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arxiv: 2605.07901 · v1 · submitted 2026-05-08 · ⚛️ physics.flu-dyn

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· Lean Theorem

Vortex ring formation from the interaction of a cavitation bubble with a confined air bubble: experiments and a timing criterion

Badarinath Karri, Charul Gupta, Harish N Dixit, Lakshmana D Chandrala, Yashwant Singh

Authors on Pith no claims yet

Pith reviewed 2026-05-11 03:13 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords vortex ringcavitation bubbleconfined air bubbletiming parameterliquid column impacthigh-speed shadowgraphyregime identification
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The pith

A dimensionless timing parameter distinguishes when vortex rings form from cavitation and confined air bubble interactions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that vortex ring formation arises specifically when a liquid column, driven by an expanding air bubble in a confined hole, impacts a collapsing cavitation bubble at the right moment. Parametric experiments varying the dimensionless stand-off distance and air fill fraction reveal three regimes, only one of which produces a ring. One-dimensional models using the Rayleigh-Plesset equation and isentropic expansion calculate the column's impact location and speed. These feed into a timing parameter Π that compares column travel time to the cavitation collapse half-period, with rings appearing only for values between roughly 1 and 1.5. This criterion lets the outcome be predicted from simple timing rather than full flow details.

Core claim

The central claim is that the dimensionless timing parameter Π = (h + R_max) / (U_lc · t_cav/2) separates the observed regimes, with vortex ring formation occurring precisely when 1 ≲ Π ≲ 1.5; this is established by high-speed imaging across stand-off and fill-fraction values together with one-dimensional predictions of liquid-column impact location and speed U_lc.

What carries the argument

The dimensionless timing parameter Π that compares the liquid-column travel time to the cavitation-bubble collapse half-period.

If this is right

  • Rings form only for small stand-off distances and low air fill fractions when the liquid column impacts during collapse.
  • Large stand-off distances produce late impacts that do not form rings.
  • Large air fill fractions allow the air bubble to bypass the column and impact directly, producing no ring.
  • Any formed ring starts at 5 m/s, slows quadratically, and breaks up via azimuthal instabilities near Re = 4500.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The timing-based separation implies that ring formation is governed more by global timing than by local three-dimensional vorticity details.
  • The reported quadratic deceleration and Re-based breakup could serve as a benchmark for simple drag models of such rings in confined geometries.
  • The same Π construction might be tested in other confined multiphase setups to see whether the 1–1.5 window holds when viscosity or wall effects are stronger.

Load-bearing premise

The one-dimensional Rayleigh-Plesset and isentropic models accurately predict liquid-column impact location and speed without corrections for three-dimensional flow, wall effects, or viscosity.

What would settle it

Measure the actual liquid-column speed and impact timing in the confined geometry and check whether rings appear only inside the predicted Π window or appear outside it.

Figures

Figures reproduced from arXiv: 2605.07901 by Badarinath Karri, Charul Gupta, Harish N Dixit, Lakshmana D Chandrala, Yashwant Singh.

