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

Matthias Kramer

TL;DR

Addresses friction prediction in high-Froude-number aerated open-channel flows where air entrainment modulates mixture density. Introduces a Favre-averaged shallow-water framework and a Darcy–Weisbach friction factor that decomposes into uniform, spatial, and temporal contributions with momentum and pressure correction factors. Demonstrates the approach on published data to show physically consistent friction estimates and provides a practical Gradually Varied Flow reduction for steadily varied flows. The work delivers a mechanistic, density-weighted modelling foundation that supports 1D SWE solvers and enables interpretation across aeration regimes, improving predictions of energy dissipation and informing design and safety analyses.

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. In this work, we introduce a density-weighted (Favre) averaging approach to rigorously account for vertical distributions of air concentration and velocity. Favre averaging naturally captures variations in mixture density induced by air entrainment, thereby enabling a density-consistent Shallow Water Equation (SWE) formulation for aerated flows. 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 from the literature demonstrates that the Favre-averaged SWE framework provides a physically consistent means of quantifying effective friction. Overall, this work establishes a mechanistic, density-weighted methodology for modelling resistance in high-Froude-number aerated flows, provides new physical insight into the role of aeration in frictional dissipation, and lays a rational foundation for future modelling of unsteady and rapidly varied aerated flows.

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

TL;DR

Addresses friction prediction in high-Froude-number aerated open-channel flows where air entrainment modulates mixture density. Introduces a Favre-averaged shallow-water framework and a Darcy–Weisbach friction factor that decomposes into uniform, spatial, and temporal contributions with momentum and pressure correction factors. Demonstrates the approach on published data to show physically consistent friction estimates and provides a practical Gradually Varied Flow reduction for steadily varied flows. The work delivers a mechanistic, density-weighted modelling foundation that supports 1D SWE solvers and enables interpretation across aeration regimes, improving predictions of energy dissipation and informing design and safety analyses.

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. In this work, we introduce a density-weighted (Favre) averaging approach to rigorously account for vertical distributions of air concentration and velocity. Favre averaging naturally captures variations in mixture density induced by air entrainment, thereby enabling a density-consistent Shallow Water Equation (SWE) formulation for aerated flows. 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 from the literature demonstrates that the Favre-averaged SWE framework provides a physically consistent means of quantifying effective friction. Overall, this work establishes a mechanistic, density-weighted methodology for modelling resistance in high-Froude-number aerated flows, provides new physical insight into the role of aeration in frictional dissipation, and lays a rational foundation for future modelling of unsteady and rapidly varied aerated flows.
Paper Structure (20 sections, 53 equations, 4 figures)

This paper contains 20 sections, 53 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Methods
  3. Favre averaging in aerated flows
  4. Definition of Favre averages
  5. Depth-integrated pressure and Favre centroid
  6. Density-weighted shallow-water framework for aerated flows
  7. Mixture continuity equation
  8. Streamwise mixture momentum equation
  9. Effective friction factor formulation
  10. General decomposition of the effective friction factor
  11. Flow-region interpretation and limiting behaviour
  12. Results: Application of the proposed framework
  13. Air-water flow properties and correction factors
  14. Streamwise development of effective friction factors
  15. Discussion
  16. ...and 5 more sections

Figures (4)

  • Figure 1: Flow regions and flow structure of an aerated high-Froude-number flow; conceptual sketch expanded from Cain1978; $d_\mathrm{eq} =$ equivalent clear-water depth; $L_i =$ upstream distance to aeration inception; $x =$ streamwise coordinate; $z=$ bed-normal coordinate; $z_0 =$ bed elevation relative to datum; $z_{90} =$ mixture flow depth; $\theta =$ bed slope angle.
  • Figure 2: Air–water flow properties, momentum and kinetic energy correction factors for aerated flows (data from Bung2009): (a) Representative air concentration profile for $Fr = 4.0$, $q = 0.11$ m$^2$/s, $\theta = 18.4^\circ$, step height $s =0.06$ m, step edge 11; (b) Corresponding interfacial velocity profile; (c) Momentum correction factors $\beta$ and kinetic energy correction factors $\alpha$ evaluated for the complete dataset of Bung2009, with $3.9 \leq Fr \leq 6.5$.
  • Figure 3: Mixture density profile, pressure distribution, and pressure correction factors $\Omega$ and $\Omega_E$ for aerated flows: (a) Representative mixture density profile for $Fr = 4.0$, $q = 0.11$ m$^2$/s, $\theta = 18.4^\circ$, step height $s = 0.06$ m, step edge 11 (data from Bung2009); (b) Corresponding mixture pressure and clear-water pressure distributions; (c) Pressure correction factors for micro-rough and macro-rough inverts. $\Omega$ data from Straub1958, Killen1968, Bung2009, Severi2018, Zhang2017DISS, and Kramer2018Transiton; $\Omega_E$ data from Bung2009.
  • Figure 4: Estimation of effective friction factors in aerated flows: (a) Streamwise development of $d_\mathrm{eq}/d_c$ for $q = 0.11$ m$^2$/s, $\theta = 18.4^\circ$, $s = 0.06$ m (data from Bung2009); (b) Streamwise decomposition of the total friction factor $f_e$ for the same dataset; (c) Streamwise variation of the ratio between spatial and uniform-flow friction contributions, $|f_{e,\mathrm{spatial}}|/f_{e,\mathrm{uniform}}$, for the same dataset; (d) Comparison of energy-based, momentum-based, and uniform friction factor estimates, with $f_{e,E}$ and $f_{e,\mathrm{uniform}}$ plotted against $f_e$. The analysis assumes an approximated gradient of $\Delta d_\mathrm{eq}/\Delta x \approx - 2 \times 10^{-3}$.