Air water flows in the presence of staggered and row boulders under macroroughness conditions

Transcription

1 WATER RESOURCES RESEARCH, VOL. 46,, doi: /2009wr008834, 2010 Air water flows in the presence of staggered and row boulders under macroroughness conditions S. Pagliara, 1 I. Carnacina, 1 and T. Roshni 1 Received 29 October 2009; revised 25 February 2010; accepted 1 March 2010; published 18 August [1] Free surface flow over a rough bed at steep slopes entrains large amounts of air bubbles due to the high interaction between the free surface and the bed materials. The assessment of mixed air water flow features in the presence of a three dimensional bed configuration from macroroughness to intermediate roughness conditions can hardly be accomplished with aeration models developed for smooth or stepped bed configurations. Two phase flow properties were measured over rough bed materials in a setup assembled at the PITLAB center at the University of Pisa, Pisa, Italy. Coarser protruding materials were incorporated over the rough bed to intensify the aeration in the mixed air water flow. Flow discharges ranging between 0.02 and 0.09 m 2 /s and slopes between 0.18 and 0.44 were tested. A detailed study of the air water inner layer and flow features over staggered and row coarser material arrangement over the base material was carried out. Results were compared with literature data on free surface flow over smooth and stepped beds. Citation: Pagliara, S., I. Carnacina, and T. Roshni (2010), Air water flows in the presence of staggered and row boulders under macroroughness conditions, Water Resour. Res., 46,, doi: /2009wr Introduction [2] Aerated flow occurs in natural or artificial hydraulics phenomena such as mountainous creeks, jet impingements, and flow transitions from supercritical to subcritical flows and, generally, in turbulent free surface flows. Air entrained in the flow leads to a positive effect in terms of flow reaeration and dissolved oxygen [Moog and Jirka, 1999], representing a valid methodology to improve the fish habitat and to reduce the pollutant content of rivers or water bodies. Indeed, air entrapped in the flow increases the free surface elevation, and correct assessment of the air content is of great importance in the correct design of hydraulic structures. Free surface flow over rough materials under macroroughness or intermediate conditions [Bathurst, 1985] and steep slopes, that is, for slope S o >0.1[Peirson and Cameron, 2006], entrains large amounts of air, owing to the high turbulence developed by the interaction between the bed material and the free surface. Numerous shear and drag vortices can be observed, with the presence of large bubble formations and droplets (Figure 1a). Basically, the presence of air entrained in steep sloped free surface flow increases with the distance from the inlet section, until quasiuniform flow is attainted with the formation of a fully aerated flow. Several theoretical models have been developed to assess the aeration features in mixed air water free surface turbulent flows in the uniform flow region. These are mainly based on the advective diffusion of air into water. Among others, Wood [1984] developed a model based on the conservation equation for the mixture density in the equilibrium region, assuming a constant diffusivity of the air water average density. Later, 1 Department of Civil Engineering, University of Pisa, Pisa, Italy. Copyright 2010 by the American Geophysical Union /10/2009WR Chanson and Toombes [2004] developed an analytical solution for the air bubble advective diffusion based on the airwater flow continuity equation and the bubble rise velocity. However, macroroughness conditions present an even more complicated flow feature in the inner region, compared to smooth or stepped channels; therefore the analytical solution presents some limitations when adapted to these conditions. Several studies have been carried out on flow characteristics over coarse materials (natural stone, strip, boulders, and cubes), which focus mainly on flow patterns, energy dissipation, the friction factor, or the material stability in steep slopes [Morris, 1959; Rouse, 1965; Hartung and Scheuerlein, 1970; Hey, 1979; Knauss, 1979; Bathurst, 1985; Rice et al., 1998; Robinson et al., 1998; Pagliara and Chiavaccini, 2006; Peirson and Cameron, 2006; Eli and Gray, 2008; Pagliara et al., 2008]. Little information is available concerning self aeration features and air