Bubbles and waves description of self-aerated spillway flow Description des écoulements à bulles et vagues sur déversoir auto-aéré

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1 Journal of Hydraulic Research Vol. 43, No. 5 (25), pp International Association of Hydraulic Engineering and Research Bubbles and waves description of self-aerated spillway flow Description des écoulements à bulles et vagues sur déversoir auto-aéré STEVEN C. WILHELMS, Engineering Research and Development Center, U.S. Army Corps of Engineers, 399 Halls Ferry Road, Vicksburg, MS 3918, USA JOHN S. GULLIVER, Joseph T. and Rose S. Ling Professor and Head, Department of Civil Engineering, University of Minnesota, St. Anthony Falls Laboratory, Mississippi River at 3rd Ave SE, Minneapolis, MN 55414, USA ABSTRACT The continuum description of self-aerated spillway flow has adequately served to describe spillway bulking, but encounters difficulties when applied to other physical phenomena, such as cavitation and gas transfer. The continuum description is adapted to separate air being transported by the flow as bubbles ( entrained air), and air transported with the flow in the roughness or waves of the water surface ( entrapped air). Results from flume experiments on aerated flow are used to develop an analysis procedure and mathematical description of entrained and entrapped air for flow along a spillway face. Entrapped air is found to be constant at a void ratio, with a vertical distribution analogous to the intermittent region of a turbulent boundary layer. Entrained air gradually increases to a maximum value depending on slope. Cain s dimensionless distance is used to collapse entrained air data from several unit discharges with the same slope to a single relationship. The analysis procedure and dimensionless parameter provide a means of analyzing a large store of additional literature data. Observations from a full-scale spillway provide verification of the procedure. RÉSUMÉ La description d un continuum d écoulement sur déversoir auto-aéré a bien servi à décrire le gonflement global, mais on rencontre des difficultés pour l appliquer à d autres phénomènes physiques, tels que le transfert de gaz et la cavitation. La description d un continuum est adaptée à de l air transporté par l écoulement sous forme de bulles (air entraîné ), différente de l air transporté par l écoulement dans la rugosité ou les vagues de la surface de l eau (air enfermé ). Les résultats des expériences en canal sur l écoulement aéré sont utilisés pour développer un procédé d analyse et une description mathématique de d air entraîné et enfermé pour l écoulement le long d un coursier de déversoir. De l air enfermé est trouvé constant pour un rapport de vide, avec une distribution verticale analogue à la région intermittente d une couche limite turbulente. L air entraîné croît graduellement jusqu à une valeur maximum dépendant de la pente. La distance sans dimensions de Cain est utilisée pour rassembler, dans une formule unique, les données d air entraîné de plusieurs débits unitaires avec la même pente. Le procédé d analyse et le paramètre sans dimensions fournissent les moyens d analyser un grand stock de données additionnelles de la littérature. Les observations d un déversoir grandeur nature fournissent la vérification du procédé. Keywords: Aerated flow, spillway, entrained air, aeration, void ratio, gas hold-up. 1 Introduction Self-aeration is a phenomenon seen in high velocity flows on spillways or in steep channels. The flow turns frothy and white with entrained air when aeration is initiated (Fig. 1). Studies of self-aerated spillway flow have shown that the turbulent boundary layer, caused by the spillway surface, initiates air entrainment when it intersects the water surface at the point of inception (Keller et al., 1974) (Fig. 2). For some distance, the flow is developing, i.e., there is a net flux of air into the water. When the air bubbles are transported to their maximum depth in the water, the flow is considered fully aerated, but continues to entrain more air and thus is still developing. At some long distance along the spillway, uniform conditions are approached. Thereafter, there is no significant change in the hydraulic or air transport characteristics. The process of self-aeration in spillways and steep chutes has historically been of interest to hydraulic engineers because of the bulking effect the entrained air has on the depth of flow (Hall, 1943). The amount of bulking is a necessary design