The nature of turbulent motion at large wave-numbers

Abstract

This paper presents some measurements which describe the distribution of turbulent energy over Fourier components of large wave-number. According to Kolmogoroff’s theory these components of the motion are in statistical equilibrium. Over part of the wave-number range the form of the spectrum is predicted by a dimensional argument, but at higher wave-numbers the spectrum depends on the manner in which energy is transferred across the spectrum. Several postulates about this transfer effect have been made. Only that made by Heisenberg leads to a spectrum function which can be consistent with the measurements. Consistency with Heisenberg’s spectrum is obtained by postulating that there is an upper limit to the range of wave-numbers containing energy. This limit evidently corresponds to a critical Reynolds number below which no energy transfer occurs and the motion is 'laminar’. Measurements describing the probability distribution of ∂ u /∂ x , ∂ 2 u /∂ x 2 and ∂ 3 u /∂ x 3 are also described. These, and oscillograms of the velocity derivatives, show that the energy associated with large wave-numbers is very unevenly distributed in space. There appear to be isolated regions in which the large wave-numbers are ‘activated’, separated by regions of comparative quiescence. This spatial inhomogeneity becomes more marked with increase in the order of the velocity derivative, i.e. with increase in the wave-number. It is suggested that the spatial inhomogeneity is produced early in the history of the turbulence by an intrinsic instability, in the way that a vortex sheet quickly rolls up into a number of strong discrete vortices. Thereafter the inhomogeneity is maintained by the action of the energy transfer.