measurements in the presence of coarse material. A field investigation carried out by Vallé and Pasternack [2006a, 2006b] related to both air concentration and hydraulic characteristics of hydraulic jump in bedrock step and pool channels. In contrast, Moog and Jirka [1999], Cokgor and Kucukali [2004], and Kucukali and Cokgor [2008] analyzed the reaeration efficiency and turbulent flow characteristics in the presence of macroroughness elements. Generally, the reaeration efficiency has been found to depend on the blockage controlled by the macroroughness elements and by the relative submergence. [3] The flow over coarse material under macroroughness conditions exhibits a flow pattern similar to that of supercritical flows on steep surfaces, either smooth or with a stepped configuration. Several studies have been carried out concerning the effect of smooth and stepped chutes under skimming flow conditions. In particular, Gonzalez et al. [2008] studied air concentration profiles over rough stepped 1of11

2 Figure 1. (a) Fully aerated flow over rough bed with bubbles and droplet formation. (b) Sketch and notation. chutes. Either turbulence characteristics or cavity recirculation features on both smooth and stepped bed configurations have been studied by Hager [1991], Chanson [1994], Hager and Blaser [1997], Boes and Hager [2003], Gonzalez and Chanson [2004], Kramer et al. [2006], and Felder and Chanson [2009]. [4] Because of the lack of information in the literature concerning the aeration features of self aerated flow over steep slopes and the numerous parameters involved in the process, this paper aims to provide a preliminary overview of the self aeration process and the air concentration distribution for flows under macro and intermediate roughness conditions over steep slopes. In addition, the influence of coarse protruding material placed over the base bed material on two phase flow features is studied. Results have been compared with experiments previously carried out over smooth, roughened, and stepped channels. 2. Experimental Apparatus and Instrumentation [5] Experiments were performed at the PITLAB center, University of Pisa, in a 7 m long and 0.3 m wide tilting channel. Figure 1b shows the experimental setup. The flow depth is measured by means of the methodology suggested by Hughes and Flack [1984] and Pagliara et al. [2008]. Hence, the air water flow depth is measured starting from the effective top (ET), which is the hypothetical plane (dashed line in Figure 1b) located 0.2D 65 below the plane interpolating the most prominent projections of the base elements (this plane represents the physical top; PT), where D 65 is the base material diameter for which 65% of the material weight is finer. Further, y is the vertical coordinate measured from the PT, y 90 is the distance normal to the bed where the air concentration C = 90%, x is the longitudinal distance from the channel entrance, and x s is the longitudinal distance relative to the stone or the boulder tips. The characteristic diameters of the rough bed elements used were D 16 = mm, D 50 = mm, and D 84 = mm, where D xx is the particle size for which xx% of the material weight is finer. Crushed angular elements were almost uniform in size, with a uniformity parameter of s p =(D 84 /D 16 ) 0.5 = 1.24 [Dey and Raikar, 2007]. Three bed configurations have been tested. The base macroroughness configuration (B MR ) was prepared gluing the rough elements over a stainless sheet. Hemispherical coarser elements of diameter D b = 55 mm (cobbles or boulders on the prototype scale) were positioned over the bed either in a row (subscript R; B C R )or in a staggered manner (subscript S; B C S ). Bed profile and vertical distances were measured with a point gauge of precision of ±0.1 mm, while water discharge was measured using a Krohne OPTIFLUX 2000 KC magnetic flow meter. Air water flow properties were measured using a USBR singletip conductivity probe air concentration meter with a 6 mmdiameter tip [Jacobs, 1997; Matos and Frizell,1997;Pagliara et al., 2009], aligned against the flow direction. The probe was carefully calibrated before each test and the probe signal was sampled at a sampling frequency of 2 khz during 30 s with the help of a NI 6009 data logger. The conductivity probe signal was processed using a threshold limit (50% of the maximum output voltage). A signal voltage higher than the threshold limit was assumed to be air, and a voltage lower than the threshold