parameter in determining the height of spillway or chute sidewalls. Engineers have also been interested in eliminating or minimizing cavitation damage caused by high velocity flow in spillways, chutes, and channels (Falvey, 199). To accomplish this, aerators have been designed to aspirate air into the flow (Rutschmann and Hager, 199). The location on the spillway, where sufficient air from the self-aeration process becomes available to prevent or reduce the damage caused by cavitation, is required by the design engineer when sitting aerators or determining if aerators are required. More recently, highly aerated flow has been recognized for its gas transfer characteristics with the transfer of atmospheric gases into the water (Gulliver and Rindels, 1993) and the Revision received July 22, 24/Open for discussion until August 31,

2 Bubbles and waves description 523 Figure 1 Self-aerated flow on spillway. Figure 2 Region of developing flow (after Keller et al., 1974). Ehrenberger s effort were (1) recognition of the significant influence that entrained air has on hydraulic characteristics and (2), a physical description of highly aerated flow, which was as follows: At the top, droplets of water interspersed through air are first noticed. Below this layer, there is a layer consisting of a mixture of air and water, which in turn covers a layer of water containing individual air bubbles, and finally there is a layer of unaerated water adjacent to the bottom. This layered description ultimately developed into the concept of a continuum of air/water from the bottom to the surface. In a benchmark article on self-aerated flow, Straub and Anderson (1958) showed measurements of air concentration (Fig. 3) that seemed to indicate that air concentration varied in a continuous fashion over the depth of flow. They conducted extensive tests in a laboratory flume at slopes from 7.5 to 75 with unit discharges from.416 to.2832 m 2 /s. In agreement with Ehrenberger s (1926) description, they conceptualized air-entrained flow as having an upper region, where water is transported with air, and a lower region, where air is transported with water. Killen (1968), however, showed in high-speed photos taken during flume experiments in the mid-195s that the water surface remained intact but very contorted (Figs 4 and 5) with a small quantity of flying droplets over the surface. Killen s measurements with an electrical surface probe also support this observation. Contrary to Ehrenberger s (1926) description, Killen showed that a well-mixed continuum of increasing air and decreasing water does not exist. Furthermore, the electrical surface probe measurements suggest that the air content in the upper region is composed primarily of air caught in the roughness of the water surface, instead of air with water globules and droplets. The measurement techniques that had been used, with the exception volatilization of pollutants. Falvey and Ervine (1988) reviewed past work, discussed the hydrodynamic processes affecting aeration, and identified areas where our understanding must be improved. The objective of this paper is to improve the description of self-aerated flow, for use in gas transfer computations and to provide a correlation of cavitation observations in the laboratory with those in the field. Because entrained air contributes greatly to absorption of oxygen and the transfer of other gases and can significantly reduce cavitation damage, a more detailed description of the amount of air entrained at various depths and the type of air entrainment is needed for spillway flows. Current conceptual descriptions of the aeration process are examined. The bubbles and waves concept is then proposed and validated regarding the conceptual definition of entrained air. Finally, data from selected past efforts are analyzed within this new framework to estimate the air entrainment at any location on a spillway flow path. Distance from Bottom, mm Review Ehrenberger (1926) is usually cited as the first study of selfaeration in open channel flow. The major contributions of 5 Air Concentration, percent Figure 3 Air concentration distribution measured by Straub and Anderson (1958). 1

