Wake-Induced Oscillatory Paths of Bodies Freely Rising or Falling in Fluids

Supplementary Data

• A collection of six supplemental videos illustrating some aspects of the wake dynamics and path of bubbles, short cylinders, and thin disks in various flow regimes.

  Supplemental Video 1: A sequence of the first four unsteady bifurcated states in the wake of a short cylinder of aspect ratio χ = 3 held fixed at normal incidence in a uniform stream. From a DNS by Auguste et al. (2010). (a) Reflectional symmetry preserving (RSP) state for Re = 182. (b) Knit-knot state with two frequencies and no reflectional symmetry for Re = 187. (c) Reflectional symmetry breaking (or ying-yang) state for Re = 195. Note that the slow pulsation observed in the previous state is not present any more. (d) Standing wave (or zigzag) state for Re = 216. Note that the symmetry plane is orthogonal to that of the RSP state.

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  Supplemental Video 1d (Download video file)
  
  

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  Supplemental Video 2: (a) An approximately oblate spheroidal air bubble with χ ≈ 2.0 and Re ≈ 760 rising in water (the sphere of same volume would have a diameter of 2.5 mm). The path is close to a planar zigzag. From experiments by Riboux et al. (2010). (b) Path and wake (illustrated with streamwise vorticity isosurfaces) of an oblate spheroidal bubble with χ = 2.5 and Ar = 138. Note the two transitions, first from a straight vertical path to a planar zigzag and much later from this zigzag to a helical path, and the associated changes in the wake structure. From a DNS by Mougin & Magnaudet (2002b).

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  Supplemental Video 3: Two perpendicular views of the wake past a zigzagging short cylinder (χ = 2, Ar = 90, Re ≈ 250). (a) Dye visualizations (Fernandes et al. 2005). (b,c) Isosurface of the λ2 criterion (Jeong & Hussain 1995) extracted from a DNS by Auguste (2010).

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  Supplemental Video 4: Planar zigzag paths of short cylinders corresponding to Ar = 90, i.e., Re ≈ 250 (from Fernandes et al. 2005). The red line (N_x) indicates the horizontal projection of a unit vector parallel to the body symmetry axis. The black (V_z') and blue (V_x') lines display the evolution of the fluctuating velocity components along the vertical and horizontal directions, respectively. (a) χ = 2. (b) χ = 10. Positions of the body center of volume (left panel) are in mm; fluctuating velocities (right panel) are in mm s^-1. Note that N_x and V_x' are almost in phase for χ = 2, whereas they are more than 90° out of phase for χ = 10.

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  Supplemental Video 5: Two perpendicular views of the wake past a short cylinder with χ = 10 and Ar = 80 rising in zigzag [from a DNS by Auguste (2010)]. The wake is visualized with an isosurface of the λ₂ criterion (Jeong & Hussain 1995), the color reflecting the sign and magnitude of the streamwise vorticity.

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  Supplemental Video 6: An infinitely thin disk with I* = 0.06 undergoing (a) a highly nonlinear fluttering motion for Ar = 83 and (b) a tumbling motion for Ar = 156 [from a DNS by Auguste (2010)]. The wake is visualized with an isosurface of the λ₂ criterion (Jeong & Hussain 1995), the color reflecting the sign and magnitude of the streamwise vorticity.

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  Supplemental Video 6b (Download video file)