limit was taken to represent water. Air concentrations C were evaluated as the percentage of time during which the signal was higher than the threshold limit (air detection). Translation of the conductivity probe along and across the channel was controlled by a fine adjustment traveling mechanism. The vertical position accuracy of the conductivity probe was ±0.1 mm. Flow visualizations were conducted using a high speed camera. Preliminary observations showed that the uniform flow zone was achieved at x/d 84 > 60. Therefore, the air concentration was surveyed at longitudinal sections x = 311 cm, x = 411 cm, x = 511 cm, and x = 611 cm for both B MR and B C S. For B C R, measurements were taken in the uniform flow zone (x > 300 cm) for four longitudinal sections, that is, for G = 0.05, where G =(N b pd b 2 )/(4BL) = boulder concentration [Pagliara and Chiavaccini, 2006], in which N b is the number of boulders on the bed length L, and B is the bed width, at x b /D b = 0.5, x b /D b =0,x b /D b = 0.5, and x b /D b = 8.2 and for G = 0.15 at x b /D b = 0.5, x b /D b =0x b /D b = 0.5, and x b /D b = 2.7, where x b = distance from the boulder top in the row configuration. For all configurations and for each longitudinal section, five transverse profiles ( 10, 5, 0, 5, and 10 cm from the channel center) were taken to check the transverse variability of the air concentration profiles, with a vertical step of 3 mm starting from the PT. Tests were carried out in the following experimental ranges: Froude number 0.67 F r = V/(gd e ) , where V = q sup /d e = average flow velocity, q sup is the surface (subscript sup ) discharge per unit width [Pagliara et al., 2008], g is the gravitational acceleration, and d e is the equivalent depth, defined as d e ¼ 0:2D 65 þ Z y¼y90 y¼0 ð1 CÞdy ¼ 0:2D 65 þ y 90 ð1 C m Þ; ð1þ 2of11

3 Figure 2. Air concentration profile for (a) S o =0.27andq =0.05m 2 /s for the base macroroughness configuration (B MR ) and the staggered configuration (B C S )and(b)s o =0.27andq =0.05m 2 /s for the row configuration (B C R ). R y¼y90 where C m ¼ 1=y 90 y¼0 C dy is the depth averaged void fraction. Channel slopes were varied in the range 0.18 S o 0.44, relative equivalent depths ranged within 0.6 d e /D , and the range for relative boulder diameters was 0 D b /D Discharges per unit width were varied between 0.02 m 2 /s q 0.09 m 2 /s or 0.64 d c /D , where d c =(q 2 /g) 1/3 = critical depth, for three bed configurations: G = 0 for B MR (base configuration), and 0.05 G 0.15 for B C S and B C R. 3. Experimental Results and Analysis 3.1. General [6] Experiments carried out for B MR, B C S, and B C R indicated substantial differences in the air concentration profiles. Air concentration measurements for B MR (G =0) and for B C S with G = 0.05 and G = 0.15 for S o = 0.27 and q = 0.05 m 2 /s are shown in Figure 2a. The figure shows C profiles versus y/y 90 measured at longitudinal section 311 cm x 611 cm for all the five transverse concentration profiles. Accordingly, concentration profiles move toward the bed as G increases, for given S o and q. High speed photography shows a combination of drag vortices and wakes downstream of the stone or the boulder tips and shear vortices in between their cavities [Pagliara et al., 2009]. The high dispersion observed in the air concentration profiles relates to the high morphological variability of the base configuration. The presence of boulders in the staggered configuration increases the spatial variability of the bed and, therefore, contributes to higher air concentration profile dispersion. Figure 2b shows C profiles at the crest of the boulder row (x b =0) and in the middle of two boulder rows for G = 0.05 and 0.15 (x b = 15 and 45 cm, respectively). For a given y, air concentration profiles at x b = 0 exhibit lower C values compared to concentration profiles between boulder rows, where stable shear vortices occur. Likewise, in skimming flow over rough stepped chutes, the presence of stable vortices between two consequent steps increases the air concentration compared to that measured over the step edge [Gonzalez et al., 2008]. It is worth observing that, even for B C R, C increases with the boulder concentration for constant S o and q. However, C profiles resulted in less data scatter compared to B C S at constant x b and G. In general, if the boulder elements are sufficiently close (G = 0.15) in the staggered or row arrangement, the wake behind each boulder element extends