3 524 Wilhelms and Gulliver of Killen s, simply could not tell the difference. This misconception was likely propagated because the flow appears to be a continuum to the naked eye, similar to Figs 1 and 5(a), while an intact but highly contorted water surface is visualized with high-speed photography, such as in Figs 4 and 5(b). The need for a more detailed description to facilitate the analysis of cavitation and gas transfer characteristics in spillway flows leads to a refinement of the continuum description. Figure 4 Section view of self-aeration flow is shown to be intact but contorted in this side view photograph taken at a shutter speed of 1/1, s (courtesy of St. Anthony Falls Laboratory, University of Minnesota). 3 Air transport through bubbles and waves Based on Killen s (1968) photographic evidence and his water surface measurements, the concepts of entrained and entrapped air, illustrated in Fig. 6 are introduced. Entrained air is transported along with the flow in the form of air bubbles that, at some point, have been pulled into the flowing water through the process of air entrainment. Entrapped air is above the water surface, located between the waves that form surface roughness. Entrapped air is not transported as air bubbles, but much of it is trapped in the surface waves, and transported along with the flow. Entrained air plus entrapped air is total conveyed air, which, for most of the literature is defined as entrained air. Most of the existing air entrainment data (DeLapp, 1947; Halbronn, et al., 1953; Viparelli, 1953; Straub and Anderson, 1958; Lai, 1968; Cain, 1978) do not separate entrapped and entrained air, because the measurements do not differentiate between them. The one exception is Killen s (1968) measurements, which also identified surface roughness. For some applications, it is important to know how much air is above the surface and how much is below the surface as air bubbles. Such a separation will also help to understand the physical phenomena associated with air entrainment on spillways. For bulking, total conveyed air is of prime importance and the differentiation of entrained and entrapped air is of no major consequence. However, for cavitation prevention, air bubbles must be present near the spillway surface and entrained air must receive stronger consideration than total conveyed air or Figure 5 Intact but contorted water surface in the St. Anthony Falls Laboratory chute: (a) taken with a shutter speed of approximately 1/3 s and (b) taken with a 1/5, s strobe light source (courtesy of St. Anthony Falls Laboratory, University of Minnesota).

4 Bubbles and waves description 525 FLOW ENTRAPPED AIR WATER SURFACE FLOW Figure 6 ENTRAINED AIR (BUBBLES) Concepts of entrained and entrapped air. entrapped air. Entrained air also is of greater significance than the entrapped air determining the gas transfer on the spillway face, because of the tremendous surface area available for transfer in a bubbly flow and the exposure of the bubbles to a high level of turbulence. When the flow plunges into a pool below the spillway, the total conveyed air would be more important since much of the entrapped air will likely become entrained at the plunge point. 4 Separation of entrained and entrapped air Most data collected in experimental studies of aerated flow consist of concentration profiles of total conveyed air (Straub and Anderson, 1958; Killen, 1968). To make use of the entrained or entrapped air concepts, each must be separated from total conveyed air. Killen was interested in the surface characteristics of aerated flow and measured surface roughness in addition to the total conveyed air concentration profile. He measured the total conveyed air content with an air concentration probe (Lamb and Killen, 195), but also used a conduction probe with a cathode in the probe and an anode at the channel bottom to measure the roughness of the free surface. The surface probe dipped in and out of the surface roughness as flow passed the probe. The signal from the probe was a maximum when it was in the water and at zero when out of the water. Our analysis of Killen s surface roughness data has shown (Wilhelms, 1997) that the time-average of these measurements can be used to determine the fraction of air contained within the surface roughness, the entrapped air. The entrained (as bubbles) air concentration can then be determined by the difference between the total conveyed air concentration and the entrapped air concentration. Killen (1968) collected these data at several locations along the length of a 5 cm wide flume for several flow rates and two slopes. An example of his observations is shown in Fig. 7 for a unit discharge, q, of.43 m 2 /s, channel slope, θ, of3, and a distance, X, along the flume of 1.37 m. All of Killen s observed profiles are presented in graphical form