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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.


Scientific Reports
| (2022) 12:1477 | https://doi.org/10.1038/s41598-022-05588-y www.nature.com/scientificreports/ cumulative log-normal profile in the upper layer. The conservation equation for the air concentration distribution was developed by characterizing the parcel movement velocity of air bubbles over the entire flow depth 10 .
Recently, as measurement and analysis methods for air-water chute flows developed, more detailed characteristics of the mixture flow were investigated [14][15][16] . For turbulent air-water flows in open channels, the air concentration fluctuation caused by the velocity fluctuation was considered in the air bubble diffusion process of air-water flows. Based on detailed interactions of bubbles with local water in the lower region where C a < 0.30 and the substantial ejections of water droplets in the upper region where C a > 0.70, the intrusive phase-detection measurements highlighted the strong air-water turbulent interactions in the intermediate cross-sectional area where 0.30 < C a < 0.70 17,18 . According to the definition of air concentration in air-water flows, the quantities of water and air are equal if the local C a = 0.50. The specific flow depth y 50 , where C a = 0.50, can be considered canonically to undergo a transition from bubbly flow to droplet flow in the cross-section. Basic theoretical analyses confirmed that the specific flow depth y 50 can be used as a hypothesis condition to describe the air concentration distribution of self-aerated open channel flows 19,20 . It can be deduced that owing to the difference in the air-water structures at different layers of the flow cross-section, there must be an interior transition flow depth that can represent the detailed air-water structure development.
However, despite the key role of the interior transition depth in the air-water turbulent mixing process, theoretical verifications concerning the specific transition depth in self-aerated open channel flows are limited. Little information about the effect of particle turbulence on the average air concentration distribution in self-aerated flows is available. This study focuses on the air-water mixing affected by turbulence to analyse the theoretical transition depth in self-aerated chute flows. The prediction of the air concentration distribution from the transition depth represents the first attempt to verify the rationality of the interior transition depth. Moreover, this study provides further insights into air-water development differentiated by the transition depth of self-aerated flows, aiming to propose quantitative estimations in practical applications.

Concentration and density fluctuations in turbulent air-water flows
For a time average air-water control volume, the volume of the control mixture parcel is V 0 , which is equal to the product of the unit area A 0 and the displacement distance per unit time. The time average air and water concentrations of the control mixture parcel are defined as C a0 = V a /V 0 and C w0 = V w /V 0 , respectively. The time average density of the control mixture volume is ρ m0 . The mass of the mixture volume M 0 is defined as M 0 = ρ m0 V 0 , where V 0 is the time average volume equal to V a + V w . For turbulent air-water flows, the time average parameter is a decomposition result as the sum of the time average mean factor and the time average fluctuation factor. To retain ρ m0 , it is hypothesized that the value of M 0 is approximately equal to The two terms can be considered as the mass rate of air contribution M a = ρ m0 C a0 V 0 and mass rate of water contribution M w = ρ m0 C w0 V 0 . In self-aerated open channel flows, there are two basic transportation patterns in the y-direction perpendicular to the streamwise direction, that is, the air phase and water phase transportations. If it is considered that the mixture fluid crosses the unit area per time, the two terms become ρ m0 C a0 u a * and ρ m0 C w0 u w  where u a * and u w * are two characteristic velocities, representing the transition of the control volume. They consist of the turbulent eddy fluctuation velocity u y ', droplet falling-back velocity u f and air bubble rising velocity u r .
The definition of the characterized air-water volume is related to the fully developed two-phase flow on the basis of the mixing length theory 21 . The air-water transfer occurs under a uniform air-water mixture, and the air-water flow behaves as a homogeneous mixture across the flow section 22,23 . This ensures sufficient timeaverage air-water control volumes for flow property analyses. Both of them are the interaction of turbulence fluctuation and gravity or buoyancy forces, resulting in volume concentration and density changes. The time average fluctuation air concentration shows an important role in drag reduction, and the production of a negative turbulent shear stress component is obtained using momentum conservation analysis 24,25 . This indicates that the air bubble interfacial transfer for the unit air-water control volume requires vital attention and that the time-average fluctuation air concentration affected by the fluctuation velocity can be used as a component part of the turbulent air-water control volume. For flow density properties, the mass transfer among air-water control particles is the accompanying appearance causing the time average fluctuation of unit volume density. The mass transfer intensity is significantly correlated with local eddy shear, fluctuation, bubble and air-water surface transfer effects in open channel turbulent flows [26][27][28][29] . For an average air-water control volume with a constant concentration, more bubbles with smaller size can significantly raise the specific surface area, compared to less bubbles with larger bubble size dominant condition. General turbulent models are established based on the microscopic air-water mass transfer and bubble density and size variations [30][31][32] . Thus, the density variation of the control volume should be considered due to microscopic air-water structures. In the present study, the deviation of the local time average mean density of air-water control particles is the time average fluctuation of particle density. The effect of the time average fluctuation of particle density is among air-water control particles, and the entire air-water mass is constant with the conservation of mass. The time-average fluctuation density affected by the flow turbulence and fluctuation factors can be used as a component part of the turbulent air-water control volume. Consequently, for air and water diffusion generalizations, the approach of mean and fluctuating decompositions for the air concentration and mixture density is used in turbulent self-aerated chute flows.
Prandtl's mixing length theory is introduced to hypothesize the time-average mean and fluctuation for concentration and density properties, which are determined by the exchange on a macroscale analogous to that of molecular motion 21 . Figure 2 schematically shows an air-water mixture volume transition between layer 1 and layer 2 affected by the characteristic velocities. For example, due to u y ′ in the − y direction, the density and air concentration of the air-water mixture volume at level 1 are retained when it just arrives at level 2, and the density and air concentration at level 2 decrease soon after by exchanging with the flow in the neighbourhood of level 2. The detailed air-water property fluctuations are shown in Table 1, where the time-average and fluctuation local densities are ρ m and ρ m '; the time-average mean and fluctuation water droplet concentration and C w and C w '; the time-average mean and fluctuation air bubble concentration are C a and C a '. In the vertical direction (y), the air-water mixture volume is presumed to retain its original time-averaged properties at the arrival position, including velocity, density, and concentration [33][34][35] . Thus, the vertical density and air concentration fluctuations of the air-water mixture parcel through level 1 are given by ρ m ′ < 0 and C a ′ > 0, respectively. Based on these definitions for density and air concentration fluctuations of air-water mixture volumes, it is deduced that both the density and air concentration fluctuations are caused by the turbulent motion of mixture parcels. For other factors, u f and u r , there are different exchange motion processes analogous to the turbulent eddy effect. The relationship where ρ w0 is the water density. Considering this related expression, the fluctuation density of the air-water mixture volume is In the present study, the mixture of air and water can be approximated as two diffusion processes, that is, the air mixing process and water mixing process. Thus, the time-average mean and fluctuation for concentration and density properties are treated, respectively.
All results are presented at normal temperatures and standard atmospheric pressure conditions. The free surface aeration and air-water mixture processes are determined by the free surface air entrainment capacity and flow turbulence intensity. Uniform equilibrium air-water flows are defined as the air concentration distribution independent of the streamwise direction. The fully aeration condition means the cross-sectional fully air diffusion across the flow section without the clear water layer existence. Detailed descriptions of the two turbulent mixing processes combined with multiple fluctuation properties are shown in the following sections.

Air turbulent mixing
The time average mass flux of air contribution m a1 per unit area transported by the local turbulent fluctuation u y1 ′ in the − y-direction is The terms C a 'u y1 ′ and ρ m 'u y1 ′ are considered turbulent diffusion rates affected by the air phase. The turbulent diffusion processes for both the air concentration and mixture density are identical, which are analogous to the Boussinesq hypothesis 31  For uniform mixing of air in the water, the mass flux of air entrained in the − y-direction equals that of air rising in the + y-direction, that is, Table 1. Fluctuation properties in the air-water mixing process. Considering that the flow density fluctuation ρ m ' is much smaller than the water density ρ w , that is, ρ m ' < < ρ w , and the higher order of multiple ρ m 'C a 'u y ' and ρ m 'C a 'u r cosα, the second term on the right term is neglected, that is, It should be noted that in the right term, (1 − C a )C a u r cosα > 0, and in the left term, K a1 + K a2 > 0 and dC a / dy > 0. The case for which the air phase continues to diffuse during the turbulent mixing process is given by the condition that the sign of the left term is positive; that is, C a < 0.5 should be maintained to ensure that Eq. (13) is satisfied. This indicates that based on the air mixing process under the turbulent diffusion effect, the limiting air concentration is C a < 0.5.

