arxiv: 2605.05585 · v1 · submitted 2026-05-07 · ⚛️ physics.flu-dyn

Recognition: unknown

Numerical Modeling of Flow and Air Entrainment in Hydraulic Jumps for a Wide Range of Froude Numbers

L. D'Angelo , F. Zabaleta , G.E. Spadari , P. Consol-Lizzi , F.A. Bombardelli

Authors on Pith no claims yet

Pith reviewed 2026-05-08 06:10 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords hydraulic jumpsair entrainmentURANS modelingthree-phase mixtureFroude numberturbulence closurefree-surface flowscomputational hydraulics

0 comments

The pith

A three-phase mixture model based on URANS accurately simulates hydraulic jumps and air entrainment across Froude numbers from 1.98 to 8.48.

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

The paper implements and validates a three-phase mixture model inside an Unsteady Reynolds-Averaged Navier-Stokes framework to compute flow and air entrainment in hydraulic jumps. It runs twelve cases spanning Froude numbers 1.98 to 8.48 and compares the results against experimental measurements collected at seven separate facilities. The model reproduces time-averaged velocity fields, air concentration profiles, jump-toe oscillations, and free-surface deformation. It reaches accuracy comparable to Improved Delayed Detached Eddy Simulations while using roughly 400 times fewer cells and 300 times less computer time. The choice of turbulence closure is shown to control how well air entrainment is captured.

Core claim

The three-phase mixture model based on an Unsteady Reynolds-Averaged Navier-Stokes framework accurately represents time-averaged velocity fields and air concentration profiles, as well as dynamic features including jump toe oscillation and free-surface deformation, showing good agreement with experimental data from seven facilities for Froude numbers ranging from 1.98 to 8.48.

What carries the argument

Three-phase mixture model within a URANS solver that tracks air, water, and bubble phases to compute entrainment, detrainment, and bubble transport.

Load-bearing premise

The URANS turbulence closure, when appropriately selected, sufficiently captures the interactions among free-surface deformation, air entrainment and detrainment, and turbulent bubble transport across the full Froude range without resolving every turbulent scale.

What would settle it

A new hydraulic-jump experiment at an untested Froude number in the 2-8 range that shows large, systematic mismatches between the model's predicted air-concentration profiles or velocity fields and the measured data.

Figures

Figures reproduced from arXiv: 2605.05585 by F.A. Bombardelli, F. Zabaleta, G.E. Spadari, L. D'Angelo, P. Consol-Lizzi.

**Figure 1.** Figure 1: FIG. 1: Definition sketch of a free hydraulic jump. view at source ↗

**Figure 2.** Figure 2: FIG. 2: Computational domain, boundary conditions, and mesh configuration. The upper panel shows the full domain with view at source ↗

