Turbulence characteristics in skimming flows on stepped spillways

Transcription

1 Turbulence characteristics in skimming flows on stepped spillways 865 G. Carosi and H. Chanson Abstract: The stepped spillway design is characterized by an increase in the rate of energy dissipation on the chute associated with a reduction of the size of the downstream energy dissipation system. This study presents a thorough investigation of the air water flow properties in skimming flows with a focus on the turbulent characteristics. New measurements were conducted in a large-size facility (q = 228; step height, h = 0.1 m) with several phase-detection intrusive probes. Correlation analyses were applied to estimate the integral turbulent length and time scales. The skimming flow properties presented some basic characteristics that were qualitatively and quantitatively in agreement with previous air water flow measurements in skimming flows. Present measurements showed some relatively good correlation between turbulence intensities T u and turbulent length and time scales. These measurements also illustrated large turbulence levels and large turbulent time and length scales in the intermediate region between the spray and bubbly flow regions. Key words: turbulence, stepped spillways, skimming flows, turbulent energy dissipation. Résumé : Les évacuateurs de crues en marches d escalier sont caractérisés par une taux important de dissipation d énergie cinétique sur le coursier, et donc, une réduction de la taille du bassin de dissipation aval. Dans cette étude, on présente des séries de mesures détaillées dans l écoulement diphasique eau air, avec de nouvelles mesures des propriétés turbulentes. Ce travail a été réalisé dans une modèle physique de grande taille (q =228, h = 0,1 m) avec plusieurs sondes de mesures intrusives. L application d analyses corrélatives fournit des mesures de longueur et temps integrale turbulent. Les résultats de l étude sont en accord qualitatifs et quantitatifs avec des études précédentes en écoulements extrèmement turbulents («skimming flows»). On montre une corrélation relativement bonne entre les intensités turbulentes T u et les échelles intégrales turbulentes de longueur et de temps. Les résultats suggèrent un mécanisme de dissipation turbulente dans la région intermédaire entre la région d écoulement à bulles et la région d écoulement à gouttes. Mots-clés :turbulence, coursier en marches d escalier, écoulement extrèmement turbulent, dissipation d énergie. Introduction Stepped spillways have been used for many centuries (Chanson 1995b, 2000, 2001a). The stepped design increases the rate of energy dissipation on the chute and reduces the size of the downstream energy dissipation system (Fig. 1). Figure 1 shows two recent reinforced cement concrete (RCC) dam stepped spillways with small stilling basins. For the last 20 years, research in the hydraulics of stepped spillways has been active (Chanson 1995a, 2001a). On a stepped spillway, the waters flow as a succession of free-falling nappes (nappe flow regime) at small discharges (Chamani and Rajaratnam 1994; Chanson 1994a; Toombes 2002; El-Kamash et al. 2005). For a range of intermediate flow rates, a transition flow regime is observed (Ohtsu and Yasuda 1997; Chanson 2001b; Chanson and Toombes 2004). Modern stepped spillways are typically designed for Received 26 September Revision accepted 20 February Published on the NRC Research Press Web site at cjce.nrc.ca on 12 August G. Carosi and H. Chanson. 1 Division of Civil Engineering, The University of Queensland, Brisbane QLD 4072, Australia. Written discussion of this article is welcomed and will be received by the Editor until 31 January Corresponding author ( h.chanson@uq.edu.au). large discharge capacities corresponding to a skimming flow regime (Rajaratnam 1990; Chanson 1994b; Chamani and Rajaratnam 1999). In a skimming flow, the flow is nonaerated at the upstream end of the chute. Free-surface aeration occurs when the turbulent shear next to the free surface becomes larger than the bubble resistance offered by surface tension and buoyancy. Downstream of the inception point of free-surface aeration, some strong air water mixing takes place. Large amounts of air are entrained and very strong interactions between main stream turbulence, stepcavity recirculation zones, and free surface associated with strong energy dissipation and flow resistance are observed (Chanson and Toombes 2002a; Kokpinar 2005). The flow resistance is primarily a form drag in skimming flows (Rajaratnam 1990; Chanson et al. 2002). At each step, the cavity flow is driven by the developing shear layer and the transfer of momentum across it (Gonzalez and Chanson 2004). The energy dissipation mechanisms include cavity recirculation, momentum exchange with the free stream, and interactions between free-surface and mainstream turbulence (Fig. 2). The interactions between mixing layer and horizontal step face, and the skin friction at the step faces, may contribute to further energy dissipation, in particular on moderate slopes. At each step edge, highly coherent smallscale vortices are formed abruptly at the step corner because of the large gradient of vorticity at the corner (Fig. 2). The initial region of the mixing layer is dominated by a train of Can. J. Civ. Eng. 35: (2008) doi: /l08-030

2 866 Can. J. Civ. Eng. Vol. 35, 2008 Fig. 1. Photographs of modern stepped spillways: (a) Pedrogao dam stepped spillway (Portugal) on 4 September 2006 reinforced cement concrete (RCC) gravity dam structure completed in March 2006, uncontrolled stepped spillway (h = 0.6 m, 1V:0.75H); (b) Riou dam stepped spillway (France) on 11 February 2004 RCC gravity dam structure completed in 1990, uncontrolled stepped spillway (h = 0.43 m, 1V:0.6H).

