Theoretical elucidation of effect of drag force and translation of bubble on weakly nonlinear pressure waves in bubbly flows

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.

Nonlinear and dissipation effects of pressure waves in water flows containing translational bubbles with a drag force

“…(i) The drag force acting on the bubbles contributed to the dissipation and increased the value of the dissipation coefficient, as our previous study [12]. (ii) The thermal conduction inside bubble contributed to the dissipation and increased the value of the dissipation coefficient.…”

“…In addition, both Π 4A and Π 4B increase with decreasing R * 0 . Furthermore, Π 4A strongly depends on both α 0 and R * 0 [12] while Π 4B is almost independent of α 0 .…”

“…We theoretically investigates weakly nonlinear (i.e., finite but small amplitude) propagation of onedimensional (plane) pressure progressive waves in a flowing compressible water uniformly containing many spherical oscillating bubbles as shown in figure 1. [12]. As in our previous papers [12,14], we newly introduce drag force and translation as the bubble dynamics.…”

“…[12]. As in our previous papers [12,14], we newly introduce drag force and translation as the bubble dynamics. The main assumptions are summarized as follows: (i) The motion of bubble is spherically symmetric; (ii) The bubbles do not coalesce, break up, extinct, and appear; (iii) At an initial state, the gas and liquid phases are flowing with a constant velocity; (iv) The temperature of liquid phase is constant; (v) For simplicity, lift, gravitation, direct interactions between bubbles, viscosity of gas phase, Reynolds stress, and phase change and mass transport across the bubble-liquid interface, are omitted.…”

“…Many theoretical, numerical, and experimental studies on drag force have been carried Figure 1. Pressure wave propagation in bubbly flows [12].…”