Flow regimes, bed morphology, and flow resistance in self-formed step-pool channels

Transcription

1 WATER RESOURCES RESEARCH, VOL. 45, W04424, doi: /2008wr007259, 2009 Flow regimes, bed morphology, and flow resistance in self-formed step-pool channels F. Comiti, 1 D. Cadol, 2 and E. Wohl 2 Received 3 July 2008; revised 28 December 2008; accepted 13 January 2009; published 25 April [1] We used a mobile bed flume with scaled grain size distribution, channel geometry, and flow to examine morphology and hydraulics of stepped channels. We hypothesized that (1) step geometry and flow resistance differs significantly as a function of the range of grain sizes present, (2) a transition from nappe to skimming flow occurs in stepped channels with mobile beds for conditions similar to stepped spillways, and (3) the partitioning of flow resistance changes significantly when flow passes from nappe to skimming conditions. Results support each of these hypotheses and help to illuminate the complexity of V-Q relationships in stepped channels, in which a dramatic decrease in flow resistance and increase in velocity accompany the transition from nappe to skimming flow near step-forming events. Therefore, a single flow resistance equation applicable to both ordinary and large floods may not be ideal in stepped channels. Nonetheless, models based on dimensionless velocity and unit discharge appear more robust compared to those based on the Darcy-Weisbach friction factor. Citation: Comiti, F., D. Cadol, and E. Wohl (2009), Flow regimes, bed morphology, and flow resistance in self-formed step-pool channels, Water Resour. Res., 45, W04424, doi: /2008wr Introduction [2] Mountain streams with bed slope >3 5% are typically characterized by a stepped longitudinal profile, where steps may be formed by bedrock outcrops, large sediment clasts, wood, or a combination of these elements [Wohl et al., 1997; Chin and Wohl, 2005], and pools form below steps by the scouring action of the flow [Comiti et al., 2005]. Following the classification of Montgomery and Buffington [1997], if each step-pool unit spans the entire channel, the reach is referred to as having a step-pool morphology, otherwise (i.e., when steps and their associated pools are shorter and more irregular across the channel width) it is labeled as cascade. Many investigations have focused on the dynamics of steppools, and several models have been proposed to explain their formation and geometry (for a summary, see Chin and Wohl [2005], Milzow et al. [2006], Church and Zimmermann [2007], and Curran [2007]). [3] Step-pool channels differ considerably in terms of mechanisms of energy dissipation from lower gradient streams characterized by plane-bed, pool-riffle, or duneripple bed forms. They feature a unique flow regime, termed tumbling flow by Peterson and Mohanty [1960]. Tumbling flow is characterized by an alternation between supercritical and subcritical flow (i.e., Froude number >1 and <1) at each step-pool unit, thus hydraulic jumps are likely to contribute a relevant component, named spill resistance [Leopold et al., 1964], to the overall flow resistance along the channel 1 Faculty of Science and Technology, Free University of Bozen-Bolzano, Bolzano, Italy. 2 Department of Geosciences, Colorado State University, Fort Collins, Colorado, USA. Copyright 2009 by the American Geophysical Union /09/2008WR [Wilcox and Wohl, 2007]. However, the extent to which spill resistance persists up to high (i.e., step forming) flows has not yet been ascertained, and the onset of a different, no longer tumbling, flow regime has been hypothesized to occur at flows sufficient to mobilize the very coarse clasts that form steps [Chanson, 1996]. If this is true, important consequences would derive in terms of both absolute values of flow resistance and its partitioning into different components, in turn affecting bed load transport efficiency and thus morphological dynamics in stepped streams [Egashira and Ashida, 1991]. Consequently, one of our objectives in this paper is to use self-formed step-pool channels in the laboratory to characterize hydraulics and resistance across a range of discharge including high flows. [4] Hypothesized changes in flow regime along stepped channels (Figure 1) derive from laboratory experiments made on stepped dam spillways, which are widely used to dissipate flow energy along steep chutes. This kind of spillway represents an analog to a natural step-pool reach in that it features a stepped longitudinal profile, although the spillway differs in having a smoother bed, extreme geometric regularity, and the absence of real pools. [5] Two flow regimes, nappe and skimming, have been identified in stepped spillways [Noori, 1984; Chanson, 1994; Chamani and Rajaratnam, 1999]. Similarly, at the channel unit scale, two broad regimes have been described for flows at single drop structures [Ohtsu and Yasuda, 1997; Wu and Rajaratnam, 1998] in which the flow dynamics parallel the transition between rollers and undular jumps [Chanson and Montes, 1995; Ohtsu et al., 2001; Comiti and Lenzi, 2006]. [6] For nappe flow in stepped spillways, flow alternates between supercritical and subcritical conditions, hydraulic jumps form at each step tread, and air pockets persist from each falling jet into the subsequent pool [Chanson, 1994]. Flow resistance decreases with increasing discharge for a W of18

