Strong Coupling in Laminar Bubbly Flows and Related Modeling Issues
Authors: Cartellier Alain 
Published: March 2009

Systems involving swarms of bubbles in an otherwise laminar continuous phase are common in industrial processes. In some cases, the gas is injected to ensure a given chemical reaction (bubble columns in oil industry) or to sustain a biochemical process (aeration tanks in waste water treatment plants). Gas inclusions can also appear due to the reaction itself (electrolysis cells, anaerobic digestion). In others circumstances, the gas phase is chemically passive and it is introduced mainly to favor mixing and/or separation (flotation devices). In these processes, it is desirable to access parameters such as the pressure drop, the mean void fraction, the bubble size distributions. In addition, their optimum functioning often depends on the transverse distribution of phasic quantities. Even if break-up/coalescence mechanisms are discarded, it happens that refined descriptions of such laminar dispersed flows has not yet reached a truly predictive status. On one hand, the Reynolds stresses reduce to the so-called bubble-induced agitation (or pseudo-turbulence) so that the interactions between inclusions and shear-induced turbulence need not to be accounted for. Yet, another complexity emerges because of strong and non-trivial couplings between phases. In particular, bubble-bubble interactions have a crucial effect on the induced agitation and consequently on the phase distribution. How to properly account for these interactions in an average description is still a matter of controversy. This presentation will highlight the importance of coupling mechanisms arising in laminar bubbly flows. Available experiments will be presented that illustrate the variety of phase organizations observed in stable Poiseuille bubbly flows [1–8]. It will be shown that some characteristics such as the mean void fraction and the wall shear stress are accessible through simplified models based on axial momentum balances [9,10]. On another hand, predictions of the phase distribution require solving transverse mechanical equilibria: the later are sensitive to many parameters, and in addition, they involve various coupling modes between phases. To overcome the corresponding modeling difficulties, a hybrid model has been developed in the spirit of approaches combining kinetic theory and classical continuum mechanics [see for example 11–13]. Compared with classical Eulerian two-fluid model, this framework provides, at least in the limit of dilute systems, a mean to derive closure laws [14–17]. These improvements will be illustrated for the interfacial momentum exchanges and the extra deformation tensor. In particular, the behavior of these coupling terms near walls will be shown to have important consequences on the phase distribution by the mediation of the continuous phase velocity profile. Concerning dispersion mechanisms, experimental information available on the bubble-induced agitation and on the dispersed phase microstructure in uniform flows will be summarized [18–20]. These observations will be connected with some characteristic features of the equations governing the perturbed liquid velocity field and the pair density distribution, and derived in the framework of the hybrid model. For finite particulate Reynolds numbers, estimates of the agitation tensors will be shown to be feasible using numerical simulations of two-body interactions [17]. Finally, the relevance of local closures for the induced-agitation for predicting phase distributions in confined systems will be debated, and the corresponding modeling issues will be underlined.