with corresponding digital profile data in Wilhelms (1997). The total conveyed air at the channel bottom represents the concentration of entrained air, since the water surface roughness does not extend to the bottom. Below the lower limit of roughness penetration, where the entrapped air concentration reaches zero, the total conveyed air consists only of entrained air bubbles. However, above this limit, DISTANCEFROM BOTTOM (mm) Figure 7 slope. Entrapped Air 5 Entrapped Plus Entrained Air 1 Measured profiles from Killen s (1968) Test No. 2 on a 3 both entrapped air and entrained air contribute to total conveyed air. Entrained air gradually decreases in proportion to entrapped air until the entrapped air and total conveyed air concentrations are equal to 1% where the probe is completely out of the water. 5 Mean concentrations Using Eq. (1), the mean concentration of entrapped and total conveyed air can be determined by integrating these two curves over depth and dividing by depth. and C = C E = Yc c(y) dy Yc dy Yc c E(y) dy Yc dy where C and C E are the mean concentrations of total conveyed air and entrapped air, respectively; c(y) and c E (y) are the concentration profiles of the total conveyed air and entrapped air as a function of depth y; and Y c is the integration limit. The mean concentration of entrained air, C e, is the difference between entrapped and total conveyed air. Because of the difficulty in defining the upper limit of flow, which should be the upper limit of integration, Straub and Anderson (1958) suggested integrating the profiles to a depth where the total conveyed air concentration was.95 or.99. Cain and Wood (1981) adopted an integration limit where total conveyed air is.95. In a later analysis, Wood (1985) used the depth where total conveyed air concentration was.9. Wilhelms and Gulliver (1994) computed the mean entrapped air concentration for several integration limits and determined that there was less (1) (2)

5 526 Wilhelms and Gulliver variability in the entrapped air concentrations as the integration limit increased (Table 1). The depth, denoted by Y 98, where total conveyed air equals.98, was easily identifiable on concentration profiles and was selected for use in calculating mean values. We will use Y 98, and adapt the analysis to Y 95 and Y 9 through a simple numerical conversion. Mean concentrations for total conveyed air and entrapped air were calculated for all of Killen s (1968) observed profiles up to an air concentration of.98 with Eqs (1) and (2). Mean entrained air concentration was the difference between total conveyed air and entrapped air. The results of integrating these profiles are given in Table 2 and shown in Fig. 8. This figure shows mean concentration as a function of distance along the flume for Killen s slopes and discharges. As expected, the total conveyed air concentration gradually increased in the developing flow region, approaching a steady-state concentration. Entrained Test No. 4,θ = 52.5 o TOTAL CONVEYED AIR Test No. 1 Test No. 2 Test No. 3 Test No. 4 ENTRAINED ENTRAPPED ENTRAPPED AIR Table 1 Variation of entrapped air concentrations from Killen s tests for various integration limits Integration limit Y 9 Y 95 Y 98 µ (mean) σ (standard deviation) Normalized σ/µ Table 2 Mean concentrations of entrapped, entrained and total conveyed air for Killen s (1968) Observations Profile Entrapped Entrained Total X (m) Test No. 1, θ = 3,q=.195 m 2 /s Test No. 2, θ = 3, q =.4 m 2 /s Test No. 3, θ = 3, q =.79 m 2 /s Test No. 4, θ = 52.5, q =.4 m 2 /s X, Meters Figure 8 Results of reanalysis of Killen s (1968) data. Mean concentrations of profiles from Test No. 4 (above). Mean concentration of entrapped air for all of Killen s data at 3 and 52.5 slopes versus distance from gate along the channel (below). air concentration followed a similar trend. It was anticipated that the entrapped air concentration would do likewise. However, the data show essentially a constant value of about 23% for the entrapped air concentration, when integrated to Y c =.98. When integrated to Y c =.95 and.9, mean entrapped air concentrations of 14.2 and 7.3%, respectively, were computed from the measurements. The standard deviations of the latter two mean-values were greater than for Y c =.98, as indicated in Table 1. The conversion for entrapped air by subtracting.23,.142, and.73 for Y C = Y 98,Y 95, and Y 9, respectively, make the analysis of literature data easily adapted to the bubbles and waves description. 