Hydrodynamic Instabilities and Turbulence

Abstract

When a classical fluid is subjected to external stress it undergoes transitions to flow states which become more and more disordered, or turbulent, as the stress is increased. Two limiting cases are of particular interest, the region of weak stress or onset of chaotic motion, and the region of large stress or fully-developed turbulence. In the first case a classical theory due to Landau and Hopf describes the onset of disorder as the pile-up of a large number of instabilities, with modes of motion at mutually incommensurate frequencies. More recently it has been realized both experimentally and theoretically, that chaotic motion can also result from non-linear interactions among a small number of modes, after the appearance of only two or three instabilities. The most striking experimental demonstration of these effects occurs in the study of Rayleigh-Bénard convection and Couette-Taylor flow. In the region of large external stress (fully developed turbulence), a statistical description of short-scale velocity correlations is sought, with the hope of finding certain universal features. Various phenomenological theories have been proposed, beginning with the famous 1941 prediction of Kolmogorov, that the energy spectrum as a function of wavenumber will vary as k^−5/3 for large k. Experimental techniques for studying fluid turbulence will be briefly surveyed.