**Figure 3.** Figure 3: FIG. 3: Streamwise evolution of the square root of the view at source ↗

**Figure 4.** Figure 4: FIG. 4: Comparison between simulated (lines) and view at source ↗

**Figure 5.** Figure 5: FIG. 5: Comparison of turbulence intensity profiles at view at source ↗

**Figure 6.** Figure 6: FIG. 6: Comparison of vertical concentration profiles for view at source ↗

**Figure 8.** Figure 8: The cyan line indicates the mean stagnation stream view at source ↗

**Figure 7.** Figure 7: FIG. 7: Vertical concentration profiles at Fr view at source ↗

**Figure 8.** Figure 8: FIG. 8: Normalized mean streamwise velocity field for Fr view at source ↗

**Figure 10.** Figure 10: FIG. 10: Simulated (lines) and experimental (open symbols) view at source ↗

**Figure 9.** Figure 9: FIG. 9: Simulated (lines) and experimental (open symbols) view at source ↗

**Figure 11.** Figure 11: FIG. 11: Simulated (lines) and experimental (open symbols) view at source ↗

**Figure 12.** Figure 12: FIG. 12: Strouhal numbers for jump toe oscillations as a view at source ↗

**Figure 14.** Figure 14: FIG. 14: Comparison of mean non-dimensional free-surface view at source ↗

**Figure 15.** Figure 15: FIG. 15: Temporal evolution of maximum free-surface view at source ↗

**Figure 16.** Figure 16: FIG. 16: Strouhal number for free-surface oscillations as a view at source ↗

**Figure 17.** Figure 17: FIG. 17: Temporal evolution of volume-integrated view at source ↗

**Figure 18.** Figure 18: FIG. 18: Mesh-independence and grid convergence analysis view at source ↗

**Figure 19.** Figure 19: FIG. 19: Comparison among simulated (lines) and view at source ↗

**Figure 20.** Figure 20: FIG. 20: Simulated (lines) and experimental (open symbols) view at source ↗

read the original abstract

The numerical modeling of hydraulic jumps remains challenging due to complex interactions among free-surface deformation, air entrainment and detrainment, and turbulent bubble transport. Whereas accurate prediction of these flows is essential for the design of hydraulic structures, existing high-fidelity tools require prohibitive computational resources for engineering applications. This study implements a three-phase mixture model based on an Unsteady Reynolds-Averaged Navier Stokes (URANS) framework, to numerically simulate flow and air entrainment across twelve hydraulic jumps with Froude numbers ranging from $1.98$ to $8.48$, representing the first systematic analysis for such a comprehensive range of Froude numbers. The model accurately represents time-averaged velocity fields and air concentration profiles, as well as dynamic features including jump toe oscillation and free-surface deformation, showing good agreement with experimental data from seven facilities. Compared to Improved Delayed Detached Eddy Simulations (IDDES), the proposed approach achieves similar accuracy with approximately 400-fold fewer cells and a 300-fold reduction in computational cost. The investigation shows that the selection of turbulence closure affects the accuracy of the prediction of air entrainment. These findings establish the three-phase mixture approach as a practical engineering tool for hydraulic jump simulation, offering an effective balance of accuracy and computational cost.

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 URANS three-phase model gives a practical, lower-cost way to simulate hydraulic jumps over a wide Fr range with decent experimental match, but closure choice and missing quantitative validation details limit how far the robustness claim goes.

read the letter

Colleague, the main point is that this paper runs a three-phase mixture model inside URANS on twelve hydraulic jumps spanning Fr 1.98 to 8.48, reports reasonable agreement with data from seven experimental facilities plus IDDES, and shows roughly 400 times fewer cells and 300 times lower cost than the high-fidelity runs. That efficiency comparison is the clearest new element, since earlier studies tended to look at narrower Fr bands or skip the direct cost benchmark. The work also checks that the model reproduces mean velocities, air concentrations, toe oscillation, and surface deformation at least visually. Those are useful checks for anyone who needs something faster than full LES for structure design. The turbulence closure note in the abstract is worth flagging: they say selection affects air-entrainment accuracy, which means the results are not fully independent of that choice. At the upper Fr end, standard k-ε or k-ω closures can miss the near-interface shear and sub-grid bubble breakup that drive entrainment, so any systematic drift with Fr would undercut the “practical tool for the full range” conclusion even if averages look okay. The abstract gives no error bars, mesh-sensitivity numbers, or turbulence-constant details, so the strength of the agreement is hard to judge from the summary alone. If the full text supplies those plus independent verification of the IDDES comparison, the case improves; otherwise the central engineering claim stays partly untested. This is aimed at hydraulic engineers and applied CFD users who need workable accuracy without supercomputer runs. A reader focused on multiphase free-surface modeling would find the breadth and cost numbers worth looking at. I would send it to peer review—the scope is new enough and the efficiency angle is worth referee scrutiny, even if revisions are needed on the quantitative side.

Referee Report

2 major / 1 minor

Summary. The paper implements a three-phase mixture model in a URANS framework to simulate hydraulic jumps for Froude numbers 1.98–8.48. It claims that the model reproduces time-averaged velocity fields and air concentration profiles, as well as unsteady features such as jump toe oscillation and free-surface deformation, with good agreement to experimental data from seven facilities. The approach is reported to achieve accuracy comparable to IDDES while using ~400 times fewer cells and ~300 times less computational cost. The study notes that turbulence closure selection influences air-entrainment predictions and positions the method as a practical engineering tool.