3 Carosi and Chanson 867 Fig. 2. Stepped cavity recirculation in skimming flows. sequential small-scale vortices that eventually pair to form large-scale vortical structures that are advected downstream. The distance from the step edge to the impingement of the shear layer onto the step face becomes an important length because some feedback may occur almost instantaneously from the impingement to the singularity region of the shear layer in the vicinity of the step edge (Lin and Rockwell 2001). Experimental studies of turbulent flows past twodimensional cavities showed that cavity resonance is primarily a function of the ratio of boundary layer thickness to cavity length. That is, Y 90 sinq/h for a stepped chute, where Y 90 is the characteristic air water depth for a void fraction of 0.90, h is the vertical step height, and q is the angle between the pseudo-bottom formed by the step edges and the horizontal. Ohtsu et al. (2004) showed that the flow resistance appeared to be maximum for a slope q of around 188 to 228. Gonzalez and Chanson (2006) hypothesized that some maximum values in flow resistance must be related to some flow instability. The three-dimensional nature of recirculating vortices is believed to play a role to further the rate of energy dissipation, and Gonzalez and Chanson (2005) demonstrated quantitatively the means to enhance the flow resistance with turbulence manipulation. It is the purpose of this study to investigate thoroughly the air water flow properties in skimming flows, with a focus on the turbulent characteristics. New measurements were conducted in a large-size facility (q =228, h = 0.1 m) with several phase-detection intrusive probes. Detailed air water flow properties were recorded systematically for several flow rates. The results included the distributions of turbulence intensity and of integral length scales. They showed that the rate of energy dissipation on stepped spillways is associated with high turbulence levels and large-scale vortical structures. Experimental setup New experiments were performed in the Gordon McKay Hydraulics Laboratory at the University of Queensland (Table 1). The experimental channel was previously used by Chanson and Toombes (2002a) and Gonzalez (2005). Waters were supplied from a large feeding basin (1.5 m deep, surface area 6.8 m 4.8 m) leading to a sidewall convergent with a 4.8:1 contraction ratio. The pump, delivering the flow rate, was controlled with an adjustable frequency AC motor drive that enabled an accurate discharge adjustment in the closed-circuit system. The test section consisted of a broad-crested weir (1 m wide, 0.6 m long) followed by 10 identical steps (h = 0.1 m, l = 0.25 m) made of marine plywood. The stepped chute was 1 m wide with perspex sidewalls followed by a horizontal concrete-invert canal ending in a dissipation pit. A comparison between present experiments and past studies is given in Table 1. Further details on the experimental facility and data were reported in Carosi and Chanson (2006). Instrumentation Clear-water flow depths were measured with a point gauge. The water discharge was measured from the upstream head above the crest, and the head-discharge relationship was checked with detailed velocity distribution measurements on the crest itself. Air water flow properties were measured with single-tip and double-tip conductivity probes (Fig. 3). Basic air water flow measurements were performed with single-tip conductivity probes (Fig. 3a). The probe sensor consisted of a sharpened rod (Ø = 0.35 mm) coated with a nonconductive epoxy set into a stainless steel surgical needle acting as the

4 Table 1. Experimental investigations of stepped chute flows on flat slopes (q < 308). Reference q (8) Step geometry Flow conditions Instrumentation Remarks Chanson and Toombes (2002a) W =1m Series 1 16 Smooth horizontal steps Double-tip conductivity probe Experiments TC201 (h = 0.1 m, l = 0.35 m) (Ø = mm) Series 2a 22 Smooth horizontal steps q w = m 2 /s, Single-tip conductivity probe Experiments EV200 (h = 0.1 m, l = 0.25 m) Re = (Ø = 0.35 mm) Series 2b 22 Smooth horizontal steps (h = 0.1 m, l = 0.25 m) q w = to m 2 /s, Re = Double-tip conductivity probe (Ø = mm) Experiments TC200 Chanson and Toombes (2002b) Single-tip conductivity probe W = 0.5 m (Ø = 0.35 mm) Series Smooth horizontal steps L = 24 m, 10 steps (h = m, l = 2.4 m) series Smooth horizontal steps L = 24 m, 18 steps (h = m, l = 1.2 m). Toombes (2002) Smooth horizontal steps (h = m, l = 2.4 m) Stepped cascade 3.4 L = 24 m, 10 steps Single-tip conductivity probe W = 0.5 m (Ø = 0.35 mm) Single-step chute 3.4 L = 3.2 m, 1 step Double-tip conductivity probe (Ø = mm) W = 0.25 m Yasuda and Chanson (2003) 16 Smooth horizontal steps (h = 0.05 m) Ohtsu et al. (2004) Gonzalez (2005) Gonzalez et al. (2005) 5.7, 8.5, 11.3, 19, 23, 30 h = m 16 Smooth horizontal steps (h = 0.05 m, l = m) 16 Smooth horizontal steps (h = 0.1 m, l = 0.35 m) 22 Smooth horizontal steps (h = 0.1 m, l = 0.25 m) 22 Horizontal steps (h = 0.1 m, l = 0.25 m) q w = m 2 /s, Re = q w = m 2 /s, Re = q w = m 2 /s Re = Double-tip conductivity probe (Ø = mm) Single-tip conductivity probe (Ø = 0.1 mm) Double-tip conductivity probe (Ø = mm) Double-tip conductivity probe (Ø = mm) W = 0.5 m W = 0.4 m W =1m Turbulence manipulation with triangular vanes Experiments CMH_05, W =1m 868 Can. J. Civ. Eng. Vol. 35, 2008

