Transition to turbulence in a statically stressed fluid system

Abstract

A mathematical picture of the transition to turbulence in statically stressed fluid systems is proposed. Systems are classified on the basis of the Hopf bifurcation theorem. One category, "inverted bifurcation," contains flows that exhibit hysteresis, finite-amplitude instabilities, and an immediate transition to turbulent behavior. A mathematical model due to Lorenz which manifests this kind of behavior is discussed. The other category, "normal bifurcation," includes flows in which, as the stress increases, a time-periodic regime precedes turbulence. A model of low-Prandtl-number convection falls into this category. The transition to nonperiodic behavior in this model is studied and found to proceed in accord with abstract mathematical proposals of Ruelle and Takens. Semiquantitative agreement with the experimental fluctuation spectrum of Ahlers is also obtained.