nearly to the next boulder element, resulting in a stable shear vortex, with a high turbulence intensity and high momentum exchange, thus entrapping large amounts of air Flow Patterns Under Macroroughness Conditions [7] According to Gangadharaiah et al. [1970] and Chanson [1993], air entrainment occurs if the turbulence intensity normal to the flow is large enough to overcome the surface tension pressure of the entrained bubble and the bubble rise velocity component. Therefore, on the basis of visual observations, three main basic flow patterns were distinguished in terms of boulder concentrations and different slopes: quasiuniform flow without aeration (M R W ), quasi uniform flow with aeration (M R A ), and skimming flow (M R SK ). Figure 3 shows a sketch of the three flow patterns: M R W,M R A, and M R SK. For small q and S o, water flows over the bed material as a succession of clear submerged nappes and the undular free surface exhibits a profile comparable to the base material profile. The velocity is not high enough to develop stable eddies between the elements or droplets from the flow, such that self aeration is achieved. For higher q and S o the continuous wake generation and fall of flow over stones and boulders allow air to be entrained in the free surface, generating self aerated flow (M R A ). Similarly to the results of Moog and Jirka [1999] and Vallé and Pasternack [2006a] for step and pool configurations in the presence of macroroughness elements, both M R W and M R A are characterized by a strong free surface undulation about the average y 90, with a continuous succession of jets and relatively low velocity flow and submerged hydraulic jumps developing all along the channel bed, in correspondence with macroroughness elements. As q or S o increases, small stable form drag and shear vortices develop over the stone or the boulders tips and between the roughness elements, respectively. Air is 3of11

4 Accordingly, for B MR the transition between M R A and M R SK relates to a higher F r as S o decreases. For a given S o, the transition curves between M R A and M R SK for B C S relates to a lower F r as long as G increases and the clearance between two successive elements decreases. Note that M R SK generally results in F r >1. Figure 3. Flow patterns: macroroughness quasi uniform flow without aeration (M R W ), macroroughness quasi uniform flow with aeration (M R A ), and macroroughness skimming flow (M R SK ). entrained by means of the high turbulence intensity of shear vortices and from the droplets impacting the free surface. As has been observed in stepped chutes, either smooth or rough, the water flows over a pseudobottom in the M R SK flow regime, which can be identified with the ET for the base configuration, indicating a quasi smooth free surface. Figure 4 classifies the flow regimes as a function of F r and S o for 0.02 m 2 /s q 0.09 m 2 /s, 0.18 S o 0.44, and 0 G 0.15 for the B MR and B C S configurations. Preliminary observations revealed that for S o < 0.09 the flow regime M R W occurs for both the base and the staggered arrangements Air Concentration Profiles [8] Detailed concentration measurements over the different rough elements are shown in Figures 5a and 5b for S o = 0.44 and S o = 0.27, at q = 0.05 m 2 /s, for B C S with G = Pagliara et al. [2009] demonstrated that both the element shape and the surface roughness affect these profiles, with a peak in the wake region for the higher slope (Figure 5a), owing to the combination of drag and shear vortex downstream of the protruding elements. Observe that a concentration peak in the inner wake region is present only for M R SK at x s /D b = 0.73 for S o = 0.44 (Figure 5b) owing to the shear vortex recirculation underneath the wake diffusion layer (dashed lines). [9] Figures 6a and 6b show the air concentration profiles for the B C R arrangement for S o = 0.27 and S o = 0.44 at q = 0.05 m 2 /s and G = Since the spacing between two boulder rows is too close here, the aeration features are affected by the wake of the flow of the previous boulder row. For S o = 0.44, owing to the presence of the wake interference and the presence of stable vortices between two boulder rows, a higher air concentration around the stone is observed compared to S o = 0.27 (Figure 6a). The higher concentration in the inner layer for S o = 0.44 and x b /D b = 0.73 (Figure 6b) is due to cavity formation downstream of the boulder tip. Figure 7a shows the normalized concentration