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Unformatted text preview: On the nature of turbulence A. S. Monin Institute of Oceanology, USSR Academy of Sciences Usp. Fiz. Nauk 125, 97-122 (May 1978) The definition of turbulence and the differences between turbulence and random wave motions of liquids or gases are discussed. The Landau scheme of the generation of turbulence with increasing Reynolds number as a result of a sequence of normal bifurcations that creates a quasiperiodic motion is considered; several examples are discussed, including flow between rotating cylinders, convection at small Prandtl numbers, and the boundary layer at a flat plate. Results obtained in recent years from the ergodic theory and associated with the discovery of strange attractors in the phase spaces of typical dynamic systems are described. Flows with inverse bifurcations are considered, including the plane Poiseuille flow and Lorenz's example with idealized three-mode convection at large Prandtl numbers. In the latter case, the results of numerical calculations are analyzed and point to the existence of a strange attractor with the structure of a Cantor discontinuum; other examples of systems with strange attractors are also considered. It becomes clear that strange attractors in the phase spaces of systems with few modes may explain their nonperiodic behavior, but cannot explain why turbulence has a continuous spatial spectrum. PACS numbers: 47.25. - c CONTENTS 1. Introduction 429 2. The definition of turbulence 429 3. Differences between turbulence and waves 430 4. Normal bifurcations and Landau turbulence 431 5. Flows with normal bifurcations 432 6. The hypothesis of strange attractors. 436 7. Flows with inverse bifurcations 437 8. The Lorenz attractor and other examples. 439 9. Discussion 440 References 442 1. INTRODUCTION According to existing conceptions the c has tic, ran- dom appearance of turbulent liquid and gas flows is explained by the excitation of a very large number of degrees of freedom in these flows. As mechanical systems, such flows represent an aggregate of a very large number of vibrating and interacting oscillators. The point representing such a system in the corres- ponding phase space (whose number of dimensions is very large, but still finite in the case of flows in limited volumes) moves during the generation of tur- bulence along a path that makes an asymptotic approach to a certain limiting cycle that can be called a quasi- periodic attractor: here the time (t) functions that de- scribe the turbulent fluctuations are quasiperiodic, i.e., they have the ίοπη/ί,ω^,..., u>Jt), where η is very large, / has a period of 2π in each argument ui k t, and the frequencies ω Ιι with different subscripts k are generally not commensurable. This concept of deve- loped turbulence was proposed already in 1944 by Landau 1 (see also § 27 of the Landau and Lifshitz book 2 ). It was used in description of data on the generation of turbulence by Monin and Yaglom 3 and in § 2 of their book, 4 which appeared in 1965.which appeared in 1965....
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