6 Constant entrapped air concentration Killen (1968) experimentally found that a Gaussian error function (cumulative normal distribution) described the surface roughness characteristics. An error function can also be used to describe the shape of the entrapped air profiles. Thus, the difference between the depths d 2 and d 98 (Fig. 9), where the entrapped air concentrations are.2 and.98, respectively, represents 4.1 σ E, where σ E is the standard deviation of the cumulative normal distribution for the entrapped air. Wilhelms (1997) showed that since the entrapped air concentration distribution is cumulative normal, then the entrapped air (numerator of Eqs (1) and (2)) is equal to a constant, K E, times σ E. This results in a mean entrapped air concentration C E of C E = K Eσ E d 98 = K E(d 98 d 2 ) 4.1d 98 (3) 15

6 Bubbles and waves description d 98 DISTANCE FROM BOTTOM (mm) C = 2 d σentrapped d 2 ENTRAPPED AIR TOTAL CONVEYED AIR C = Figure 9 Entrapped air concentration distribution (percent by volume), definition of Gaussian error function parameters. For the entrapped air concentration to be constant, the ratio of σ E or (d 98 d 2 ) to depth of flow d 98 must be constant, implying that the surface roughness is related to the overall depth of flow. This is reasonable when one considers the cause of the surface roughness: turbulent eddies being generated by shear at the floor of the channel. In a steep channel, the strength of these eddies is sufficient to deform and contort the water surface. Furthermore, the size of these eddies determines the magnitude of the surface roughness and the largest length scale of the turbulent eddies is a function of depth in a turbulent open channel flow. We believe that surface roughness in a steep open channel flow is analogous to the distribution of turbulence at the outer edge of a turbulent boundary layer, illustrated in Fig. 1. The turbulence generated within the boundary layer causes the interface of the outer flow and the boundary layer to be highly contorted (Tennekes and Lumley, 1972). Measurements of turbulence in this region have resulted in the concept of an intermittency factor, which is the proportion of time at some location in the interfacial region that the fluid is turbulent. Hinze (1959) reported that the distribution of the intermittency factor across this interfacial region was described by a Gaussian error function. The intermittency parameter also defines the concentration by volume of non-turbulent fluid within the boundary layer and is directly analogous to the entrapped air concentration. Using the distributions given by Hinze (1959), the mean concentration of non-turbulent fluid in a boundary layer between intermittency parameter values of.2 and.98 is constant at approximately 25%, which compares well with the 23% entrapped air concentration found from an analysis of Killen s (1968) water surface data. Moreover, the thickness of the intermittent region of turbulent and nonturbulent fluid is proportional to the boundary layer thickness, which corresponds to the depth of flow. Figure 1 Boundary layer turbulence analogy to entrapped air by surface roughness. We can thus describe the surface roughness and entrapped air as follows: The highly energized boundary layer that is generated by the spillway propagates through the depth of flow and manifests itself as the roughened water surface. Thus, the characteristics of the water surface are a direct reflection of the turbulence generated by the spillway. Through analogy with the outer region of a turbulent boundary layer, it can be concluded that the surface roughness should be related to the depth of flow. With this in mind, a constant entrapped air concentration is expected once the boundary layer of the spillway reaches the free surface. The simple numerical conversion from the continuum description to the bubbles and waves description of self-aerated flow is a natural outcome of a highly energized, free-surface flow. 7 Non-dimensional parameters for developing aerated flow Non-dimensional parameters are used for their convenience in illustrating similarities in fluid flow. Past efforts related to self-aerating flow proposed non-dimensional terms that have included a distance Reynolds Number (Keller et al., 1974), where the critical dimension was distance along the direction of flow. Cain (1978) used the depth of flow at the point of inception Y i to introduce a dimensionless distance parameter X /Y i, where X is the distance from the point of inception X i to the measurement location X or X = X X i. This particular variable has the convenient characteristic of implicitly including the unit discharge with Y i. At the point of inception, regardless of

