ALBERT

Description: In a previous paper, it was shown that abrupt changes in the surface conditions under a very deep boundary layer cause changes of mean velocity and temperature that satisfy the dynamical conditions for self-preserving development. Here the theory is extended to predict the development of the modified flow in the moderately deep layers that occur in nature and the laboratory. The problems considered are the changes in the velocity profile produced by an abrupt change of surface roughness and also by a line of concentrated roughness such as a fence, the changes in temperature produced by change of roughness combined with changes of heat flux at the surface, and diffusion of heat or a scalar pollutant from a line source at or near ground level. The predictions are compared with observations by Rider (1952) of the flow downwind of a hedge, by Rider, Philip & Bradley (1963) of temperature and humidity downwind of a change in surface, and of vertical diffusion from a line-source at ground level. © 1965, Cambridge University Press. All rights reserved.

Description: A considerable quantity of observations and measurements exists concerning the phenomenon of intermittency which is connected closely with the entrainment process in free turbulent flows. A number of these are described in the first part of the paper and conclusions are drawn about the shape and motion of the bounding surface that separates turbulent and non-turbulent fluid. The salient features are that indentations of the surface grow and decay cyclically, that each cycle leads to substantial entrainment of ambient fluid into the turbulent region, that the indentations move at a considerable speed relative to the free stream, and that the surface has a comparatively simple form. The growth–decay cycle of the indentations suggests that a critical condition for growth exists, but the pressure field consequent on the convection velocity of the indentations makes for a Helmholtz type of instability that is unlikely to be stabilized by purely viscous behaviour of the turbulent fluid. It is known that the initial response of turbulent fluid to distortion is elastic in character, with incremental Reynolds stress proportional to increment of total strain, and sufficient rigidity could stabilize the bounding surface. A simple flow model–an inviscid stream flowing over an elastic jelly—is examined and the condition for marginal stability is compared with the observed properties of the flow. The model leads to the conclusion that indentations of more than a critical wave-number are stable, and provides reasons for the comparatively simple form of the surface and for the occurrence of indentations in groups of about three. The relative values of entrainment constants in different flows of uniform density do not depend critically on the nature of the entrainment process provided that the main turbulent motion remains geometrically similar, but the correlation between entrainment constant and relative depth of the indentations found by Gartshore (1966) appears as a consequence of the ‘elastic’ control of the growth–decay cycle. Lastly, the properties of the engulfment mechanism are used to show that the entrainment constant for a jet is proportional to the square root of the ratio of ambient density to the average density inside the jet. In contrast, the corresponding result for engulfment controlled by an eddy viscosity is variation as the ratio of the mean of the ambient and inside density to the inside density. Observations of high-speed jets of water in air and air in water give some support to the ‘elastic’ hypothesis. © 1966, Cambridge University Press. All rights reserved.

Description: The changes of surface stress in a deep boundary layer passing from a surface of one roughness to another of different roughness are described fairly accurately by theories that assume self-preserving development of the flow modifications. It has been shown that the dynamical conditions for self-preserving flow can be satisfied if the change in friction velocity is small and if log l0/z0 is large (l0 is the depth of the modified flow and z0 is the roughness length of the surface). In this paper it is shown that, if the change of friction velocity is not small, the dynamical conditions can be satisfied to a good approximation over considerable fetches if log l0/z0 is large. The flow modification is then locally self-preserving, that is, the fields of mean velocity and turbulence are in a moving equilibrium but one which changes very slowly with fetch and depends on the ratio of the initial to the current friction velocity. In the limit of a very large increase in friction velocity, the moving equilibrium is essentially that of a boundary layer developing in a non-turbulent free stream. Equations describing the flow development are derived for all changes of friction velocity, and the form of the velocity changes is discussed. For large increases of friction velocity, the depth of the modified layer is substantially less than would be expected from the theories of Elliott and of Panofsky & Townsend. © 1966, Cambridge University Press. All rights reserved.

Description: The rate of generation of internal waves by a thin turbulent boundary layer was calculated in a previous paper for a stably-stratified atmosphere with no significant wind-shear outside the boundary layer by considering the excitation of normal modes of wave propagation. By using the concept of wave-packets propagating upwards from the boundary layer, the effects of wind-shear can be included. Conditions for the validity of the approximation are given. In general, the spectral distribution of wave-energy at a particular height takes large values in two bands of horizontal wave-number, one band deriving from wave-packets undergoing internal reflexion near that height and the other from wave-packets of very small local frequency that accumulate there. The ‘reflexion’ wave-numbers are dominant if the wind increases with height and the ‘accumulation’ wave-numbers if the wind initially decreases with height. The spectral energy distributions and intensities of the wave-motion are discussed in more detail for an atmosphere of uniform stability and unidirectional wind-shear. The accumulation process may lead to instability or overturning of the waves, and estimates are made of the probable scale and intensity of the ‘clear-air’ turbulence produced. An interesting point is that the rate of energy loss from the boundary layer by radiation of internal waves turns out to be comparable with the rate of production in the outer nine-tenths of the layer, both for atmospheric boundary layers and for the surface layer of the ocean. It seems likely that radiation limits the layer thickness to some extent.

