Joël Tchoufag

Linear stability and sensitivity of the flow past a fixed oblate spheroidal bubble

The stability properties of the wake past an oblate spheroidal bubble held fixed in a uniform stream are studied in the framework of a global linear analysis. In line with previous studies, provided the geometric aspect ratio of the bubble, χ, is large enough, the wake is found to be unstable only within a finite range of Reynolds number, Re. The neutral curves corresponding to the occurrence of the first two unstable modes are determined over a wide range of the (χ, Re) domain and the structure of the modes encountered along the two branches of each neutral curve is discussed. Then, using an adjoint-based approach, a series of sensitivity analyses of the flow past the bubble is carried out in the spirit of recent studies devoted to twodimensionaland axisymmetric rigid bodies. The regions of the flow most sensitiveto an external forcing are found to be concentrated in the core or at the periphery of the standing eddy, as already observed with bluff bodies at the surface of which the flow obeys a no-slip condition. However, since the shear-free condition allows the fluid to slip along the bubble surface, the rear half of this surface turns out to be also significantly sensitive to disturbances originating in the shear stress, a finding which may be related to the well-known influence of surfactants on the structure and stability properties of the flow past bubbles rising in water.

A global stability approach to wake and path instabilities of nearly oblate spheroidal rising bubbles

A global Linear Stability Analysis (LSA) of the three-dimensional flow past a nearly oblate spheroidal gas bubble rising in still liquid is carried out, considering the actual bubble shape and terminal velocity obtained for various sets of Galilei (Ga) and Bond (Bo) numbers in axisymmetric numerical simulations. Hence, this study extends the stability analysis approach of Tchoufag et al. [“Linear stability and sensitivity of the flow past a fixed oblate spheroidal bubble,” Phys. Fluids 25, 054108 (2013) and “Linear instability of the path of a freely rising spheroidal bubble,” J. Fluid Mech. 751, R4 (2014)] (which considered perfectly spheroidal bubbles with an arbitrary aspect ratio) to the case of bubbles with a realistic fore-aft asymmetric shape (i.e., a flatter front and a more rounded rear). The critical curve separating stable and unstable regimes for the straight vertical path is obtained both in the (Ga,Bo) and the (Re,χ) planes, where Re is the bubble Reynolds number and χ its aspect ratio (i.e., the major-to-minor axes length ratio). This provides new insight into the effect of the shape asymmetry on the wake instability of bubbles held fixed in a uniform stream and on the path instability of freely rising bubbles, respectively. For the range of Ga and Bo explored here, we find that the flow past a bubble with a realistic shape is generally more stable than that past a perfectly spheroidal bubble with the same aspect ratio. This study also provides the first critical curve for the onset of path instability that can be compared with experimental observations. The tendencies revealed by this critical curve agree well with those displayed by available data. The quantitative agreement is excellent for O(1) Bond numbers. However, owing to two simplifying assumptions used in the LSA scheme, namely, the steadiness of the base state and the uncoupling between the bubble shape and the flow disturbances, quantitative discrepancies (up to 20%–30%) with experimental threshold values of the Galilei number remain for both small and large Bond numbers.

Weakly Nonlinear Model with Exact Coefficients for the Fluttering and Spiraling Motion of Buoyancy-Driven Bodies

Gravity- or buoyancy-driven bodies moving in a slightly viscous fluid frequently follow fluttering or helical paths. Current models of such systems are largely empirical and fail to predict several of the key features of their evolution, especially close to the onset of path instability. Here, using a weakly nonlinear expansion of the full set of governing equations, we present a new generic reduced-order model based on a pair of amplitude equations with exact coefficients that drive the evolution of the first pair of unstable modes. We show that the predictions of this model for the style (e.g., fluttering or spiraling) and characteristics (e.g., frequency and maximum inclination angle) of path oscillations compare well with various recent data for both solid disks and air bubbles.