Research ArticleAPPLIED SCIENCES AND ENGINEERING

Self-determined shapes and velocities of giant near-zero drag gas cavities

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Science Advances  08 Sep 2017:
Vol. 3, no. 9, e1701558
DOI: 10.1126/sciadv.1701558
  • Fig. 1 Formation of a gas cavity around an impacting 20-mm-diameter steel sphere.

    Snapshots from movie S2 showing (A and B) the formation of a gas cavity around a hot 20-mm Leidenfrost steel sphere at sphere temperature TS = 400°C as it enters a 2-m-tall tank containing water at 95°C and (C and D) the development and trajectory of the sphere-in-cavity structure at the indicated depths and times after entry. (E) A close-up of the steady-state gas cavity of length L and maximum diameter D formed around the 20-mm Leidenfrost steel sphere at TS = 400°C in 95°C water. (F) The steady-state gas cavity formed around a cold 20-mm superhydrophobic steel sphere at TS = 21°C in 21°C water.

  • Fig. 2 Cavity volume variation.

    (A) Dependence of the cavity-to-sphere volume ratio VC/VS on the sphere-to-liquid density ratio ρS/ρ. Data are for 10-mm (blue circles), 15-mm (green triangles), and 20-mm (red squares) steel (ρS = 7.7 g/cm3) or tungsten carbide (TC) (ρS = 14.9 g/cm3) spheres falling in PP1 (ρ = 1.7 g/cm3) and 95°C water (ρ = 0.96 g/cm3). The dotted line corresponds to neutral buoyancy of the sphere-in-cavity structure. (B) Cavity volume, as a function of (LD2), cavity diameter D, and length L, for the same sphere sizes and liquid combination as in (A). The dotted line is the best linear fit to the data that gives the relation VC ≈ 0.45 LD2.

  • Fig. 3 Comparison of drag coefficients of cavities and solid projectiles of the same shape.

    Comparison of the variation of the drag coefficient CD with Reynolds number Re for (i) lower data set: gas cavity in 21°C water around superhydrophobic steel spheres of diameter DS = 15, 20, and 25 mm (solid red squares) and around Leidenfrost steel spheres at TS = 400°C in 95°C water of diameter DS = 10, 15, 20, and 25 mm (solid red triangles), and (ii) upper data set: the drag on solid replica projectiles: DP = 25 mm, LP/DP = 4.5 containing different weights in 21°C water (solid blue squares) and 95°C water (solid blue triangles). Side images are snapshots of the falling projectile and sphere-in-cavity from movie S4. Fins on the solid projectiles were added to ensure rectilinear free fall. Their effect on the magnitude of the drag coefficient is estimated to be less than 10%.

  • Fig. 4 Pressure variation on the cavity surface.

    Variation of the hydrodynamic pressure, ρu2/2 (symbols), obtained from potential flow and the gravitational, ρgz (solid line), components of the pressure on the surface of the gas cavity around 20-mm-diameter heated metal spheres in the Leidenfrost state (see Eq. 2) and the pressure around a sphere (dashed line) (see Eq. 3) for the systems: (A) steel sphere in 95°C water, (B) tungsten carbide sphere in 95°C water, (C) steel sphere in fluorocarbon PP1 liquid, and (D) tungsten carbide sphere in fluorocarbon PP1 liquid.

  • Fig. 5 Dependence of cavity velocity on sphere size and density ratio.

    Variation of the sphere-in-cavity velocity U with sphere diameter DS and the sphere-to-fluid density ratio ρS/ρ. The dotted line is a linear best fit to the data that resulted in Eq. 4.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/9/e1701558/DC1

    section S1. Theory: Achieving near-zero drag at high Reynolds numbers

    section S2. Stable streamlined cavity experiments

    section S3. Solid projectile experiments

    section S4. Determination of the drag coefficients

    fig. S1. The experimental apparatus.

    fig. S2. 3D-printed solid projectiles.

    fig. S3. Comparison of shapes of sphere-in-cavities and solid projectiles.

    fig. S4. Three-piece fitting function for the sphere-in-cavity.

    fig. S5. Velocity versus depth data.

    fig. S6. Dependence of the drag coefficient on the aspect ratio.

    table S1A. Physical parameters of the sphere-in-cavities in 95°C water.

    table S1B. Physical parameters of the sphere-in-cavities in 21°C water.

    table S1C. Physical parameters of the sphere-in-cavities in PP1 liquid.

    movie S1. Free fall of sphere-in-cavity for 10-mm Leidenfrost steel sphere in PP1.

    movie S2. Impact and cavity formation by 20-mm steel sphere in 21° and 95°C water.

    movie S3. Close-up of sphere-in-cavity for 20-mm steel sphere in 21° and 95°C water.

    movie S4. Comparison of solid projectiles and sphere-in-cavity free fall.

    Reference (29)

  • Supplementary Materials

    This PDF file includes:

    • section S1. Theory: Achieving near-zero drag at high Reynolds numbers
    • section S2. Stable streamlined cavity experiments
    • section S3. Solid projectile experiments
    • section S4. Determination of the drag coefficients
    • fig. S1. The experimental apparatus.
    • fig. S2. 3D-printed solid projectiles.
    • fig. S3. Comparison of shapes of sphere-in-cavities and solid projectiles.
    • fig. S4. Three-piece fitting function for the sphere-in-cavity.
    • fig. S5. Velocity versus depth data.
    • fig. S6. Dependence of the drag coefficient on the aspect ratio.
    • table S1A. Physical parameters of the sphere-in-cavities in 95°C water.
    • table S1B. Physical parameters of the sphere-in-cavities in 21°C water.
    • table S1C. Physical parameters of the sphere-in-cavities in PP1 liquid.
    • Legends for movies S1 to S4
    • Reference (29)

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    Other Supplementary Material for this manuscript includes the following:

    • movie S1 (.mov format). Free fall of sphere-in-cavity for 10-mm Leidenfrost steel sphere in PP1.
    • movie S2 (.mov format). Impact and cavity formation by 20-mm steel sphere in 21° and 95°C water.
    • movie S3 (.mov format). Close-up of sphere-in-cavity for 20-mm steel sphere in 21° and 95°C water.
    • movie S4 (.mov format). Comparison of solid projectiles and sphere-in-cavity free fall.

    Files in this Data Supplement: