Abstract
Theoretical investigation of the effects of a translation of bubbles and a drag force acting on bubbles on the wave propagation in bubbly flows has long been lacking. In this study, we theoretically and numerically investigate the weakly nonlinear (i.e., finite but small amplitude) propagation of plane progressive pressure waves in compressible water flows that contain uniformly distributed spherical gas bubbles with translation and drag forces. First, we assume that the gas and liquid phases flow at independent velocities. Then, the drag force and virtual mass force are introduced in an interfacial transport across the bubble–liquid interface in the momentum conservation equations. Furthermore, we consider the translation and spherically symmetric oscillations as bubble dynamics and deploy a two-fluid model to introduce the translation and drag forces. Bubbles do not coalesce, break up, extinct, or appear. For simplicity, the gas viscosity, thermal conductivities of the gas and liquid, and phase change and mass transport across the bubble–liquid interface are ignored. The following results are then obtained: (i) Using the method of multiple scales, two types of Korteweg–de Vries–Burgers equations with a correction term due to the drag force are derived. (ii) The translation of bubbles enhances the nonlinear effect of waves, and the drag force acting on bubbles contributes the nonlinear and dissipation effects of waves. (iii) The results of long-period numerical analysis verify that the temporal evolution of the wave (not flow) dissipation due to the drag force differs from that caused by the acoustic radiation.
Highlights
Many forces act on gas bubbles, such as drag, lift, gravity, and virtual mass force
The following results are obtained: (i) Using the method of multiple scales, two types of Korteweg–de Vries–Burgers equations with a correction term due to the drag force are derived. (ii) The translation of bubbles enhances the nonlinear effect of waves, and the drag force acting on bubbles contributes the nonlinear and dissipation effects of waves. (iii) The results of long-period numerical analysis verify that the temporal evolution of the wave dissipation due to the drag force differs from that caused by the acoustic radiation
As in nonlinear acoustics or nonlinear wave theory for pure water flows, the pressure wave evolves into a shock wave owing to the competition between the nonlinear effect and dissipation effect of the waves
Summary
Many forces act on gas bubbles, such as drag, lift, gravity, and virtual mass force. The drag force acting on translational bubbles, in particular, is one of the most important factors in terms of the dynamics of single- or multi-bubble flows. It is important to explore the relative ratios of the nonlinear, dissipation, and dispersion effects, because the pressure wave in bubbly flows may evolve into a shock wave or a soliton,[9] which have quite different properties. Based on the two-fluid model equations,[17] our group theoretically investigated the linear[17,18] and weakly nonlinear[19,20] waves and derived the KdVB equation[19] under the assumption that the translation and drag forces are negligible. Our group numerically solved the original KdVB equation[19] and studied the temporal evolution of waveforms taking into account the nonlinear, dissipation, and dispersion effect; the translation and drag forces were not considered.[22].
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