^{1}, J. Magnaudet

^{1,2}and D. Fabre

^{1}

### Abstract

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 two-dimensional and axisymmetric rigid bodies. The regions of the flow most sensitive to 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.

We warmly thank Flavio Giannetti for valuable comments and suggestions and Jérôme Mougel for his help with FreeFem++ and with the implementation of the shear-free boundary condition. This work was supported by the Agence Nationale de la Recherche under Grant No. ANR-09-BLAN-0132 OBLIC.

I. INTRODUCTION

II. PROBLEM CONFIGURATION AND NUMERICAL APPROACH

III. SOLUTION PROCEDURE

A. Base flow

B. Perturbations

IV. DISCUSSION OF RESULTS

A. Neutral curves

B. Structure of the unstable modes

V. RECEPTIVITY AND SENSITIVITY TO EXTERNAL DISTURBANCES

A. Adjoint modes

B. Receptivity to a localized feedback

C. Sensitivity to base flow modifications

D. Effects of a localized quadratic forcing

VI. SUMMARY

### Key Topics

- Flow instabilities
- 30.0
- Eddies
- 18.0
- Reynolds stress modeling
- 17.0
- Eigenvalues
- 13.0
- Boundary value problems
- 11.0

## Figures

Problem configuration. The minor axis of the bubble is parallel to the flow at infinity.

Problem configuration. The minor axis of the bubble is parallel to the flow at infinity.

Sketch of the computational domain .

Sketch of the computational domain .

Base flow around a bubble with χ = 2.5: (a) Re = 155; (b) Re = 2000. Top half: axial velocity and streamlines; bottom half: azimuthal vorticity.

Base flow around a bubble with χ = 2.5: (a) Re = 155; (b) Re = 2000. Top half: axial velocity and streamlines; bottom half: azimuthal vorticity.

Variation of (a) the growth rate and (b) the Strouhal number λ i /(2π) as a function of the Reynolds number for several bubble aspect ratios. Solid (resp. dashed) lines are associated with stationary (resp. oscillating) modes. In grayscale, the one-to-one correspondence between curves and aspect ratios may be established by noting that the larger the χ, the larger the maximum growth rate and Strouhal number of a given mode.

Variation of (a) the growth rate and (b) the Strouhal number λ i /(2π) as a function of the Reynolds number for several bubble aspect ratios. Solid (resp. dashed) lines are associated with stationary (resp. oscillating) modes. In grayscale, the one-to-one correspondence between curves and aspect ratios may be established by noting that the larger the χ, the larger the maximum growth rate and Strouhal number of a given mode.

Phase diagram (χ,Re) showing the neutral curves (in red/solid lines) corresponding to the onset of the stationary (squares) and oscillating (diamonds) modes. Open (resp. closed) circles correspond to DNS results from Ref. 9 in which the wake was observed to be stable (resp. unstable). The dashed line is also from Ref. 9 and was determined by linearly interpolating the growth rates of the neighboring data points. The open square and diamond (green online), respectively, correspond to the threshold of the stationary and oscillating modes determined in Ref. 10 for Re = 660.

Phase diagram (χ,Re) showing the neutral curves (in red/solid lines) corresponding to the onset of the stationary (squares) and oscillating (diamonds) modes. Open (resp. closed) circles correspond to DNS results from Ref. 9 in which the wake was observed to be stable (resp. unstable). The dashed line is also from Ref. 9 and was determined by linearly interpolating the growth rates of the neighboring data points. The open square and diamond (green online), respectively, correspond to the threshold of the stationary and oscillating modes determined in Ref. 10 for Re = 660.

(a) Axial velocity (upper half) and vorticity (lower half) of the stationary mode observed at Re = 155 for a bubble with χ = 2.5. (b) Energy distribution along the wake: the dotted, dashed, and dashed-dotted lines correspond to the radial, azimuthal, and axial contributions, respectively, while the solid line corresponds to the total energy.

(a) Axial velocity (upper half) and vorticity (lower half) of the stationary mode observed at Re = 155 for a bubble with χ = 2.5. (b) Energy distribution along the wake: the dotted, dashed, and dashed-dotted lines correspond to the radial, azimuthal, and axial contributions, respectively, while the solid line corresponds to the total energy.

(a) Axial velocity (upper half) and vorticity (lower half) of the oscillating mode observed at Re = 215 for a bubble with χ = 2.5. (b) Energy distribution along the wake (same convention as in Figure 6 ).

(a) Axial velocity (upper half) and vorticity (lower half) of the oscillating mode observed at Re = 215 for a bubble with χ = 2.5. (b) Energy distribution along the wake (same convention as in Figure 6 ).

Axial velocity (upper half) and vorticity (lower half) in the wake of a bubble with χ = 2.5. (a) mode at Re = 700; (b) mode at Re = 2000. The high values reached by the velocity and vorticity of these modes indicate that the lift force acting on the bubble (used to define their normalization) is quite small in this high-Re regime.

Axial velocity (upper half) and vorticity (lower half) in the wake of a bubble with χ = 2.5. (a) mode at Re = 700; (b) mode at Re = 2000. The high values reached by the velocity and vorticity of these modes indicate that the lift force acting on the bubble (used to define their normalization) is quite small in this high-Re regime.

(Top) Axial velocity (upper half) and vorticity (lower half) of the adjoint modes: (a) stationary mode computed at and (b) oscillating mode computed at . (Bottom) Energy distribution of the adjoint modes along the wake: (c) stationary mode and (d) oscillating mode (see Figure 6 for caption).

(Top) Axial velocity (upper half) and vorticity (lower half) of the adjoint modes: (a) stationary mode computed at and (b) oscillating mode computed at . (Bottom) Energy distribution of the adjoint modes along the wake: (c) stationary mode and (d) oscillating mode (see Figure 6 for caption).

