The role of big eddies in homogeneous turbulence

Abstract Recent experimental work on the decay of isotropic turbulence has shown that big eddies play an important part in the motion. There is a range of eddy sizes which, during the initial period of decay, contains a negligible proportion of the total energy and is excluded from the similarity possessed by the smaller eddies. This paper examines the motion associated with this small range of large wave-lengths in the more general case of homogeneous turbulence. For this purpose it is convenient to introduce a spectrum tensor, defined as the three-dimensional Fourier transform of the double-velocity correlation tensor. This spectrum function is also suitable for the application of similarity hypotheses, unlike the conventional one-dimensional spectrum function. The properties of the spectrum as a function of the wave-number vector k, are discussed with particular reference to small values of the magnitude k. When k is small the energy per unit interval of wave-number magnitude varies as k4. The rate of change of the spectrum function is obtained from the Navier-Stokes equations in terms of Fourier transforms of the triple-velocity and pressure-velocity mean values. After taking into account the continuity condition it is found that the terms of the first and second degree in the expansion of the spectrum function in powers of components of k are constant throughout the decay. The biggest eddies of the turbulence are therefore permanent, being determined wholly by the initial conditions, and are dominant in the final period when the smaller eddies have decayed. The action of smaller eddies on the invariant big eddies is equivalent to that of a turbulent viscosity, the value of which may vary with direction. The implications of the analysis for similarity hypotheses are discussed briefly.