ALBERT

Description: Fluctuations of velocity and temperature which occur in a turbulent flow in a stably-stratified atmosphere far from restraining boundaries are discussed using the equations for the turbulent intensity and for the mean square temperature fluctuation. From these, an equation is derived for the flux Richardson number in terms of the ordinary Richardson number and some non-dimensional ratios connected with the turbulent motion. It is shown that the interaction between the temperature and velocity fields imposes on the flux Richardson number an upper limit of 0·5, and on the ordinary Richardson number a limit of about 0·08. If these values are exceeded, no equilibrium value of the turbulent intensity can exist and a collapse of the turbulent motion would occur. Although the analysis applies strictly only to a homogeneous non-developing flow, it should have approximate validity for effectively homogeneous, developing flows, and the predictions are compared with some recent observations of these flows.

Description: To determine experimentally the mean value of a randomly fluctuating quantity, it may be necessary to measure the average value over a considerable interval of time. This problem arose in a recent study of the temperature fluctuations over a heated horizontal plate, and a system was used that depended on the counting of electrical pulses generated at a rate proportional to the quantity being measured. The advantage of this technique is that mean values may be measured over time intervals of almost unlimited length with little added difficulty for the experimenter. Circuits are described which measure: (a) the mean square of a fluctuating quantity and of its time-derivative, (b) the statistical distribution of the fluctuations, (c) the mean frequency of the fluctuation assuming a particular value, and (d) the mean product of two fluctuating quantities. Over the range of use, the stability and linearity of the calibrations is better than 1%, more than sufficient for work on natural convection. In its present form, the equipment responds uniformly to all frequencies below 100 c/s, but it would not be difficult to extend this range of response to higher frequencies.

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: Measurements have been made of the Couette flow in the annular space between concentric cylinders with a radius ratio of 1.5, the outer cylinder being held stationary and the inner one rotated at speeds to give Taylor numbers in the range 1.0 × 104−2.3 × l06 times the critical value for first instability of the steady viscous flow. Mean velocities have been measured both with Pitot tubes and with linearized hot-wire anemometers, and turbulent intensities and stresses, frequency spectra and space-time correlations have been obtained using single hot-wire anemometers of X-form and linear arrays of eight single-wire anemometers. For Taylor-number ratios to the critical number less than 3 × l05, the most prominent feature of the flow is a system of toroidal eddies, encircling the inner cylinder and uniformly spaced in the axial direction with nearly the separation of the Taylor vortices of the viscous instability. They are superimposed on a background of irregular motion and, except within the thin wall layers, the toroidal eddies contribute more to the total intensity. With increase of rotation speed, the toroidal eddies lose their regularity, and they cannot be clearly distinguished at Taylor-number ratios beyond 5 × l05.The change of flow type from quasi-regular toroidal to fully irregular turbulent takes place over an extensive range of Taylor-number ratio centred near 3 × l05, and it may be linked with changes in the thin wall layers that separate the flow boundaries from the central region of nearly constant circulation. For ratios over 5 × l05, an appreciable part of the wall layers is comparatively unaffected by flow curvature and has a logarithmic distribution of mean velocity similar to that found in channel flows. It is suggested that the motion in the wall layers changes from a set of Gortler vortices characteristic of curved-wall flow to the more irregular motion found on plane walls, causing the toroidal eddies to break into sections of length ranging from a considerable fraction of the flow perimeter to nearly the separation of the cylinders. Changes in the frequency spectra of the radial and azimuthal velocit'y fluctuations are consistent with such a change.