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

Tchoufag, Joël; Magnaudet, Jacques; Fabre, David

doi:10.1063/1.4804552

Cited by 23 publications

(35 citation statements)

References 46 publications

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“…be found in appendices A and B as well as in previous related studies (Meliga, Chomaz & Sipp 2009b;Assemat et al 2012;Tchoufag, Magnaudet & Fabre 2013). …”

Section: Strategy For the Lsa And Numerical Implementationsupporting

confidence: 48%

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

2014

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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 11158 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, Magnaudet & Fabre (J. Fluid Mech., vol. 719, 2013, pp. 388-405), 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.

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“…be found in appendices A and B as well as in previous related studies (Meliga, Chomaz & Sipp 2009b;Assemat et al 2012;Tchoufag, Magnaudet & Fabre 2013). …”

Section: Strategy For the Lsa And Numerical Implementationsupporting

confidence: 48%

“…We already discussed the sensitivity of the computational approach used to solve the generalized eigenvalue problem to grid characteristics in previous papers (Assemat et al 2012;Tchoufag et al 2013). Here we present some additional tests performed to select the grid employed in the present study.…”

Section: Appendix a Grid Sensitivitymentioning

confidence: 99%

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

2014

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“…The reader is referred to these two references, especially the first one, for all technical details. As shown in Tchoufag et al (2013), dealing with the shear-free condition in the framework of a variational formulation requires the normal stress, Σ = n · T · n, to be considered as an additional unknown. The formulation described in Tchoufag et al (2014) is modified accordingly and the vector state corresponding to the fluid unknowns is Q f = [V, P, Σ], where all variables depend on space and time.…”

Section: Parameterization and Formulation Of The Global Stability Promentioning

confidence: 99%

“…The corresponding LSA problem is solved thanks to the finite element solver FreeFem++ already used by Tchoufag et al (2014) for freely moving disks and by Tchoufag, Magnaudet & Fabre (2013) for fixed oblate bubbles. The reader is referred to these two references, especially the first one, for all technical details.…”

Section: Parameterization and Formulation Of The Global Stability Promentioning

confidence: 99%

Linear instability of the path of a freely rising spheroidal bubble

2014

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Path and wake instabilities of buoyancy-driven oblate spheroidal bubbles with a prescribed shape rising freely in a viscous fluid otherwise at rest are studied using global stability analysis, following the technique recently developed for a coupled fluid + body system by Tchoufag, Fabre & Magnaudet (J. Fluid Mech. vol. 740, 2014, pp. 278-311). The essential role of the wake on the path instability is evidenced by comparing the shape of the global stability diagram with that obtained in the case of a fixed bubble. However, dramatic differences are also found, since the critical curve of the coupled system mostly involves low-and high-frequency oscillating modes, whereas that of a fixed bubble only involves stationary modes. Comparison of the present predictions with results obtained through direct numerical simulation is achieved in several regimes, confirming the predictions of the linear approach but also highlighting some of its limitations when the system successively encounters several unstable modes.

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“…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].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2016

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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 16174 (2016)] in which the neutral curve associated with this transition was obtained by considering realistic but frozen bubble shapes. Depending on the dimensionless parameters that characterize the system, various paths geometries are observed by letting an initially spherical bubble starting from rest rise under the effect of buoyancy and adjust its shape to the surrounding flow. These include the well-documented rectilinear axisymmetric, planar zigzagging, and spiraling (or helical) regimes. A flattened spiraling regime that most often eventually turns into either a planar zigzagging or a helical regime is also frequently observed. Finally, a chaotic regime in which the bubble experiences small horizontal displacements (typically one order of magnitude smaller than in the other regimes) is found to take place in a region of the parameter space where no standing eddy exists at the back of the bubble. The discovery of this regime provides evidence that path instability does not always result from a wake instability as previously believed. In each regime, we examine the characteristics of the path, bubble shape, and vortical structure in the wake, as well as their couplings. In particular, we observe that, depending on the fluctuations of the rise velocity, two different vortex shedding modes exist in the zigzagging regime, confirming earlier findings with falling spheres. The simulations also reveal that significant bubble deformations may take place along zigzagging or spiraling paths and that, under certain circumstances, they dramatically alter the wake structure. The instability thresholds that can be inferred from the computations compare favorably with experimental data provided by various sets of recent experiments guaranteeing that the bubble surface is free of surfactants.

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

Cited by 23 publications

References 46 publications

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

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

Linear instability of the path of a freely rising spheroidal bubble

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

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