Some preliminary results on the behaviour of controlled wave disturbances introduced artificially into turbulent channel flow are reported. Weak plane-wave disturbances are introduced by vibrating ribbons near each wall. The amplitude and relative phase of the streamwise component of the induced wave is educed from a hot wire signal, allowing the wave speed and attenuation characteristics and the wave shape to be traced downstream. The normal component and wave Reynolds stress have been inferred from these data. It appears that Orr–Sommerfeld theories attempted to date are inadequate for description of these waves.

An experimental study of the interaction of a turbulent boundary layer with a wavy wall was conducted in a wind tunnel. A smooth neoprene rubber sheet comprising a portion of the floor of the tunnel was mechanically deformed into 12 sinusoidal waves which progressed upwind or down at controlled speed. The turbulent layer thickness was a little less than the wavelength. The mean velocity profile was linear on a semi-log plot over a substantial range of vertical height.

The wall pressure was observed to be asymmetrical about the wave profile, resulting in a pressure drag. Flow separation was not the cause of the drag. The drag was found to be larger than that predicted by the inviscid wave generation theory. The measurements indicate that the waves strongly modulate the turbulent structure. The phase of the turbulent stresses with respect to the waves varies with wave speed, indicating that the dynamical reaction time of the turbulence is not negligible in comparison with the wave period.

The outer intermittent region of a fully developed turbulent boundary layer with zero pressure gradient was extensively explored in the hope of shedding some light on the shape and motion of the interface separating the turbulent and non-turbulent regions as well as on the nature of the related large-scale eddies within the turbulent regime. Novel measuring techniques were devised, such as conditional sampling and conditional averaging, and others were turned to new uses, such as reorganizing in map form the space-time auto- and cross-correlation data involving both the U and V velocity components as well as I, the intermittency function. On the basis of the new experimental results, a conceptual model for the development of the interface and for the entrainment of new fluid is proposed.

The two-dimensional incompressible mixing layer was investigated by using constant-temperature, linearized hot wire anemometers. The measurements were divided into three categories: (1) the conventional average measurements; (2) time-average measurements in the turbulent and the non-turbulent zones; (3) ensemble average measurements conditioned to a specific location of the interface. The turbulent energy balance was constructed twice, once using the conventional results and again using the turbulent zone results. Some differences emerged between the two sets of results. It appears that the mixing region can be divided into two regions, one on the high velocity side which resembles the outer part of a wake and the other on the low velocity side which resembles a jet. The binding turbulent–non-turbulent interfaces seem to move independently of each other. There is a strong connexion between the instantaneous location of the interface and the axial velocity profile. Indeed the well known exponential mean velocity profile never actually exists at any given instant. In spite of the complexity of the flow the simple concepts of eddy viscosity and eddy diffusivity appear to be valid within the turbulent zone.

This paper surveys the current state of analytical attempts at a theory of turbulence. The formulation of the problem in terms of moments is discussed. The difficulty posed by the closure problem is examined in detail using the quasinormal approximation as an example. The notion of dynamical relaxation by non-linear scrambling leads to the introduction of eddy relaxation times and the direct-interaction approximation. The properties of the direct-interaction approximation are indicated. Finally, a comparison is made between numerical solution of the equations of turbulence their and direct numerical simulation of the Navier–Stokes equations.

Meecham and his co-workers have developed a theory of turbulence involving a truncated Wiener–Hermite expansion of the velocity field. The randomness is taken up by a white-noise function associated, in the original version of the theory, with the initial state of the flow. The mechanical problem then reduces to a set of coupled integro-differential equations for deterministic kernels. We have solved numerically an analogous set for Burgers's model equation and have computed, for the sake of comparison, actual random solutions of the Burgers equation. We find that the theory based on the first two terms of the Wiener–Hermite expansion predicts an insufficient rate of energy decay for Reynolds numbers larger than two, because the equations for the kernels contain no convolution integrals in wave-number space and therefore permit no cascade of energy. An energy cascade in wave-number space corresponds to a cascade up through successive terms of the Wiener-Hermite expansion. Pictures of the Gaussian and non-Gaussian components of an actual solution of the Burgers equation show directly that only higher-order terms in the Wiener–Hermite expansion are capable of representing shocks, which dissipate the energy. Higher-order terms would be needed even for a nearly Gaussian field of evolving three-dimensional turbulence. ‘Gaussianity’, in the experimentalist's sense, has no bearing on the rate of convergence of a Wiener–Hermite expansion whose white-noise function is associated with the initial state. Such an expansion would converge only if the velocity field and its initial state were joint-normally distributed. The question whether a time-varying white-noise function can speed the convergence is treated in the paper following this one.

