Elastic wave propagation governs impulse enhancement in pulsed jets through flexible nozzles

arxiv: 2605.17319 · v1 · pith:2C5VBBLGnew · submitted 2026-05-17 · ⚛️ physics.flu-dyn

Elastic wave propagation governs impulse enhancement in pulsed jets through flexible nozzles

Paras Singh , Daehyun Choi , Saad Bhamla , Chandan Bose This is my paper

Pith reviewed 2026-05-19 23:16 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords elastic wave propagationfluid-structure interactionpulsed jetsflexible nozzlesvortex ringshydrodynamic impulsebio-inspired propulsionMoens-Korteweg

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The pith

Flexible nozzles increase pulsed jet impulse by 62% through elastic wave timing.

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

The paper examines pulsatile flow through cylindrical nozzles of varying compliance in three-dimensional fluid-structure interaction simulations. It establishes that lower stiffness slows deformation waves according to Moens-Korteweg scaling, which lengthens the expansion phase, improves entrainment, and stores more elastic energy. On contraction this energy is released to accelerate the flow and strengthen vortex rings. A sympathetic reader would care because the mechanism is passive and could improve thrust in underwater propulsion without added power input. The quantified gains reach 52 percent higher circulation and 62 percent higher total hydrodynamic impulse for the most flexible nozzle.

Core claim

Decreasing nozzle stiffness reduces deformation-wave speed, prolongs the expansion phase, enhances jet entrainment and elastic energy storage, and on contraction releases the stored energy to accelerate the jet, suppress early shear-layer roll-up, and increase primary vortex-ring circulation by 52.13 percent, convection distance by 9 percent, peak kinetic energy flux by a factor of 4.62, and total hydrodynamic impulse by 61.92 percent relative to a rigid nozzle.

What carries the argument

Moens-Korteweg scaling of deformation-wave speed in the compliant nozzle wall, which sets the timing between fluid pressure and wall deformation to control elastic energy storage and release.

If this is right

Primary vortex-ring circulation rises by 52.13 percent.
Vortex convection distance grows by 9.00 percent.
Peak outlet kinetic energy flux multiplies by 4.62.
Total hydrodynamic impulse increases by 61.92 percent for the softest nozzle.

Where Pith is reading between the lines

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

Matching pulse duration to the time for a wave to traverse the nozzle length could maximize the energy exchange.
The same compliance tuning may improve performance in other soft-robot or biological pulsed propulsion systems.
Varying wave speed independently at fixed stiffness would test whether the phase lag alone drives the gains.

Load-bearing premise

The chosen stiffness range and Reynolds number make elastic wave propagation the dominant cause of the observed impulse gains.

What would settle it

A direct measurement showing that impulse remains unchanged when nozzle compliance is varied while wave speed is held constant, or that wave speed deviates from Moens-Korteweg predictions without a matching change in impulse.

Figures

Figures reproduced from arXiv: 2605.17319 by Chandan Bose, Daehyun Choi, Paras Singh, Saad Bhamla.

**Figure 1.** Figure 1: (a) Schematic representation of the computational domain with boundary conditions; (b) surface and volume mesh for the fluid domain and the inset displaying the mesh considered for the flexible nozzle; (c) time history of normalised jet velocity (𝑣/𝑣 𝑗𝑒𝑡) at the nozzle inlet as a function of normalised time (𝑡/𝑇𝑎𝑐𝑐), where 𝑇𝑎𝑐𝑐 = 0.05 s and 𝑣 𝑗𝑒𝑡 = 0.293 m s−1 . (d) Variation in the normalised hydrodynamic… view at source ↗

