Self-aerated Flows Research Articles

Numerical modeling of self-aerated flows: Turbulence modeling and the onset of air entrainment

Numerical modeling of self-aerated flows is essential for understanding water systems and designing hydraulic structures. This work discusses and extends a theoretical/numerical model to represent air entrainment in self-aerated flows, which includes a criterion to define the occurrence of air entrainment, based on a balance between disturbing and stabilizing energies. The impact of the turbulence closure on the modeling of the onset of air entrainment and the distribution of bubble concentration is studied with particular emphasis. This work shows that uniform-density formulations of Reynolds-averaged Navier–Stokes closures lead to severe overprediction of air entrainment due to unphysical transport of turbulent kinetic energy generated in the air phase to the water phase. In contrast, combining variable-density turbulence closures with the energy balance criterion enables accurate prediction of the regions where air is entrained for stepped spillways and plunging jets. The choice of turbulence closure significantly influences the proposed criterion, emphasizing the importance of selecting an appropriate closure for an accurate description of the air entrainment process. Based on the conducted tests, the standard k−ε and buoyancy modified k−ε models better predict the onset of air entrainment and bubble distribution in stepped spillways, while the k−ω SST model proves to be more effective in capturing air entrainment at the impingement point in plunging jets. This study expands the capabilities of numerical models in predicting air entrainment and provides valuable insights into the effects and interrelations of the turbulence modeling, the air entrainment occurrence criteria, and the bubble transport equations.

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

The characterisation and the modelling of air concentration distributions in self-aerated free-surface flows has been subject to sustained research interest since the 1970s. Recently, a novel two-state formulation of the structure of a self-aerated flow was proposed by Kramer & Valero (J. Fluid Mech., vol. 966, 2023, A37), which physically explains the air concentration through the weak interaction of two canonical flow momentum layers, comprising a turbulent boundary layer and a turbulent wavy layer (TWL). The TWL was modelled using a Gaussian error function, assuming that the most dominant contribution are wave troughs. Here, it is shown that air bubbles form an integral part of the TWL, and its formulation is expanded by adopting a superposition principle of entrapped air (waves) and entrained air (bubbles). Combining the superposition principle with the two-state formulation, an expression for the depth-averaged (mean) air concentration is derived, which allows us to quantify the contribution of different physical mechanisms to the mean air concentration. Overall, the presented concepts help to uncover new flow physics, thereby contributing fundamentally to our understanding of self-aerated flows.

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

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.

A novel three-phase mixture approach for the numerical modeling of self-aerated flows

This work presents a novel theoretical/numerical model for the simulation of self-aerated flows under a Reynolds-Averaged Navier-Stokes (RANS) framework. The new formulation is based on a three-phase mixture approach composed of a continuous air phase, a bubble phase, and a continuous water phase. A mass transfer mechanism that does not depend on an entrainment function and does not require calibration accounts for the incorporation of air into the flow. A modification in the formulation of the Volume-of-Fluid algorithm (used to track the free surface) allows one to capture the increase in water depth due to the presence of bubbles. The proposed formulation recovers the traditional Volume-of-Fluid formulation for free surface flows in the absence of bubbles, allowing the model to represent simultaneously the aerated and not aerated regions of a flow. Governing equations for the mixture are derived from mass and momentum conservation equations for each phase, and a numerical algorithm that ensures the boundedness of the numerical solution is proposed. The model is tested and validated using four experimental cases: a degassing tank, a bubble plume, a plunging jet, and a stepped spillway, showing very satisfactory results. The new methodology provides a significant advance in the current capabilities for simulating self-aerated flows.

Free surface aeration and development dependence in chute flows

The microscopic description of the mixture behaviour of air–water flow remains a challenge. It is not clear how to represent a complex two-phase interaction with the turbulent air–water structure development process. In this study, based on the air–water mixing fluctuation properties in self-aerated chute flows, a prediction model of air concentration distribution related to a theoretical transition depth is developed. The air–water turbulent mixing analysis reveals that the mixture flow depth, at which the local air concentration is 0.5, represents the interior transition boundary. The agreement of the calculated results with the test data confirms that local water and air turbulent mixing through the free surface area should be equally considered. On the basis of the interior transition depth, the development of self-aeration mainly manifests as the air turbulent mixing process in the low aerated region, while the water turbulent mixing in the high aerated region remains mostly unchanged. A series of relationships concerning the development of self-aerated flows is proposed to enable quantitative estimations in practical applications of water engineering.