Figure 1
Figure 1. Figure 1: (a) Schematic of the experimental setup; numbered components are described in the text. The inset [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Regime map of all experimental data in the [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Image sequence for Case 1 (H = 0.44, B = 0.37, 𝑡cav = 3300 𝜇s). Image 1: bubble nucleation. Image 2: onset of air bubble compression (downward yellow arrow). Image 3: entry of the penetrating bubble into the blind hole. Image 4: maximum penetrating bubble depth. Image 5: onset of air bubble expansion (upward yellow arrow) and exit of the penetrating bubble (maroon box). Image 6: maximum cavitation bubble r… view at source ↗
Figure 4
Figure 4. Figure 4: Cavitation time map for the three representative cases. The [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Image sequence for Case 2 (H = 0.72, B = 0.37, 𝑡cav = 2200 𝜇s). Image 1: bubble nucleation. Image 2: onset of air bubble compression. Image 3: onset of air bubble expansion. Image 4: maximum cavitation bubble radius. Image 5: complete collapse; the red dashed box highlights the toroidal deformation of the liquid column caused by the collapsing jet. No vortex ring forms. See supplementary movie 2 for the te… view at source ↗
Figure 6
Figure 6. Figure 6: Image sequence for Case 3 (H = 0.24, B = 0.63, 𝑡cav = 3750 𝜇s). Image 1: bubble nucleation. Image 2: onset of air bubble compression. Image 3: entry of the penetrating bubble. Image 4: onset of air bubble expansion and exit of the penetrating bubble. Image 5: maximum cavitation bubble radius. Image 6: direct impact of the expanding air bubble on the far cavitation boundary (B-B impact; red dashed box). Ima… view at source ↗
Figure 7
Figure 7. Figure 7: Growth trajectory of the cavitation bubble: experiment (blue circles) versus the Rayleigh–Plesset [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Space–time trajectories of three boundaries for each case. Cyan curves: far boundary of the [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Variation of the height difference of the air bubble due to compression with H. Here, the 𝑥-axis denotes the dimensionless height difference of the air bubble, normalized by the height of the blind hole, while the 𝑦-axis denotes H. Data are shown for three key cases, along with a linear fit (𝑅 2 = 0.99) with a slope of -5.04 and an intercept of 1.06. 4.2.2. Expansion model and liquid column speed Once comp… view at source ↗
Figure 10
Figure 10. Figure 10: Air bubble boundary position 𝐿 = 𝑧 − 𝑑hole as a function of time. Blue circles: experimental data (compression then expansion). Black dashed curve: prediction of equation 4.5, applied from the point of maximum compression (𝑧 = 𝑧in). Red dotted line: maximum slope of the model curve, from which the terminal liquid column speed 𝑈lc is extracted. Negative 𝐿 indicates the interface lies inside the blind hole.… view at source ↗
Figure 11
Figure 11. Figure 11: Space–time map for Case 1 (H = 0.44, B = 0.37), showing all observed processes on a single plot. The 𝑦-axis is distance from the top surface (𝑥) of the block (negative values are inside the blind hole); 𝑡 = 0 is bubble generation. Grey circles: far boundary of the cavitation bubble during growth; maroon circles: during collapse; cyan circles: air bubble upper boundary; grey circles: penetrating bubble bou… view at source ↗
Figure 12
Figure 12. Figure 12: Vortex-ring characteristics: (a) Variation of the vertical distance traveled by the vortex ring with [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Schematic summarizing the three interaction outcomes between a growing cavitation bubble and [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
read the original abstract

We study vortex ring formation arising from the interaction between a cavitation bubble and a confined air bubble in a cylindrical blind hole, using high-speed shadowgraphy imaging. As the cavitation bubble grows above the hole, it drives a downward flow that compresses the air bubble at the base. The air bubble subsequently expands, expelling the overlying liquid column upward as a coherent slug; impact of this slug on the far boundary of the collapsing cavitation bubble produces a vortex ring. Parametric experiments across the dimensionless stand-off distance $\mathcal{H} = h/R_{\max}$ and the air bubble fill fraction $\mathcal{B} = (d_\text{hole} - d_\text{top})/d_\text{hole}$ identify three regimes: (i) liquid column impact during collapse, producing a vortex ring ($\mathcal{H} \lesssim 0.5$, $\mathcal{B} \lesssim 0.5$); (ii) late impact near the end of collapse (large $\mathcal{H}$); and (iii) direct air bubble impact after bypassing the liquid column (large $\mathcal{B}$), with neither (ii) nor (iii) producing a ring. Two one-dimensional models, based on the Rayleigh-Plesset equation and isentropic air bubble expansion, predict the liquid column impact location and its speed $U_\text{lc}$, respectively. A dimensionless timing parameter $\Pi = (h + R_{\max}) / (U_\text{lc} \cdot t_\text{cav}/2)$, comparing the liquid column travel time to the cavitation collapse half-period, distinguishes the three regimes: ring formation occurs for $1 \lesssim \Pi \lesssim 1.5$. The ring propagates from the hole at an initial speed of $5$ m/s, decelerating quadratically, and breaks apart via azimuthal instabilities at $Re \approx 4500$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript uses high-speed shadowgraphy to examine vortex ring formation from the interaction of a cavitation bubble with a confined air bubble in a cylindrical blind hole. Parametric experiments in dimensionless stand-off distance H = h/R_max and air-bubble fill fraction B identify three regimes: liquid-column impact during collapse (producing a ring for H ≲ 0.5, B ≲ 0.5), late impact, and direct air-bubble impact. One-dimensional Rayleigh-Plesset and isentropic-expansion models predict liquid-column impact location and speed U_lc; the resulting dimensionless timing parameter Π = (h + R_max)/(U_lc · t_cav/2) separates the regimes, with rings forming only for 1 ≲ Π ≲ 1.5. Ring propagation speed and breakup at Re ≈ 4500 are also reported.