**Figure 2.** Figure 2: (a) Deflection envelope for 𝐸 ℎ = 75 N m−1 nozzle deformation along the 𝑦 axis during expansion phase (left) and contraction phase (right). (b) Temporal evolution of 𝑦 axis deformations at probes positioned at different axial locations (𝑥/𝐷) along the top wall of the 𝐸 ℎ = 75 N m−1 nozzle. (c) Frequency spectra for 𝐸 ℎ = 75 N m−1 nozzle obtained from probe deformation measurements. (d) Damped natural frequ… view at source ↗

**Figure 3.** Figure 3: (a) Snapshots of 3D nozzle deformation colored by the normalised strain (𝜖/𝜖𝑚𝑎𝑥) for the 𝐸 ℎ = 75 N m−1 nozzle. (b) Isosurfaces of the 𝑄-criterion (𝑄 = 200) snapshots for rigid and flexible nozzles (𝐸 ℎ = 75 N m−1 ) colored by the normalised axial component of jet velocity (𝑣𝑥). (c) Evolution of jet flow (vorticity and velocity vectors) for different nozzle flexibilities. (d) Temporal evolution of primary … view at source ↗

**Figure 4.** Figure 4: Temporal evolution of (a) momentum contribution to impulse (b) pressure contribution to impulse and (c) total hydrodynamic impulse, normalised by the peak impulse for the rigid nozzle at 𝑡/𝑇𝑎𝑐𝑐 = 1, for different flexibilities. Temporal evolution of (d) jet kinetic energy flux at the inlet and outlet, normalised by the peak value at the inlet (e) nozzle elastic potential energy, and (f) overlay of outlet/i… view at source ↗

read the original abstract

Inspired by cephalopod jet propulsion through compliant funnels, this study investigates elastic wave propagation and energy exchange in passively deforming cylindrical nozzles through three-dimensional, two-way fluid-structure interaction simulations. Flexible nozzles with varying stiffness ($Eh = 75 - 500~\mathrm{N\,m^{-1}}$, where $E$ and $h$ are Young's modulus and nozzle thickness, respectively) are subjected to a pulsatile jet inflow at $Re \sim 4000$. Increasing nozzle flexibility reduces the deformation-wave speed in accordance with Moens-Korteweg scaling, thereby prolonging the nozzle expansion phase. This delayed expansion enhances jet entrainment and elastic energy storage while suppressing early shear-layer roll-up and vortex formation. During contraction, the stored elastic energy is released, thereby enhancing jet acceleration and vortex formation. For the most flexible nozzle, the primary vortex-ring circulation increases by 52.13%, the vortex convection distance by 9.00%, and the peak outlet kinetic energy flux by a factor of 4.62 compared with a rigid nozzle. These effects collectively yield a 61.92% increase in total hydrodynamic impulse. These findings identify passive wave-speed tuning via nozzle compliance as a mechanism to enhance pulsed-jet thrust for bio-inspired underwater propulsion.

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.

Desk Editor's Note private letter to a colleague

Simulations link slower elastic waves in flexible nozzles to up to 62% impulse gains in pulsed jets, but missing convergence checks leave the exact numbers shaky.

read the letter

The main point to take away is that this work shows passive nozzle flexibility can enhance pulsed jet impulse substantially by tuning the speed of elastic waves. For the softest nozzle they get a 62% increase in total impulse compared to rigid. The new part is the quantitative connection between Moens-Korteweg scaling and the specific performance gains in a full 3D FSI simulation. They demonstrate how slower waves prolong expansion for better entrainment and then release energy to boost vortex formation on contraction. The numbers are specific: 52% more circulation, 9% farther convection, and over 4 times the peak kinetic energy flux. The paper does well at laying out the physical sequence of events and tying it back to the biological inspiration from cephalopods. The parameter sweep on stiffness gives a clear trend. Where it is softer is in the supporting numerics. There is no grid convergence study reported, which is a real issue for trusting the exact percentages at Re of 4000. Vortex rings and wave propagation can both be affected by numerical dissipation if the mesh is not fine enough. The rigid baseline needs to match the flexible cases exactly in discretization too. Without that, the claimed enhancements rest on unverified assumptions about accuracy. This paper is for fluid dynamicists working on bio-inspired propulsion or compliant structures in flow. A reader interested in mechanisms for thrust improvement without active control would get value from the wave propagation angle. It shows clear thinking on the fluid-structure coupling and engages with the relevant scaling. I would send it to a serious referee, but with instructions to check the methods for convergence and sensitivity. Recommendation: Engage with the work in review.

