Research ArticlePHYSICS

Turbulence generation through an iterative cascade of the elliptical instability

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Science Advances  28 Feb 2020:
Vol. 6, no. 9, eaaz2717
DOI: 10.1126/sciadv.aaz2717
  • Fig. 1 Vortex ring collisions.

    (A) Schematic side-view showing the formation and collision of dyed vortex rings in experiments. Fluorescent dye (Rhodamine B) is injected into the core of the vortex via a thin gap in the orifice of the vortex cannon. The dashed horizontal line denotes the symmetry axis. (B) Vortex ring radius versus rescaled time for collisions at various Reynolds numbers. Both cores are dyed, the core centerlines are extracted from 3D reconstructions, and the centerlines are fitted to circles with a fixed center point. The initial time begins when the vortex rings enter the scanning volume and ends when the vortex cores break down. Inset: All experimental curves shifted by t to collapse. The radial growth of the rings coincides with the Biot-Savart prediction.

  • Fig. 2 Antisymmetric perturbations in vortex ring collisions.

    A montage of core centerline trajectories for vortex ring collisions in both (A) experiment and (B) DNS. The top (z > 0) cores are indicated by the red lines, and the bottom (z < 0) cores are indicated by the blue lines. For the experimental collision, Re = 7000, SR = 2, and R0 = 17.5 mm. For the DNS collision, ReΓ = Γ/ν = 4500 and σ = 0.1R0. The cores are segmented from the 3D flow visualization in the experimental collision and from the pressure distribution in the simulation. Mean core separation distance versus rescaled time for the same (C) experimental and (D) numerical collisions. The blue circles correspond to the trajectories in (A) and (B), and the red dashed lines correspond to the visualizations in the insets. (C, inset) 3D visualization of the dyed vortex cores in the experimental collision. (D, inset) 3D visualization of the dyed vortex rings in the simulation, showing both the dye in the cores (dark) and surrounding them (light).

  • Fig. 3 Formation of perpendicular secondary filaments in an experimental vortex ring collision.

    3D reconstruction of two fully dyed vortex rings colliding head-on, viewed from overhead (top) and from the side (bottom). Re = 6000 and SR = 2.5. (A to C) As the rings grow, they interdigitate as the dye from the upper ring is wrapped around the lower ring and vice versa. (D and E) The colliding rings form an array of secondary vortex filaments that are perpendicular to the vortex cores. (F) The cores and perpendicular filaments break down into a fine-scale turbulent cloud.

  • Fig. 4 Generation of perpendicular secondary filaments in DNS.

    (A to C) Vorticity modulus for simulated interacting tubes where ReΓ = 4500, σ = 0.06ℒ, b = 2.5σ, and t* = Γt/b2. The vorticity modulus is normalized by the maximum vorticity modulus during the simulation, ∣ω∣max. (A) The initial antisymmetric perturbations of the cores develop as the tips of the perturbations locally flatten (0.103 ≤ ∣ω∣/∣ω∣max ≤ 0.117). (B) At the same time, low-vorticity perpendicular filaments form as a result of the perturbations (0.046 ≤ ∣ω∣/∣ω∣max ≤ 0.092). (C) Once the secondary filaments form, their vorticity amplifies (0.076 ≤ ∣ω∣/∣ω∣max ≤ 0.114). (D) Vorticity distribution in the z direction along the center plane (z = 0) indicated by the dashed line in (C). Adjacent secondary filaments counter-rotate.

  • Fig. 5 The development of a turbulent cascade.

    (A to F) Vorticity modulus for simulated interacting tubes where ReΓ = 6000, σ = 0.06ℒ, b = 2.5σ, and t* = Γt/b2. Each panel shows the front view of the full cores (left) and a close-up top view of the interacting secondary filaments indicated in the full view (right). (A) The antisymmetric perturbations of the cores develop. (B) Perpendicular secondary filaments form between the cores. (C) Secondary filaments begin to interact with each other and break down. (D) Tertiary filaments begin to form perpendicular to the secondary filaments. (E) Tertiary filaments are fully formed. (F) The flow breaks down into a disordered tangle of vortices. The vorticity thresholds are 0.079 ≤ ∣ω∣/∣ω∣max ≤ 0.099 for (A) and 0.110 ≤ ∣ω∣/∣ω∣max ≤ 0.211 for (B to F), where ∣ω∣max is the maximum vorticity modulus for the entire simulation. (G) Normalized shell-to-shell energy transfer spectra indicate whether a mode is an energy source [T(k) > 0] or sink [T(k) < 0]. At early times (t* = 65.56 and 74.44), the secondary filaments switch from energy sinks to sources as they are generated and then interact to form new vortices. At late times, the spectra flatten as energy is transferred more uniformly across the scales of the flow. (G, inset) Normalized kinetic energy dissipation rate as a function of time. The energy dissipation rate increases with the development of the secondary filaments and peaks as the secondary filaments and residual cores break down into a tangle of fine-scale vortices. (H) Normalized kinetic energy spectra show the rapid development of a sustained turbulent state with Kolmogorov scaling—as indicated by the black line—around the peak dissipation rate.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/9/eaaz2717/DC1