profiles C ~ = C/C max versus ~Y = y/y max, with C max = inner layer peak concentration and y max = y(c = C max ), of the wake mixing layer and their streamwise diffusion. It is worth observing that only data that presented a concentration peak are plotted in the graph. Similar to the diffusion equation adopted by Chanson and Brattberg [2000] and Murzyn et al. [2005] for direct hydraulic jumps, the nondimensional air concentration distribution can be approximated by the Gaussian distribution as h 2 i ~C ¼ exp 0:5 ~Y 1 = : ð2þ The nondimensional coefficient x results from wake airconcentration profiles by means of a standard regression of Figure 4. Flow pattern classification in uniform flow, M R W (diagonally striped area), M R A (white area), and M R SK (hatched area), for (a) B MR and (b) B C S for G = 0.05 and (c) B C S for G = of11

5 Figure 5. Air concentration profiles over (a) stone and (b) boulder for B C S. profiles at constant x s /D 84 or x s /D B. As for hydraulic jump, x depends on the distance from the element tip and can be considered a measure of the wake air water diffusion. According to Figure 7, x = 0.36 for x s /D 84 = 0.43 and x = 0.26 for x s /D 84 = 0.85 and x s /D b = In the presence of the direct hydraulic jump, equation (2) gives relatively good results, as the concentration is measured in the mixing layer (i.e., ~Y < 1), as shown by Murzyn et al. [2005]. The slight dispersion around equation (2) for ~Y < 1 is due to local differences in the aeration process and the local interaction between the wake and the shear vortices between cavities. As observed by high speed photography, air is entrained at the free surface and the wake, generated by stones or boulders protruding in the outer flow region, continuously entrapping bubbles that are convected downstream (Figure 7b). The flow is not accelerated by the presence of the recirculation vortex present at the top of the mixing layer, as at the toe of hydraulic jump, but is accelerated by the presence of stone or boulder obstruction [Cokgor and Kucukali, 2004; Tritico and Hotchkiss, 2005; Kucukali and Cokgor, 2008] and its interaction with the flow surface. Concentration profiles rapidly break away from equation (2) as the data approach the transition between the wake and the outer layer, that is, for ~Y > 1. The transition between the wake mixing layer and the free surface region is comparable with either the airconcentration profile of hydraulic jumps observed by Chanson and Brattberg [2000] and Murzyn et al. [2005] or the mixing layer concentration profile observed between step cavities by Chanson and Toombes [2004]. In the present study, experiments showed that for d e /D 84 >1,C max follows C max ¼ 1:22ðd e =D 84 Þ 4:5 ; ð3þ and the corresponding vertical elevation can be averaged by y max =0.5d e, with values in the range 0.31d e < y max <0.61d e. [10] Figures 8a and 8b show the bubble frequency F ab of the wake region as a function of y/y 90 and C, respectively, where F ab is the number of bubbles measured per second. It is noteworthy to highlight the main differences versus the F ab from the hydraulic jump. According to Chanson and Brattberg [2000] and Murzyn et al. [2005], the frequency of the hydraulic jump shear region first increases, to a Figure 6. Air concentration profiles over (a) stone and (b) boulder for B C R. 5of11

6 maximum in the shear layer, and then decreases toward the hydraulic jump free surface. In contrast, the wake diffusion layer bubble frequency generally has two peaks: the first F ab peak corresponds with the wake region, then the second peak is in the transition zone between the outer flow and the wake diffusion layer (Figure 8a). F ab then decreases toward the free surface, where droplets and air pockets dominate the aeration process, as observed in skimming flow over stepped chutes. Moreover, as observed by Chanson and Toombes [2004], F ab distributions as a function of C can be approximated by a parabolic law. However, the presence of the wake affects the frequency distribution, which differs slightly from the parabolic law (Figure 8b), resulting in a relatively low F ab in the wake region compared to the same C measured in the outer region. Figure 7. (a) Normalized concentration profiles over stone (S) or boulder (B) tips for B C S and (b) wake sketch Aeration Features for the B C R Configuration [11] Morris [1959] analyzed the flow features between rough elements of a row configuration. The interaction