Wake instability of a fixed spheroidal bubble

Abstract

Direct numerical simulations of the flow past a fixed oblate spheroidal bubble are carried out to determine the range of parameters within which the flow may be unstable, and to gain some insight into the instability mechanism. The bubble aspect ratio χ (i.e. the ratio of the major axis length over the minor axis length) is varied from 2.0 to 2.5 while the Reynolds number (based on the upstream velocity and equivalent bubble diameter) is varied in the range 102 ≤ Re ≤ 3 × 103. As vorticity generation at the bubble surface is at the root of the instability, theoretical estimates for the maximum of the surface vorticity and the surface vorticity flux are first derived. It is shown that, for large aspect ratios and high Reynolds numbers, the former evolves as χ8/3 while the latter is proportional to χ7/2Re−1/2. Then it is found numerically that the flow first becomes unstable for χ = χc ≈ 2.21. As the surface vorticity becomes independent of Re for large enough Reynolds number, the flow is unstable only within a finite range of Re, this range being an increasing function of χ − χc. An empirical criterion based on the maximum of the vorticity generated at the body surface is built to determine whether the flow is stable or not. It is shown that this criterion also predicts the correct threshold for the wake instability past a rigid sphere, suggesting that the nature of the body surface does not really matter in the instability mechanism. Also the first two bifurcations of the flow are similar in nature to those found in flows past rigid axisymmetric bluff bodies, such as a sphere or a disk. Wake dynamics become more complex at higher Reynolds number, until the Re−1/2-dependency of the surface vorticity flux makes the flow recover its steadiness and eventually its axisymmetry. A qualitative analysis of the azimuthal vorticity field in the base flow at the rear of the bubble is finally carried out to make some progress in the understanding of the primary instability. It is suggested that the instability originates in a thin region of the flow where the vorticity gradients have to turn almost at right angle to satisfy two different constraints, one at the bubble surface, the other within the standing eddy.