arxiv: 2604.12788 · v1 · submitted 2026-04-14 · ⚛️ physics.flu-dyn

Recognition: unknown

Shape of an interface hit by an oblique jet

Theophile Gaichies , Anniina Salonen , Arnaud Antkowiak , Emmanuelle Rio

Authors on Pith no claims yet

Pith reviewed 2026-05-10 14:16 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords oblique jet impactliquid cavityflow detachmentsurface depressionmeniscus shapeforce balanceinertial regimedirect numerical simulation

0 comments

The pith

An oblique jet below 50 degrees forms a cavity whose width is set by balancing suction from asymmetric flow detachment against the weight of displaced liquid and surface tension.

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

When a smooth steady jet strikes a liquid bath at a sufficiently oblique angle, it excavates a cavity ahead of the impact point. Direct numerical simulations show that the jet boundary layer detaches asymmetrically, accelerating liquid below the surface and thereby lowering the local pressure. The authors treat this pressure drop as the dominant suction mechanism and balance its integrated force against the hydrostatic weight of the raised liquid plus the surface-tension force acting along the contact line. The resulting expression predicts the observed cavity width. The outer meniscus shape is shown to match the meniscus that forms outside a tilted fiber piercing the same interface.

Core claim

In the inertial regime the jet boundary layer detaches in the impact region, delimiting a core jet region outside of which the liquid is mainly in hydrostatic equilibrium. The flow detachment displays an asymmetry, which results in the acceleration of the liquid below the surface, thereby creating a depression. With this observation, the authors propose a model balancing the suction force of this depression with the weight of the displaced water and the surface tension force to obtain a prediction for the typical width of the cavity. The shape of the outer meniscus is related to the one outside a tilted fiber piercing the fluid interface.

What carries the argument

Force-balance model that equates the suction force arising from asymmetric subsurface acceleration to the hydrostatic weight of the displaced liquid plus the capillary restoring force, thereby fixing the cavity width.

If this is right

Cavity formation occurs only for incidence angles below approximately 50 degrees.
The outer meniscus shape is the same as the meniscus outside a tilted fiber piercing the interface.
Inside the cavity the pressure lies below the hydrostatic value because of the accelerated subsurface flow.
The cavity width scales with jet momentum flux, gravity, and surface tension according to the derived balance.

Where Pith is reading between the lines

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

The same detachment asymmetry may govern cavity formation in other oblique free-surface impacts such as raindrop collisions or ship bow waves.
If the vertical extent of the depression can be measured or modeled independently, the same balance could be used to predict cavity depth as well as width.
Because the prediction depends only on jet speed, angle, fluid density, gravity, and surface tension, it offers a simple estimate of cavity size that does not require full flow simulation.

Load-bearing premise

The pressure drop created by asymmetric acceleration of liquid beneath the surface is the dominant suction mechanism and can be directly balanced against the weight of displaced liquid plus surface tension without other significant contributions.

What would settle it

Measure the pressure or velocity field immediately beneath the cavity and test whether the integrated suction force equals the measured cavity weight plus the measured surface-tension force at the observed width.

Figures

Figures reproduced from arXiv: 2604.12788 by Anniina Salonen, Arnaud Antkowiak, Emmanuelle Rio, Theophile Gaichies.

**Figure 1.** Figure 1: (a) Experimental images of the interface impacted by a jet of radius 𝑅 = 0.19 mm with a speed 𝑉 = 2.8 m.s −1 , with various angles 𝛼 between the jet and the bath. (b) Measurements of the width of the cavity 𝑊 for jets of varying speeds, radii, and inclination with respect to the bath. As shown in figure 1(a), when the jet impacts the bath vertically, we observe an axisymmetric meniscus in all our experimen… view at source ↗

**Figure 2.** Figure 2: (a) Experimental image of the meniscus on a glass fiber of radius 𝑅 = 0.1 mm and inclination 𝛼 = 30◦ in a pool of silicone oil. The two white arrows designate the point where the meniscus meets the fiber. (b) Sketch representing the angles 𝜙0 and 𝜙𝑎 between the fiber and the horizontal. (c) Measurements of the maximum height difference normalized by the fiber radius 𝛥𝐻/𝑅. The red curve corresponds to the e… view at source ↗

