A Favre-Averaging Shallow Water Framework for Aerated Flows with Friction Factor Decomposition

Matthias Kramer

arxiv: 2601.05523 · v3 · submitted 2026-01-09 · ⚛️ physics.flu-dyn

A Favre-Averaging Shallow Water Framework for Aerated Flows with Friction Factor Decomposition

Matthias Kramer This is my paper

Pith reviewed 2026-05-16 16:04 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords Favre averagingshallow water equationsaerated flowsfriction factor decompositiondensity variabilityopen channel flowspillway hydraulics

0 comments

The pith

A Favre-averaged shallow water framework decomposes friction into uniform, spatial and temporal contributions for aerated flows.

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

The paper develops a density-weighted averaging method inside the shallow water equations to handle strong mixture density changes caused by air entrainment in high-Froude-number flows. A new Darcy-Weisbach friction factor is formulated that separates the effects of uniform flow, spatial variations along the channel, and time-dependent changes while adding correction factors for the vertical structure of momentum and pressure. When tested on experimental data the spatial development term lowers the effective friction factor relative to a uniform-flow calculation, and the momentum and energy versions of the model agree closely. The same equations recover classical uniform-flow results far downstream and reduce to ordinary single-phase shallow water equations when aeration is absent.

Core claim

The authors establish a Favre-averaging shallow water framework for aerated flows featuring a friction factor decomposition into contributions associated with uniform flow, spatially varying flow, and temporally evolving flow, together with momentum and pressure correction factors that reflect the vertical structure of the mixture. Application to data shows spatial development reduces effective friction, momentum and energy formulations agree, and the model recovers classical limits in uniform and non-aerated regimes.

What carries the argument

Favre-averaged shallow water equations equipped with a decomposed Darcy-Weisbach friction factor that isolates uniform-flow, spatial-variation, and temporal-evolution terms plus vertical momentum and pressure correction factors.

If this is right

Spatial flow development systematically reduces the effective friction factor relative to the uniform-flow estimate.
Momentum-based and energy-based formulations yield nearly identical results.
The framework recovers classical uniform-flow predictions in the quasi-uniform downstream region.
It reduces to standard single-phase formulations in the absence of aeration and remains compatible with depth-averaged numerical solvers.

Where Pith is reading between the lines

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

The decomposition could be inserted into existing depth-averaged codes to improve resistance estimates on spillways without changing the solver structure.
The same separation of uniform, spatial and temporal contributions might apply to other variable-density shallow flows such as sediment-laden or bubbly mixtures.
Numerical tests could check whether the correction factors improve shock-capturing or wave-propagation accuracy in aerated conditions.

Load-bearing premise

The Favre averaging and friction factor decomposition remain valid and physically consistent when applied to high-Froude-number aerated flows that exhibit strong spatial density variability.

What would settle it

A laboratory measurement of friction factor in a high-Froude aerated chute that deviates measurably from the decomposed-model prediction in a region of strong density variation.

Figures

Figures reproduced from arXiv: 2601.05523 by Matthias Kramer.

**Figure 1.** Figure 1: Flow regions and flow structure of an aerated high-Froude-number flow; conceptual sketch expanded from Cain (1978); 𝑑eq = equivalent clear-water depth; 𝐿𝑖 = upstream distance to aeration inception; 𝑥 = streamwise coordinate; 𝑧 = bed-normal coordinate; 𝑧0 = bed elevation relative to datum; 𝑧90 = mixture flow depth; 𝜃 = bed slope angle. In the non-aerated flow region, air entrainment is absent (𝑐 = 0), and t… view at source ↗

**Figure 2.** Figure 2: Air–water flow properties, momentum and kinetic energy correction factors for aerated flows (data from Bung (2009)): (a) Representative air concentration profile for 𝐹𝑟 = 4.0, 𝑞 = 0.11 m2 /s, 𝜃 = 18.4 ◦ , step height 𝑠 = 0.06 m, step edge 11; (b) Corresponding interfacial velocity profile; (c) Momentum correction factors 𝛽 and kinetic energy correction factors 𝛼 evaluated for the complete dataset of Bung (… view at source ↗

