Shock Propagation in Polydisperse Bubbly Flows

The effect of distributed bubble nuclei sizes on shock propagation in dilute bubbly liquids is computed using a continuum two-phase model. An ensemble-averaging technique is employed to derive the statistically averaged equations and a finitevolume method is used to solve the model equations. The bubble dynamics are incorporated using a Rayleigh-Plesset-type equation which includes the effects of heat transfer, liquid viscosity and compressibility. The numerical model is verified by computing linear wave propagation and comparing to the acoustic theory of dilute bubbly liquids. It is known that for the case of monodisperse mixtures, relaxation oscillations occur behind the shock due to the bubble dynamics. The present computations show that bubble size distributions lead to additional damping of the average shock dynamics. If the distribution is sufficiently broad, the effect of polydispersity dominates over the singlebubble-dynamic damping and the shock profile is smoothed out. The size distribution effect on bubble screen problems is also discussed. INTRODUCTION A fundamental understanding of the dynamics of bubbly flows is of great importance in underwater explosions [1, 2], a spallation neutron source [3,4], turbomachinery [5–7] and shock wave lithotripsy [8–10]. In such flows, cavitation frequently occurs due to tension in the liquid phase. Shock dynamics provide a canonical example where cavitation and bubble dynamics have ∗Address all correspondence to this author. a large effect on the shock structure and propagation speed. Most of the previous studies [11–19] have focused on shock propagation in monodisperse bubbly liquids (i.e. all the bubbles initially have the same size.). However, in flows of practical interest, the nuclei size is broadly distributed; thus, the size distributions need to be included for more realistic modeling. We treat the liquid and disperse phases as a continuum medium and solve statistically averaged equations to determine the average shock structure. First, we present the continuum and bubble-dynamic models with their assumptions and discuss the model limitations. Next, we formulate and verify the numerical scheme developed to solve the system. Then, we solve one-dimensional wave propagation in dilute bubbly liquids with bubble size distributions (i.e. polydisperse mixtures) and describe the effects of the size distributions on linear and nonlinear wave propagation. We also conduct parameter studies of probable bubble sizes, void fractions and shock strengths to investigate the effects of these parameters on shock structure. Finally, we examine shock propagation through polydisperse bubble screens. MODEL EQUATIONS Continuum model We use an ensemble-averaging technique [20, 21] to derive the averaged mixture equations. The mixture model assumes that (a) the bubbles are spherical; (b) mutual interactions among the bubbles are negligible except through their effect on the mixtureaveraged flow; (c) wavelengths in the mixture are large compared 1 Copyright © 2009 by ASME to the mean bubble spacing; (d) the bubbles advect with the ambient liquid velocity (no slip); and (e) density and velocity fluctuations in the liquid phase are uncorrelated. Assumption (a) implies that fission and coalescence of the bubbles are not permitted, so that the bubble number is conserved in time. Assumptions (b) to (d) are generally valid in the dilute limit, which is used for the model closure. Relative motion between the phases has been shown to have minimal impact on linear wave propagation [22] and also plays a minor role in shock propagation [18]. Assumption (e) is reasonable due to the fact that the velocity fluctuations caused by the bubble dynamics concentrate in the vicinity of the bubbles, where the liquid is effectively incompressible [23]. Under these assumptions, we may write the one-dimensional conservation equations (mass, momentum and bubble number conservation) as ∂ρ ∂t + ∂ρu ∂x = 0, (1) ∂ρu ∂t + ∂ ∂x ( ρu2 + pl )