Parameters Density Concentration
The alternative formulation for the final rising velocity of air parcel under a local concentration condition is introduced: u r = (u r ) 0 (1 − C a ) 2 , where (u r ) 0 is the final velocity of an individual air parcel in pure water 36 . This is a generalized expression that assumes that the control volume movement is affected by mixture flow density variation and does not interact with each other. A dimensionless air turbulent diffusivity K a and a dimensionless distance y * are normalized by a flow depth y 50 , where local C a = 0.5, given as Then, Eq. (13) yields Integrating Eq. (15) in the y-direction yields where D a is the integration constant. This relation is a function of the air concentration distribution in the region where C a ≤ 0.5. The relationship between K a and D a can be deduced from C a = 0.5 for y * = 1, that is,

Water turbulent mixing
The time average mass flux of water contribution m w1 per unit area transported by the local turbulent fluctuation u y2 ′ in the + y-direction is The terms C w 'u y2 ' and ρ m 'u y2 ' are considered the turbulent diffusion rate affected by the water phase. The turbulent diffusion processes for both the water concentration and mixture density are identical, and they are proportional to the gradient of the time-average concentration, given by The terms C w 'u f cosα and ρ m 'u f cosα are considered the diffusion rates by the gravity effect, given by (11) m a1 =m a2 For uniform mixing of water in the air, the mass flux of the water contribution expelled in the + y-direction equals that falling back in the − y-direction, that is, Then, it is obtained as This is analogous to the previous analysis; the second term on the right term is neglected, that is, The alternative velocity formulation for the local concentration condition in the mixture fluid is introduced as u f = (u f ) 0 (1 − C w ) 2 , where (u f ) 0 is the final velocity of an individual water parcel. A dimensionless turbulent diffusivity K w is defined as Then, Eq. (28) with the dimensionless y * yields Integrating Eq. (30) in the y-direction yields where D w is the integration constant. This relation is a function of the water concentration distribution. The air concentration distribution can be obtained by C a = 1 − C w , that is, Considering y 90 as the total mixture flow depth where the local water droplet concentration is C w = 0.1 (and C a = 0.9), the relationship between K w and D w can be deduced under two boundary conditions: C w = 0.5 for y * = 1 and C w = 0.1 for y * = y 90 /y 50 , which yields Comparing the air and water turbulent mixing Eqs. (15) and (30), it is found that the concentration gradient variation is continuous for the water phase, while it is not continuous for the air phase in air-water flows. Thus, this theoretical analysis reveals that y 50 , where C a = 0.5, represents the interior boundary for air turbulent mixing and water turbulent mixing in which the water phase can be considered a homogenous phase.

Air concentration distributions
The turbulent diffusion coefficients K a and K w are determined by the relationship between the aeration level and y 50 /y 90 . The average cross-sectional air concentration C mean is defined as the integration of the local C a over the mixture flow depth between the channel bottom at y = 0 and y 90 : www.nature.com/scientificreports/ The statistical relationships between y 50 /y 90 and C mean are shown in Fig. 3 for different model and prototype chute flows. For a constant channel slope condition, y 50 /y 90 decreases with increasing C mean , and it is independent of other hydraulic conditions, such as approach flow conditions, smooth or stepped channel type, and slope. For C mean values smaller than approximately 0.40, a linear trend is followed with a constant gradient k 1 ≈ − 0.8. The gradient changes to k 2 ≈ − 2.4 for C mean greater than approximately 0.40, for which the mixture flow crosssection is fully aerated and the local air concentration is greater than 0.10. On the basis of C mean = 0.40, the two relationships can be expressed as y 50 /y 90 = 1-0.8C mean for C mean < 0.40 and y 50 /y 90 = 1.7-2.4C mean for C mean > 0. 40. This indicates that the interior transition boundary of turbulence diffusion changes suddenly if the aeration development of the flow cross-section exceeds C mean = 0.40.
