Two-way coupling simulations of instabilities in a plane bubble plume

Abstract

In the present study we aim at investigating the instabilities in a plane bubble plume by means of two-way coupling simulations. The continuous phase motion is obtained by direct numerical solution of the Navier–Stokes equations forced by the presence of the bubbles. The collective effects induced by the presence of the bubbles are modeled by a spatiotemporal distribution of momentum. Time evolution of the dispersed phase is solved by Lagrangian tracking of all the bubbles. In the present study, the motion of the carrying fluid is initiated and driven by the induced buoyancy of bubbles released from a source located in an initially quiescent fluid layer. A quantitative analysis of the flow transition is thus investigated for several plume widths and for various fluid viscosities over a range of Grashof numbers based on the injection conditions. An analogy is drawn with buoyant single-phase flows for the steady laminar region. Following the similarity formulation of Fujii [Int. J. Heat Mass Transfer 6, 597 (1963)] under boundary layer approximations for free thermal plumes, the velocity profiles can be collapsed to a single self-similar plot. Nevertheless, this analogy with single-phase flow shows some discrepancies in the description of the transition. Numerical data emphasize that the key parameter controlling the height of transition is the Grashof number, which is based on injection conditions of the dispersed phase. Our results concur with the recent experiments of Alam and Arakeri [J. Fluid Mech. 254, 363 (1993)]. Although the Grashof number also determines the transition in thermal plumes [Wakitani and Yosinobu, Fluid Dyn. Res. 2, 363 (1988)], the two-phase configuration is more unstable. These new results underline the important role played by the slip velocity of the bubbles in plume stability. Indeed, it tends to delay the plume transition when the slip velocity increases and approaches the buoyancy-induced velocity. This feature should also be related to the lack of diffusion by the bubble cloud in the Lagrangian transport of the density gradient.