Significance. If the timing criterion holds, the work supplies a compact, experimentally grounded predictor for vortex-ring generation in confined bubble interactions that builds directly on standard bubble-dynamics equations without ad-hoc fitting. The experimental regime classification is independent of the models, and the use of established Rayleigh-Plesset and isentropic formulations is a clear strength. The result is relevant to cavitation-driven flows, vortex dynamics, and confined multiphase systems.

major comments (1)
  1. [Modeling section (one-dimensional models for U_lc)] The timing parameter Π is computed from U_lc obtained via the one-dimensional isentropic model for air-bubble expansion (described after the experimental methods). No direct experimental measurement or validation of the predicted liquid-column speed is presented for the confined cylindrical geometry; wall shear, possible 3D recirculation, and viscous boundary layers omitted from the model could systematically shift actual arrival time and speed, thereby affecting the precise boundaries of the 1 ≲ Π ≲ 1.5 interval even if the experimental regime classification remains correct.
minor comments (1)
  1. [Abstract] The abstract states that the ring breaks apart at Re ≈ 4500 but does not indicate how the Reynolds number is defined or which characteristic length and viscosity are used.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [Modeling section (one-dimensional models for U_lc)] The timing parameter Π is computed from U_lc obtained via the one-dimensional isentropic model for air-bubble expansion (described after the experimental methods). No direct experimental measurement or validation of the predicted liquid-column speed is presented for the confined cylindrical geometry; wall shear, possible 3D recirculation, and viscous boundary layers omitted from the model could systematically shift actual arrival time and speed, thereby affecting the precise boundaries of the 1 ≲ Π ≲ 1.5 interval even if the experimental regime classification remains correct.

    Authors: We agree that the one-dimensional isentropic model for air-bubble expansion omits wall shear, three-dimensional recirculation, and viscous boundary layers in the confined cylindrical geometry, and that no direct experimental measurement of the liquid-column speed U_lc is presented. The three flow regimes are identified experimentally from high-speed shadowgraphy and are therefore independent of the model. The timing parameter Π is offered as a simple predictor constructed from the established Rayleigh-Plesset equation and isentropic expansion; in the reported parametric data this predictor correctly delineates the interval in which vortex rings form. We will add a concise paragraph in the modeling section that states the model assumptions, acknowledges the neglected effects, and notes that the precise numerical limits of the Π interval carry some uncertainty as a result. revision: partial

Circularity Check

0 steps flagged

No circularity: models are standard and unfitted; regime separation is an independent empirical correlation

full rationale

The experimental regimes are classified directly from high-speed imaging using the geometric parameters H and B, without reference to the timing parameter. The Rayleigh-Plesset and isentropic models are the standard formulations with no parameters adjusted to the vortex-ring observations or to the target Π values; they supply U_lc and impact location as forward predictions from bubble dynamics alone. Π is then computed for each experimental case and observed to cluster the ring-forming cases in a narrow band. This is a post-hoc correlation between an independent model output and independent experimental labels, not a self-definition, fitted prediction, or self-citation chain. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard bubble dynamics equations and experimental imaging; no new entities are postulated and no parameters are fitted specifically to the vortex ring data.

axioms (2)
  • standard math Rayleigh-Plesset equation governs the dynamics of the cavitation bubble
    Invoked to model collapse and compute t_cav and impact timing.
  • domain assumption Air bubble expansion is isentropic
    Used to predict liquid column speed U_lc from rebound.

pith-pipeline@v0.9.0 · 5672 in / 1389 out tokens · 29881 ms · 2026-05-11T03:13:39.036811+00:00 · methodology

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