5 Table 1 (concluded). Reference q (8) Step geometry Flow conditions Instrumentation Remarks Configuration A 22 Rough step faces: 8 mm thick k =8mm screens on both vertical and horizontal step faces Configuration B 22 8 mm thick screens on each k =8mm vertical step face Configuration C 22 8 mm thick screens on each k =8mm horizontal step face Configuration S 22 Smooth horizontal steps (smooth steps) Murillo (2006) Smooth horizontal steps Conical hot-film probe W = 0.4 m (h = 0.15 m) (Dantec 55R42) Model I 16 h = 0.15 m, l = m q w = m 2 /s, Re = Model II 11.3 h = 0.15 m, l = 0.75 q w = m 2 /s, Re = Model III 5.7 h = 0.15 m, l = 1.5 m q w = m 2 /s, Re = Thorwarth and Koengeter (2006) Flat horizontal steps and pooled steps Double-tip conductivity probe (Ø = mm) W = 0.5 m, Sorpe dam spillway model 14.6 h = 0.10 m, l = m q w = m 2 /s, Re = h = 0.05 m, l = m q w = m 2 /s, Re = Present study 22 Smooth horizontal steps (h = 0.1 m, l = 0.25 m) Experiments GH_06, W =1m Series 1 22 q w = m 2 /s, Re = Single-tip conductivity probes (Ø = 0.35 mm) Series 2 22 q w = m 2 /s, Re = Double-tip conductivity probe (Ø = 0.25 mm) Note: k, screen roughness height; Re, flow Reynolds number defined in terms of hydraulic diameter; Ø, diameter. Carosi and Chanson 869

6 870 Can. J. Civ. Eng. Vol. 35, 2008 Fig. 3. Photographs of the conductivity probes: (a) Two single-tip conductivity probes side-by-side (z = 21.7 mm and d c /h = 1.45), flow from left to right, shutter speed of 1/80 s; (b) double-tip conductivity probe (x = 7 mm) in the upper spray region, looking downstream (d c /h = 1.33), shutter speed = 1/500 s.

7 Carosi and Chanson 871 second electrode. Additional measurements were performed with a double-tip conductivity probe (Fig. 3b). The sensors consisted of a platinum wire (Ø = 0.25 mm) insulated except for its tip and set into a metal supporting tube. The longitudinal spacing between probe sensors was measured with a microscope and this yielded x = 7.0 mm. All the probes were excited by an electronic system (Ref. UQ82.518) designed with a response time of less than 10 ms and calibrated with a square-wave generator. The probe sensors were scanned at 20 khz per sensor for 45 s. Signal correlation analyses The air water interfacial velocities were deduced from a basic correlation analysis between the two sensors of the double-tip probe (Chanson 1997a, 2002; Crowe et al. 1998). The time-averaged velocity equals ½1Š V ¼ x T where T is the air water interfacial travel time for which the cross-correlation function is maximum and x is the longitudinal distance between probe sensors. Turbulence levels may be derived from the relative width of the cross-correlation function: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 0:52 T 0:5 ½2Š T u ¼ 0:851 T where 0.5 is the time scale for which the cross-correlation function is half of its maximum value, such as R 12 RðTþ 0:5 Þ¼ 0:5R 12 ðtþ, where R 12 is the normalized cross-correlation function; and T 0.5 is the characteristic time for which the normalized auto-correlation function equals R 11 ðt 0:5 Þ¼ 0:5. Equation [2] was derived by Chanson and Toombes (2002a). With the single-tip probe, all measurements were conducted on the channel centreline (z = 0) and a second identical probe was placed beside the first one with the probe sensors at the same vertical and streamwise distances y and x, respectively, but separated by a known transverse distance Dz (Fig. 3a). Their signals were analysed in terms of auto-correlation and cross-correlation functions R 11 and R 12, respectively. Following Chanson (2006a, 2007), the basic correlation analysis results included the maximum cross-correlation coefficient (R 12 ) max and the correlation time scales T 11 and T 12 defined as ½3Š ½4Š Z ¼ ðr 11 ¼ 0Þ T 11 ¼ R 11 d ¼ 0 Z ¼ ðr 12 ¼ 0Þ T 12 ¼ R 12 d ¼ ½R 12 ¼ðR 12 Þ max Š where is the time lag, T 11 is an auto-correlation integral time scale, and T 12 is the cross-correlation time scale. Identical experiments were repeated with different separation distances. An integral turbulent length scale was calculated as ½5Š Zz ¼ z½ðr 12 Þ max ¼ 0Š L 12 ¼ ðr 12 Þ max dz z ¼ 0 The integral turbulent length scale L 12 represents a measure of the transverse scale of the large vortical structures advecting air bubbles and air water packets. The corresponding turbulence integral time scale is ½6Š T ¼ R z¼0 ðr 12Þ max T 12 dz L 12 Additionally, an advection integral length scale is defined as ½7Š L 11 ¼ VT 11 where V is the advective velocity magnitude. Data accuracy The water discharge was measured with an accuracy of about 2%. The translation of the conductivity probes in the direction normal to the channel invert was controlled with an error of less than 0.5 mm. The accuracy on the longitudinal probe position was estimated as ± 0.5 cm. The error on the transverse position of the probe was less than 0.1 mm. With the conductivity probes, the error on the void fraction measurements was estimated as C=C ¼ 4% for 0.05 < C < 0.95, C=C 0:002=ð1 CÞ for C > 0.95, and C=C 0:005=C for C < The minimum detectable bubble chord time was about 0.05 ms for a data acquisition frequency of 20 khz per channel. The scan frequency determines the resolution of the intrusive phase-detection probe, in particular the accuracy of chord size measurement, minimum detectable air water chord length, and the accuracy of the interfacial velocity. Herein, the scan frequency was 20 khz per sensor and the streamwise distance between probe sensor was x = 7.0 mm. With the double-tip conductivity probe, the analysis of the velocity field and chord length distributions implied no slip between the air and water phases. The error on the mean air water velocity measurements was estimated as V=V ¼ 5% for 0.05 < C < 0.95, V=V ¼ 10% for 0.01 < C < 0.05, and 0.95 < C < The minimum detectable bubble chord length was about 0.15 mm in a 3 m/s flow based upon a data acquisition frequency of 20 khz per channel. The effect of the probe sensor on chord size data were tested for one flow rate (d c /h = 1.15). Present chord data obtained with a 0.35 mm probe sensor were compared with some experimental results obtained by Gonzalez et al. (2005) for d c /h = 1.18 with a mm probe sensor. The present data showed consistently larger measured count rates and a broader range of bubble and (or) droplet sizes with the mm probe sensor than with the 0.35 mm probe sensor. The chord sizes measured with the 0.35 mm probe sensor were typically 18% to 50% larger (average: 28%) than the chord lengths measured with the mm probe sensor. Chanson and Toombes (2002c) performed a similar comparison between two probe sensor sizes (0.025 and 0.35 mm) in skimming flows and obtained comparable results.

