Linking turbulent waves and bubble diffusion in self-aerated open-channel flows: two-state air concentration

Abstract

High-Froude-number flows become self-aerated when the destabilizing effect of turbulence overcomes gravity and surface tension forces. Traditionally, the resulting air concentration profile has been explained using single-layer approaches that invoke solutions of the advection–diffusion equation for air in water, i.e. bubbles’ dispersion. Based on a wide range of experimental evidence, we argue that the complete air concentration profile shall be explained through the weak interaction of different canonical turbulent flows, namely a turbulent boundary layer (TBL) and a turbulent wavy layer (TWL). Motivated by a decomposition of the streamwise velocity into a pure wall flow and a free-stream flow (Krug et al., J. Fluid Mech., vol. 811, 2017, pp. 421–435), we present a physically consistent two-state formulation of the structure of a self-aerated flow. The air concentration is mathematically built upon a modified Rouse profile and a Gaussian error function, resembling vertical mass transport in the TBL and the TWL. We apply our air concentration theory to over 500 profiles from different data sets, featuring excellent agreement. Finally, we show that the turbulent Schmidt number, characterizing the momentum-mass transfer, ranges between 0.2 and 1, which is consistent with previous mass-transfer experiments in TBLs. Altogether, the proposed flow conceptualization sets the scene for more physically based numerical modelling of turbulent mass diffusion in self-aerated flows.

Turbulent free-surface in self-aerated flows: superposition of entrapped and entrained air

“…This section provides a brief summary of the governing equations of the two-state convolution model, while more details are presented in Kramer & Valero (2023). The air concentration of the TBL (c TBL ) is reflected through a solution of the advection-diffusion equation for air in water, whereas the air concentration of the TWL (c TWL ), encompassing bubbles and waves, was found to follow a Gaussian error function (Kramer & Valero 2023)…”

“…y 90 = y(c = 0.9). The two-state model assumes a fluctuating interface that separates the TBL and the TWL, and a convolution of the two states with a Gaussian interface probability leads to the following expression for the mean air concentration (Krug, Philip & Marusic 2017;Kramer & Valero 2023)…”

“…For more information on the development of (2.1) to (2.4), as well as on the definition and determination of associated physical model parameters, the reader is referred to Kramer & Valero (2023).…”

“…It is argued that single-layer approaches are unable to represent these different flow processes, while good data-driven agreement between single-layer models and measurements has been achieved, which is, however, at the expense of empirically fitted coefficients. Recently, Kramer & Valero (2023) presented a physically based two-state convolution formulation for the air concentration that is built upon a turbulent boundary layer (TBL) and a turbulent wavy layer (TWL).…”

“…The key novelty of the present work is the introduction of a superposition principle, which explicitly accounts for entrapped air (waves) and entrained air (bubbles), allowing us to quantify the importance of different physical mechanisms to the mean air concentration. In the following, the two-state formulation of Kramer & Valero (2023) is briefly summarised ( § 2.1). Thereafter, the superposition principle for the air concentration of the TWL is proposed, demonstrating that entrapped air and entrained air follow a Gaussian error function and a normal distribution, respectively ( § 2.2).…”