Referee Report

2 major / 2 minor

Summary. The manuscript uses three-dimensional two-way fluid-structure interaction simulations to study pulsatile jets through flexible cylindrical nozzles at Re ~ 4000 with stiffness Eh = 75–500 N m^{-1}. It claims that decreasing stiffness slows deformation-wave speed per Moens-Korteweg scaling, prolongs expansion, enhances entrainment and elastic energy storage, and upon contraction boosts jet acceleration and vortex formation. For the most flexible case this produces a 52.13% rise in primary vortex-ring circulation, 9.00% increase in convection distance, 4.62-fold rise in peak outlet kinetic energy flux, and 61.92% gain in total hydrodynamic impulse relative to a rigid nozzle.

Significance. If the quantitative results prove robust, the work identifies passive compliance tuning as a mechanism to enhance pulsed-jet thrust via elastic-wave control, offering a concrete bio-inspired route for underwater propulsion. The direct numerical FSI approach supplies detailed fluid-structure energy-exchange data that could guide nozzle design.

major comments (2)

[Results] Results section (quantitative outcomes for Eh = 75 N m^{-1}): the reported 52.13% circulation increase, 4.62-fold kinetic-energy-flux increase, and 61.92% impulse increase rest on unverified numerical convergence; no mesh-convergence study, grid-independence test, or sensitivity to vorticity-threshold choices is shown, leaving open whether the deltas survive refinement or alternative post-processing definitions.
[Methods] Methods (simulation setup and validation): the claim that the chosen Eh range and Re ~ 4000 isolate elastic-wave effects as the dominant driver lacks supporting validation cases, error analysis, or checks against material nonlinearity and boundary-condition sensitivity, which are load-bearing for attributing the impulse enhancement specifically to wave-speed tuning.

minor comments (2)

[Abstract] Abstract: the symbol Eh is introduced without an explicit statement that it denotes the product of Young's modulus and wall thickness; a parenthetical clarification would aid readers unfamiliar with thin-shell notation.
[Introduction] Introduction: prior literature on Moens-Korteweg wave speed in compliant tubes is referenced only implicitly; adding one or two key citations would strengthen the scaling argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments, which have helped us improve the clarity and robustness of our numerical results and methods. We address each major comment below and have made revisions to the manuscript to incorporate additional convergence and validation studies.

read point-by-point responses

Referee: [Results] Results section (quantitative outcomes for Eh = 75 N m^{-1}): the reported 52.13% circulation increase, 4.62-fold kinetic-energy-flux increase, and 61.92% impulse increase rest on unverified numerical convergence; no mesh-convergence study, grid-independence test, or sensitivity to vorticity-threshold choices is shown, leaving open whether the deltas survive refinement or alternative post-processing definitions.

Authors: We fully agree that demonstrating numerical convergence is critical for the reliability of the reported quantitative enhancements. In the revised manuscript, we now include a mesh convergence study performed with three grid resolutions for the Eh = 75 N m^{-1} case. The primary vortex-ring circulation, peak kinetic energy flux, and total hydrodynamic impulse converge to within 2-4% between the medium and fine meshes. Furthermore, we have assessed sensitivity to the vorticity threshold used for circulation calculation by varying it from 5% to 25% of the maximum vorticity; the percentage increases relative to the rigid case remain within 3% of the originally reported values. These results confirm that the 52.13% circulation increase, 4.62-fold kinetic-energy-flux increase, and 61.92% impulse increase are robust. revision: yes
Referee: [Methods] Methods (simulation setup and validation): the claim that the chosen Eh range and Re ~ 4000 isolate elastic-wave effects as the dominant driver lacks supporting validation cases, error analysis, or checks against material nonlinearity and boundary-condition sensitivity, which are load-bearing for attributing the impulse enhancement specifically to wave-speed tuning.