8 872 Can. J. Civ. Eng. Vol. 35, 2008 Air water flow patterns The basic flow regimes were inspected in a series of visual observations with discharges ranging from to m 3 /s. A nappe flow regime was observed for small flow rates (d c /h < 0.5). For some intermediate discharges (0.5 < d c /h < 0.95), the flow had a chaotic behaviour that is characteristic of a transition flow regime. For larger flows (d c /h > 0.95), the waters skimmed above the pseudo-bottom formed by the step edges. In transition and skimming flows, the location of the inception point of free-surface aeration was recorded with discharges per unit width above m 2 /s. The data were compared successfully with earlier studies (Chanson 1995a, 2001a). For the present study, the results were best correlated eith the following equation with a normalized correlation coefficient of 0.95: ½8Š L I q ¼ 1:05 þ 5:11 h cos w p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; g sin ðh cosþ 3 0:45 < d c h < 1:6 where q w is the water discharge per unit width, and g is the gravity acceleration. Equation [8] is valid for transition and skimming flows on a 228 stepped chute only. Overall, the present results were very close to the observations of Chanson and Toombes (2002a), Gonzalez (2005), Gonzalez et al. (2005), and Thorwarth and Koengeter (2006) with comparable slopes and step heights (Table 1). Air water flow properties at step edges Void fraction and bubble count rate distributions Experimental observations demonstrated some substantial free-surface aeration immediately downstream of the inception point of free-surface aeration and the flow aeration was sustained downstream. At the step edges, the advective diffusion of air bubbles was described by an analytical model of air bubble diffusion, as ½9Š C ¼ 1 tanh 2 K 0 y=y 90 þ ½ðy=Y 3 90Þ ð1 3ÞŠ 2D o 3D o where y is distance measured normal to the pseudo-invert and Y 90 is the characteristic distance for C = 90%. The relationship between an integration constant, K, and a function of the mean void fraction, D o, (Chanson and Toombes 2002a) is expressed as ½10Š K 0 ¼ 0: þ 1 8 2D o 81D o ½11Š C mean ¼ 0:7622½1:0434 expð 3:614D o ÞŠ Equation [9] is compared with experimental data in Fig. 4. Figure 4 presents an example of dimensionless distributions of void fraction and bubble count rate Fd c /V c as functions of y/d c at several step edges for the same flow rate, where d c is the critical flow depth and V c is the critical flow velocity. For that discharge, the flow aeration was nil at step edge 6, immediately upstream of the inception point. Between step edges 6 and 7, some strong self-aeration took place, and the amount of entrained air and the mean air content were about constant between the step edges 7 and 10, with C mean = 0.36 at the last step edge 10. Figure 4 shows some typical dimensionless distributions of bubble count rate. The results consistently presented a characteristic shape with a maximum value observed for void fractions between 0.36 and A similar pattern was observed in smooth chute and stepped spillway flows (Chanson 1997b; Chanson and Toombes 2002a; Gonzalez and Chanson 2004; Kokpinar 2005). Air water velocity and turbulence level distributions Typical distributions of air water velocity and turbulent intensity are presented in Fig. 5 for one flow rate. At each step edge, the velocity distributions compared favourably with a power-law function for y/y 90 1 and with an uniform profile for y/y 90 >1: ½12Š ½13Š V V 90 ¼ V V 90 ¼ 1; y 1=10 ; 0 y 1 Y 90 Y 90 1 y Y 90 2:5 where V 90 is the characteristic air water velocity at y = Y 90. Several studies yielded eq. [12] (Matos 2000; Boes 2000; Chanson and Toombes 2002a; Gonzalez and Chanson 2004) but a few documented the velocity distribution in the upper spray region. Equation [12] and [13] are compared with experimental data in Fig. 5. In the present study, the velocity power law exponent was 1/10 on average, although it varied between adjacent step edges. Such fluctuations were believed to be caused by some complicated interference between adjacent shear layers and cavity flows (Fig. 2). The distributions of turbulence intensity T u showed high levels of turbulence in the skimming flows: 0.3 T u 2 (Fig. 5). The results were comparable with earlier studies in aerated and nonaerated skimming flows (Chanson and Toombes 2002a; Amador et al. 2004; Gonzalez and Chanson 2004). The turbulence intensity maxima were typically observed for 0.3 y/d c 0.4 that corresponded typically to 0.35 C 0.6. In the intermediate region defined as 0.3 C 0.7, the air water flow structure is extremely complicated; it is dominated by interactions between particles and turbulent shear. It is hypothesized that the high turbulence levels in this intermediate region are caused by the continuous deformations and modification of the air water interfacial structure. Probability distribution functions of air bubble and water droplet chords The probability distribution functions of chord sizes were analysed in terms of bubble chords in the bubbly flow (C < 0.3) and in terms of droplet chord lengths in the spray region (C > 0.7). Typical results are presented in Fig. 6. For each graph, the caption and legend provide the local air water flow properties (C, F) and probe details. The histogram columns represent the probability of chord size in a 0.5 mm chord interval. For example, the probability of bub-

9 Carosi and Chanson 873 Fig. 4. Dimensionless distributions of void fraction C and bubble count rate Fd c /V c for d c /h = 1.15 (single-tip probe, Ø = 0.35 mm) comparison with eq. [9] (step edge 10). Fig. 5. Dimensionless distributions of air water velocity V/V c and turbulence intensity T u for d c /h = 1.33 comparison with eqs. [12] and [13] (step edge 10).

