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

Bifurcations and symmetry breaking in the wake of axisymmetric bodies

We consider the generic problem of wake instabilities past fixed axisymmetric bodies, and focus on the extreme cases of a sphere and a flat disk. Numerical results reveal that the wakes of these two bodies evolve differently as the Reynolds number is increased. Especially, two new vortex shedding modes are identified behind a disk. To interpret these results, we introduce a model based on the theory of mode interactions in presence of O(2) symmetry. This model, which was initially developed for the Taylor–Couette system, allows us to explain the structural differences observed in the evolution of the two types of wakes and to accurately predict the evolution of the lift force

The onset of unsteadiness of two-dimensional bodies falling or rising freely in a viscous fluid: a linear study

We consider the transition between the steady vertical path and the oscillatory path of two-dimensional bodies moving under the effect of buoyancy in a viscous fluid. Linearization of the Navier–Stokes equations governing the flow past the body and of Newton’s equations governing the body dynamics leads to an eigenvalue problem, which is solved numerically. Three different body geometries are then examined in detail, namely a quasi-infinitely thin plate, a plate of rectangular cross-section with an aspect ratio of 8, and a rod with a square cross-section. Two kinds of eigenmodes are observed in the limit of large body-to-fluid mass ratios, namely ‘fluid’ modes identical to those found in the wake of a fixed body, which are responsible for the onset of vortex shedding, and four additional ‘aerodynamic’ modes associated with much longer time scales, which are also predicted using a quasi-static model introduced in a companion paper. The stability thresholds are computed and the nature of the corresponding eigenmodes is investigated throughout the whole possible range of mass ratios. For thin bodies such as a flat plate, the Reynolds number characterizing the threshold of the first instability and the associated Strouhal number are observed to be comparable with those of the corresponding fixed body. Other modes are found to become unstable at larger Reynolds numbers, and complicated branch crossings leading to mode switching are observed. On the other hand, for bluff bodies such as a square rod, two unstable modes are detected in the range of Reynolds number corresponding to wake destabilization. For large enough mass ratios, the leading mode is similar to the vortex shedding mode past a fixed body, while for smaller mass ratios it is of a different nature, with a Strouhal number about half that of the vortex shedding mode and a stronger coupling with the body dynamics

Falling styles of disks

We numerically investigate the dynamics of thin disks falling under gravity in a viscous fluid medium at rest at infinity. Varying independently the density and thickness of the disk reveals the influence of the disk aspect ratio which, contrary to previous belief, is found to be highly significant as it may completely change the route to non-vertical paths as well as the boundaries between the various path regimes. The transition from the straight vertical path to the planar fluttering regime is found to exhibit complex dynamics: a bistable behaviour of the system is detected within some parameter range and several intermediate regimes are observed in which, although the wake is unstable, the path barely deviates from vertical. By varying independently the body-to-fluid inertia ratio and the relative magnitude of inertial and viscous effects over a significant range, we set up a comprehensive map of the corresponding styles of path followed by an infinitely thin disk. We observe the four types of planar regimes already reported in experiments but also identify two additional fully three-dimensional regimes in which the body experiences a slow horizontal precession superimposed onto zigzagging or tumbling motions

The steady oblique path of buoyancy-driven disks and spheres

We consider the steady motion of disks of various thicknesses in a weakly viscous flow, in the case where the angle of incidence $\alpha$ (defined as that between the disk axis and its velocity) is small. We derive the structure of the steady flow past the body and the associated hydrodynamic force and torque through a weakly nonlinear expansion of the flow with respect to $\alpha$. When buoyancy drives the body motion, we obtain a solution corresponding to an oblique path with a non-zero incidence by requiring the torque to vanish and the hydrodynamic and net buoyancy forces to balance each other. This oblique solution is shown to arise through a bifurcation at a critical Reynolds number ReSO which does not depend upon the body-to-fluid density ratio and is distinct from the critical Reynolds number ReSS corresponding to the steady bifurcation of the flow past the body held fixed with $\alpha=0$. We then apply the same approach to the related problem of a sphere that weakly rotates about an axis perpendicular to its path and show that an oblique path sets in at a critical Reynolds number ReSO slightly lower than ReSS , in agreement with available numerical studies

A quasi-static approach to the stability of the path of heavy bodies falling within a viscous fluid

We consider the gravity-driven motion of a heavy two-dimensional rigid body freely falling in a viscous fluid.We introduce a quasi-static linear model of the forces and torques induced by the possible changes in the body velocity,or by the occurrence of a nonzero incidence angle or a spanwise rotation of the body. The coefficients involved in this model are accurately computed over a full range of Reynolds number by numerically resolving the Navier 13Stokes equations, considering three elementary situations where the motion of the body is prescribed. The falling body is found to exhibit three distinct eigenmodes which are always damped in the case of a thin plate with uniform mass loading or a circular cylinder,but may be amplified for other geometries,such as in the case of a square cylinder