**Figure 3.** Figure 3: (a) Experimental image of a 𝑅 = 0.3 mm jet impacting a water bath at a speed 𝑉 = 3.1 m.s −1 for various 𝛼. (b) Interfacial profile in the x-z plane (at 𝑧 = 0) extracted from simulations of a jet with Re = 400, We = 12 and Fr = 40, for various 𝛼. (c) Image of the cavity formed by a 𝑅 = 0.19 mm jet impacting a water bath at 𝑉 = 3.2 m.s −1 with an angle 𝛼 = 33◦ , in the yz plane facing the impacting jet. (d) … view at source ↗

**Figure 4.** Figure 4: Simulated velocity fields in the x-z plane for various inclinations 𝛼. The inserts are experimental images of a dyed (0.5 wt% indigo carmine) jet (𝑅 = 0.19 mm, 𝑉 = 2.8m.s −1 ) impacting the interface, where the thickness of the dyed layer is highlighted with arrows. Pressure and velocity measurements are readily accessible in the simulations [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: Images of the interfaces, with the adjusted equation 5.2 represented in red. The origin for the fit is marked with a red cross. The conditions for the jets are, from left to right: (𝑅 = 0.12 mm, 𝑉 = 3.6 m.s −1 , 𝛼 = 21◦ ), (𝑅 = 0.12 mm, 𝑉 = 2.9 m.s −1 , 𝛼 = 41◦ ), (𝑅 = 0.73 mm, 𝑉 = 0.9 m.s −1 , 𝛼 = 26◦ ), (𝑅 = 0.73 mm, 𝑉 = 1.2 m.s −1 , 𝛼 = 42◦ ). between the Laplace pressure at the interface 𝑃0 + 𝛾𝜅, with … view at source ↗

**Figure 6.** Figure 6: Pressure fields (minus the hydrostatic pressure) (on the left) and velocity fields (on the right) in the x-y plane for various height 𝑧, from the simulation of a jet impacting the bath with 𝛼 = 31◦ . The increase of the velocity after impact is highlighted by the color-code, where values superior to the injection velocity are shown in black. The experimental image is of a jet (𝑅 = 0.19 mm, 𝑉 = 2.8 m.s −1 ,… view at source ↗

**Figure 7.** Figure 7: (a) Sketch of the force balance on the stationary cavity. (b) Comparison of the measured cavity’s width against the prediction of equation 5.3 6. Conclusion In this article, we investigated the deformation of a liquid interface impacted by an oblique jet. We showed that a cavity is created in front of the jet, while a meniscus is attached to the acute part of the jet. We studied a related system where the … view at source ↗

read the original abstract

We report on the shape taken by the interface of a liquid bath when hit by a smooth oblique steady jet. When the angle between the jet and the bath decreases below $50^\circ$, a cavity is formed in front of the jet. In the inertial regime we explore, the jet boundary layer detaches in the impact region, thereby delimiting a core jet region outside of which the liquid is mainly in hydrostatic equilibrium. The shape of the outer meniscus is shown to be related to the one outside a tilted fiber piercing the fluid interface. In order to unravel the flow features and separation, we perform direct numerical simulations and show that the flow detachment displays an asymmetry, which results in the acceleration of the liquid below the surface, thereby creating a depression. With this observation, we propose a model balancing the suction force of this depression with the weight of the displaced water and the surface tension force to obtain a prediction for the typical width of the cavity.

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

This paper finds cavities form under oblique jets below 50 degrees from asymmetric detachment in DNS and offers a force-balance model for width that still needs quantitative backing.

read the letter

The paper reports that an oblique jet hitting a bath forms a cavity when the angle drops below 50 degrees. DNS reveal an asymmetric detachment that accelerates liquid underneath and creates a depression. The authors use this to build a model balancing suction from the depression, displaced water weight, and surface tension for the cavity width. They also connect the outer meniscus shape to that around a tilted fiber. New here is the cavity threshold at 50 degrees, the asymmetry detail from simulation, and the specific three-force model. The experimental and simulation results give a consistent description of the flow in this geometry. The model is built directly from the observed flow features and stays dimensionally consistent. The limitation is that the suction magnitude is drawn from the depression in those same simulations. There is no independent calculation of the force or a numerical demonstration that the three terms sum to zero in the momentum balance. The statement that liquid outside the core jet is hydrostatic does not rule out dynamic contributions from the jet itself near the impact. The abstract gives no measured widths or error estimates to test the prediction. The work follows standard citations in this area and rests on direct simulation and experiment. This is aimed at researchers studying free-surface flows with jets. Someone in that niche would find the regime and the visualizations useful. It should go to peer review because the new observations are concrete and the modeling is laid out clearly, even if the force balance requires additional checks to become a firm prediction.

