On the linearized dynamics of two-dimensional bubbly flows over wave-shaped surfaces

Abstract

In the last decades the technological importance or bubbly flows has generated considerable efforts to achieve a better understanding of their properties, [1],[2]. However, the presence or two interacting phases so much increases the complexity or the problem that a satisfactory mathematical model of these flows has been possible only in special cases under fairly restrictive simplifying assumptions. The main purpose of the present note is to investigate the effects due to the inclusion or bubble dynamic response in two-dimensional flows over wave-shaped surfaces. The earlier studies of bubbly flows based on space averaged equations for the mixture in the absence or relative motion between the two phases, [5], [6], do not consider bubble dynamic effects. This approach simply leads to an equivalent compressible homogeneous medium and has been used to analyze the behaviour or one-dimensional bubbly flows through converging-diverging nozzles. In order to account for bubble dynamic response, in a classical paper by Foldy, [7], each individual bubble is described as a randomly distributed point scatterer. Assuming that the system is ergodic, the collective effect of bubble dynamic response on the flow is then obtained by taking the ensemble average over all possible configurations. An alternative way to account for bubble dynamic effects would be to include the Rayleigh-Plesset equation in the space averaged equations. Both methods have been successfully applied to describe the propagation or one-dimensional perturbances through liquids containing small gas bubbles, [8], [9], [10], [11]. However, because of their complexity, there are not many reported examples of the application to specific flow geometries of the space averaged equations which include the effects of bubble response, [12]. In an earlier note, [13], we considered the one-dimensional time dependent linearized dynamics or a spherical cloud of bubbles. The results clearly show that the motion of the cloud is critically controlled by bubble dynamic effects. Specifically, the dominating phenomenon consists of the combined response of the bubbles to the pressure in the surrounding liquid, which results in volume changes leading to a global accelerating velocity field. Associated with this velocity field is a pressure gradient which in turn determines the pressure encountered by each individual bubble in the mixture. Furthermore, it can be shown that such global interactions usually dominate any pressure perturbations experienced by one bubble due to the growth or collapse or a neighbor (see section 5). In the present note the same approach is applied to the two-dimensional case or steady flows over wave-shaped surfaces (for which there exist well established solutions for compressible and incompressible flow), With the aim, as previously stated, of assessing the effects due to the introduction or bubble dynamic response. Despite its intrinsic limitations, the following linear analysis indicates some of the fundamental phenomena involved in such flows and provides a useful basis for the study of the same flows with non-linear bubble dynamics, which we intend to discuss in a later publication. The present extention to the case of bubbly flows over arbitrarily shaped surfaces also constitutes the starting point for the investigation or such flows, a problem of considerable technical interest, for example in cavitating flows past lifting surfaces.

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