate. The flow develops from an initial boundary layer of finite thickness δ0 to a similar free-mixing layer far downstream. The wave is characterized by a dimensionless frequency parameter β0 formed from the wave frequency β*, ue, and δ0. Convection speeds, in general, increase in the downstream direction. They are subsonic initially for Mach numbers Me < 2 and remain subsonic for Me < 1.5. For Me > 2 peaking in the local intensity levels occurs when convection speeds are supersonic and this may explain the observed supersonic far-field radiation a t the higher jet speeds. Induced wave patterns in the ambient region are determined by the complex instability-wave speed rather than the real convection speed alone, consequently ambient wave patterns exist even a t subsonic convection speeds, but are more heavily damped near the origin and fan out laterally downstream for a given β0. According to the present model, if the waves are given an upstream excitation level about 10−3–10−2 times that of Ue, resembling, for instance, the levels of the upstream wall turbulent boundary-layer fluctuations over a wide, low-frequency, spectrum or other possible disturbances a t the nozzle exit, the development of the calculated near noise field as a function of downstream distance bears striking resemblances to the observed near jet noise field and is thus fully sufficient to explain such observations. This comparison leads to the suggestion that the essential energetics of the large-scale wavelike eddies are that they are formed at the origin, amplified and subsequently decay in a developing mean turbulent free shear flow. Therefore this leads also to a most important indication of shear-layer instabilities and noise control. If their historical evolution can be controlled so can the noise from the damaging wavelike eddies. Methods of control on the basis of this model are discussed.

Plate tectonics provides a remarkably accurate kinematic description of the motion of the earth's crust but a fully dynamical theory requires an understanding of convection in the mantle. Thus the properties of plates and of the mantle must be related to a systematic study of convection. This paper reviews both the geophysical information and the fluid dynamics of convection in a Boussinesq fluid of infinite Prandtl number. Numerical experiments have been carried out on several simple two-dimensional models, in which convection is driven by imposed horizontal temperature gradients or else by heating either internally or from below. The results are presented and analysed in terms of simple physical models. Although the computations are highly idealized and omit variation of viscosity and other major features of mantle convection, they can be related to geophysical measurements. In particular, the external gravity field depends on changes in surface elevation; this suggests an observational means of investigating convection in the upper mantle.

A calculation procedure for three-dimensional parabolic flows is applied to predict the velocity and temperature fields in helically coiled pipes. The curvature produces a secondary flow and causes departures from the symmetric velocity profile of Poiseuille flow. Predictions are presented of flow and heat transfer in the developing and fully developed regions. Comparisons of the developing and fully developed velocity profiles with experimental data exhibit good agreement. The development of the wall temperature for the case of axially uniform heat flux with an isothermal periphery has been compared with experimental data and the agreement is good. Predictions for fully developed temperature profiles and heat-transfer coefficients also exhibit good agreement with experimental data. Effects of the Dean number on the friction factor and heat transfer are presented.

Stokes’ infinitesimal-wave expansion for steady progressive free-surface waves has been extended to high order using a computer to perform the coefficient arithmetic. Stokes’ expansion has been found to be incapable of yielding the highest wave for any value of the water depth since convergence is limited by a square-root branch-point some distance short of the maximum. By reformulating the problem using a different independent parameter, the highest waves are obtained correctly. Series summation and analytic continuation are facilitated by the use of Padé approximants. The method is valid in principle for any finite value of the wavelength and solutions of high accuracy can be obtained for most values of the wave height and water depth. An alternative expansion procedure proposed by Havelock for the computation of waves short of the highest has been reconsidered and found to be defective.

The onset of convection is observed in a cylindrical annulus which is heated from the outside, cooled from the inside and rotating about its vertical axis of symmetry. The dynamical constraint exerted by the dominating Coriolis force inhibits the instability when the top and bottom boundaries of the annulus are conical so as to make the vertical height vary with distance from the axis. The experimental observations are in good agreement with the theoretical predictions by Busse (1970a). This confirmation indicates the absence of subcritical finite amplitude instabilities and suggests that the annulus experiment provides a close dynamical model for convection in the liquid core of the earth.

Data are presented for the extensional flow of suspensions containing 0.1−1 % of fibres by volume. The aspect ratio of the fibres was varied from 280 to 1260. The observed stress levels were between one and two orders of magnitude greater than in shearing flow, in agreement with the a priori predictions of Batchelor (1971).

General dimensional and similarity arguments are applied to derive a heat and mass transfer law for fully turbulent flow along a rough wall. The derivation is quite analogous to Millikan's (1939) derivation of a skin-friction law for smooth-and rough-wall flows and to the derivation of the heat and mass transfer law for smooth-wall flows by Fortier (1968a, b) and Kader & Yaglom (1970, 1972).

The equations derived for the heat or mass transfer coefficient (Stanton number) Ch and Nusselt number Nu include the constant term β of the logarithmic equation for the mean temperature or concentration of a diffusing substance. This term is a function of the Prandtl number, the dimensionless height of wall protrusions and of the parameters describing the shapes and spatial distribution of the protrusions. The general form of the function β is roughly estimated by a simplified analysis of the eddy-diffusivity behaviour in the proximity of the wall (in the gaps between the wall protrusions). Approximate values of the numerical coefficients of the equation for β are found from measurements of the mean velocity and temperature (or concentration) above rough walls. The equation agrees satisfactorily with all the available experimental data. It is noted that the results obtained indicate that roughness affects heat and mass transfer in two ways: it produces the additional disturbances augmenting the heat and mass transfer and simultaneously retards the fluid flow in the proximity of the wall. This second effect leads in some cases to deterioration of heat and mass transfer from a rough wall as compared with the case of a smooth wall at the same values of the Reynolds and Prandtl numbers.