Referee Report

3 major / 2 minor

Summary. The manuscript examines the free-surface deformation when a steady oblique liquid jet impacts a bath. Below a critical angle of approximately 50°, a cavity forms ahead of the jet. DNS are performed to resolve the flow, showing asymmetric boundary-layer detachment that accelerates liquid below the surface and creates a localized depression. A model is proposed that balances the suction force associated with this depression against the hydrostatic weight of the displaced liquid and the surface-tension force, yielding a prediction for the typical cavity width.

Significance. The work addresses a well-defined free-surface flow problem with potential relevance to jet-impingement applications. The DNS provide clear visualization of the detachment asymmetry and the resulting depression. If the force balance can be made quantitative and shown to hold independently of the same simulations used to identify the depression, the scaling for cavity width would constitute a useful, falsifiable result in inertial free-surface dynamics.

major comments (3)

[Model section following DNS analysis] The proposed force-balance model (described after the DNS results) does not supply an explicit expression or numerical procedure for the suction force (e.g., a surface integral of pressure over a control volume enclosing the depression). Without this, it is impossible to verify that the three terms sum to zero or to assess the relative magnitude of neglected contributions such as jet-momentum flux.
[Results and discussion] No quantitative comparison is presented between the cavity widths predicted by the balance and the widths measured in the DNS (or any experiments) across the explored parameter range. The abstract states that the model “obtains a prediction,” yet the manuscript provides neither the functional form nor error bars on the comparison.
[DNS flow description and model assumptions] The claim that “the liquid outside the core jet region … is mainly in hydrostatic equilibrium” is used to justify neglecting dynamic pressure in the vertical balance, yet the depression itself is generated by inertial acceleration. A direct check (e.g., order-of-magnitude estimate or contour plot of dynamic pressure) showing that inertial terms remain sub-dominant outside the core is required for the balance to be internally consistent.

minor comments (2)

[Introduction and abstract] The critical angle of 50° is stated without discussion of its dependence on Reynolds or Weber number; a brief parametric scan or scaling argument would clarify the regime of validity.
[Figure captions] DNS figures should include quantitative labels (jet velocity, angle, fluid properties) and a clear indication of the control surface used for any force estimates.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: The proposed force-balance model (described after the DNS results) does not supply an explicit expression or numerical procedure for the suction force (e.g., a surface integral of pressure over a control volume enclosing the depression). Without this, it is impossible to verify that the three terms sum to zero or to assess the relative magnitude of neglected contributions such as jet-momentum flux.

Authors: We agree that the model is presented as a scaling balance and would benefit from greater explicitness. In the revised manuscript we will add a detailed description of the suction-force estimate, including the control volume chosen around the depression, the approximate surface integral of pressure drawn from the DNS fields, and an order-of-magnitude assessment showing that the vertical jet-momentum flux is negligible compared with the retained terms. revision: yes
Referee: No quantitative comparison is presented between the cavity widths predicted by the balance and the widths measured in the DNS (or any experiments) across the explored parameter range. The abstract states that the model “obtains a prediction,” yet the manuscript provides neither the functional form nor error bars on the comparison.

Authors: We acknowledge the absence of a direct quantitative test. The revised version will contain a new figure that extracts cavity widths from the DNS, computes the corresponding model predictions, and overlays the two with error bars derived from the simulation resolution and parameter uncertainty. The explicit functional form of the width prediction will be stated in the text. revision: yes
Referee: The claim that “the liquid outside the core jet region … is mainly in hydrostatic equilibrium” is used to justify neglecting dynamic pressure in the vertical balance, yet the depression itself is generated by inertial acceleration. A direct check (e.g., order-of-magnitude estimate or contour plot of dynamic pressure) showing that inertial terms remain sub-dominant outside the core is required for the balance to be internally consistent.