2 W04424 COMITI ET AL.: FLOW RESISTANCE IN STEP-POOL CHANNELS W04424 Additionally, the upper limit for the persistence of a nappe flow is given by (experimental range 0.1 < S < 1.2) h c z ¼ 0:714 ð 1:4 S Þ0:26 ð3þ Figure 1. Possible analogies in flow regimes between stepped spillways and step-pool channels [see also Chanson, 1996; Church and Zimmermann, 2007]. In the nappe regime, hydraulic jumps occur in pools, in contrast to the skimming flow where critical to supercritical conditions are established along the entire channel. See notation section for definitions. given geometry of stepped spillway. Above a certain discharge threshold, jumps and air pockets disappear and the entire flow (skimming flow) becomes supercritical, with the formation of recirculation eddies attached to the main flow stream at step corners and a large air concentration [Toombes, 2002; Chanson and Toombes, 2002]. Flow resistance in this regime is independent of discharge and is dominated by form losses and cavity recirculation [Yasuda and Ohtsu, 1999; Chanson et al., 2002], even though step roughness (i.e., skin friction) affects aeration processes, which may interact with flow energy dissipation in a complex way [Gonzalez et al., 2008]. [7] The discharge threshold, expressed as critical flow depth h c on each step, for the onset of skimming flow is related to geometry of the stepped spillway. Chanson [1994] proposed a threshold step submergence which can be expressed, by recalling that z = SL, with z step height, S channel slope (= sin q tan q with q bed inclination) and L step length, as h c z ¼ 1:057 0:465S ð1þ Yasuda and Ohtsu [1999] found that the lower limit of step submergence for the establishment of skimming flow is (in the range 0.1 < S < 1.43) h c z ¼ 0:862S 0:165 ð2þ The transition regime lies between the two curves defined by equations (2) and (3). [8] To date, no field investigation has observed the passage from nappe to skimming flow regimes in self-formed stepped channels because velocity measurements have been carried out only up to moderate (i.e., approximately bankfull) flows which did not show any visual evidence of skimming flow [Lee and Ferguson, 2002; Comiti et al., 2007; Zasso and D Agostino, 2007]. Laboratory studies on self-formed steppools [Maxwell and Papanicolau, 2001; Lee and Ferguson, 2002; Aberle and Smart, 2003] also have not provided any quantitative evidence of the existence of skimming flow, although scale effects and/or incomplete modeling of bed complexity in stepped channels might be the reason for this. In particular, we suggest that the use of an adequately wide grain size distribution is crucial for proper modeling of hydraulics and sediment processes in poorly sorted steep channels. We believe that the presence of a few large (i.e., of the order of the D 90 ) clasts may have dramatic effects on bed morphology (e.g., by creating higher steps) and consequently on the variation of flow resistance with discharge. In addition, several authors [see Chanson et al., 2002] have shown that large-scale models (i.e., <1/10) are necessary to adequately reproduce under a Froude similitude the air entrainment processes, which are not negligible in stepped channels. Consequently, the laboratory experiments summarized in this paper incorporate very large clasts and large-scale models. [9] The key control exerted by hydraulic jumps in determining the flow regimes described above is also encountered when attempting to establish a partitioning of flow resistance. In lower gradient alluvial channels, the total flow resistance or channel roughness is traditionally partitioned into components of grain f grain,formf form, and bed load transport f bl, where f grain results from skin friction at the channel boundary, f form results from internal friction associated with macroscale turbulent structures stemming from, for example, bed forms, flow curvature, large clasts, and wood elements, and f bl results from the energy expended to move sediment particles [Yen, 2002; Recking et