Path instability of deformable bubbles rising in Newtonian liquids: A linear study

The first stages of the path instability phenomenon known to affect the buoyancy-driven motion of gas bubbles rising in weakly or moderately viscous liquids are examined thanks to a recently developed numerical tool designed to assess the global linear stability of incompressible flows involving freely-evolving interfaces. Predictions for the critical bubble size and frequency of the most unstable mode are found to agree well with reference data obtained in ultrapure water and in several silicone oils. By varying the bubble size, stability diagrams are built for several specific fluids, revealing three distinct regimes characterized by different bifurcation sequences. The spatial structure of the corresponding unstable modes is analysed, together with the variations of the bubble shape, position and orientation. For this purpose, displacements of the bubble surface are split into rigid-body components and volume-preserving deformations, allowing us to determine how the relative magnitude of the latter varies with the fluid properties and bubble size. Predictions obtained with freely-deformable bubbles are compared with those found when the bubble shape determined in the base state is constrained to remain frozen during the stability analysis. This comparison reveals that deformations leave the phenomenology of the first bifurcations unchanged in low-viscosity fluids, especially water, only lowering the critical bubble size and increasing the frequency of path oscillations. In contrast, they introduce a dramatic change in the nature of the primary bifurcation in oils slightly more viscous than water, whereas, somewhat surprisingly, they leave the near-threshold phenomenology unchanged in more viscous oils.Comment: 35 pages, 16 figure

Dynamics of a gas bubble in a straining flow: Deformation, oscillations, self-propulsion

We revisit from a dynamical point of view the classical problem of the deformation of a gas bubble suspended in an axisymmetric uniaxial straining flow. Thanks to a recently developed Linearized Arbitrary Lagrangian-Eulerian approach, we compute the steady equilibrium states and associated bubble shapes. Considering perturbations that respect the symmetries of the imposed carrying flow, we show that the bifurcation diagram is made of a stable and an unstable branch of steady states separated by a saddle-node bifurcation, the location of which is tracked throughout the parameter space. We characterize the most relevant global mode along each branch, namely, an oscillatory mode that becomes neutrally stable in the inviscid limit along the stable branch, and an unstable nonoscillating mode eventually leading to the breakup of the bubble along the unstable branch. Next, considering perturbations that break the symmetries of the carrying flow, we identify two additional unstable nonoscillating modes associated with the possible drift of the bubble centroid away from the stagnation point of the undisturbed flow. One of them corresponds merely to a translation of the bubble along the elongational direction of the flow. The other is counterintuitive, as it corresponds to a drift of the bubble in the symmetry plane of the undisturbed flow, where this flow is compressional. We confirm the existence and characteristics of this mode by computing analytically the corresponding leading-order disturbance in the inviscid limit, and show that the observed dynamics are made possible by a specific self-propulsion mechanism that we explain qualitatively

Studying Sound Production in the Hole-Tone Configuration Using Compressible and Incompressible Global Stability Analyses

We study the jet passing through two successive circular holes, also known as hole-tone configuration. Such flow is relevant to many applications like human whistling, wind instruments and tea kettles. Recently, Fabre et al. [1] investigated this flow configuration adopting a global stability approach, showing that the whistling is linked to a purely incompressible instability of the jet between the two holes. In this work, we focus our attention on a little different and more realistic geometry, known as birdcall configuration, consisting into two successive holes in curved thick plates. Although the whistle is related to compressible phenomena, the incompressible approach can provide some useful information, at least in the region near the hole, where, in some conditions, the flow can be considered incompressible. We thus initially perform a purely incompressible stability approach. We identify the critical conditions, the global frequencies and discuss the structure of the resulting global eigenmodes. In order to reintroduce and evaluate compressible effects, which can be relevant into the cavity between the two holes, we model the cavity as a Helmholtz resonator and couple it to the incompressible simulation. Finally, a full compressible stability analysis is performed in order to check the accuracy of these simplified approaches in term of critical conditions, global frequencies and structure of the modes

Application of global stability approaches to whistling jets and wind instruments

We discuss the application of the so-called global approaches, arising from the field of hydrodynamical instabilities, to aeroacoustic resonators. We illustrate the potential of the approach for the case of a jet passing through two successive holes (”hole-tone” configuration) which is relevant to human whistling, birdcalls and tea kettles. First, treating the hydrodynamic system as locally incompressible and linearized around a base flow, we compute the conductance of the double aperture and show that this one can provide positive energy feedback to an external system. Secondly, introducing the coupling with an acoustical resonator through convenient impedances imposed as boundary conditions and solving an eigenvalue problem, we show that the full system is e ectively linearly unstable and able to support self-sustained oscillations. The results compare favorably with recent experiments, and the analysis yields novel insight into the nature of the feedback mechanism responsible for the whistling. The application to the related situation of a corrugated pipe, and to more realistic instruments such as ocarinas and free reeds, will also be discussed

A Practical Review on Linear and Nonlinear Global Approaches to Flow Instabilities

This paper aims at reviewing linear and nonlinear approaches to study the stability of fluid flows. We provide a concise but self-contained exposition of the main concepts and specific numerical methods designed for global stability studies, including the classical linear stability analysis, the adjoint-based sensitivity, and the most recent nonlinear developments. Regarding numerical implementation, a number of ideas making resolution particularly efficient are discussed, including mesh adaptation, simple shift-invert strategy instead of the classical Arnoldi algorithm, and a simplification of the recent nonlinear self-consistent (SC) approach proposed by Mantic-Lugo et al. (2014, Self- Consistent Mean Flow Description of the Nonlinear Saturation of the Vortex Shedding in the Cylinder Wake, Phys. Rev. Lett., 113(8), p. 084501). An open-source software implementing all the concepts discussed in this paper is provided. The software is demonstrated for the reference case of the two-dimensional (2D) flow around a circular cylinder, in both incompressible and compressible cases, but is easily customizable to a variety of other flow configurations or flow equations