Study of Pressure Wave Propagation in a Two-Phase Bubbly Mixture

Abstract

(1) Background. Bubbly flows are used in a wide variety of applications and require accurate modeling. In this paper, three modeling approaches are investigated using the geometrically simple configuration of a gas bubble strongly oscillating in a bubbly medium. (2) Method of approach. A coupled Eulerian-Lagrangian, a multicomponent compressible, and an analytical approach are compared for different void fractions. (3) Results. While the homogeneous mixture models (analytical and multicomponent) compare well with each other, the Eulerian-Lagrangian model captures additional features and inhomogeneities. The discrete bubbles appear to introduce localized perturbations in the void fraction and the pressure distributions not captured by homogeneous mixture models. (4) Conclusions. The bubbly mixture impedes the growth and collapse of the primary bubble while wavy patterns in the velocity, pressure, and void fraction fields propagate in space and time.

Spherical bubble dynamics in a bubbly medium using an Euler–Lagrange model

“…During this collapse process the primary bubble recovers back most of its potential energy (but for viscous, and acoustic losses) and another cycle starts following bubble rebound (Figure 9f). All these observations are in line with classical bubble dynamics and with our previously published experiments of spark generated bubbles in bubbly media (Jayaprakash et al, 2011), and preliminary Euler-Lagrange coupled modeling work (Raju et al, 2011).…”

“…The grid in the radial direction is clustered near the bubble surface. This grid has been shown to be fine enough to provide gridconverged solutions for problems similar to that studied here (Raju et al, 2011). In the far field a discretized boundary is used at r= A R 0 and a zero-gradient condition is imposed.…”

“…To apply the Gilmore equations for a two phase medium, the dependencies of the density and the sound speed on the void fraction (Brennen, 1995) are incorporated in the solution of the equations (Raju et al, 2011).…”

“…While homogeneous models, which are useful for low void fraction and very small bubble oscillations conditions, ignore bubble dynamics and do not require Rayleigh-Plesset equation solutions, the other two approaches do require such modeling. Eulerian-Lagrangian approaches are more appropriate for higher void fractions (Spelt & Biesheuvel, 1997, Balachandar & Eaton, 2010, Raju et al, 2011, Shams et al, 2011. In a recent work by Raju et al (2011) comparing a continuum homogeneous model (Gilmore, 1952) , an Eulerian multicomponents model (Wardlaw & Luton, 2000, Wardlaw & Luton, 2003, and an Eulerian-Lagrangian model , Chahine, 2009, Hsiao et al, 2013b it was found that high-frequency local fluctuations were only captured when an Eulerian viscous solver was coupled with a Lagrangian discrete bubble dynamics and when the microscale behavior of the field bubbles was well resolved.…”

“…Initially, the void fraction was defined in each computational cell as the ratio of the total volume of bubbles in the cell by the cell volume. However, this was found not totally satisfactory because it often resulted in difficulty with differentiation due to the non-continuous character of α across cells (Wu et al, 2010, Raju et al, 2011 Here, to smooth the discontinuities and improve stability, we apply a Gaussian gas volume distribution scheme to smoothly "spread" a bubble volume across neighboring cells as in (Kitagawa et al, 2001, Darmana et al, 2006, Shams et al, 2011, Capecelatro & Desjardins, 2013. As illustrated in Figure 1, the void fraction computation scheme for a representative cell i is obtained using the following expression: where |X i,j | is the distance between bubble j and the center of cell i, and S is the standard deviation.…”