Evolution of energy spectrum of an eddy system

Abstract

The turbulent flow of an ideal fluid is represented by a discrete model, in which the turbulence is concentrated at individual points and the cloud of these point eddies determines the turbulent flow. In the given mathematical model of turbulence (in contrast to other models of finite dimensions), the accurate hydrodynamic equation is not replaced by a finite system of coupled ordinary differential equations, and singular initial data are used. Advantages of such an approach are that the interactions between the point eddies are obvious, and their dynamics can be described by comparatively simple equations. The present work is a numerical investigation of the interactions between 100 identical point eddies which are initially distributed uniformly over a circle. This model is used to trace the tendency to directional energy transfer from small scale to large scale in two-dimensional turbulence. This phenomenon, which has been shown to be possible in several theoretical works and numerical investigations [1–3], may be attributed to the static irreversibility of turbulence [4]. A similar effect can be observed directly in the earth's atmosphere, in that the general circulation of the atmosphere is fed by cyclonic energy; this is referred to as “negative viscosity” [5]. A spectral analysis is carried out for such a system of point eddies, using a new procedure based on the formula obtained in [6]. In the evolution of the spectrum, sections close to the “−5/3 law” and the “−3 law” are obtained. Secondary instability of the eddy system is noted. At the end of the numerical experiment, a quasisteady state is established in the system.