    Section S1. PIV analysis of vortex ring geometry

    Section S2. Simulating vortex ring collisions using the Biot-Savart approximation

    Section S3. Nonlinear development of the elliptical instability

    Section S4. Interactions of secondary vortex filaments

    Section S5. Analysis of the transfer of energy in a turbulent flow

    Section S6. Emergence of turbulence from the elliptical instability with increasing Reynolds number

    Section S7. Supplementary movie descriptions

    Fig. S1. Vortex ring tracking and measurement through PIV.

    Fig. S2. Vortex ring and core geometry.

    Fig. S3. Modeling radial growth of colliding vortex rings.

    Fig. S4. Onset of the elliptical instability for colliding vortex rings.

    Fig. S5. Formation of a secondary vortex filament.

    Fig. S6. Alternating structure of secondary filaments.

    Fig. S7. Evolution of circulation.

    Fig. S8. Interactions of secondary vortex filaments.

    Fig. S9. Transition to turbulence in DNS of interacting vortex tubes.

    Fig. S10. Vorticity evolution for DNS of interacting vortex tubes.

    Movie S1. Head-on collision of vortex rings.

    Movie S2. Experimental vortex ring collision with dyed cores.

    Movie S3. DNS of dyed vortex ring collision.

    Movie S4. Experimental fully dyed vortex ring collision.

    Movie S5. DNS of vortex tube interaction: ReΓ = 4500.

    Movie S6. DNS of vortex tube interaction: ReΓ = 3500.

    Movie S7. Interaction and splitting of secondary vortex filaments.

    Movie S8. DNS of vortex tube interaction: ReΓ = 6000.

    Movie S9. Iterative cascade of elliptical instabilities.

    Movie S10. Gaussian fit to vortex core PIV data.

    Movie S11. DNS of vortex tube interaction: ReΓ = 2000.

    References (3743)

  • Supplementary Materials

    The PDF file includes:

    • Section S1. PIV analysis of vortex ring geometry
    • Section S2. Simulating vortex ring collisions using the Biot-Savart approximation
    • Section S3. Nonlinear development of the elliptical instability
    • Section S4. Interactions of secondary vortex filaments
    • Section S5. Analysis of the transfer of energy in a turbulent flow
    • Section S6. Emergence of turbulence from the elliptical instability with increasing Reynolds number
    • Section S7. Supplementary movie descriptions
    • Fig. S1. Vortex ring tracking and measurement through PIV.
    • Fig. S2. Vortex ring and core geometry.
    • Fig. S3. Modeling radial growth of colliding vortex rings.
    • Fig. S4. Onset of the elliptical instability for colliding vortex rings.
    • Fig. S5. Formation of a secondary vortex filament.
    • Fig. S6. Alternating structure of secondary filaments.
    • Fig. S7. Evolution of circulation.
    • Fig. S8. Interactions of secondary vortex filaments.
    • Fig. S9. Transition to turbulence in DNS of interacting vortex tubes.
    • Fig. S10. Vorticity evolution for DNS of interacting vortex tubes.
    • Legends for movies S1 to S11
    • References (3743)

    Download PDF

    Other Supplementary Material for this manuscript includes the following:

    • Movie S1 (.mp4 format). Head-on collision of vortex rings.
    • Movie S2 (.mp4 format). Experimental vortex ring collision with dyed cores.
    • Movie S3 (.mp4 format). DNS of dyed vortex ring collision.
    • Movie S4 (.mp4 format). Experimental fully dyed vortex ring collision.
    • Movie S5 (.mp4 format). DNS of vortex tube interaction: ReΓ = 4500.
    • Movie S6 (.mp4 format). DNS of vortex tube interaction: ReΓ = 3500.
    • Movie S7 (.mp4 format). Interaction and splitting of secondary vortex filaments.
    • Movie S8 (.mp4 format). DNS of vortex tube interaction: ReΓ = 6000.
    • Movie S9 (.mp4 format). Iterative cascade of elliptical instabilities.
    • Movie S10 (.mp4 format). Gaussian fit to vortex core PIV data.
    • Movie S11 (.mp4 format). DNS of vortex tube interaction: ReΓ = 2000.

    Files in this Data Supplement:

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