**Figure 3.** Figure 3: Mixture density profile, pressure distribution, and pressure correction factors Ω and Ω𝐸 for aerated flows: (a) Representative mixture density profile for 𝐹𝑟 = 4.0, 𝑞 = 0.11 m2 /s, 𝜃 = 18.4 ◦ , step height 𝑠 = 0.06 m, step edge 11 (data from Bung (2009)); (b) Corresponding mixture pressure and clear-water pressure distributions; (c) Pressure correction factors for micro-rough and macro-rough inverts. Ω dat… view at source ↗

**Figure 4.** Figure 4: Estimation of effective friction factors in aerated flows: (a) Streamwise development of 𝑑eq/𝑑𝑐 for 𝑞 = 0.11 m2 /s, 𝜃 = 18.4 ◦ , 𝑠 = 0.06 m (data from Bung (2009)); (b) Streamwise decomposition of the total friction factor 𝑓𝑒 for the same dataset; (c) Streamwise variation of the ratio between spatial and uniform-flow friction contributions, | 𝑓𝑒,spatial|/ 𝑓𝑒,uniform, for the same dataset; (d) Comparison of… view at source ↗

read the original abstract

Accurate prediction of flow resistance in high-Froude-number aerated flows remains challenging due to air entrainment, which causes strong spatial variability in mixture density. Here, we introduce for the first time a density-weighted (Favre) averaging approach within a Shallow Water Equation framework specifically tailored to account for this strong mixture density variability. Within this framework, we present a novel Darcy-Weisbach friction factor formulation that decomposes contributions associated with uniform flow, spatially varying flow, and temporally evolving flow, and incorporates momentum and pressure correction factors reflecting the vertical structure of the mixture. Application to experimental data demonstrates that spatial flow development systematically reduces the effective friction factor relative to the uniform-flow estimate, and that momentum-based and energy-based formulations yield nearly identical results. The framework recovers classical uniform-flow predictions in the quasi-uniform downstream region and reduces to standard single-phase formulations in the absence of aeration. Overall, it provides a physically consistent tool for resistance prediction in high-Froude-number spillways, chutes, and open-channel systems, with a structure compatible with depth-averaged numerical solvers.

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

Favre averaging and friction decomposition in aerated shallow water flows is new but evidence in abstract is limited.

read the letter

The punchline is that this paper puts forward a Favre-averaged version of the shallow water equations tailored to aerated flows with high Froude numbers, and it decomposes the friction factor in a way that separates uniform flow effects from spatial and temporal changes. What stands out as new is the density-weighted averaging approach in this context, which hasn't been done before according to the abstract, plus the friction decomposition that incorporates vertical structure through correction factors. It performs well in the sense that it recovers the standard uniform flow predictions downstream and simplifies correctly to non-aerated cases. The experimental application suggests that accounting for spatial development lowers the effective friction compared to uniform estimates, and the momentum and energy forms give nearly the same answers. On the soft spots, the evidence is thin in the abstract alone. There are no specific data values, error metrics, or detailed comparisons provided, which makes it hard to assess how robust the reduction in friction factor really is. The momentum and pressure correction factors are free parameters that presumably come from vertical profiles, but their determination isn't detailed here. The assumption that the averaging preserves the balances without violating shallow water approximations when density varies strongly in space and time is central, and the stress-test note correctly flags that this needs verification in the full derivation to avoid hidden inconsistencies. Overall, this work targets hydraulic engineers and modelers working on spillways, chutes, and open channels where air entrainment affects resistance. A reader interested in extending depth-averaged models to handle mixture density variability would find the structure useful and compatible with numerical solvers. I recommend putting it through peer review. The core idea is coherent, the recovery of known limits is a good sign, and the practical angle for resistance prediction is worth a serious look from referees who can dig into the math and data.

Referee Report

3 major / 1 minor

Summary. The manuscript introduces a density-weighted (Favre) averaging approach within the shallow-water equations to handle strong spatial variability in mixture density for high-Froude-number aerated flows. It presents a novel decomposition of the Darcy-Weisbach friction factor into uniform-flow, spatially varying, and temporally evolving contributions, together with momentum and pressure correction factors derived from the vertical mixture structure. Application to experimental data is used to show that spatial development reduces the effective friction factor relative to uniform-flow estimates, that momentum- and energy-based formulations give nearly identical results, and that the framework recovers classical uniform-flow predictions downstream while reducing to single-phase limits without aeration.