Drag Reduction in Aerated Chute Flow: Role of Bottom Air Concentration

Wall shear stress modification in self-aerated flows has been observed in boundary layer–type flows mainly in the form of drag reduction. Due to its complexity, drag reduction has predominantly been studied using experimental techniques and empirical analyses, whereas the development of conceptual approaches is limited. In this study, two-phase open-channel flow equations with variable density are revisited and bottom air concentration is identified as a key parameter for drag reduction in air-water flows. The application of this new theory to smooth and stepped chute flows is introduced, and implications for the determination of friction factors are discussed. A modified version of the Manning-Strickler formula is proposed to model friction factors across a wide range of roughness and bottom air concentration levels.

Air Diffusion and Velocity Characteristics of Self-Aerated Developing Region in Flat Chute Flows

Self-aerated flows in flat chutes are encountered downstream of the bottom outlet, in spillways with a small slope and in storm waterways. In the present study, the development of self-aeration in flat chute flow is described and new experiments are performed in a long flat chute with a pressure outlet for different flow discharge rates. The distribution of air concentration, time mean velocity and velocity fluctuation in flow direction in the self-aerated developing region—where air bubbles do not diffuse next to the channel bottom—were measured and analyzed. The region of self-aeration from free surface was about 27.16% to 51.85% of the water depth in the present study. The analysis results revealed that the maximum distance of air bubble diffusion to the channel bottom increased with the development of self-aeration along the flow direction. This indicates that for flat chute flow, the process of air bubble diffusion from free surface to channel bottom was relatively long. Cross-section velocities increased along the flow direction in the self-aerated developing region, and this trend was much more remarkable in the area near water free surface. The velocity fluctuations in flow direction in cross-sections flattened and increased with the development of self-aerated flow. Higher velocity fluctuations in flow direction corresponded to the presence of much stronger turbulence, which enhanced air bubble diffusion from the water free surface to channel bottom along the flow direction.

Simulation of self-aerated flows by switching interface closures

This work presents a numerical study of a skimming flow regime in a stepped spillway structure, which is a complex two-phase flow with self-aeration phenomenon. The numerical models employed consist in a hybrid approach capable of switching between interface resolving and interface modelling closures on a local level, and a state-of-the-art volume-of-fluid (VOF) method. A new switch criterion for the hybrid approach is proposed, allowing calculations on irregular grid cells. A detached eddy simulation (DES) turbulence model is selected in order to directly resolve the turbulent flow structures triggering the onset of self-aeration. The main flow properties computed in the non-aerated and aerated region are confronted with experimental measurements. Compared to the VOF method, the hybrid approach provides an overall better prediction of time-averaged air concentration, highlighting the positive outcome of modelling sub-grid interface structures.

Free Surface Air Entrainment and Single-Bubble Movement in Supercritical Open-Channel Flow

AbstractThere has been little study on the microscopic bubble entrainment and diffusion process on high-speed self-aerated flows, although the problem under investigation is theoretically important...

Best practices for velocity estimations in highly aerated flows with dual-tip phase-detection probes

Dual-tip phase-detection probes can be used to measure flow properties in gas-liquid flows. Traditionally, time-averaged interfacial velocities have been obtained through cross-correlation analysis of long time-series of phase fraction signals. Using small groups of detected particles, a recently developed adaptive window cross-correlation (AWCC) technique enables the computation of pseudo-instantaneous interfacial velocities and turbulence quantities in highly aerated flows, albeit subject to some smoothing which is due to the use of a finite window duration. This manuscript provides guidance on the selection of optimum processing parameters for the AWCC technique, additionally addressing shortcomings such as velocity bias correction and extrapolation of turbulence levels to single particles. The presented technique was tested for three self-aerated flows: smooth and rough-wall boundary layers (tunnel chute and stepped spillway), as well as breaking shear layer flows of a hydraulic jump. Robust estimations of mean velocities and velocity fluctuations were obtained for all flow situations, either using dual-tip conductivity or fiber optical probe data. The computation of integral time scales and velocity spectra is currently limited by the data rate and must be treated with caution.

Experimental study of air-water interface properties in self-aerated flows

The microscopic air-water structures and the interface area properties in the self-aerated flows are important research issues in the high-speed self-aerated flows. The present experimental study concerns the mean and medium air chord length distributions in the self-aerated chute flows for different flow Reynolds numbers and air concentration conditions. The relationship between the microscopic and the macroscopic aerated properties in the air-water mixture region is analyzed. The distribution of the microscopic specific air-water interface area with the macroscopic air concentration variation is self-similar in the self-aerated region. In view of the difference of the air-water structure in high and low aerated regions, a new relationship is proposed for predicting the distributions of a specific air-water interface area, and the agreement between the measured and predicted results is satisfactory.