Wiener proposed that the turbulent velocity be expanded in Hermite functionals of a Gaussian white noise random function advected by the fluid. This paper describes the mechanics of converting his suggestion into a computable model, and assesses its range of validity as an approximation for incompressible, homogeneous, and isotropic turbulence. The terms retained are a linear term, v1, representing an arbitrary Gaussian velocity, and a quadratic term, v2, representing a non-Gaussian contribution to the velocity needed for energy transfer. The requirement that advection by the dependent velocity v = v1 + v2 does not alter the statistics of the base necessitates a further truncation of the base to antisymmetric quadratic basis elements. Realizability of any statistics of v is common to all Wiener–Hermite expansions. The projected equations for the Lagrangian expansion conserve energy by non-linear interaction, preserve the inviscid Gaussian equipartition ensemble, and are invariant to random Galilean transformations. Numerical calculations with an approximate form of these equations reveal that irreversible relaxation to the inviscid equipartition solution is not a property of the Lagrangian model, and that the rapid convergence advanced as the original motivation for studying Wiener–Hermite expansions does not survive closure by truncation. The dynamics of the model is not inconsistent with the existence of an inertial range. A simple numerical search routine failed to produce a solution corresponding to such an equilibrium ensemble.

In an attempt to explain the failure of the various pure homogeneous strain experiments to reach equilibrium (and consequently to support the contention of Townsend of an equilibrium structure of the Reynolds stress dependent only on geometry), the nature of the general Reynolds stress-mean velocity relation is examined. It is shown that if homogeneous flows become asymptotically independent of initial conditions, and if the Reynolds stress bearing structure can be characterized by a single time scale (i.e.–at sufficiently high Reynolds number) then these flows behave like classical non-linear viscoelastic media, with the Reynolds stress structure dependent on the (strain-rate) (time scale) product. Thus, the existence of an equilibrium structure implies the existence of an equilibrium time scale and a universal value of the product. The ideas permitting Reynolds stress and mean velocity to be related are applied to the dissipative structure in homogeneous flows, and it is found that in such flow the time scale never ceases to grow, so that these flows can never reach an equilibrium structure. With the aid of an ad-hoc assumption these flows are examined in some detail, and the results of experiments are predicted with considerable accuracy. It is suggested that (inhomogeneous) flows having an equilibrium time scale may, in the homogeneous limit, be expected to display a universal structure. The small departure from universality induced by the large eddies associated with inhomogeneity may be adequately predicted by this same ad-hoc model.

The effect of turbulence on a magnetic field whose length-scale L is initially large compared with the scale l of the turbulence is considered. There are no external sources for the field, and in the absence of turbulence it decays by ohmic dissipation. It is assumed that the magnetic Reynolds number Rm = u0l/λ (where u0 is the root-mean-square velocity and λ the magnetic diffusivity) is small. It is shown that to lowest order in the small quantities l/L and Rm, isotropic turbulence has no effect on the large-scale field; but that turbulence that lacks reflexional symmetry is capable of amplifying Fourier components of the field on length scales of order Rm−2l and greater. In the case of turbulence whose statistical properties are invariant under rotation of the axes of reference, but not under reflexions in a point, it is shown that the magnetic energy density of a magnetic field which is initially a homogeneous random function of position with a particularly simple spectrum ultimately increases as t−½exp (α2t/2λ3) where α(= O(u02l)) is a certain linear functional of the spectrum tensor of the turbulence. An analogous result is obtained for an initially localized field.

The three-dimensional, primitive equations of motion have been integrated numerically in time for the case of turbulent, plane Poiseuille flow at very large Reynolds numbers. A total of 6720 uniform grid intervals were used, with sub-grid scale effects simulated with eddy coefficients proportional to the local velocity deformation. The agreement of calculated statistics against those measured by Laufer ranges from good to marginal. The eddy shapes are examined, and only the u-component, longitudinal eddies are found to be elongated in the downstream direction. However, the lateral v eddies have distinct downstream tilts. The turbulence energy balance is examined, including the separate effects of vertical diffusion of pressure and local kinetic energy.

It is concluded that the numerical approach to the problem of turbulence at large Reynolds numbers is already profitable, with increased accuracy to be expected with modest increase of numerical resolution.