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

Abstract

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.

Paths and wakes of deformable nearly spheroidal rising bubbles close to the transition to path instability

“…In parallel, the development of LSA codes in which the bubble surface is allowed to deform is needed to understand the possible role of transient shape deformations, especially regarding the threshold Ga c (Bo) and the nature and characteristics of the first linearly unstable modes, through their coupling with the surrounding flow and the translational and rotational movements of the bubble. Once such codes will be available, it will become possible to develop weakly nonlinear models in the spirit of that of [22] to determine which path geometry is actually selected by the system close to the threshold.…”

“…The initial model used in the computational approach was that of an oblate spheroidal bubble with a prescribed aspect ratio (the length ratio of the major to minor axes, hereinafter denoted by χ ). The direct numerical simulation (DNS) and linear stability analysis (LSA) approaches were applied to this simplified geometry, with the bubble maintained fixed in a uniform stream [18][19][20] or allowed to freely translate and rotate under the action of the buoyancy force [21,22]. However, under real conditions, millimeter-size bubbles with a given volume do not exhibit a strict fore-aft symmetric shape: Compared to an oblate spheroid with the same aspect ratio, their front is somewhat flatter and their rear part is more rounded, owing to viscous effects [9,14].…”

The flow past a freely rotating sphere

“…Terms of order 2 and 3 are the solution of linear inhomogeneous problems arising from the expansion of (1) at the corresponding order. Details about the mathematical structure of these problems and the numerical procedure used to solve them are given in the Supplemental Material in [12] where the weakly nonlinear analysis has been performed for the more general case of an unsteady mode. It suffices here to say that at order 2 , the flow is modified by higher-order harmonics which obey the inhomogeneous linear system of equations…”

“…where r denotes the position vector relative to the body center of inertia and T = −PI + Re −1 (∇V + T ∇V) the stress tensor. Note that in the present case, the coupling only involves the torque M while the force F is not coupled to the motion of the sphere, unlike in the more general case considered in [12]. Finally, this set of equations is completed by the boundary condition V = U 0 x for r → ∞.…”

Passive control of a falling sphere by elliptic-shaped appendages

“…The direct numerical simulation of such complex bodies falling/rising freely is very challenging, and it is likely to be -at least initially -modeled by a more sophisticated model than the one presented in [5]. One possible direction is to extend the model of Lācis et al [5] from 2D to 3D, while another possibility is to investigate the weakly non-linear model by Tchoufag et al [27]. The latter model is able to predict oblique falling paths of disks and bubbles and could be extended to more complex bodies in order to exploit the IPL instability.…”