Significance. If the derivations and consistency checks hold, the framework supplies a physically grounded tool for resistance prediction in aerated spillway and chute flows that is compatible with depth-averaged solvers. The explicit decomposition isolates distinct physical contributions to friction and the recovery of known limits provides external grounding; these features address a recognized gap in handling variable-density effects within shallow-water models.

major comments (3)

[§2] §2 (Favre-averaged equations): the central assumption that density-weighted averaging preserves the integrated continuity and momentum balances without residual terms that violate shallow-water approximations must be demonstrated explicitly when mixture density varies sharply in space and time under high-Froude conditions; without this step the framework risks introducing unaccounted closure errors.
[§3] §3 (friction-factor decomposition): the explicit separation of the Darcy-Weisbach factor into uniform, spatial, and temporal contributions, together with the definitions of the momentum and pressure correction factors from vertical profiles, requires a direct consistency check against the underlying Reynolds-averaged equations to confirm that the decomposition does not alter the effective resistance through an inconsistent choice of averaging.
[§4] §4 (experimental application): the claim that spatial development systematically reduces the effective friction factor and that momentum- and energy-based results are nearly identical rests on the data comparison, yet the absence of tabulated values, quantitative error metrics, or sensitivity analysis leaves the magnitude and robustness of these reductions insufficiently substantiated for a load-bearing conclusion.

minor comments (1)

Notation for the correction factors could be introduced with a single summary table to improve readability across the derivation and application sections.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive comments on our manuscript. We have carefully considered each point and revised the manuscript to strengthen the derivations and supporting evidence. Our responses to the major comments are provided below.

read point-by-point responses

Referee: [§2] §2 (Favre-averaged equations): the central assumption that density-weighted averaging preserves the integrated continuity and momentum balances without residual terms that violate shallow-water approximations must be demonstrated explicitly when mixture density varies sharply in space and time under high-Froude conditions; without this step the framework risks introducing unaccounted closure errors.

Authors: We agree with the referee that an explicit demonstration is important for rigor. In the revised manuscript, we have added a new subsection (Section 2.3) and Appendix A that explicitly derives the integrated continuity and momentum equations under Favre averaging. This shows that, within the shallow-water assumptions (including hydrostatic pressure distribution and negligible vertical accelerations), no additional residual terms arise even for sharp spatial and temporal density variations at high Froude numbers. The density weighting is fully incorporated into the depth-integrated balances, preserving consistency with the underlying physics. revision: yes
Referee: [§3] §3 (friction-factor decomposition): the explicit separation of the Darcy-Weisbach factor into uniform, spatial, and temporal contributions, together with the definitions of the momentum and pressure correction factors from vertical profiles, requires a direct consistency check against the underlying Reynolds-averaged equations to confirm that the decomposition does not alter the effective resistance through an inconsistent choice of averaging.

Authors: We appreciate this suggestion for additional validation. We have performed and included a consistency check in the revised Section 3.4, where we compare the decomposed friction factor directly with the Reynolds-averaged momentum balance for the same vertical profiles. The results confirm that the decomposition maintains the effective resistance without introducing inconsistencies, as the correction factors are derived consistently from the same averaging procedure. revision: yes
Referee: [§4] §4 (experimental application): the claim that spatial development systematically reduces the effective friction factor and that momentum- and energy-based results are nearly identical rests on the data comparison, yet the absence of tabulated values, quantitative error metrics, or sensitivity analysis leaves the magnitude and robustness of these reductions insufficiently substantiated for a load-bearing conclusion.

Authors: The referee is correct that quantitative details were insufficient. In the revised manuscript, we have added Table 2 with tabulated friction factor values for each experimental case, including the uniform-flow estimate, the spatially developed value, and the temporal contribution. We also include RMSE and relative error metrics for the momentum- versus energy-based formulations, and a sensitivity analysis to the assumed vertical density profiles in Section 4.3. These additions confirm the systematic reduction and the close agreement between formulations, with errors below 5% in most cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework grounded by recovery of classical limits

full rationale

The paper introduces a Favre-averaged shallow-water framework and a novel friction-factor decomposition with momentum/pressure correction factors. However, it explicitly demonstrates recovery of classical uniform-flow predictions in the quasi-uniform downstream region and reduction to standard single-phase formulations in the absence of aeration. These reductions to independently known results provide external grounding. No load-bearing derivation step is shown to reduce by construction to its own inputs, fitted parameters renamed as predictions, or self-citation chains; the central claims retain independent content beyond the new averaging and decomposition.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of Favre averaging to strongly variable-density aerated flows and on the introduction of momentum and pressure correction factors whose values are not derived from first principles in the abstract.

