Low-frequency dispersion in bubbly liquids

Low-frequency dispersion in bubbly liquids
Preston S.Wilson
Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas 78712-1063
pswilson@mail.utexas.edu
Abstract: The low-frequency sound speed in bubbly liquids is often modeled
with an approximate equation [Wood, ATextbook of Sound, 1st ed. (Mac-
Millan, New York, 1930)] that depends primarily on the void fraction. More
recent models, developed for use at the individual bubble resonance frequency,
predict a small amount of dispersion in Wood’s ‘‘dispersionless’’ regime.
We present low-frequency data and compare it to a broadband model by
Commander and Prosperetti [J. Acoust. Soc. Am. 85(2), 732–746 (1989)].
The purpose is to illustrate that dispersion of the order 5% is expected and
occurs inWood’s regime, and to point out a discrepancy between the Ruggles
and the Commander and Prosperetti models regarding void fraction.
© 2005 Acoustical Society of America Polytechnic
PACS numbers: 43.30.Es, 43.35.Bf
Date Received: January 12, 2005 Date Accepted: April 29, 2005
1. Introduction
The low-frequency sound speed in bubbly liquids is often approximated as dispersionless using
an equation attributed toWood1 that depends only on the volume fraction of the gas phase, and
the density and compressibility of the gas and the host liquid. In this approximation, the
excitation frequency is constrained to be much lower than the resonance frequency of any
bubble in the mixture, and the bubble size must be large enough that the Laplace pressure due to
surface tension is negligible compared to the ambient pressure in the liquid. Wood’s
approximation is widely accepted in the literature,2–5 yet recent and more detailed analyses exist
that predict a small amount of dispersion well below bubble resonance frequency.
These newer models6,7 were developed in part to describe propagation and attenuation
in bubbly liquids at the individual bubble resonance frequency (IBRF), and hence account for
energy losses in a detailed way. Accordingly, most of the experimental work has been devoted to
the IBRF regime. In this work we present low-frequency data obtained by Ruggles8 that have not
appeared in the acoustics literature and compare them to a model by Commander and
Prosperetti.6 The primary purpose is to illustrate that dispersion of the order of 5% is expected
and observed in Wood’s regime, and that dispersionless behavior does not occur until the
excitation frequency falls below typical acoustic frequencies. For the size ranges discussed here
(bubble radii near 1 mm), this dispersion is due to the frequency-dependent variation of the
polytropic exponent that describes the compression of the gas inside the bubbles, and is not due
to surface tension effects.
The secondary purpose is to point out a discrepancy between the model of Ruggles
et al.7,8 and the model described in Ref. 6. The former arose from the two-phase flow
community, and considers the gas and liquid as separate fluid phases that can exchange
momentum. The latter considers the bubbly liquid as an effective homogeneous medium.
In Sec. II, Wood’s equation is reviewed. In Sec. III, the broadband model of
Commander and Prosperetti (C&P) is reviewed, and it is shown that low-frequency dispersion is
predicted and is a function of bubble size. In Sec. IV, Ruggles’ model and experiment is
described in brief. Measured phase speed data are presented and compared to the C&P model,
and the discrepancy is discussed.
Preston S. Wilson: Acoustics Research Letters Online [DOI: 10.1121/1.1903024] Published Online 24 June 2005
188 ARLO 6(3), July 2005 1529-7853/05/6(3)/188/7/$22.50 © 2005 Acoustical Society of America 188