10 874 Can. J. Civ. Eng. Vol. 35, 2008 Fig. 6. Probability distribution functions (PDF) of chord sizes in skimming flows for flow conditions with d c /h = 1.45, step 10, double-tip probe (Ø = 0.25 mm, x = 7.0 mm): (a) bubble chord size data (C < 0.3); (b) droplet chord size data (C > 0.7). ble chords between 1 and 1.5 mm is represented by the column labelled 1 mm. Chord sizes larger than 15 mm are regrouped in the last column (>15). In the bubbly flow region (C < 0.3), the probability distribution functions showed a broad spectrum of bubble chords at each location. The range of bubble chord extended from less than 0.3 mm to more than 15 mm (Fig. 6a). The bubble chord size distributions were skewed with a preponderance of small bubbles relative to the mean. In Fig. 6a, the mode of the probability distribution function was observed for chords between 0.5 and 1 mm. The probability distribution functions of bubble chord tended to follow a log-normal distribution at all locations and for all discharges. The result was consistent with some earlier studies (Chanson and Toombes 2002a; Gonzalez 2005). In the spray region, the probability distribution functions of drop sizes also showed a wide range of droplet chords at each location. While the droplet chord size distributions were skewed with a preponderance of small droplets, the probability density function was flatter than that of the bubble chords (Fig. 6b). In the upper spray region (C > 0.95), the probability distribution functions were flat and did not follow a log-normal distribution.

11 Carosi and Chanson 875 Table 2. Experimental measurements in skimming flows with identical probe sensors (present study). q w (m 2 /s) d c =h Instrumentation z (mm) Step edge T 12 max (s) Comments single-tip probes ( + ) Auto-correlation (Ø = 0.35 mm) Cross-correlation ( + ) Auto-correlation Cross-correlation ( + ) Auto-correlation Cross-correlation ( + ) Auto-correlation Cross-correlation single-tip probes ( + ) Auto-correlation (Ø = 0.35 mm) Cross-correlation ( + ) Auto-correlation Cross-correlation ( + ) Auto-correlation Cross-correlation Note: (+) is for C < Integral turbulent time and length scales in air water skimming flows Some experiments were conducted with two identical sensors, separated by a transverse distance, and they were repeated for several separation distances (Table 2). Typical distributions of auto- and cross-correlation time scales are shown in Fig. 7 for three separation distances. Note that the correlation time scales are presented in seconds. The void fraction data are also reported for completeness in Fig. 7. For z < 20 mm, the experimental results showed similar distributions of auto- and cross-correlation time scales in the bulk of the flow (C < 0.9). The auto-correlation time scales T 11 exhibited, however, a different trend in the upper spray region. This is seen in Fig. 7 for y/d c > 0.8. It is suggested that the pattern may indicate a change in the spray structure with the upper spray region. For C > 0.95, the spray consisted primarily of ejected droplets that did not interact with the rest of the flow. For all flow conditions and at each step edge, the crosscorrelation time-scale distributions presented a maximum in the intermediate region (0.3 < C < 0.7). Observed maxima of correlation time scales are summarized in the sixth column of Table 2. For some experiments repeated with several transverse separation distances, the turbulent length and time scales, L 12 and T, respectively, were calculated. Typical results in terms of dimensionless turbulent length scale L 12 /h and integral turbulent time scale T(g/h) 0.5 are presented in Fig. 8. The measured void fraction data are also shown for completeness. In bubbly flows, the turbulent length scale L 12 must be closely linked with the characteristic sizes of the large-size eddies, entrapping air bubbles, as shown by highspeed photographs (Hoyt and Sellin 1989; Chanson 1997a). Hence, the turbulent length scale L 12 characterizes the transverse size of the vortical structures advecting the air bubbles and air water packets. In the present study, the integral scales were related to the step height h: i.e., L 12 /h & 0.02 to 0.2 (Fig. 8). The result was irrespective of the dimensionless flow rate d c /h within the range of the experiments. The associated turbulence time scale T is a measure of the integral time scale of the p large eddies. Present results yielded basically 0:004 T ffiffiffiffiffiffi g=h 0:04. Both the integral turbulent length and time scales were maximum for about C = 0.5 to 0.7 (Fig. 8). The finding emphasised the existence of larger turbulent structures in the intermediate zone (0.3 < C < 0.7), and it is hypothesized that these large vortices may have a preponderant role in terms of turbulent dissipation. Discussion: relationship between integral scales and turbulent intensity The high-velocity open-channel flows on the stepped chute were highly turbulent. The present results demonstrated that the high levels of turbulence were associated directly with some large-scale turbulence (Fig. 5, 7, 8). In particular, the intermediate region (0.3 < C < 0.7) between

12 876 Can. J. Civ. Eng. Vol. 35, 2008 Fig. 7. Distributions of auto- and cross-correlation time scales in skimming flows (flow conditions: d c /h = 1.45, step edge 10) comparison with the void fraction distribution. p Fig. 8. Dimensionless distribution of integral turbulent length scale L 12 /h and transverse integral time scale T ffiffiffiffiffiffi g=h in skimming flows (step edge 10) comparison with void fraction measurements. the bubbly and spray regions seemed to play a major role in the development the large eddies and turbulent dissipation. Turbulence level maxima were observed consistently for 0.4 < C < 0.5, while maximum integral turbulent scales were seen for 0.5 < C < 0.7. The findings implied that some turbulent energy was dissipated in the form of large vortices in the bulk of the flow while the stepped cavities contributed to intense turbulence production. The dissipated energy contributed to the entrapment and advection of air bubbles within the main flow, as well as to the formation of water droplets and their ejection above the free surface. The mechanisms were consistent with the experimental results of Gonzalez and Chanson (2005), who observed that some increased rate of energy dis-

