J. Fluid Mech. (1998), vol. 377, pp. 313–345. Printed in the United Kingdom c  1998 Cambridge University Press 313 Direct numerical simulations of bubbly flows. Part 1. Low Reynolds number arrays By ASGHAR ESMAEELI AND GR ´ ETAR TRYGGVASON Department of Mechanical Engineering and Applied Mechanics, The University of Michigan, Ann Arbor, MI 48109, USA (Received 23 January 1996 and in revised form 30 July 1998) Direct numerical simulations of the motion of two- and three-dimensional buoyant bubbles in periodic domains are presented. The full Navier–Stokes equations are solved by a finite difference/front tracking method that allows a fully deformable interface between the bubbles and the ambient fluid and the inclusion of surface tension. The governing parameters are selected such that the average rise Reynolds number is O(1) and deformations of the bubbles are small. The rise velocity of a regular array of three-dimensional bubbles at different volume fractions agrees rela- tively well with the prediction of Sangani (1988) for Stokes flow. A regular array of two- and three-dimensional bubbles, however, is an unstable configuration and the breakup, and the subsequent bubble–bubble interactions take place by ‘drafting, kiss- ing, and tumbling’. A comparison between a finite Reynolds number two-dimensional simulation with sixteen bubbles and a Stokes flow simulation shows that the finite Reynolds number array breaks up much faster. It is found that a freely evolving array of two-dimensional bubbles rises faster than a regular array and simulations with different numbers of two-dimensional bubbles (1–49) show that the rise velocity increases slowly with the size of the system. Computations of four and eight three- dimensional bubbles per period also show a slight increase in the average rise velocity compared to a regular array. The difference between two- and three-dimensional bubbles is discussed. 1. Introduction Bubbly flows are central to many industrial processes. Heat transfer through boiling is the preferred mode in most power plants and bubble-driven circulation systems are used in metal processing operations such as steel making, ladle metallurgy, and the secondary refining of aluminium and copper. Similarly, many natural processes involve bubbles. Bubbles play a major role in the interactions of the oceans with the atmosphere, for example, and both air bubbles near a free surface and cavitation bubbles are of major importance for detection of submarines in naval applications. In this paper we present results from direct numerical simulations of a system of a few two- and three-dimensional bubbles (or light drops, since the density is finite) at a low but finite Reynolds numbers, O(1), in an initially quiescent homo- geneous flow, modelled by periodic domains. While the bubbles are deformable, the actual deformations are small due to the low Reynolds number. The study has two goals: to compare the evolution of freely evolving bubbles to fixed arrays, and to examine the utility of two-dimensional simulations for the understanding of a fully three-dimensional system. Several authors have used regular arrays to compute the