Description: The development of a turbulent boundary layer in a strong adverse pressure gradient can be described by the two-layer model proposed by Stratford (1959), in which the outer part of the flow is assumed to be unmodified by the pressure-rise and the inner part described by two local parameters, the surface stress and the pressure gradient. The description suggests that the modification of the original flow is in some sense self-preserving, and it is shown here that self-preserving development of the modification is consistent with the Reynolds equations of turbulent flow in particular pressure distributions. For these distributions, the predictions of the two-layer model are confirmed without any need to make the sharp and arbitrary distinction between the two parts of the boundary layer. © 1965, Cambridge University Press. All rights reserved.

Description: If a thick, turbulent boundary layer is disturbed near the rigid boundary, the flow changes are confined initially to a thin layer adjacent to the boundary. Elliott (1958) and Panofsky & Townsend (1964) have attempted to calculate the flow disturbance caused by an abrupt change in surface roughness by assuming special velocity distributions which are consistent with a logarithmic velocity variation near the boundary. Inspection of their distributions shows that the deviations from the upstream distribution are self-preserving in form, and it is shown that self-preserving development is dynamically possible if log / being depth of modified flow roughness length) is fairly large and if is small compared with the total thickness of the layer. Other kinds of surface disturbance may lead to self-preserving changes of the original flow and the theory is developed also for flow downstream of a line roughness, for the temperature distribution downstream of a boundary separating an upstream region of uniform roughness and heat-flux from a region of different or possibly varying roughness and heat-flux, and for the return of a complete boundary layer to self-preserving development after a disturbance. The requirement that the distributions of velocity and temperature should conform to the logarithmic, equilibrium forms near the surface makes the predictions of surface stress and surface flux nearly independent of the exact nature of the turbulent transfer process, and the profiles of velocity and temperature are determined within narrow limits by the surface fluxes. To provide explicit profiles, the mixing-length transfer relation is used. Its validity for the self-preserving flows is discussed in an appendix. © 1965, Cambridge University Press. All rights reserved.

Description: Fluid impacts on the base of a stably stratified region of fluid cause internal-wave ripples whose spread is predominantly horizontal if the duration of the impacts is long compared with the natural period of the stratified fluid. The development of a single ripple in a slightly viscous fluid is calculated, first with a constant vertical gradient of potential density and then with a gradient varying linearly with height. The single-ripple results are used to find the intensity of the statistically steady wave motion generated by impacts which are randomly distributed in space and time. Above a critical height, dependent on the viscosity and stability of the fluid and on the time and length scales of the impacts, wave energy falls off as the −5/3 power of the height with a constant density gradient and as the −25/6 power with a linearly varying gradient. The predictions are compared with observations of temperature fluctuations in the stable region of an ice-water convection system and with observations of ‘clear-air turbulence’ over strato-cumulus cloud. Reasonable numerical agreement can be obtained with plausible values for the scales of the convective motion which provides the impacts. © 1966, Cambridge University Press. All rights reserved.

Description: In a barotropic fluid, a free turbulent flow induces a fluctuating potential flow which is determined by the instantaneous flow near the edge of the turbulent flow. If the surrounding fluid is stably stratified, internal wave-motions are possible and, in general, wave-energy accumulates until it is sufficient to modify the turbulent flow. Here the growth of wave-motion from rest is examined with particular reference to the atmospheric problem of wave excitation by the surface boundary layer. Wind shear is supposed negligible outside the turbulent flow and the disturbances from the boundary layer are assumed to travel with a convection velocity V relative to the upper air. For times large compared with {−g/ρ(dρ/dz)}−½ (ρ is the potential density), most of the wave-energy resides in components of phase-velocity near the convection velocity. For a model atmosphere with increased stability above a tropopause, this resonance mechanism leads to the formation of wave-groups with crests at right-angles to the convection velocity and wavelengths near 2πV[−g/ρ(dρ/dz)]−½. Using likely values for the surface disturbances, wave-amplitudes of order 100 m can develop within several hours of the initiation of the boundary layer but sufficient amplitude to produce overturning or breaking is unlikely within a reasonable time. © 1965, Cambridge University Press. All rights reserved.