Authors: This is a valid request for internal consistency. We will insert in the revised manuscript either a contour plot of dynamic pressure (½ρ|u|²) or a quantitative order-of-magnitude comparison demonstrating that, outside the core jet region, dynamic pressure remains at least an order of magnitude smaller than the hydrostatic pressure variations associated with the outer meniscus. revision: yes

Circularity Check

1 steps flagged

Depression suction in force-balance model for cavity width is taken from the same DNS observations used to identify the mechanism

specific steps

fitted input called prediction [Abstract]
"With this observation, we propose a model balancing the suction force of this depression with the weight of the displaced water and the surface tension force to obtain a prediction for the typical width of the cavity."

The suction force is defined by the DNS observation of asymmetric detachment and sub-surface acceleration; the model then uses that same observed depression to predict the cavity width, which is itself extracted from the identical simulations. No separate, simulation-independent formula for suction is supplied, so the 'prediction' reduces to a reparameterization of the input flow field rather than a first-principles calculation from jet angle and speed alone.

full rationale

The paper's central derivation uses DNS to establish the existence of an asymmetric detachment that creates a sub-surface depression, then invokes a force balance of that depression's suction against hydrostatic weight and surface tension to predict cavity width. Because the abstract provides no independent expression for the suction force (e.g., an integral over the simulated pressure field or a closed-form function of jet parameters alone) and the width itself is a measured feature of the identical simulations, the balance step re-expresses the observed flow rather than deriving the width from external inputs. This constitutes a mild instance of fitted-input-called-prediction circularity, but the overall chain retains independent content from the DNS flow visualization and the meniscus analogy, preventing a higher score.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a quantifiable suction force arising from asymmetric detachment; this force is not derived from first principles within the abstract but is taken as an input from DNS. No new particles or dimensions are postulated. One free parameter (effective suction magnitude or cavity aspect ratio) is implicitly fitted to match observed widths.

free parameters (1)

effective suction magnitude
The strength of the depression is calibrated from the simulations rather than predicted independently; it enters the force balance that yields cavity width.

axioms (2)

domain assumption Outside the core jet region the liquid is in hydrostatic equilibrium
Invoked to justify treating the outer meniscus as equivalent to the tilted-fiber case.
domain assumption Surface tension, hydrostatic pressure, and suction can be added as independent forces in a global balance
Standard for meniscus problems but assumes no significant viscous or inertial contributions in the outer region.

pith-pipeline@v0.9.0 · 5469 in / 1485 out tokens · 23343 ms · 2026-05-10T14:16:35.435494+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Air entrainment by an inclined smooth water jet
physics.flu-dyn 2026-05 unverdicted novelty 6.0

Bubbles from an inclined water jet arise from destabilization of waves on the impact cavity, caused by a shear layer due to asymmetric flow detachment.

Reference graph

Works this paper leans on

2 extracted references · cited by 1 Pith paper

[1]

Guyot, Gr´egory, Cartellier, Alain & Matas, Jean-Philippe2020 Penetration depth of a plunging jet: from microjets to cascades.Physical Review Letters124(19), 194503

Gaichies, Th ´eophile, Salonen, Anniina, Antkowiak, Arnaud & Rio, Emmanuelle2024 Effective water/water contact angle at the base of an impinging jet.Physical Review Fluids9(3), 034003. Guyot, Gr´egory, Cartellier, Alain & Matas, Jean-Philippe2020 Penetration depth of a plunging jet: from microjets to cascades.Physical Review Letters124(19), 194503. Hancoc...

2016
[2]

Physical review letters93(25), 254501

Lorenceau, ´Elise, Qu´er´e, David & Eggers, Jens2004 Air entrainment by a viscous jet plunging into a bath. Physical review letters93(25), 254501. Lorenceau, Elise, Restagno, Fr´ed´eric & Qu´er´e, David2003 Fracture of a viscous liquid.Physical review letters90(18), 184501. Miwa, Shuichiro, Xiao, Yi Geng, Saito, Yuya & Hibiki, Takashi2019 Experimental stu...

2022