between the element wake and the element immediately downstream affects vortex generation. Different flow patterns have been distinguished, in terms of the relative distance between the semispherical elements, that is, isolated roughness flow and wake interference flow. Likewise, for the B C R arrangement over the base material, the air entrainment is strongly affected by the element wake characteristics and the relative distance between rows. [12] The transversely averaged air concentrations measured between two boulder rows at different boulder concentrations, G = 0.05 and G = 0.15, are compared with field data of Vallé and Pasternack [2006a] for a step pool configuration in Figures 9a and 9b. Here C t is the transverse averaged concentration based on five transverse concentration profiles. For submerged and unsubmerged jumps the C m data relate to the thalweg section, D 84 = m [Vallé and Pasternack, 2006b], while x b = 0 corresponds to the jump toe. In both cases, C t for B C R first decreases from x b /D b = 0.5 to x b /D b = 0 and then increases in the wake region of the rows, that is, at x b /D b = 0.5. However, for G = 0.05 (Figure 9a) C t decreases beyond x b /D b = 0.5 as the wake moves downstream from Figure 8. Wake s F ab analysis for G = 0.15: (a) F ab vertical distribution and (b) F ab versus C. F ab is the number of bubbles per second. 6of11

7 Figure 9. Comparison of C t between two rows of boulders and Vallé and Pasternack [2006a] data for (a) G = 0.05 and (b) G = the boulder tip. Since the gap between two boulders is relatively wide, the wake interaction with the element downstream is less prominent and results in air detrainment as the turbulent intensity is not high enough to overcome the bubble buoyancy. Small stable vortices are present in between the cavities of the base material elements, which further contributes to the self aeration process. The figure also confirms the increase in C with slope. Figure 9b compares the data for two discharges, S o = 0.18 and G = The wake interference between the boulder rows results in the formation of stable vortices, which leads to a larger air entrainment in the center of the boulder rows, confirming the important role of the relative distance between boulders on the self aeration process. [13] Similarly, the Vallé and Pasternack [2006a] field data for unsubmerged hydraulic jumps showed a first maximum C m and a strong detrainment action. Although less sloped, field data in case of unsubmerged jump showed higher C m (about 50% against 31% for B C R ) as compared to B C R. Note that C m field data relate to the thalweg section and that field data showed a high transverse dispersion due to the high spatial variability of natural step and pool. Vice versa, the submerged jump showed maximum C m values of the same magnitude of maximum C t for both G = 0.05 and G = 0.15, whilst the transverse C m presented less spatial distribution as observed in B C R Average Air Concentration [14] Average air concentration profiles have been compared with the air diffusion model proposed by Chanson and Toombes [2004]: " # C ¼ 1 tanh 2 K 0 y=y 90 ð þ y=y 90 1=3Þ 2D 0 3D 0 ; ð4þ where D 0 is a dimensionless diffusivity coefficient and K is the dimensionless integration coefficient. The average Figure 10. Air concentration profiles at S o = 0.18 (open circles), S o = 0.27 (filled squares), and S o = 0.44 (open triangles) for q = 0.09 m 2 /s and G = 0.15 in (a) the B C S and (b) the B C R configurations. 7of11

8 Figure 11. (a) Relation between average air concentration and relative equivalent depth for different slopes for the B MR and B C S configurations; (b) comparison of C mmeas versus equation (6). concentration C m is defined as a function of D 0 and is given by C m ¼ 0:76½1:04 expð 3:61D 0 ÞŠ: ð5þ The profiles shown in Figures 10a and 10b were obtained for slopes 0.18 S o 0.44 for B C S and B C R with a boulder concentration G = 0.15 for q = 0.09 m 2 /s. For the B C S configuration the average concentration curve relates to the average of all the vertical sections measured under uniform flow conditions, that is, for 66 x/d Generally, as the slope increases, the relative equivalent depth is reduced and, thereby, results in higher interaction between the bed elements and the free surface, causing greater air entrainment. In contrast, for B C R arrangements data have been plotted for the four longitudinal sections ( 0.5 x b /D b 2.7) measured in the uniform flow condition