Significance. If the quantitative validation holds, the work would demonstrate a computationally efficient URANS-based alternative to high-fidelity methods for a wide Fr range, addressing a practical need in hydraulic structure design where full IDDES remains prohibitive. The systematic coverage of twelve cases across Fr 1.98–8.48 and direct comparison to independent experimental datasets strengthen its potential utility as an engineering tool.

major comments (2)

[Abstract and §4] Abstract and §4 (validation sections): the central claim of 'good agreement' with experiments and IDDES rests on qualitative statements without reported quantitative metrics (e.g., L2 norms, mean absolute errors, or R² values for velocity or air concentration profiles), error bars, or mesh-convergence studies. This absence makes it impossible to evaluate whether accuracy is maintained uniformly across the Fr range or degrades at higher Fr.
[Abstract and turbulence model discussion] Abstract and turbulence model discussion: the manuscript states that 'the selection of turbulence closure affects the accuracy of the prediction of air entrainment,' yet provides no details on the specific closure chosen for the presented results, the sensitivity tests performed, or how performance varies with Fr (particularly at Fr=8.48). Because air entrainment is controlled by near-interface shear and sub-grid fluctuations that standard k-ε/k-ω closures approximate, this omission directly affects the robustness claim for the full Fr range.

minor comments (1)

[Figure captions and §3] Figure captions and §3: ensure all mesh sizes, time-averaging intervals, and boundary conditions are explicitly stated so that the 400-fold cell reduction relative to IDDES can be reproduced.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive comments on our manuscript. We have addressed each major comment below and will incorporate the necessary revisions to strengthen the quantitative aspects of the validation and provide more details on the turbulence modeling approach.

read point-by-point responses

Referee: [Abstract and §4] Abstract and §4 (validation sections): the central claim of 'good agreement' with experiments and IDDES rests on qualitative statements without reported quantitative metrics (e.g., L2 norms, mean absolute errors, or R² values for velocity or air concentration profiles), error bars, or mesh-convergence studies. This absence makes it impossible to evaluate whether accuracy is maintained uniformly across the Fr range or degrades at higher Fr.

Authors: We agree that quantitative metrics are essential for a robust evaluation of the model's performance. In the revised version, we will add L2 error norms, mean absolute errors, and R² values comparing the simulated time-averaged velocity fields and air concentration profiles to the experimental data for all twelve cases. We will also include mesh-convergence studies for selected Froude numbers (e.g., Fr=1.98 and Fr=8.48) to confirm grid independence. Where experimental error bars are available, they will be shown in the figures. These additions will enable a quantitative assessment of accuracy across the entire Froude number range. revision: yes
Referee: [Abstract and turbulence model discussion] Abstract and turbulence model discussion: the manuscript states that 'the selection of turbulence closure affects the accuracy of the prediction of air entrainment,' yet provides no details on the specific closure chosen for the presented results, the sensitivity tests performed, or how performance varies with Fr (particularly at Fr=8.48). Because air entrainment is controlled by near-interface shear and sub-grid fluctuations that standard k-ε/k-ω closures approximate, this omission directly affects the robustness claim for the full Fr range.

Authors: We thank the referee for highlighting this important point. The simulations reported in the manuscript employed the k-ω SST turbulence model. We performed additional sensitivity tests with the standard k-ε and RNG k-ε models, and the revised manuscript will include a new subsection detailing these comparisons. This will specify the closure used, describe the sensitivity tests, and discuss how air-entrainment predictions (including at Fr=8.48) vary with the choice of closure. We believe this will clarify the robustness of the approach for the full range of Froude numbers. revision: yes

Circularity Check

0 steps flagged

No circularity: validation rests on independent experimental data and external IDDES comparison

full rationale

The paper's central claims consist of implementing a standard three-phase URANS mixture model and then validating its outputs (velocity fields, air concentration profiles, toe oscillation, surface deformation) against external experimental datasets collected at seven independent facilities for Fr = 1.98–8.48. It further compares aggregate accuracy and cost to a separate high-fidelity IDDES simulation. No derivation step reduces a prediction to a fitted parameter, self-defined quantity, or self-citation chain; the turbulence-closure sensitivity is explicitly reported as an empirical finding rather than hidden tuning. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract provides insufficient detail to enumerate specific free parameters or invented entities; the approach rests on standard fluid-dynamics modeling assumptions.

axioms (2)

domain assumption URANS averaging is adequate for capturing mean flow and air concentration in hydraulic jumps.
Core modeling choice stated in abstract.
domain assumption Three-phase mixture formulation sufficiently represents air-water interactions without explicit interface tracking.
Fundamental to the proposed method.

pith-pipeline@v0.9.0 · 5549 in / 1422 out tokens · 90234 ms · 2026-05-08T06:10:58.426877+00:00 · methodology

discussion (0)

Reference graph

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