Mathematics – Complex Variables
Scientific paper
Jul 1975
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1975phrva..12..186m&link_type=abstract
Physical Review A - General Physics, 3rd Series, vol. 12, July 1975, p. 186-203.
Mathematics
Complex Variables
98
Flow Stability, Mathematical Models, Static Loads, Transition Flow, Turbulent Flow, Channel Flow, Complex Variables, Eigenvalues, Free Convection, Laminar Flow, Landau Damping, Periodic Variations, Pipe Flow, Prandtl Number, Transition Points
Scientific paper
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.
Martin C. P.
McLaughlin J. B.
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