region. Therefore, C m results in the combination of the higher aeration due to the wake downstream of the boulder row and that measured in the middle section of the row arrangement. Concentration profile showed a higher dispersion toward the averaged air concentration profile compared to the B C S arrangement, while, also as in staggered boulders, a higher C m relates to a higher S o B MR and B C S Configurations [15] On the basis of the experimental results, the average concentration C m for B MR and B C S for different boulder concentrations for the selected range of slopes can be expressed as a function of G, S o, and the relative equivalent depth d e /D 84 : Cokgor [2008], in which the reaeration efficiency decreased as d e /D 84 increased and increased as the area occluded by the macroroughness elements increased. This is due to the lesser interaction between the free surface and the bed material as d e /D 84 increases. It is worth observing that, given q, the boulders presence strongly affected the free surface, which results in a rise in the relative equivalent water depth. Figure 11b shows a comparison between measured C mmeas and calculated C mcalc average air concentration (equation (6)) for the B C S configuration. It can be observed that the points are confined within the range ±20% with respect to the perfect agreement line, showing a good agreement with equation (6) for both the B MR and the B C S configuration B C R Configuration [16] Figure 12 illustrates C m in terms of d e /D 84 for 0.18 S o 0.44 and 0.05 G Unlike the B C S arrangement, the air concentration measured around the boulder profile and in between the boulder rows results in a lower C m for! C m ¼ ð0:12 0:9GÞþ 1:69 þ 2GS 2:15 S 2:27 d 0:9 e : o o D 84 ð6þ Figure 11a illustrates that the mean air concentration decreases with the increase in d e /D 84, for all configurations, and the effect of G becomes negligible at higher d e /D 84. These results are in agreement with that of Kucukali and Figure 12. Relation between C m and d e /D 84 for the B C R configuration. 8of11

9 Table 1. Literature Experimental Ranges and Average Air Concentration Laws Experimental Range Air Concentration Law (C m ) Note d e /D 84 or d c /h G R 10 4 Fr Reference(s) Rock base material channel up to 0.35 m, Hartung and Scheuerlein [1970] data on flow on rock fill channel bottom Knauss [1979] sin() 0.08 Hager [1991] sin() 0.75 D 50 = 0.7 mm, B = 0.46 m Data on artificially roughened channel: Data on artificially roughened channel and prototype and large spillway model Chanson [1993] sin() Skimming flow over stepped spillways with different step heights Chamani and Rajaratnam [1999] a log[sin() 0.1 /q 0.3 ] Stepped channels with uniform channel slope and step height Ohtsu et al. [2004] a Present study ( G) + [( G) S o ]So (de /D 84 ), where, S o = tan() Block ramp with boulders D 0.3exp[ 5(h/dc) 2 4h/dc], where, D = 0.3 for D = , for a h/d c = relative step height. q = discharge per unit width (m 2 /s). G = 0.05 compared to the B MR configuration for S o = 0.27 and S o = However, for G = 0.15, C m again shows higher values compared to B MR. Indeed, the aeration mechanics in the presence of boulders in rows is originated by a complete different self aeration process compared to both the base configuration and the staggered arrangement. Since boulders are arranged in rows over the base configuration at a low boulder concentration (G = 0.05), self aeration reflects the wake generation of the boulders in rows that showed an air detrainment action as far as the flow moves from the wake (also shown in Figure 9). Therefore, the mechanism of air entrainment in self aerated flow is affected by the rows rather than by the air entrapped in the base material cavities. In contrast, at G = 0.15 air is entrapped and recirculated by the presence of a stable vortex in between the rows, showing a higher C m compared to B MR. 4. Literature Comparison [17] The measured C m has been compared with previous models developed for free surface flow. Table 1 reports the range of validity of studies carried out on a smooth channel, stepped chute, and roughened stepped chute, in terms of d e /D 84 or relative step height d c /h, where h is the step height, is the channel angle from the horizontal, R =4q/n is the Reynolds number, in which n is the water viscosity, and F r. Table 1 also reports the equation proposed by the authors to predict C m. Figure 13 shows a comparison between the measured air