Hydrodynamic Instabilities and the Transition to Turbulence

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Abstract
For many flows the first one or two instabilities reached as the Reynolds number is increased are well understood both theoretically and experimentally. However, the successive instabilities leading ultimately to turbulent flow are not well understood for any single system, and it is not known to what extent there exist universal characteristics of the transition to turbulence for different flows. The theoretical and experimental approaches that have been used in the past to study the first one or two instabilities cannot usually be extended in practice to study higher instabilities. However, in recent years computer data acquisition and analysis techniques have been developed which make it possible to distinguish experimentally between different dynamical regimes of a time-dependent flow, and these results can be compared with newly developed concepts in the theory of nonlinear nonequilibrium systems. In this paper we describe a laser Doppler velocimetry experiment conducted at City College on the transition to turbulence in a fluid contained between concentric cylinders with the inner cylinder rotating. Results will also be presented from a related study of Rayleigh-Benard convection by Gollub and Benson at Haverford College. In these experiments digital records of the velocity are Fourier-transformed to obtain velocity power spectra. In both experiments are velocity spectra indicate that beyond the time-independent flow regimes there are three successive distinct dynamical regimes: periodic flow, quasiperiodic flow with two frequencies and chaotic flow. The chaotic regime initially has both broad and narrow peaks, but as the Reynolds or Rayleigh number is increased further the narrow peaks disappear, leaving only a broad peak and a background continuum. Studies of nonlinear models with a few degrees of freedom show a behavior siilar in most respects to that observed in these experiments.

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