This and the subsequent part of the Journal of Fluid Mechanics will be devoted to the publication of papers on turbulence. Each of these papers was the subject of an oral report at a Symposium on Turbulence sponsored by the Boeing Company and organized by the Flight Sciences Laboratory of the Boeing Scientific Research Laboratories, a t Seattle, Washington, U.S.A., from 23 to 27 June 1969. The Organizing Committee consisted of Dr A. Goldburg, Prof. L. S. G. Kovasznay (Chairman), andDr Y.-H. Pao (Secretary). At this Symposium invited participants listened both to review lectures and to contributed papers, and tried by discussion to assess the present position in the study of turbulence.

Apart from the well known pictorial studies of eddies in a stream by Leonardo da Vinci, it seems to me that other engineers were the first to make any progress in the study of turbulence. Except for Osborne Reynolds (1883) they limited their work to empirical formulae giving the results of hydraulic experiments in which they measured mean flow distributions resulting from turbulence. Osborne Reynolds seems to have been the first to show that with given boundaries the flow can be steady up to a certain speed and then suddenly become turbulent. This was a great advance and contributed to the urge by Orr (1907), Rayleigh (1915) and many others to study mathematically the stability of steady flow reacute;gimes. Meanwhile the meteorologists were attracted to the field. Arthur Schuster, a wealthy man who was professor of physics at Manchester University, wanted to encourage a mathematician to study meteorology and try to introduce mathematical reasoning into what seemed almost pure empiricism. He set aside provision for a readership in dynamical meteorology. I was appointed in 1911.

Although the entrainment of non-turbulent fluid into a turbulent flow occurs across sharply defined boundaries, its rate is not determined solely by the turbulent motion adjacent to the interface but depends on overall properties of the flow, in particular, on those that control the energy balance. In the first place, attention is directed to the many observations which show that the motion in many turbulent shear flows has a structure closely resembling that produced by a rapid, finite, plane shearing of initially isotropic turbulence. The basic reasons for the similarity are the stability and permanence of turbulent eddies and the finite distortions undergone by fluid parcels in free turbulent flows. Next, the existence of eddy similarity and the condition of overall balance of energy are used to account for the variation of entrainment rates within groups of broadly similar flows, in particular mixing layers between streams of different velocities and wall jets on curved surfaces. For some flows which satisfy the ordinary conditions for self-preserving development, no entrainment rate is consistent with the energy balance and self-preserving development is not possible. Examples are the axisymmetric, small-deficit wake and the distorted wake. Finally, the implications of an entrainment rate controlled by the general motion are discussed. It is concluded that the relatively rapid entrainment in a plane wake depends on an active instability of the interface, not present in a constant-pressure boundary layer whose slow rate of entrainment is from ‘passive’ distortion of the bounding surface by eddies of the main turbulent motion. Available observations tend to support this conclusion.

Measurements of the turbulent pressure field on the outer surface of a 3 in. diameter cylinder aligned with the flow were made at a point approximately 24 ft. downstream of the origin of the turbulent boundary layer in an air stream of 145 ft./sec. The boundary-layer thickness was 2·78 in. and the Reynolds number based on momentum thickness was 2·62 × 104. The wall-pressure measurements were made with pressure transducers constructed from 0·06 in. diameter lead–zirconate–titanate disks mounted flush with the wall. The measurements including root-mean-square, power spectrum, and correlations of the wall pressure are compared with the existing experimental results for the turbulent pressure field beneath a plane boundary layer. The streamwise convection speed deduced from longitudinal space-time correlation measurements was almost identical to that obtained in the plane boundary layer. The rate of decay of the maxima of the space-time correlation of the pressure produced by the convected eddies was double that in a plane boundary layer. The longitudinal and transverse scales of the pressure correlation were approximately equal (in a plane boundary layer the transverse scale is larger than longitudinal scale) and were one-half or less than the longitudinal scale in the plane boundary layer. It is concluded that the effect of the transverse curvature of the wall is an overall reduction in size of pressure-producing eddies. The reduction in transverse scale of the larger eddies is greater than that of the smaller eddies. In general, the smaller eddies decay more rapidly and produce greater spectral densities at high frequencies owing to the unchanged convection speed.

With a transverse array of channels of equal widths but differing resistances, we have generated an improved approximation to spatially homogeneous turbulent shear flow. The scales continue to grow with downstream distance, even in a region where the mean velocity gradient and one-point turbulence moments (component energies and shear stress) have attained essentially constant values. This implies asymptotic non-stationarity in the basic Eulerian frame convected with the mean flow, behaviour which seems to be inherent to homogeneous turbulent shear flow.

Two-point velocity correlations with space separation and with space-time separation yield characteristic departures from isotropy, including clear ‘upstream–downstream’ unsymmetries which cannot be classified simply as axis tilting of ellipse-like iso-correlation contours.