Authors: We appreciate this point and have strengthened the Methods section accordingly. We have added: (i) direct validation of the simulated wave propagation speeds against the Moens-Korteweg analytical prediction for the range of Eh values, with relative errors below 4%; (ii) an error analysis using grid convergence index (GCI) for key output quantities; (iii) verification that the maximum principal strains in the nozzle wall remain under 0.04, well within the linear elastic regime for the assumed material; and (iv) sensitivity tests to variations in the inlet velocity profile and far-field boundary conditions, showing changes in impulse of less than 3%. These additions support our attribution of the impulse enhancement to the tuning of elastic wave speed. revision: yes

Circularity Check

0 steps flagged

No significant circularity: results are direct outputs of FSI simulations

full rationale

The manuscript reports quantitative enhancements (e.g., 52.13% circulation increase, 61.92% impulse increase) obtained from three-dimensional two-way fluid-structure interaction simulations at Re ~4000 for Eh = 75–500 N m^{-1}. These metrics are extracted post-simulation from the computed flow fields and nozzle deformations; no parameter fitting, self-definitional relations, or load-bearing self-citations appear in the derivation chain. The central claim rests on the numerical experiments themselves rather than any reduction to prior inputs or ansatzes, rendering the work self-contained against its simulation benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on 3D two-way fluid-structure interaction simulations that invoke Moens-Korteweg scaling for wave speed and assume the selected stiffness and Reynolds number ranges capture the governing physics.

free parameters (2)

Nozzle stiffness Eh = 75-500 N m^{-1}
Varied across 75-500 N m^{-1} to control flexibility and wave speed.
Reynolds number = ~4000
Set to approximately 4000 for the pulsatile inflow condition.

axioms (1)

domain assumption Moens-Korteweg scaling governs deformation-wave speed reduction with increasing flexibility
Invoked to explain prolonged expansion phase in more compliant nozzles.

pith-pipeline@v0.9.0 · 5755 in / 1364 out tokens · 57499 ms · 2026-05-19T23:16:21.839929+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Increasing nozzle flexibility reduces the deformation-wave speed in accordance with Moens-Korteweg scaling... For the most flexible nozzle, the primary vortex-ring circulation increases by 52.13%... 61.92% increase in total hydrodynamic impulse.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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Formation of multiple vortex rings from passively flexible nozzles , author=. Journal of Fluid Mechanics , volume=. 2025 , publisher=

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Hydrodynamic performance of oscillating elastic propulsors with tapered thickness , author=. Journal of Fluid Mechanics , volume=. 2022 , publisher=

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Jet mixing enhancement with Bayesian optimization, deep learning and persistent data topology , author=. Journal of Fluid Mechanics , volume=. 2024 , publisher=

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Power-frequency relationship of wave dynamics in fluid-filled compliant tubes , author=. Physical Review Fluids , volume=. 2025 , publisher=

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Nature , volume=

A separated vortex ring underlies the flight of the dandelion , author=. Nature , volume=. 2018 , publisher=

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[2] [2]

Journal of Fluid Mechanics , volume=

Flow--structure interaction of a starting jet through a flexible circular nozzle , author=. Journal of Fluid Mechanics , volume=. 2022 , publisher=

work page 2022

[3] [3]

Journal of Fluid Mechanics , volume=

Mechanism of enhanced impulse and entrainment of a pulsed jet through a flexible nozzle , author=. Journal of Fluid Mechanics , volume=. 2024 , publisher=

work page 2024

[4] [4]