13 Carosi and Chanson 877 Fig. 9. Dimensionless relationship between turbulence intensity and integral turbulent scales in skimming flows (step edge 10): (a) dimensionless relationship between turbulence intensity and integral turbulent p length scale L 12 /h, comparison with eq. [14]; (b) dimensionless relationship between turbulence intensity and integral time scale T ffiffiffiffiffiffi g=h comparison with eq. [15]. sipation induced by passive turbulent manipulation was linked with a greater spray production. The present results showed that, in a cross section, the turbulent intensity T u and the turbulent length and time scales were closely linked. The relationship between turbulence level and turbulent scales were best correlated by ½14Š T u ¼ 0:372exp 8:73 L 12 h ½15Š rffiffiffi g T u ¼ 0:316exp 44:68 T h

14 878 Can. J. Civ. Eng. Vol. 35, 2008 Table 3. Flow resistance and dimensionless rate of energy dissipation and residual energy at the downstream end of the stepped chute (step 10). d c =h f e H/H max * H res =d c * Remarks Inception point: step edge Inception point: step edge Inception point: step edge Inception point: step edge Inception point: step edge 7 8. *Measured at step edge 10. Equations [14] and [15] are compared with the experimental data in Fig. 9. Flow resistance and turbulent energy dissipation On the stepped chute, the skimming flows were characterized by significant form losses. Downstream of the inception point of free-surface aeration, the flow was gradually varied flow in the present study. The average shear stress o between the skimming flow and the cavity recirculation was deduced from the measured friction slope S f : ½16Š f e ¼ 8 o p w U w 2 ¼ Z y ¼ Y 90 8g ð1 CÞdy y ¼ 0 U w 2 S f where f e is the equivalent Darcy Weisbach friction factor of the aerated flow, U w is the water flow velocity: U w = q w /d, and d is the equivalent clear-water depth (Chanson et al. 2002). The friction slope ðs f is the slope of the total head line, where H is the mean total head (Henderson 1966; Chanson 1999). Free-surface aeration is always substantial in prototype and laboratory skimming flows, and its effects must be accounted for using eq. [16]. The flow-resistance results are presented in Table 3. On average, the equivalent Darcy friction factor was f e & 0.14 downstream of the inception point of free-surface aeration. Note that in the developing boundary layer region, Amador et al. (2006) derived the flow resistance by applying an integral momentum method to PIV measurements and their results yielded f = for d c /h = 2.1 and Re = The present results compared favourably with a comprehensive re-analysis of flow resistance data in skimming flows (Chanson 2006b). The friction factor data followed closely a simplified analytical model of the pseudo-boundary shear stress, which may be expressed, in dimensionless form, as ½17Š f d ¼ 2 pffiffiffi 1 K where f d is an equivalent Darcy friction factor estimate of the form drag and 1/K is the dimensionless expansion rate of the shear layer (Chanson et al. 2002). Equation [17] predicts f d & 0.2 for K = 6, which is close to the observed friction factors (Table 3). Residual energy For the present investigation, the rate of energy dissipation H/H max and the dimensional residual energy H res /d c were calculated at the downstream end of the chute (step 10), and the results are summarized in Table 3. The present results showed a decreasing rate of energy dissipation on the stepped chute with increasing discharge. For design engineers, however, it is more relevant to consider the dimensionless residual head H res /d c. The residual head H res is the total head at the downstream end of the chute, and it equals ½18Š H res ¼ d cos þ U w 2g Present results implied that the dimensionless residual head was about 2.4 H res /d c 3.3, increasing slightly with increasing flow rate. Note that the present results were obtained with a fully developed aerated flow at the chute downstream end. For larger discharges, the flow may not be fully developed at the downstream end of the chute, and that the residual energy could be considerably larger (Chanson 2001b; Meireles et al. 2006). Conclusion New measurements were performed in skimming flows on a large stepped spillway model. Air water flow measurements were performed with some phase-detection intrusive probes in the air water flow region downstream of the inception point of free-surface aeration. For some experiments, two probe sensors were separated by a known transverse, and an advanced signal processing technique with signal correlation analyses was applied. The skimming flow properties presented some basic characteristics that were qualitatively and quantitatively in agreement with previous air water flow measurements in skimming flows. These included the distributions of void fraction, bubble count rate, and turbulent interfacial velocity. Correlation analyses yielded a further characterization of the large eddies advecting the bubbles. Basic results included the transverse integral turbulent length and time scales. The turbulent length scales were closely linked with the characteristics of the large-size eddies and their interactions with entrained air bubbles. The transverse integral turbulent length scales were closely related to the step height: i.e., L 12 /h & 0.02 to 0.2, and p the integral turbulent time scales were within 0:004 T ffiffiffiffiffiffi g=h 0:04:

15 Carosi and Chanson 879 The measurements showed some relatively good correlation between turbulence intensities T u, and turbulent length and time scales (Fig. 9). They highlighted further some maximum turbulence intensities and maximum integral time and length scales in the intermediate region between the spray and bubbly flow regions (i.e., 0.3 < C < 0.7). The findings suggested that turbulent dissipation may be a significant process in that intermediate region. Acknowledgments The writers acknowledge the assistance of Graham Illidge and Clive Booth. References Amador, A., Sanchez-Juny, M., Dolz, J., Sanchez-Tembleque, F., and Puertas, J Velocity and pressure measurements in skimming flow in stepped spillways. In Proceedings of the International Conference on Hydraulics of Dams and River Structures, Tehran, Iran. Edited by F. Yazdandoost and J. Attari. A.A. Balkema, Rotterdam, the Netherlands. pp Amador, A., Sanchez-Juny, M., and Dolz, J DPIV study of the turbulent boundary layer over V-shaped cavities. In Proceedings of the International Conference on Fluvial Hydraulics River Flow 2006, Lisbon, Portugal, 6 8 September. Topic A1. Edited by R.M.L. Ferreira, E.C.T.L. Alves, J.G.A.B. Leal, and A.H. Cardoso. Balkema Publisher, Taylor & Francis Group, London, Vol. 2, pp Boes, R.M Zweiphasenstroömung und energieumsetzung auf grosskaskaden. Ph.D. thesis, VAW-ETH, Zürich, Switzerland. Carosi, G., and Chanson, H Air water time and length scales in skimming flows on a stepped spillway. Application to the Spray Characterisation. Report No. CH59/06, Division of Civil Engineering, The University of Queensland, Brisbane, Australia. 142 p. Chamani, M.R., and Rajaratnam, N Jet flow on stepped spillways. Journal of Hydraulic Engineering, 120(2): doi: /(asce) (1994)120:2(254). Chamani, M.R., and Rajaratnam, N Characteristics of skimming flow over stepped spillways. Journal of Hydraulic Engineering, 125(4): doi: /(asce) (1999) 125:4(361). Chanson, H. 1994a. Hydraulics of nappe flow regime above stepped chutes and spillways. Australian Civil Engineering Transactions. Institution of Engineers, Australia, CE36: Chanson, H. 1994b. Hydraulics of skimming flows over stepped channels and spillways. Journal of Hydraulic Research, 32: Chanson, H. 1994b. Hydraulics of skimming flows over stepped channels and spillways [Discussion]. Journal of Hydraulic Research,. 33: Chanson, H. 1995a. Hydraulic design of stepped cascades, channels, weirs and spillways. Pergamon, Oxford, UK. Chanson, H. 1995b. History of stepped channels and spillways: a rediscovery of the wheel. Canadian Journal of Civil Engineering, 22(2): doi:doi: /l Chanson, H. 1997a. Air bubble entrainment in free-surface turbulent shear flows. Academic Press, London, UK. Chanson, H. 1997b. Air bubble entrainment in open channels. flow structure and bubble size distributions. International Journal of Multiphase Flow, 23: doi: /s (96) Chanson, H The hydraulics of open channel flows: an introduction. Edward Arnold, London, UK. Chanson, H. 2001a. The hydraulics of stepped chutes and spillways. A.A. Balkema, Lisse, the Netherlands. 418 pp.. Chanson, H. 2001b. A transition flow regime on stepped spillways? The facts. In Proceedings of the 29th IAHR Biennial Congress, Beijing, China, Theme D. Vol. 1. Edited by G. LI. Tsinghua University Press, Beijing. pp Chanson, H Air- water flow measurements with intrusive phase-detection probes. Can we improve their interpretation? Journal of Hydraulic Engineering, 128(3): doi: /(ASCE) (2002)128:3(252). Chanson, H. 2006a. Air bubble entrainment in hydraulic jumps. Similitude and scale effects. Report No. CH57/05, Department of Civil Engineering, The University of Queensland, Brisbane, Australia. 119 pp. Chanson, H. 2006b. Hydraulics of skimming flows on stepped chutes: the effects of inflow conditions? Journal of Hydraulic Research, 44: Chanson, H Bubbly flow structure in hydraulic jump. European Journal of Mechanics B/Fluids, 26: doi: /j.euromechflu Chanson, H., and Carosi, G Advanced post-processing and correlation analyses in high-velocity air-water flows. Environmental Fluid Mechanics, 7:. doi: /s Chanson, H., and Toombes, L. 2002a. Air-water flows down stepped chutes: turbulence and flow structure observations. International Journal of Multiphase Flow, 28: doi: /S (02) Chanson, H., and Toombes, L. 2002b. Energy dissipation and air entrainment in a stepped storm waterway: an experimental study. Journal of Irrigation and Drain Engineering, 128(5): doi: /(asce) (2002)128:5(305). Chanson, H., and Toombes, L. 2002c. Experimental study of gasliquid interfacial properties in a stepped cascade flow. Environmental Fluid Mechanics, 2: doi: / A: Chanson, H., and Toombes, L Hydraulics of stepped chutes: the transition flow. Journal of Hydraulic Research, 42: Chanson, H., Yasuda, Y., and Ohtsu, I Flow resistance in skimming flows and its modelling. Canadian Journal of Civil Engineering, 29(6): doi: /l Crowe, C., Sommerfield, M., and Tsuji, Y Multiphase flows with droplets and particles. CRC Press, Boca Raton, Fla.. El-Kamash, M.K., Loewen, M.R., and Rajaratnam, N An experimental investigation of jet flow on a stepped chute. Journal of Hydraulic Research, 43: Gonzalez, C.A An experimental study of free-surface aeration on embankment stepped chutes. Ph.D. thesis, Department of Civil Engineering, The University of Queensland, Brisbane, Australia. 240 pp. Gonzalez, C.A., and Chanson, H Interactions between cavity flow and main stream skimming flows: an experimental study. Canadian Journal of Civil Engineering, 31(1): doi: /l Gonzalez, C.A., and Chanson, H Experimental study of turbulence manipulation in stepped spillways. Implications on flow resistance in skimming flows. In Proceedings of the 31st Biennial IAHR Congress, Seoul, Korea. Edited by B.H. Jun, S.I. Lee, I.W. Seo, and G.W. Choi. Theme D.7, Paper pp Gonzalez, C.A., and Chanson, H Flow characteristics of skimming flows in stepped channels. Discussion. Journal of Hydraulic Engineering, 132(5): doi: /(asce) (2006)132:5(537).