Analysis on Link Between the Macroscopic and Microscopic Air–Water Properties in Self-Aerated Flows

Self-aeration in high-speed free surface flows occurs commonly and is of interest to ocean engineering, hydraulic engineering, and environmental engineering. For two-phase air–water flows, macroscopic air–water flow properties develop gradually, accompanied by the change of microscopic air–water structures. In this article, representational experimental studies on macroscopic and microscopic characteristics of self-aerated open-channel flows are summarized and compared. The isolated effect of the flow Reynolds number and air quantity on the differences in air count rate and chord size are analyzed and discussed. The results show that the characterized flow depth y50, affected by the turbulence transfer, is a specific criterion to distinguish the interior air–water structure development. Two distinct linear trends of self-aeration are found, depending on the y50/y90 variation with a breaking point at Cmean = 0.50. The air count rate and size scale in self-aerated flows are affected by the air quantity of self-aerated flows, even with identical flow Reynolds numbers. Thus, a specific parameter is proposed to assess the air–water structures and a series of self-similarity relationships in self-aeration properties are obtained. The link between macroscopic and microscopic air–water properties results in significant scale effect on air–water structures in self-aerated flows.

Free-surface air entrainment in open-channel flows

In hydraulic engineering, free-surface aeration is a natural phenomenon occurring in smooth channel flows. In self-aerated flows, a key aspect that has not yet been well understood is the formation mechanism of free-surface air entrainment. In this research, the process of free-surface entrapped deformation is analyzed theoretically and the critical radius of curvature for air entrainment is obtained, affected by flow mean velocity and depth. When the severity of local free-surface deformation exceeds the critical condition, the entrapped free surface encounters closure in the unstable deformation movement process, resulting in air entrainment. This inference agrees well with observed experimental results that are obtained from the processes of surface entrapped deformation and air entrainment captured by a high-speed camera-based data acquisition system. This agreement indicates that self-aeration occurs in low-velocity open-channel flows. It is also confirmed that free-surface turbulent deformation provides a mechanism for air entrainment.

Total pressure fluctuations and two-phase flow turbulence in self-aerated stepped chute flows

Current knowledge in high-velocity self-aerated flows continues to rely upon physical modelling. Herein a miniature total pressure probe was successfully used in both clear-water and air-water flow regions of high-velocity open channel flows on a steep stepped channel. The measurements were conducted in a large size facility (θ=45°, h=0.1m, W=0.985m) and they were complemented by detailed clear-water and air-water flow measurements using a Prandtl-Pitot tube and dual-tip phase-detection probe respectively in both developing and fully-developed flow regions for Reynolds numbers within 3.3×105 to 8.7×105. Upstream of the inception point of free-surface aeration, the clear-water developing flow was characterised by a developing turbulent boundary layer and an ideal-flow region above. The boundary layer flow presented large total pressure fluctuations and turbulence intensities, with distributions of turbulence intensity close to intermediate roughness flow data sets: i.e., intermediate between d-type and k-type. The total pressure measurements were validated in the highly-aerated turbulent shear region, since the total pressure predictions based upon simultaneously-measured void fraction and velocity data agreed well with experimental results recorded by the total pressure probe. The results demonstrated the suitability of miniature total pressure probe in both monophase and two-phase flows. Both interfacial and water phase turbulence intensities were recorded. Present findings indicated that the turbulence intensity in the water phase was smaller than the interfacial turbulence intensity.

Phase-detection measurements in free-surface turbulent shear flows

High-velocity self-aerated flows are described as 'white waters' because of the entrained air bubbles. The air entrainment induces a drastic change in the multiphase flow structure of the water column and this leads to significant bubble-turbulence interactions, turbulence modulation and associated mixing processes impacting on the bulk flow properties. In these high-velocity free-surface turbulent flows, the phase-detection needle probe is a most reliable instrumentation. The signal processing of a phase-detection probe is re-visited herein. It is shown that the processing may be performed on the raw probe signal as well as the thresholded data. The latter yields the time-averaged void fraction, the bubble count rate, the particle chord time distributions and the particle clustering properties within the particulate flow regions. The raw probe signal analysis gives further the auto-correlation time scale and the power spectrum density function. Finally dimensional considerations are developed with a focus on the physical modelling of free-surface flows in hydraulic structures. It is argued that the notion of scale effects must be defined in terms of some specific set of air–water flow properties within well-defined testing conditions, while a number of free-surface flow characteristics are more prone to scale effects than others, even in large-size physical facilities.

