A Chaotic Attractor in a Periodically Forced Two-Phase Flow System

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Abstract
Motivated by the enhancement of heat transfer under oscillating flow conditions in single-phase heated channels and by stability problems in two-phase systems such as those in boiling water reactors, density-wave oscillations have been analyzed by numerically solving the nonlinear, variable delay, functional, ordinary integrodifferential equations that result from integrating the nonlinear partial differential equations for the single- and two-phase heated channel regions along characteristics and along channel length for axially uniform heat fluxes. The cases of constant pressure drop ΔPex across the channel (steady-state feed pump operation), exponentially decaying ΔPex (feed pump coastdown), and periodic ΔPex (feed pump oscillations) were studied. In the constant ΔPex case, the system undergoes a supercritical Hopf bifurcation from a stable fixed point to a stable limit cycle as the parameters are moved into the linearly unstable region. In the exponentially decaying ΔPex case, depending on the initial and final pressures, the system travels along a hysteresis curve, jumps at the first turning point to another stable branch, and eventually evolves to a stable limit cycle. In the periodic ΔPex case when the system is in the linearly unstable region, it usually evolves asymptotically to one of several different attracting sets, depending on the frequency of ΔPex: stable period-N limit cycles, stable invariant tori, and a chaotic (or strange) attractor. The nature of the strange attractor was analyzed quantitatively by calculating its correlation dimension —an estimate of its fractal dimension—and the dimension of the phase space in which it can be embedded. These calculations indicate that the strange attractor is indeed a fractal object of fractional dimension 2.048 ± 0.003 and embedding dimension 6. The results of these numerical studies suggest that the heated channel model can operate safely in the linearly unstable region in a dynamically stable mode without excessively large excursions when driven at many frequencies; however, at many other frequencies it cannot. The trajectories that do remain in bounded regions of phase space can be, depending on the forcing frequency, periodic with a short or very long period, very near periodic, or completely aperiodic or chaotic. Hence, it is possible to enhance heat transfer while maintaining safety in two-phase flow systems by operating them in an oscillatory mode.