concentration C mmeas = for the B MR,B C S, and B C R configurations and the average air concentration C mcalc calculated with equation (6) and with the relations reported in Table 1. The slight differences shown in the C mmeas and equation 6 relate to the B C R data, whose self aeration mechanics differs from that of both the base and the staggered configuration, as already mentioned. However, even in the case of the B C R configuration, equation (6) seems to predict C mmeas well, with errors of generally less than 25%. Seemingly, the equation proposed by Chanson [1993] and Hager [1991] for an artificially roughened channel showed a good agreement with C mmeas, although based only on the chute slope. It is worth noting that the Hager [1991] and Figure 13. Table 1). Comparison between C mmeas and C mcalc (see 9of11

10 Chanson [1993] equations have been calibrated on tests performed under low roughness conditions with a high F r. In contrast, for a similar F r value and close relative equivalent depth, but under the stepped chute condition [i.e., Ohtsu et al., 2004], the equation performance deteriorates, showing a higher C mcalc with respect to C mmeas. Also, the models of Knauss [1979], Chamani and Rajaratnam [1999], and Ohtsu et al. [2004] greatly overpredict C mmeas. However, further investigation of aeration structures and of the air concentration should highlight scale effects and differences that actually occurred between macroroughness skimming flow and the chute or stepped chute self aeration process. 5. Conclusions [18] The flow properties under macro and intermediate roughness conditions with different boulder arrangements and concentrations in an open channel flow with steep slopes have been investigated. Three flow patterns have been distinguished. Air concentration measurements and flow patterns were compared with the basic configuration. Detailed air concentration profiles around the rough bed elements were studied. The measurements were compared with those over smooth stepped and roughened channels in earlier studies. The results demonstrate that the staggered boulder arrangement exhibits features similar to those of the base configuration. In addition, the average air concentration increases with boulder concentration for the selected test range. But for G = 0.05 in the row arrangement of boulders, the average air concentration was found to be lower than that with the base configuration because of flow detrainment action and free surface rise. A detailed study of the flow properties in the inner layers of air water mixed flow exhibited that the coarseness of the bed materials enhances air entrainment, with its influence indicated in the concentration profile. The proposed analytical model for calculating the mean air concentration for the staggered boulder arrangement fits well with the experimental data in the selected experimental range. Notation B Channel width [L] C Air concentration [ ] ~C C/C max [ ] C m Depth averaged air concentration in terms of y 90 [ ] Maximum air concentration in the inner layer [ ] Calculated depth averaged air concentration in terms of y 90 [ ] Measured depth averaged air concentration in terms of y 90 [ ] C t Average concentration along a transverse section [ ] D 0 Dimensionless diffusivity coefficient [ ] D b Median size of boulders [L] d c Critical flow depth [L] d e Equivalent depth [L] D xx Characteristic diameter of the bed material for which xx%, in weight, of material is finer [L] F r Froude number [ ] F ab bubble frequency [T 1 ] C max C mcalc C mmeas g Gravitational acceleration [L T 2 ] h Step height [L] K Dimensionless integration coefficient [ ] L Length of the bed [L] N b Number of boulders [ ] q Total discharge per unit width [L 2 T 1 ] q sup Superficial discharge per unit of width [L 2 T 1 ] R Reynolds number [ ] S o Channel slope [ ] V Average flow velocity over the rock [L T 1 ] x Longitudinal distance from channel entrance [L] x b Distance from boulder row [L] x s Longitudinal distance relative to the stone or boulder tip [L] y Vertical coordinate measured from the PT [L] y 90 Depth at which the air concentration C equals 90% [L] y max Vertical coordinate for C max in the inner layer [L] ~Y y/y max [ ] Greek Symbols Ø Diameter [L] x Coefficient [ ] G Percentage coverage of boulders [ ] Channel slope [deg] s p Particle uniformity parameter [ ] n Water kinematics viscosity [L 2 T] References Bathurst, J. 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