Self-aeration in the rapidly- and gradually-varying flow regions of steep smooth and stepped spillways

In high-velocity chute flows, free-surface aeration is often observed. The phenomenon is called self-aeration or white waters. When the turbulent shear stresses next to the free-surface are large enough, air bubbles are entrained throughout the entire air–water column. A rapidly-varied flow region is observed immediately downstream of the inception point of free-surface aeration. An analytical solution of the air diffusion equation is proposed and the results compare well with new experimental data. Both experiments and theory indicate that the flow bulking spans over approximately 3–4 step cavities downstream of the inception point of free-surface aeration on a stepped chute. Further downstream the void fraction distributions follow closely earlier solutions of the air diffusion equation. The application of the diffusion equation solution to prototype and laboratory data shows air bubble diffusivities typically larger than the momentum transfer coefficient. The result highlights however a marked decrease in the ratio of air bubble diffusivity to eddy viscosity with increasing Reynolds number. The finding might indicate some limitation of laboratory investigations of air bubble diffusion process in self-aerated flows and of their extrapolation to full-scale prototype applications.

Briefing: Experimental study on air-phase frequency in self-aerated flow

The distribution and development of air frequency in self-aerated open channel flows were experimentally investigated. A series of experiments was conducted in a rectangular flume with various initial flow velocities (3·50–6·35 m/s) and the test area was in the self-aerated developing region. Analysis of the results shows that the position of maximum air frequency is generally in the relatively low air concentration region in the self-aerated developing region of an open channel flow, and the air concentration with the maximum air frequency becomes greater and gradually approaches 0·50 with the development of self-aeration downstream. Further analysis reveals that the positions of maximum air frequency remain around the centre of the self-aerated region and, with an increase in initial flow velocity, the air frequency distributions over the cross-section gradually tend to a simple normal distribution. The reason for the distribution and development of air frequency in the self-aerated developing region is mainly from the free surface into the limit of self-aeration; the distribution of an air–water flow structure is a developing process involving the coexistence of entrapped air and entrained air, and this air–water structure leads to the difference in air frequency distributions and development in the self-aerated developing region.

A numerical model for air concentration distribution in self-aerated open channel flows

The self-aeration in open channel flows, called white waters, is a phenomenon seen in spillways and steep chutes. The air distribution in the flow is always an important and fundamental issue. The present study develops a numerical model to predict the air concentration distribution in self-aerated open channel flows, by taking the air-water flow as consisting of a low flow region and an upper flow region. On the interface between the two regions, the air concentration is 0.5. In the low flow region where air concentration is lower than 0.5, air bubbles diffuse in the water flow by turbulent transport fluctuations, and in the upper region where air concentration is higher than 0.5, water droplets and free surface roughness diffuse in the air. The air concentration distributions obtained from the diffusion model are in good agreement with measured data both in the uniform equilibrium region and in the self-aerated developing region. It is demonstrated that the numerical model provides a reasonable description of the self-aeration region in open channel flows.

Mass-based depth and velocity scales for gravity currents and related flows

Gravity driven flows on inclines can be caused by cold, saline or turbid inflows into water bodies. Another example are cold downslope winds, which are caused by cooling of the atmosphere at the lower boundary. In a well-known contribution, Ellison and Turner (ET) investigated such flows by making use of earlier work on free shear flows by Morton, Taylor and Turner (MTT). Their entrainment relation is compared here with a spread relation based on a diffusion model for jets by Prandtl. This diffusion approach is suitable for forced plumes on an incline, but only when the channel topography is uniform, and the flow remains supercritical. A second aspect considered here is that the structure of ET’s entrainment relation, and their shallow water equations, agrees with the one for open channel flows, but their depth and velocity scales are those for free shear flows, and derived from the velocity field. Conversely, the depth of an open channel flow is the vertical extent of the excess mass of the liquid phase, and the average velocity is the (known) discharge divided by the depth. As an alternative to ET’s parameterization, two sets of flow scales similar to those of open channel flows are outlined for gravity currents in unstratified environments. The common feature of the two sets is that the velocity scale is derived by dividing the buoyancy flux by the excess pressure at the bottom. The difference between them is the way the volume flux is accounted for, which—unlike in open channel flows—generally increases in the streamwise direction. The relations between the three sets of scales are established here for gravity currents by allowing for a constant co-flow in the upper layer. The actual ratios of the three width, velocity, and buoyancy scales are evaluated from available experimental data on gravity currents, and from field data on katabatic winds. A corresponding study for free shear flows is referred to. Finally, a comparison of mass-based scales with a number of other flow scales is carried out for available data on a two-layer flow over an obstacle. Mass-based flow scales can also be used for other types of flows, such as self-aerated flows on spillways, water jets in air, or bubble plumes.

Closure to “ Self‐Aerated Flows on Chutes and Spillways ” by H. Chanson (February, 1993, Vol. 119, No. 2)

The writer thanks the discusser (ANWAR 1994) for his interesting comments and information. The writer wishes to clarify three points : free-surface instabilities, calculations of the inception point of air entrainment and analogy between self-aerated flows and sediment-laden flows.