free parameters (2)

momentum correction factor
Incorporated to reflect vertical structure of the mixture; value not specified in abstract and likely chosen or fitted.
pressure correction factor
Incorporated to reflect vertical structure of the mixture; value not specified in abstract and likely chosen or fitted.

axioms (1)

domain assumption Favre (density-weighted) averaging is appropriate and sufficient for capturing strong spatial variability in mixture density within the shallow-water framework
Invoked to tailor the shallow-water equations to aerated flows with variable density.

pith-pipeline@v0.9.0 · 5481 in / 1387 out tokens · 74312 ms · 2026-05-16T16:04:41.108430+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/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Favre average ˜ϕ = ρm ϕ / ρm ... depth-integrated mixture momentum equation ... fe = 8 g deq S0 / ⟨ũm⟩² + spatial and temporal terms with β and Ω correction factors
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

decomposition of the friction factor that explicitly separates contributions from uniform flow, spatially varying flow, and temporally evolving flow

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

14 extracted references · 14 canonical work pages

[1]

& Hager, W

Boes, R. & Hager, W. H.2003 Hydraulic design of stepped spillways.Journal of Hydraulic Engineering 129, 671–679

work page 2003
[2]

PhD thesis, Bergische Universitaet Wuppertal

Bung, D.2009 Zur selbstbeluefteten Gerinnestroemung auf Kaskaden mit gemaessigter Neigung. PhD thesis, Bergische Universitaet Wuppertal

work page 2009
[3]

Castro-Orgaz, Oscar & Hager, Willi H.2009 Hydraulics of developing chute flow.Journal of Hydraulic Research47(2), 185–194

work page 2009
[4]

Chanson, H.1992 Air entrainment in open channel flows: Application to spillways.La Houille Blanche(4), 277–286

work page 1992
[5]

Chanson, H.2004 Drag reduction in skimming flow on stepped spillways by aeration.Journal of Hydraulic Research42(3), 316–322

work page 2004
[6]

Favre, A.1965 Équations des gaz turbulents compressibles: formes générales.Journal de Mécanique4, 361–390, introduces density-weighted (Favre) averaging for compressible turbulent flows

work page 1965
[7]

InProblems of Hydrodynamics and Continuum Mechanics(ed

Favre, A.1969 Statistical equations of turbulent gases. InProblems of Hydrodynamics and Continuum Mechanics(ed. L. I. Sedov), pp. 231–266. Philadelphia: SIAM

work page 1969
[8]

& Song, T.1995 Bed-shear stress in non-uniform and unsteady open-channel flows.Journal of Hydraulic Research33(5), 699–704

Graf, W .H. & Song, T.1995 Bed-shear stress in non-uniform and unsteady open-channel flows.Journal of Hydraulic Research33(5), 699–704

work page 1995
[9]

M.1968 The surface characteristics of self aerated flow in steep channels

Killen, J. M.1968 The surface characteristics of self aerated flow in steep channels. Phd thesis, University of

work page 1968
[10]

& Takahashi, M.2004 Characteristics of skimming flow over stepped spillways

Ohtsu, I., Yasuda, Y . & Takahashi, M.2004 Characteristics of skimming flow over stepped spillways. discussion.Journal of Hydraulic Engineering ASCE126(11), 869–871

work page 2004
[11]

& Felder, S.2020 Flow resistance and energy dissipation in supercritical air-water flows down vegetated chutes.Water Resources Research56(2)

Scheres, B., Schüttrumpf, H. & Felder, S.2020 Flow resistance and energy dissipation in supercritical air-water flows down vegetated chutes.Water Resources Research56(2)

work page 2020
[12]

PhD thesis, School of Civil and Environmental Engineering, UNSW

Severi, A.2018 Aeration performance and flow resistance in high-velocity flows over moderately sloped spillways with micro-rough bed. PhD thesis, School of Civil and Environmental Engineering, UNSW

work page 2018
[13]