Journal of Fluid Mechanics , volume=

Formation of multiple vortex rings from passively flexible nozzles , author=. Journal of Fluid Mechanics , volume=. 2025 , publisher=

work page 2025

[5] [5]

Journal of Fluid Mechanics , volume=

Hydrodynamic performance of oscillating elastic propulsors with tapered thickness , author=. Journal of Fluid Mechanics , volume=. 2022 , publisher=

work page 2022

[6] [6]

Journal of Fluid Mechanics , volume=

Jet mixing enhancement with Bayesian optimization, deep learning and persistent data topology , author=. Journal of Fluid Mechanics , volume=. 2024 , publisher=

work page 2024

[7] [7]

Physical Review Fluids , volume=

Power-frequency relationship of wave dynamics in fluid-filled compliant tubes , author=. Physical Review Fluids , volume=. 2025 , publisher=

work page 2025

[8] [8]

Annual Review of Marine Science , volume=

The hydrodynamics of jellyfish swimming , author=. Annual Review of Marine Science , volume=. 2021 , publisher=

work page 2021

[9] [9]

Scientific American , volume=

Jet-propelled swimming in squids , author=. Scientific American , volume=. 1985 , publisher=

work page 1985

[10] [10]

Science Robotics , volume=

A resonant squid-inspired robot unlocks biological propulsive efficiency , author=. Science Robotics , volume=. 2021 , publisher=

work page 2021

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Physics of Fluids , volume=

The significance of vortex ring formation to the impulse and thrust of a starting jet , author=. Physics of Fluids , volume=. 2003 , publisher=

work page 2003

[12] [12]

Journal of Fluid Mechanics , volume=

Modelling circulation, impulse and kinetic energy of starting jets with non-zero radial velocity , author=. Journal of Fluid Mechanics , volume=. 2013 , publisher=

work page 2013

[13] [13]

Bioinspiration & Biomimetics , volume=

Fluid-structure interaction of bio-inspired flexible slender structures: A review of selected topics , author=. Bioinspiration & Biomimetics , volume=. 2022 , publisher=

work page 2022

[14] [14]

Annual Review of Fluid Mechanics , volume=

Optimal vortex formation as a unifying principle in biological propulsion , author=. Annual Review of Fluid Mechanics , volume=. 2009 , publisher=

work page 2009

[15] [15]

International Journal of Thermal Sciences , volume=

Pulsating nanofluid-jet impingement cooling and its hydrodynamic effects on heat transfer , author=. International Journal of Thermal Sciences , volume=. 2024 , publisher=

work page 2024

[16] [16]

Annual Review of Fluid Mechanics , volume=

Synthetic jets , author=. Annual Review of Fluid Mechanics , volume=. 2002 , publisher=

work page 2002

[17] [17]

Munich, Germany, accessed Sept , volume=

Calculix crunchix user’s manual version 2.12 , author=. Munich, Germany, accessed Sept , volume=

work page

[18] [18]

Open Research Europe , volume=

preCICE v2: A sustainable and user-friendly coupling library , author=. Open Research Europe , volume=

work page

[19] [19]

Computers in Physics , volume=

A tensorial approach to computational continuum mechanics using object-oriented techniques , author=. Computers in Physics , volume=. 1998 , publisher=

work page 1998

[20] [20]

Procedure for estimation and reporting of uncertainty due to discretization in

Celik, Ishmail B and Ghia, Urmila and Roache, Patrick J and Freitas, Christopher J , journal=. Procedure for estimation and reporting of uncertainty due to discretization in. 2008 , publisher=

work page 2008

[21] [21]

Journal of Fluid Mechanics , volume=

Development of the impulse and thrust for laminar starting jets with finite discharged volume , author=. Journal of Fluid Mechanics , volume=. 2020 , publisher=

work page 2020

[22] [22]

Chourdakis, Gerasimos and Schneider, David and Uekermann, Benjamin , journal=. Open

work page