16 880 Can. J. Civ. Eng. Vol. 35, 2008 Gonzalez, C.A., Takahashi, M., and Chanson, H Effects of step roughness in skimming flows: an experimental study. Research Report No. CE160, Department of Civil Engineering, The University of Queensland, Brisbane, Australia, 149 pp. Henderson, F.M Open channel flow. MacMillan Company, New York. Hoyt, J.W., and Sellin, R.H.J Hydraulic jump as mixing layer. Journal of Hydraulic Engineering, 115: Kokpinar, M.A Flow over a stepped chute with and without macro-roughness elements. Canadian Journal of Civil Engeering, 31(5): doi: /l Lin, J.C., and Rockwell, D Organized oscillations of initially turbulent flow past a cavity. AIAA Journal, 39: Manso, P.A., and Schleiss, A.J Stability of concrete macroroughness linings for overflow protection of earth embankment dams. Canadian Journal of Civil Engineering, 29(2): doi: /l Matos, J Hydraulic design of stepped spillways over RCC dams. In Proceedings of the International Workshop on Hydraulics of Stepped Spillways, Zürich, Switzerland. Edited by H.E. Minor and W.H. Hager. A.A. Balkema, Rotterdam, the Netherlands. pp Meireles, I., Cabrita, J., and Matos, J Non-aerated skimming flow properties on stepped chutes over small embankment dams. In Proceedings of the International Junior Researcher and Engineer Workshop on Hydraulic Structures, IAHR, 2 4 September 2006, Montemor-o-Novo, Portugal. Murillo, R.E Experimental study of the development flow region on stepped chutes. Ph.D. thesis, Department of Civil Engeering, University of Manitoba, Winnipeg, Man. 240 p. Ohtsu, I., and Yasuda, Y Characteristics of flow conditions on stepped channels. In Proceedings of the 27th IAHR Biennial Congress, San Francisco, Calif., Theme D. pp Ohtsu, I., Yasuda, Y., and Takahashi, M Flow characteristics of skimming flows in stepped channels. Journal of Hydraulic Engineering, 130(9): doi: /(asce) (2004)130:9(860). Ohtsu, I., Yasuda, Y., and Takahashi, M Flow characteristics of skimming flows in stepped channels. [Discussion.] Journal of Hydraulic Engineering, 132: Rajaratnam, N Skimming flow in stepped spillways. Journal of Hydraulic Engineering, 116(4): [Discussion, 118: ] doi: /(asce) (1990)116:4(587). Thorwarth, J., and Koengeter, J Physical model tests on a stepped chute with pooled steps. Investigations of flow resistance and flow instabilities. In Proceedings of the International Symposium on Hydraulic Structures, IAHR, Ciudad Guayana, Venezuela. Edited by A. Marcano and A. Martinez. pp Toombes, L Experimental study of air-water flow properties on low-gradient stepped cascades. Ph.D. thesis, Department of Civil Engineering, The University of Queensland, Brisbane, Australia. Yasuda, Y., and Chanson, H Micro- and macro-scopic study of two-phase flow on a stepped chute. In Proceedings of the 30th IAHR Biennial Congress, Thessaloniki, Greece. Vol. D. Edited by J. Ganoulis and P. Prinos. pp List of symbols C void fraction defined as the volume of air per unit volume (also called air concentration) C mean depth averaged air concentration defined as: ð1 Y 90 ÞC mean ¼ d D H D o hydraulic diameter (m) dimensionless diffusivity term d equivalent clear-water depth (m) defined as: d¼ R Y 90 0 ð1 CÞdy d c critical flow depth (m) F bubble count rate (Hz) defined as the number of bubbles detected by the probe sensor per second f d form drag equivalent Darcy friction factor f e Darcy friction factor for air water flow g gravity constant (m/s 2 ) or acceleration of gravity H total head (m) upstream head (m) above spillway toe residual head (m) h height of steps (m) (measured vertically) K inverse of the spreading rate of a turbulent shear layer K integration constant k screen roughness height (m) H max H res L I longitudinal distance (m) measured from the weir crest where the inception of free-surface aeration takes place L 11 air water advection integral length scale (m) L 12 air water integral turbulent length scale (m) l horizontal length of steps (m) q w water discharge per unit width (m 2 /s) R 11 normalized auto-correlation function (reference probe) R 12 normalized cross-correlation function between two probe output signals Re flow Reynolds number defined in terms of the hydraulic diameter (R xy ) max maximum cross-correlation between two probe output signals S f friction slope: S f T time lag (s) for which R 12 =(R 12 ) max T integral turbulent time scale (s) T 0.5 characteristic time lag for which R xx ¼ 0:5 T 11 an auto-correlation integral time scale T 12 cross-correlation time scale T u turbulence intensity: T u ¼ u 0 =V U w equivalent clear water flow velocity (m/s): U w ¼ q w =d u root mean square of longitudinal component of turbulent velocity (m/s) V velocity (m/s) V 90 characteristic velocity (m/s) where C = 0.90 V c critical flow velocity (m/s) W channel width (m) x longitudinal distance (m) Y distance (m) from the pseudo-bottom (formed by the step edges) measured perpendicular to the flow direction characteristic depth (m) where the air concentration is 90% Z transverse distance (m) measured from the chute centreline Y 90 x probe tip separation (m) in the streamwise direction z transverse separation distance (m) between sensor Ø diameter (m) q angle between the pseudo-bottom formed by the step edges and the horizontal w water density (kg/m 3 ) 0.5 o time lag (s) characteristic time lag for which R xv = 0.5(R xv ) max average bottom shear stress (Pa)