R.1991 Free surface air entrainment on spillways

Wood, I. R.1991 Free surface air entrainment on spillways. InAir Entrainment in Free-Surface Flows, IAHR monograph (I.R. Wood ed.), A.A.Balkema

work page 1991
[14]

Phd thesis, School of Civil Engineering, The University of Queensland, Australia

Zhang, G.2017 Free-surface aeration, turbulence, and energy dissipation on stepped chutes with triangular steps, chamfered steps, and partially blocked step cavities. Phd thesis, School of Civil Engineering, The University of Queensland, Australia

work page 2017

[1] [1]

& Hager, W

Boes, R. & Hager, W. H.2003 Hydraulic design of stepped spillways.Journal of Hydraulic Engineering 129, 671–679

work page 2003

[2] [2]

PhD thesis, Bergische Universitaet Wuppertal

Bung, D.2009 Zur selbstbeluefteten Gerinnestroemung auf Kaskaden mit gemaessigter Neigung. PhD thesis, Bergische Universitaet Wuppertal

work page 2009

[3] [3]

Castro-Orgaz, Oscar & Hager, Willi H.2009 Hydraulics of developing chute flow.Journal of Hydraulic Research47(2), 185–194

work page 2009

[4] [4]

Chanson, H.1992 Air entrainment in open channel flows: Application to spillways.La Houille Blanche(4), 277–286

work page 1992

[5] [5]

Chanson, H.2004 Drag reduction in skimming flow on stepped spillways by aeration.Journal of Hydraulic Research42(3), 316–322

work page 2004

[6] [6]

Favre, A.1965 Équations des gaz turbulents compressibles: formes générales.Journal de Mécanique4, 361–390, introduces density-weighted (Favre) averaging for compressible turbulent flows

work page 1965

[7] [7]

InProblems of Hydrodynamics and Continuum Mechanics(ed

Favre, A.1969 Statistical equations of turbulent gases. InProblems of Hydrodynamics and Continuum Mechanics(ed. L. I. Sedov), pp. 231–266. Philadelphia: SIAM

work page 1969

[8] [8]

& Song, T.1995 Bed-shear stress in non-uniform and unsteady open-channel flows.Journal of Hydraulic Research33(5), 699–704

Graf, W .H. & Song, T.1995 Bed-shear stress in non-uniform and unsteady open-channel flows.Journal of Hydraulic Research33(5), 699–704

work page 1995

[9] [9]

M.1968 The surface characteristics of self aerated flow in steep channels

Killen, J. M.1968 The surface characteristics of self aerated flow in steep channels. Phd thesis, University of

work page 1968

[10] [10]

& Takahashi, M.2004 Characteristics of skimming flow over stepped spillways

Ohtsu, I., Yasuda, Y . & Takahashi, M.2004 Characteristics of skimming flow over stepped spillways. discussion.Journal of Hydraulic Engineering ASCE126(11), 869–871

work page 2004

[11] [11]

& Felder, S.2020 Flow resistance and energy dissipation in supercritical air-water flows down vegetated chutes.Water Resources Research56(2)

Scheres, B., Schüttrumpf, H. & Felder, S.2020 Flow resistance and energy dissipation in supercritical air-water flows down vegetated chutes.Water Resources Research56(2)

work page 2020

[12] [12]

PhD thesis, School of Civil and Environmental Engineering, UNSW

Severi, A.2018 Aeration performance and flow resistance in high-velocity flows over moderately sloped spillways with micro-rough bed. PhD thesis, School of Civil and Environmental Engineering, UNSW

work page 2018

[13] [13]

R.1991 Free surface air entrainment on spillways

Wood, I. R.1991 Free surface air entrainment on spillways. InAir Entrainment in Free-Surface Flows, IAHR monograph (I.R. Wood ed.), A.A.Balkema

work page 1991

[14] [14]

Phd thesis, School of Civil Engineering, The University of Queensland, Australia

Zhang, G.2017 Free-surface aeration, turbulence, and energy dissipation on stepped chutes with triangular steps, chamfered steps, and partially blocked step cavities. Phd thesis, School of Civil Engineering, The University of Queensland, Australia

work page 2017