7 528 Wilhelms and Gulliver 8 Test No. 4,θ = 52.5 o TOTAL CONVEYED AIR 6 ENTRAINED 4 2 ENTRAPPED X*/Y I Figure 11 Non-dimensional mean concentrations of profiles from Killen s (1968) Test No. 4 on a 52.5 slope. MEAN AIRCONTENT (PERCENT) Straub andanderson1958 Killen 1968 C e = exp (-.11 X*/Y I ) X*/Y I Figure 13 Mean concentration of entrained air (percent by volume) versus dimensionless distance from the point of inception on a 3 slope for Killen s (1968) tests and computed from Straub and Anderson s (1958) tests assuming 23% entrapped air Test Nos. 1-3,θ = 3. o TOTAL CONVEYED AIR ENTRAPPED ENTRAINED X*/Y I Figure 12 Non-dimensional mean concentrations of profiles from Killen s (1968) tests on a 3 slope. discharge, the hydraulic character of the flow is completely developed. Hence, the discharge can be described in terms of the depth of flow, channel slope, and the friction factor or another resistance coefficient such as Chezy s or Manning s coefficient. It follows that the dimensionless distance parameter X /Y i may be an appropriate term for the analysis of developing flow. This term is compared against Killen s (1968) experimental data in Figs 11 and 12. The data for three discharges at the 3 slope seem to fall onto one curve when plotted with X /Y i, which indicates the success of Cain s non-dimensional parameter. 8 Analysis of other data Straub and Anderson (1958) made extensive measurements of aerated flow for a large variety of slopes and discharges in the same flume that Killen (1968) used. Their observations, however, consisted only of total conveyed air concentration profiles. By subtracting the entrapped air content of 23% from a re-analysis of these data to Y 98, the entrained entrapped air concept can be extended to conditions where only total conveyed air measurements were made. Furthermore, Straub and Anderson intended to measure the entrained air characteristics for uniform flow conditions. Wood (1983), however, showed that many of the Straub and Anderson measurements were in non-uniform aerated flow. In addition, Straub and Anderson collected more profiles at and 1.67 m along the flume that were not published. Wilhelms (1997) presents these data, which will be used in this analysis. By using Cain s dimensionless distance to locate these profiles, all of the profiles for one slope can be analyzed as profiles of developing flow at different locations, X /Y i, along the dimensionless flume. A methodology was developed to estimate the point of inception X i (Wilhelms, 1997), based upon the normal depth of flow Y i and the flume gate opening. Using this methodology, a dimensionless distance X /Y i along the flume can be calculated for Straub andanderson s and m observations. Figure 13 shows the developing nature of entrained air concentration from observations on a 3 slope from Straub andanderson s and Killen s tests. This analysis of Straub and Anderson s experiments, when compared to Killen s observations of developing flow, indicates the validity of the approach. Figure 13 implies that all spillway air entrainment data with a given bottom roughness and slope will plot on one curve. To verify this conclusion, Cain s (1978) measurements can be added to the Straub and Anderson flume measurements. Cain measured total conveyed air concentration profiles at several locations along the flow path on the Aviemore spillway, which has a 45 slope, representing measurements of developing flow. Cain s observed profiles of total conveyed air concentration were integrated to determine mean concentrations at Y 98 of total conveyed air. Then 23% was subtracted for mean entrapped air, leaving mean concentrations of entrained air. Figure 14 shows the mean entrained air concentrations as a function of dimensionless distance along the spillway face for the two discharges Cain tested. Because air concentration was measured along the flow path, a good estimate of the air entrainment inception location, X i, and the equivalent depth at inception, Y i, could be obtained. Also given in Fig. 14 are the Straub and Anderson (1958) measurements on a 45 slope at 1 different discharges. Considering the scale-up difficulties encountered when laboratory measurements are compared to full-scale and the wall-roughness differences between the flume and Aviemore spillway, the two sets of data are remarkably close.