As aeration develops through the turbulent free surface, the air-water mixing layer diffuses into the water flow, protruding to the channel bottom. For some cases, the entire flow is partially aerated, where a clear water layer exists at the bottom area of the entire flow. For the other cases, the entire flow section becomes fully aerated without the presence of clear water. Here, the air concentration distributions obtained from the prototype Aviemore dam spillway 37 are used for detailed analysis. For partially aerated flows, the measured data showed that the local C a at the bottom area was mostly smaller than 0.05 for C mean < 0.40. If K w and D w are obtained from Eq. (33), the comparison between the measured data and Eq. (32) in Fig. 4a shows that the theoretical prediction is acceptable only for C a > 0.5, while for C a < 0.5, Eq. (32) overestimates the air diffusion in the low aerated region. This indicates that the presence of a clear water layer can reduce the air bubble penetration caused by water turbulent diffusion. For this case, it is suggested that K a = K w /3. For fully aerated flows, the local C a mainly exceeds 0.10 at the channel bottom when C mean exceeds 0.40. The determinations of K w and D w are obtained from Eq. (33). The comparison between the measured data and Eq. (32) is shown in Fig. 4b. The good agreement indicates that homogenous water turbulent mixing in self-aerated open channel flows can predict the air concentration distribution for both high aerated regions (C a > 0.5) and low aerated regions (C a < 0.5). Under this situation, the  www.nature.com/scientificreports/ sudden change in y 50 /y 90 represents the difference in interior boundary variation and the capacity reduction of air diffusion for C mean < 0.40. Because the ratio of k 1 /k 2 = 1/3, K w needs to be three times greater. The air turbulent diffusivity K a can be set as K a = K w /9 to obtain good agreement on the air concentration distribution in the low aerated region (C a < 0.5). Consequently, the characterized flow depth y 50 , at which the local air concentration is 0.5, represents the interior transition boundary in self-aerated flows C mean < 0.40, and the air-water mixture cross-section can be divided into water turbulent mixing and air turbulent mixing. For C mean > 0.40, the full crosssection can be seen as the water mixing process, and the y < y 50 region also fits the air turbulent mixing process. Based on Eqs. (16) and (32) with boundary conditions, the calculated air concentration profiles are compared with the experimental and prototype data 12,20,[37][38][39] , as shown in Fig. 5. For each case, the determinations of K w and K a are deduced from each C mean within the calibrated procedure proposed above. For C mean < 0.40 within clear water, the air concentration distributions agree well for air and water turbulent mixing areas, and the theoretical results are applicable for both smooth and stepped spillways. Thus, it is reasonable to relate the diffusion behaviour of air turbulent mixing with water turbulent mixing. Air diffusion within the presence of clear water is verified to change K a and restrain air diffusion according to the sudden change in the y 90 /y 50 gradient for C mean ≈ 0.40. Moreover, the air concentration profiles for high C mean up to 0.65 agree with the theoretical results obtained from water turbulent mixing. This indicates that the local water turbulent mixing in the free-surface area plays an important role in the air-water mixture process. The air-water structures in the upper spray region, where macroscopic water roughness and microscopic water particles are predominant, are of various types, such as individual droplets, water projections, mixture clusters, and foam transported with the free surface. Owing to the individual water structures affected by the interaction between the free surface and the local turbulence, the satisfactory predicted results indicate that the movement of complex water-predominant structures can be simplified as local individual water particle movements affected by turbulence.
Most theoretical and observation descriptions of the aerated region of classical air-water mixtures in selfaerated flows involve a "layered" description: (1) individual water droplets in air in the top layer; (2) a mixture of water and air in the middle layer; and (3) individual air bubbles in water in the bottom layer. The layer is a generalization concept on the basis of the dominant air-water structures. In the present theoretical derivation process, there is no flow depth limit until the discussion of the satisfied condition for establishing the equation. The specific interior flow depth y 50 is obtained because the boundary condition of particle concentration is 0.5 in the air or water mixing process affected by turbulent diffusion. The interior transition depth y 50 is considered the transition layer from the top layer, where the water phase dominates, to the bottom layer, where the effect of the air phase increases. Consequently, the characterized water parcels and air parcels should be considered equally in high-speed chute flows, and the movement of complex air-water structures can be simplified as local particle turbulent diffusion. This promotes the further understanding and description of layer theory in selfaerated open channel flows.

