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Linear and nonlinear instability in vertical counter-current laminar
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We consider the genesis and dynamics of interfacialinstability
in vertical gas-liquidflows, using
as a model the two-dimensional channel flow of a thin falling film sheared by
counter-current gas. The methodology is linear stability theory (Orr-Sommerfeld
analysis) together with direct numerical simulation of the two-phase flow in the case of
disturbances. We investigate the influence of two main flow parameters on the
interfacialdynamics, namely the film thickness and pressure drop applied to
drive the gas stream. To make contact with existing studies in the literature, the
effect of various density contrasts is also examined. Energy budget analyses based on
the Orr-Sommerfeld theory reveal various coexisting unstable modes (interfacial, shear,
internal) in the case of high density contrasts, which results in mode coalescence
and mode competition, but only one dynamically relevant unstable interfacial mode for low
density contrast. A study of absolute and convective instability for low
density contrast shows that the system is absolutely unstable for all but two narrow
regions of the investigated parameter space. Direct numerical simulations of the same
system (low density contrast) show that linear theory holds up remarkably well upon
the onset of large-amplitude waves as well as the existence of weakly nonlinear waves. For high
density contrasts, corresponding more closely to an air-water-type system, linear
stability theory is also successful at determining the most-dominant features in the
interfacial wave dynamics at early-to-intermediate times.
Nevertheless, the short waves selected by the linear theory undergo secondary
instability and the wave train is no longer regular but rather
exhibits chaotic motion. The same linear stability theory predicts when the direction
of travel of the waves changes — from downwards to upwards. We outline the practical
implications of this change in terms of loading and flooding. The change in direction
of the wave propagation is represented graphically in terms of a flow map based on the
rates and the prediction carries over to the nonlinear regime with
only a small deviation.
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