8 Bubbles and waves description Straub and Anderson(1958) Cain (1978) X*/Y I C e = exp (-.1 X*/Y I ) Figure 14 Mean concentration of entrained air versus dimensionless distance from the point of inception, computed by subtracting 23% entrapped air from Straub and Anderson s (1958) tests on a 45 slope and Cain s (1978) observations on Aviemore Spillway. This comparison demonstrates the appropriateness of the arguments made in earlier sections regarding Cain s non-dimensional terms and shows that the relationships established provide reasonably accurate predictions of air concentrations at any location in the developing flow region. The comparison also indicates that the uniform aerated flow concentration is accurately predicted. It seems likely that this formulation should be applicable to most spillways because, even though the relationships were developed from observations made in a laboratory flume, the results reasonably predicted observations from a full-scale project. 8 Equilibrium Entrained Air content (void fraction) Table 3 Regression coefficients for Straub and Anderson s and Killen s observations on 3 75 slopes Slope ( ) C e, void ratio α r N/A a N/A a N/A a N/A a a Insufficient data to determine rate coefficient. Ce =.656 (1 e.356(θ 1.9) ) Slope, degrees Figure 15 Equilibrium entrained air concentration versus channel slope for Straub and Anderson s (1958) and Killen s (1968) observations. 8 9 Prediction of spillway air entrainment The Straub and Anderson (1958) data are the most comprehensive with regard to variation of slope and, for that reason, a predictive equation for air entrainment into a spillway flow will be fit to these data, after subtracting the 23% entrapped air. Also included are Killen s data for a 52.5 slope, because of a shortage of high slope, developing aerated flow data. The normalized data tend to asymptotically approach an equilibrium value C e, described by C e = C e ( 1 e αx /Yi ) (4) where C e is the mean entrained air concentration and α is the spatial dimensionless rate at which the entrained air approaches equilibrium. A non-linear, multi-variant regression analysis provided estimates of C e and α. Table 3 shows these results. The alpha coefficient showed little variation with no apparent relationship to slope. Equation (4) was therefore revised to incorporate α equal to.1 to describe the mean entrained air content at any location in aerated flow. C e = C e ( 1 e.1x /Yi ) (5) Equation (5) is straight-forward in solution, the dimensionless distance implicitly includes the important variable of distance and unit discharge, and appears to be applicable over relatively wide range of slopes and discharges. However, it is possible that the alpha coefficient will increase or decrease depending upon channel roughness. To apply Eq. (5), the relationship must be known between mean equilibrium air concentration, C e and slope as shown in Fig. 15, along with the fitted relationship for slopes between 11 and 75 C e =.656 ( 1 e.356(θ 1.9)) (6) where the units of θ is degrees and C e is provided as a fraction of Application The following steps illustrate a method to apply Eq. (5) to estimate the entrained air concentration and total conveyed air at any location along the flow path of a spillway with a slope between 11 and 75 : 1. Estimate the depth of flow Y i and the location X i of surface roughness inception with procedures outlined, for example, by Keller et al. (1974). 2. Determine the dimensionless distance to the point of interest with X /Y i = (X X i )/Y i. 3. Determine the equilibrium concentration with Eq. (6). 4. Calculate the entrained air concentration with Eq. (5). 5. Add the entrapped air concentration of 23, 14.2, and 7.3% for the 98, 95, and 9% TCA integration levels, respectively. 6. If computing the spillway side wall height, choose the preferred percent exceedence level and add freeboard.