Self-aeration streamwise development
When self-aeration of the free surface occurs in high-speed chute flows, the air-water mixes with an increase in the macroscopic C mean , as shown in Fig. 6a, where x is the distance from the self-aeration inception point along the streamwise direction, and y I is the initial water flow depth at the inception point. x/y I is used as the nondimensional parameter to illustrate the development of self-aerated flows. The water flow discharge per unit width ranges from 0.15 to 3.16 m 2 /s with α = 4°-52.5°4 0-43 . Owing to the differences between water and air turbulent mixing areas, the mean cross-sectional air concentration can be divided into two parts demarcated by y 50 , separately defined as (C mean ) Ca>0.5 and (C mean ) Ca<0.5 : In Fig. 6b, the values of (C mean ) Ca>0. 5 do not change notably in the entire development process, remaining at (C mean ) Ca>0.5 ≈ 0.75 in the incipient distance for 0 < x/y I < 50 and slightly changing to (C mean ) Ca>0.5 ≈ 0.70 for x/y I > 50. The development of (C mean ) Ca<0.5 is similar to the variation in C mean as self-aeration develops downstream (Fig. 6c). This is because the air concentration in the upper free-surface area results from the growth of surface waves and entrapped air, which develops rapidly into small free-surface roughness and water droplets in the following incipient distance 4 . Altogether, once self-aeration occurs, the local water turbulent mixing affected by the free-surface turbulence is relatively stable. The development of an air-water mixture in a self-aerated open channel flow manifests mainly as air diffusion in the water. Compared with the two developments of C mean and (C mean ) Ca<0.5 with other identical conditions in Fig. 6d, the approximately same trends confirm that as the air turbulent mixing approaches near full development, the entire air-water mixture of self-aerated flows reaches a uniform equilibrium level.
Furthermore, self-aeration typically reaches the near-fully developed region with an approximately identical development process under the same channel slope conditions, such as α = 30°, 45° and 52.5°, as shown in Fig. 6a and Fig. 6d. For uniform air-water mixture flow conditions, the mean cross-sectional air concentration (C mean ) u is dependent only on the channel slope. Hager 4 analyzed the uniform (C mean ) u for 7.5° ≤ α ≤ 75° and defined it as This equation results in (C mean ) u being approximately 0.7 for a steep channel slope α = 75°, and it is in agreement with the y 50 /y 90 variation within C mean . This indicates that y 50 /y 90 decreases to zero for extremely developed aeration near C mean = 0.7. In the present analysis, it is considered that the air-water flow reaches a near-fully developed region when C mean ≥ 0.85(C mean ) u (reference lines in Fig. 6a), owing to differences in the experimental  www.nature.com/scientificreports/ measurement. The streamwise distance length x L from the inception point to the fully developed flow crosssection is defined as the self-aeration development length, as shown in Fig. 6d. For a specific channel slope condition, the average value is used, which is deduced from the experimental data. A larger channel slope corresponds to a longer distance of self-aeration development. In Fig. 7a, the relationship between x L /y I and sinα can be expressed as The developments of C mean and (C mean ) Ca<0. 5 can be applied as where (C mean ) Ca<0.5u is the mean cross-sectional air concentration in the air turbulent mixing area (y < y 50 ) for the fully developed uniform region. The streamwise normalization of Eq. (39) includes x/x L . Figure 7b compares all datasets with the data pertaining to Eq. (39), resulting in a coefficient of determination R 2 = 0.847. The data trend indicates that self-aeration in high-speed chute flows develops rapidly in the incipient half of the streamwise distance length x/x L < 0.5 and reduces gradually until the air-water uniform condition is reached. As Eqs. (38) and (39) describe C mean using y I and these relationships in self-aerated chute flows can be estimated for a given initial configuration, it is expected that the quantitative expressions are comparable for both model and prototype applications. The present study proposes an analysis of air-water turbulent diffusion based on a theoretical model. Recent experimental and theoretical analyses confirm that a reasonable model for particle groups in air-water flows can result in reliable two-phase properties, including time-average mixture-velocity and air concentration distributions 44 . According to the present analysis results, the air-water mixture in high-speed chute flows is a result of the interaction of air and water under turbulence. Microscopic water and air turbulent mixing through the free surface plays an important role in air-water flow development. Further research on self-aeration onset should consider the formation of water ejection and air entrainment equally. The detailed differences in water ejection and air entrainment are required for a complete understanding of air-water mixture development, thereby helping promote hydraulic and environmental applications. Moreover, the effects of individual particle distribution and movement in different flow conditions, including flow turbulence intensity and mixture levels, should be further studied. The limitations and explanations should be clarified, which can contribute to the relation of air-water mixtures and a continuous phase analysis model. (38) x L y I = 280 · tanh(1.2 sin α) + 80 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.