Linear stability of disks falling or rising freely in a viscous fluid

The problem of the instability of a solid body moving in a viscous fluid has been extensively studied in the case where the body is fixed and its wake undergoes instabilities when some control parameter is increased (e:g: 1;2 for two-dimensional configurations and 3;4 for three-dimensional bodies). On the other hand, the deeply different and frequent case of a body freely falling or rising under buoyancy in a fluid otherwise at rest hasreceived much less attention. Yet, results from experiments5;6 and direct numerical simulations7;8 have evidenced a large variety of motion styles ranging from steady oblique to chaotic, though zig-zag and tumbling. To gain more insight into the nature of the instability that drives the departure from a straight vertical path, we recently carried out a linear study of the coupled fluid-body problem in two dimensions9. Here we extend this work to axisymmetric bodies. The linear stability analysis is performed for a disk of finite thickness with an aspect ratio (defined as the ratio of the diameter over thickness) X = 3 and another disk with X = 10. We select these two values because past studies10 led us to consider the corresponding two bodies as prototypes of general thick and thin bodies. This talk mainly focuses on the parametric modal stability analysis of the coupled fluid-disk problem. We present neutral stability curves in the phase space (pb=pf ,Re) and (pb=pf ; St), where pb=pf is the body-to-fluid density ratio, St the non-dimensional frequency of the unstable mode and Re the Reynolds number based on the disk diameter and the relative velocity between the fluid and body. These curves reveal how rich the dynamics of the problem are, including features such as destabilization-restabilization regions and abrupt jumps of the marginal frequency. The types and thresholds of the linear instability show close agreement with existing DNS and experimental results, as well as with theoretical predictions from asymptotic theories11;12. Finally, we consider the structure of the global modes. This allows us to extract information such as the origin of the instability (i.e. we disentangle \body-related modes" from \ fluid-related modes") and the characteristic footprints of both low-frequency and high-frequency modes

Weakly nonlinear analysis of the flutter motion of thin cylinders

The nature overflows with examples which prove that buoyancy-driven objects in a viscous medium can result in diverse and exotic trajectories. Among them is found the so-called Zig-Zag (ZZ)path or flutter which we investigate in the present work. The configuration is that of a thin cylinder initially rising/falling vertically in an unbounded fluid otherwise at rest. The problem is parametrized by the aspect ratio (diameter to thickness) X, the moment of inertia I* and Archimedes number (gravitational-velocity-based Reynolds) Ar. For small I*, past studies have reported asupercritical transition from the vertical to the ZZ path at a critical Arc. We show by means of linear and weakly nonlinear analyses that the observed flutter results from the nonlinear saturation of an unstable global mode of the coupled fluid+disk problem

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

Global linear stability analysis of the wake and path of buoyancy-driven disks and thin cylinders

The stability of the vertical path of a gravity- or buoyancy-driven disk of arbitrary thickness falling or rising in a viscous fluid, recently studied through direct numerical simulation by Auguste et al. (2013), is investigated numerically in the framework of global linear stability. The disk is allowed to translate and rotate arbitrarily and the stability analysis is carried out on the fully coupled system obtained by linearizing the Navier-Stokes equations for the fluid and Newton’s equations for the body. Three disks with different diameter-to-thickness ratios are considered: one is assumed to be infinitely thin, the other two are selected as archetypes of thin and thick cylindrical bodies, respectively. The analysis spans the whole range of body-to-fluid inertia ratios and considers Reynolds numbers (based on the fall/rise velocity and body diameter) up to 350. It reveals that four unstable modes with an azimuthal wavenumber m = ±1 exist in each case. Three of these modes result from a Hopf bifurcation while the fourth is associated with a stationary bifurcation. Varying the body-to-fluid inertia ratio yields rich and complex stability diagrams with several branch crossings resulting in frequency jumps; destabilization/restabilization sequences are also found to take place in some subdomains. The spatial structure of the unstable modes is also examined. Analyzing differences between their real and imaginary parts (which virtually correspond to two different instants of time in the dynamics of a given mode) allows us to assess qualitatively the strength of the mutual coupling between the body and fluid. Qualitative and quantitative differences between present predictions and known results for wake instability past a fixed disk enlighten the fact that the first non-vertical regimes generally result from an intrinsic coupling between the body and fluid and not merely from the instability of the sole wake