J. Fluid Mech. (2002), vol. 466, pp. 17–52. c  2002 Cambridge University Press DOI: 10.1017/S0022112002001179 Printed in the United Kingdom 17 Dynamics of homogeneous bubbly flows Part 1. Rise velocity and microstructure of the bubbles By BERNARD BUNNER 1 AND GR ´ ETAR TRYGGVASON 2 1 Coventor, Inc., Cambridge, MA 02138, USA 2 Mechanical Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609-2280, USA (Received 14 March 2000 and in revised form 5 March 2002) Direct numerical simulations of the motion of up to 216 three-dimensional buoy- ant bubbles in periodic domains are presented. The full Navier–Stokes equations are solved by a parallelized finite-difference/front-tracking method that allows a de- formable interface between the bubbles and the suspending fluid and the inclusion of surface tension. The governing parameters are selected such that the average rise Reynolds number is about 12–30, depending on the void fraction; deformations of the bubbles are small. Although the motion of the individual bubbles is unsteady, the simulations are carried out for a sufficient time that the average behaviour of the system is well defined. Simulations with different numbers of bubbles are used to explore the dependence of the statistical quantities on the size of the system. Exam- ination of the microstructure of the bubbles reveals that the bubbles are dispersed approximately homogeneously through the flow field and that pairs of bubbles tend to align horizontally. The dependence of the statistical properties of the flow on the void fraction is analysed. The dispersion of the bubbles and the fluctuation characteristics, or ‘pseudo-turbulence’, of the liquid phase are examined in Part 2. 1. Introduction Bubbly flows have been studied for a long time. Although the dynamics of a single bubble has attracted considerable attention and is now well understood, many practical applications require predictions of the behavior of a large number of bubbles. Examples include boiling flows, bubble columns for diverse chemical processes, air entrainment at the air/ocean interface, and many others. Engineering predictions of multiphase flows rely on conservation equations for the averaged properties of the mixture and closure laws to relate subgrid processes to the averaged behavior of the system. For turbulent flows, direct numerical simulations, where the unsteady Navier– Stokes equations are solved on grids fine enough to fully resolve all flow scales, have had a major impact on the current understanding of turbulence in single-phase flows. In two-phase flows, additional complexity arises from the presence of a second phase with significantly different physical properties. The need for direct numerical simulations in the study of multiphase flows has been apparent for some time. However, the challenge of simulating the unsteady motion of moving fluid interfaces has led investigators to use simplified models. For dispersed flows, where bubbles,