The total flow around a bubble with χ = 2.5 at Re = 155: (a) iso-contours of the axial velocity and streamlines; (b) iso-contours of the azimuthal vorticity.

The total flow around a bubble with χ = 2.5 at Re = 155: (a) iso-contours of the axial velocity and streamlines; (b) iso-contours of the azimuthal vorticity.

Sensitivity to a localized feedback expressed through the quantity β(r, x). (a) (resp. (c)) stationary (resp. oscillating) mode for a bubble with χ = 2.5 and Re = 155 (resp. Re = 215); (b) stationary mode for a solid sphere at Re = 212.9; (d) stationary mode for a bubble with the critical aspect ratio χ = χ Csta = 2.21 at Re = 400. The solid line marks the separation line.

Sensitivity to a localized feedback expressed through the quantity β(r, x). (a) (resp. (c)) stationary (resp. oscillating) mode for a bubble with χ = 2.5 and Re = 155 (resp. Re = 215); (b) stationary mode for a solid sphere at Re = 212.9; (d) stationary mode for a bubble with the critical aspect ratio χ = χ Csta = 2.21 at Re = 400. The solid line marks the separation line.

Spatial distribution of the magnitude of the sensitivity function .

Spatial distribution of the magnitude of the sensitivity function .

Distribution of the magnitude of the sensitivity function : (a) sensitivity of for a bubble with χ = 2.5 at Re = 155; (b) same for a rigid sphere at Re = 212.9; (c) sensitivity of for a bubble with χ = 2.5 at Re = 215; (d) same for .

Distribution of the magnitude of the sensitivity function : (a) sensitivity of for a bubble with χ = 2.5 at Re = 155; (b) same for a rigid sphere at Re = 212.9; (c) sensitivity of for a bubble with χ = 2.5 at Re = 215; (d) same for .

Profiles of the sensitivity functions (blue/solid line) and (green/dashed line) along the surface for a bubble with χ = 2.5: (a) at Re = 155; (b) (resp. (c)) real (resp. imaginary) part for at Re = 215. The polar angle α goes from 0° at the rear of the bubble to 180° at its front; the arrow indicates the position of the separation angle.

Profiles of the sensitivity functions (blue/solid line) and (green/dashed line) along the surface for a bubble with χ = 2.5: (a) at Re = 155; (b) (resp. (c)) real (resp. imaginary) part for at Re = 215. The polar angle α goes from 0° at the rear of the bubble to 180° at its front; the arrow indicates the position of the separation angle.

Growth rate and frequency variations due to the forcing by a small body located on the axis of the base flow: (a) for a bubble with χ = 2.5 close to Re = 155; (b) same for a rigid sphere close to Re = 212.9; (c) ; and (d) for a bubble with χ = 2.5 close to Re = 215. Blue/thick dashed line: base flow; green/thin dashed-dotted line: disturbance flow; gray/solid line: total flow. The bubble (resp. sphere) stands in the interval x 0 = ±0.271 = (2χ2/3)−1 (resp. x 0 = ±0.5).

Growth rate and frequency variations due to the forcing by a small body located on the axis of the base flow: (a) for a bubble with χ = 2.5 close to Re = 155; (b) same for a rigid sphere close to Re = 212.9; (c) ; and (d) for a bubble with χ = 2.5 close to Re = 215. Blue/thick dashed line: base flow; green/thin dashed-dotted line: disturbance flow; gray/solid line: total flow. The bubble (resp. sphere) stands in the interval x 0 = ±0.271 = (2χ2/3)−1 (resp. x 0 = ±0.5).

Adjoint base flow around: (a) a bubble with χ = 2.5 at Re = 155; (b) a rigid sphere at Re = 212.9. (Top) Velocity field and streamlines; (bottom) azimuthal vorticity. Note that, along the upstream part of the symmetry axis, the velocities are from right to left in (a) and from left to right in (b).

Adjoint base flow around: (a) a bubble with χ = 2.5 at Re = 155; (b) a rigid sphere at Re = 212.9. (Top) Velocity field and streamlines; (bottom) azimuthal vorticity. Note that, along the upstream part of the symmetry axis, the velocities are from right to left in (a) and from left to right in (b).

## Tables

Drag coefficient C D for a steady axisymmetric flow past the bubble in the range 102 ⩽ Re ⩽ 103. Comparison between present study, results from DNS reported in Ref. 27 (only bubbles with χ ⩽ 1.95 were considered in that study), and predictions from an approximate correlation proposed in Ref. 28 . The drag coefficient is defined as C D = 8D/π with . Note that in Ref. 27 , distances were normalized by the length of the major axis, 2b, instead of the equivalent diameter, 2(b 2 a)1/3, so that the corresponding Reynolds number and drag coefficient had to be multiplied by χ−1/3 and χ2/3, respectively, to obtain the values reported in the table.

Drag coefficient C D for a steady axisymmetric flow past the bubble in the range 102 ⩽ Re ⩽ 103. Comparison between present study, results from DNS reported in Ref. 27 (only bubbles with χ ⩽ 1.95 were considered in that study), and predictions from an approximate correlation proposed in Ref. 28 . The drag coefficient is defined as C D = 8D/π with . Note that in Ref. 27 , distances were normalized by the length of the major axis, 2b, instead of the equivalent diameter, 2(b 2 a)1/3, so that the corresponding Reynolds number and drag coefficient had to be multiplied by χ−1/3 and χ2/3, respectively, to obtain the values reported in the table.

Influence of grid characteristics on some quantities of interest slightly above the first two thresholds for a bubble with χ = 2.5.

Influence of grid characteristics on some quantities of interest slightly above the first two thresholds for a bubble with χ = 2.5.

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