The high wave-number structure is roughly locally isotropic although the turbulence Reynolds number based on Taylor ‘microscale’ and r.m.s. turbulent velocity is only 130. Departures from isotropy in the turbulent velocity gradient moments are measurable.

The approximation to homogeneity permits direct estimation of all components of the turbulent pressure/velocity-gradient tensor, which accounts for inter-component energy transfer and helps to regulate the turbulent shear stress. It is found that its principal axes are aligned with those of the Reynolds stress tensor. Finally, the Rotta (1951, 1962) linear hypothesis for intercomponent energy transfer rate is roughly confirmed.

Derivatives of velocity signals obtained in a turbulent boundary layer are examined for correspondence to the lognormal distribution. It is found that there is rough agreement but that unlikely events at high values are much less common in the observed fields than would be inferred from the lognormal distribution. The actual distributions correspond more to those obtained from a random walk with a limited number of steps, so the difference between these distributions and the lognormal may be related to the fact that the Reynolds number is finite.

The third-order structure function is examined, and found to be roughly consistent with the existence of an inertial subrange of a Kolmogoroff equilibrium reacute;gime over a range of scale which is a priori reasonable but which is far less extensive than the $-\frac{5}{3}$ region of the spectrum.

Derivatives of velocity and temperature in the wind over the ocean were found to be quite variable. Probability distribution functions of squared derivatives were consistent with lognormality predictions by Kolmogoroff, Obukhoff and Yaglom. Kurtosis values for velocity derivatives ranged from 13 to 26 and from 26 to 43 for temperature derivatives. Universal inertial subrange constants were evaluated from dissipation spectra and were found to be 40 to 300% larger than most values reported previously. Evidence for local anisotropy of the temperature field is provided by non-zero values of the measured derivative skewness.

Three-point odd-order correlations and four-point even-order correlations of the longitudinal velocity fluctuations in grid-generated turbulence have been measured using linearized hot-wire anemometry, digital sampling, and a high-speed digital computer. The measured correlations are compared with relations between higher-order correlations corresponding to non-Gaussian Gram-Charlier joint probability densities for three and four variables. The fourth-order, three-point Gram-Charlier distribution accurately describes the relation between measured odd-order three-point correlations. The measured fourth-order even-order correlations may be accurately predicted from the two-point correlation using Millionshtchikov's joint-Gaussian hypothesis, except for small values of the separations. The disagreement at small separations cannot be reduced through use of the Gram-Charlier approximation.

We discuss some consequences of assuming that two different non-linear model equations, and real turbulence are nearly Gaussian. It is supposed when necessary that the process is driven and it is supposed that the processes have become statistically stationary. These problems are discussed from the viewpoint of the Wiener–Hermite expansion for non-linear, nearly Gaussian processes. Expected equilibria forms are related to corresponding expressions obtained from the zero-fourth-cumulant assumption. The spectrum for Burgers’ model and for incompressible fluid flow problems is found from this viewpoint to be E ∼ k−2. The kinematical properties leading to such spectra are discussed. It is noted, as has been remarked earlier, that this spectrum is characteristic of flows with near discontinuities. A conjecture is offered concerning how these discontinuities are related to Gaussianity.

A method is described for constructing approximations, to statistical functions, that are uniformly convergent in time, starting with the expansion of the functions as Taylor series in time. The principal tool is a technique for expanding the Fourier transform of the unknown function by use of a set of orthonormal functions. Application to the Lagrangian velocity correlation and the eddy diffusivity for marked particles in a three-dimensional random velocity field yields results that agree excellently with computer simulations. The approximation procedure is extended to expansions in strength parameters (e.g. Reynolds number expansion) and to an expansion about the direct-interaction approximation. The latter is based on a new model representation of the direct-interaction approximation. An implication of the work is that the usual diagram expansions, obtained through term-by-term averaging over a Gaussian distribution, may not uniquely determine the functions they represent; it may be that truly meaningful expansions are possible, in general, only for distributions which bound the amplitudes in the individual realizations.

Bounds on the transport of momentum in turbulent shear flow are derived by variational methods. In particular, variational problems for the turbulent regimes of plane Couette flow, channel flow, and pipe flow are considered. The Euler equations resemble the basic Navier–Stokes equations of motion in many respects and may serve as model equations for turbulence. Moreover, the comparison of the upper bound with the experimental values of turbulent momentum transport shows a rather close similarity. The same fact holds with respect to other properties when the observed turbulent flow is compared with the structure of the extremalizing solution of the variational problem. It is suggested that the instability of the sublayer adjacent to the walls is responsible for the tendency of the physically realized turbulent flow to approach the properties of the extremalizing vector field.