9 53 Wilhelms and Gulliver The greatest potential for error in this procedure is item 1, calculating the point of inception. If available, a picture of the spillway flow at a known discharge is recommended to verify or calibrate the equations to estimate point of inception. If the spillway is exceptionally rough, such as a stepped spillway, the coefficient α in Eq. (5) could be significantly larger than determined herein. The equations developed herein, with a larger α value should still be applicable. 11 Conclusions The description of air entrainment on a steep spillway or chute, developed in this paper, provides an estimate of entrained and entrapped air transport for application to engineering concerns. With these new descriptions, the entrained air at any location in a high velocity spillway flow can be estimated, which could improve the methodology of reducing localized cavitation damage on spillway surfaces. The volume of entrained air in high velocity flows can also be estimated as a basis for computing the surface area available for gas transfer, which can increase by several orders of magnitude when air is entrained. The most important results of this research are: The descriptions of entrained, entrapped, and total conveyed air provide a more accurate physical description of self-aerated flow on a spillway. The recognition that Killen s (1968) surface measurements represent entrapped air and clearly indicate the need for two measurements (one for total conveyed air and one for entrapped air) to determine the true character of self-aerated flow. The separation of entrapped air from total conveyed air shows that entrapped air dominated the distribution of total conveyed air in the upper region of aerated flow. The entrapped air concentration was constant at about 23% by volume when averaged over the depth up to an air concentration of.98 (Y c = Y 98 ). The similarity of the surface of aerated flow to the momentum surface of a turbulent boundary layer corroborates the cumulative normal distribution of entrapped air and provides a theory to justify a constant entrapped air concentration. An analysis procedure was developed that enabled analysis of several sets of experimental data. A simple non-dimensional mathematical description of developing self-aerated flow was presented for application to self-aerated spillway flow. We recommend that any future experimental work in this area include measurements of both entrained and entrapped air. This requires the use of two instruments: an air concentration probe (Lamb and Killen, 195) for the total conveyed air contained in the bubbles and waves and a surface conduction probe (Killen, 1968) for entrapped air contained only in between the surface waves. The results of this effort have application to at least these three areas of concern to engineers: bulking of flow, cavitation damage to spillway surfaces, and the air water transfer of oxygen, nitrogen, and other gases, as discussed in a companion paper. Acknowledgments The authors wish to recognize the US Army Corps of Engineers, Research and Development Center Waterways Experiment Station for its support in this effort. Notation C e = Mean entrained air concentration C e = Entrained air equilibrium value C = Mean concentration of total conveyed air C E = Mean entrapped air concentration σ E = Standard deviation of the cumulative normal distribution for the entrapped air c(y) = Concentration profile of the total conveyed air as a function of distance y from the spillway surface c E (y) = Concentration profile of the entrapped air as a function of distance y from the spillway surface d 2 = Depth where the entrapped air concentration is.2 d 98 = Depth where the entrapped air concentration is.98 K = Constant q = Unit discharge X = Distance along flume or spillway to location of interest X = Distance from the point of inception to the measurement location X = X X i X /Y i = Dimensionless distance parameter X i = Location of point of inception along flume or spillway y = Depth Y 98 = Integration limit where total conveyed air equals.98 Y c = Integration limit Y i = Depth of flow at the point of inception α = Spatial dimensionless rate at which the entrained air approaches equilibrium θ = Channel slope ( ) References 1. Cain, P. (1978). Measurements within Self-Aerated Flow on a Large Spillway. PhD Thesis, University of Caterbury, Christchurch, New Zealand. 2. Cain, P. and Wood, I.R. (1981). Measurements of Self- Aerated Flow on a Spillway. J. Hydraul. Div. (ASCE) 17(HY11), DeLapp, W.W. (1947). High Velocity Flow of Water in a Small Rectangular Channel. PhD Thesis, University of Minnesota, Minneapolis, MN. Available at safl.umn.edu. 4. Ehrenberger, R. (1926). Flow of Water in Steep Chutes with Special Reference to Self-Aeration. Translated by E.F. Wilsey from Wasserbewegung in Steilen Rinnen (Schusstennen) mit besonder Berucksichtigung der Selbstbeluftung, Osterreichischer Ingenieur und Architektverein, No. 15/16 and 17/ Falvey, H.T. (199). Cavitation in Chutes and Spillways. Engineering Monograph No. 42, Bureau of Reclamation, Dept of Interior, Denver, CO.

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