## Abstract

Organized structures in turbulent shear flow have been observed both in the laboratory and in the atmosphere and ocean. Recent work on modelling such structures in a temporally developing, horizontally homogeneous turbulent free shear layer (Liu & Merkine 1976b) has been extended to the spatially developing mixing layer, there being no available rational transformation between the two nonlinear problems. We consider the kinetic energy development of the mean flow, large-scale structure and fine-grained turbulence with a conditional average, supplementing the usual time average, to separate the non-random from the random part of the fluctuations. The integrated form of the energy equations and the accompanying shape assumptions are used to derive 'amplitude' equations for the mean flow, characterized by the shear layer thickness, the non-random and the random components of flow (which are characterized by their respective energy densities). The closure problem was overcome by the shape assumptions which entered into the interaction integrals: the instability-wavelike large-scale structure was taken to be two-dimensional and the local vertical distribution function was obtained by solving the Rayleigh equation for various local frequencies; the vertical shape of the mean stresses of the fine-grained turbulence was estimated by making use of experimental results; the vertical shapes of the wave-induced stresses were calculated locally from their corresponding equations. The general energy-transfer mechanisms among the mean flow, large-scale structure and fine-grained turbulence are as follows: the large-scale structure gains energy from the mean flow and pumps energy into the fine-grained turbulence. The fine-grained turbulence gains energy from both the mean flow and the large-scale structure and converts energy to heat by viscous dissipation. The viscous dissipation rate of the mean flow and the large-scale motion is negligible compared to that of the fine-grained turbulence. The growth of the mixing layer is due to the energy transfer from the mean flow to both the large-scale structure and the fine-grained turbulence. Repeated pronounced 'bursts' of certain modes of the large-scale structure close to the initially neutral modes were obtained in the previous work (Liu & Merkine 1976b); these are primarily attributable to the fact that the local fine-grained turbulence was taken to be in equilibrium with the mean flow, rather than to the temporal problem itself. In addition to making an extension to the spatial problem from the temporal problem of Liu & Merkine (1976b), the present work also removes the previous simplifying restriction that the mean turbulence was in equilibrium with mean flow, so that here the fine-grained turbulence production from the mean flow is not necessarily in instantaneous balance with its viscous dissipation rate. Consequently the repeated pronounced 'bursts' in the spatial direction were not obtained for modes of the large-scale structure close to the initially neutral modes. It is found that the maximal amplitude of the large-scale structure is attained by the initially most amplified mode. As the large-scale structure amplifies and decays, the fine-grained turbulence is enhanced by the efficient intermediary action of the large-scale structure in extracting energy from the mean flow and subsequently giving it to the fine-grained turbulence. The relative enhancement of the fine-grained turbulence is achieved by both the magnitude of the large-scale structure and its streamwise lifetime. Thus a greater enhancement of the turbulence is achievable by the lower-frequency modes which have longer lifetimes. It is shown that the large-scale structure can be controlled by increasing the initial level of turbulence which renders its decay more rapid. Comparisons of the equilibrium spreading rates for the most amplified mode with observations show reasonably good agreement.