al., 2008]. In step-pool channels, the component f spill (i.e., resistance due to hydraulic jumps, which could be included within the form factors) deserves to be considered separately in order to analyze how the possible flow regime transition influences the sources of flow resistance. [10] Even though a clear analytical derivation is lacking, the different resistance components are usually assumed to be linearly additive [Meyer-Peter and Muller, 1948; Einstein and Barbarossa, 1952; Millar, 1999]. Consequently, when total flow resistance is expressed through the Darcy- Weisbach friction factor f as f tot ¼ 8gR hs e V 2 8gR hs V 2 with g acceleration due to gravity, R h hydraulic radius, S e energy slope, S channel slope, and V mean flow velocity, ð4þ 2of18

3 W04424 COMITI ET AL.: FLOW RESISTANCE IN STEP-POOL CHANNELS W04424 the following relationship for stepped channels (eliminating the dissipation due to bed load movement) can be written f tot ¼ f grain þ f form þ f spill ð5þ The practical relevance of establishing a correct flow resistance partitioning lies in the prediction of bed load transport rate in the streamwise direction along a reach, which is eventually driven by skin friction only and therefore may be greatly overestimated using the total shear stress (i.e., associated with f tot ) if the other components prevail, as could occur in steep mountain rivers [Meyer-Peter and Muller, 1948; Yager et al., 2007]. [11] Several efforts to model total flow resistance in alluvial, self-formed stepped channels have been carried out, both in the lab [Maxwell and Papanicolau, 2001; Lee and Ferguson, 2002; Aberle and Smart, 2003] and in the field [Marcus et al., 1992; Lee and Ferguson, 2002; Comiti et al., 2007; Ferguson, 2007]. Few attempts, however, have been made to determine the components listed in equation (5) [Curran and Wohl, 2003; MacFarlane and Wohl, 2003]. The experiments presented by Wilcox et al. [2006] and Wilcox and Wohl [2006], although not dealing with self-formed steps, provide valuable insights into the complex hydraulics of stepped channels, and demonstrate that partitioning estimates for stepped channels are highly sensitive to the order in which components are calculated and that linearly additive approaches inflate the values of difficult-to-measure components that are calculated by subtraction from measured components. One of their main findings is that discharge is the most important factor controlling the interaction among the different flow resistance components f x. We can summarize this by setting f x ¼ a x f tot ; with X a x ¼ 1 where for a given channel reach, the coefficients a x are a function of the water discharge Q and sediment transport rate Q s. In many channels, Q s is itself a function of Q, but steep mountain channels are typically supply limited with respect to sediment and thus Q alone is not enough to model sediment transport rates in these systems [Blizard and Wohl, 1998; Lenzi et al., 2004a]. Therefore, the relative importance of each energy dissipation mechanism varies with changing discharge and sediment transport conditions. [12] This paper presents results from a laboratory investigation on flows in steep alluvial channels carried out with the aim of (1) assessing how large sediment fractions affect development of bed morphology; (2) evaluating the interaction between bed morphology and flow resistance; (3) detecting the possible transition from nappe to skimming flow at high flows; and (4) quantifying the contribution of each flow resistance source (grain, form, spill). We hypothesize that (1) a step geometry and total flow resistance differ significantly as a function of the range of grain sizes present; (2) a transition from nappe to skimming flow occurs in stepped channels, analogous to stepped spillways; and (3) the partitioning of total flow resistance changes ð6þ Figure 2. (a) The Rio Cordon (Italian Alps), which was a rough prototype for (b) these flume experiments. significantly when flow changes from nappe to skimming conditions. 2. Experimental Setup and Methods 2.1. Flume Configuration and Sediment Mixtures [13] A mobile bed physical model of a natural stepped channel was developed at the Colorado State University Engineering Research Center. An 8-m-long, 0.6-m-wide wooden flume was used. In our experiments the flume was inclined at a constant 14% slope and no sediment feed was provided from upstream (clear water conditions). [14] Two sediment mixtures for the bed were tested, mixes A and B (Table 1), which differ only for the largest fraction. The grain size distribution of mix A was derived by using a 1:8 geometric scaling of the sediment distribution measured in the Rio Cordon (Italian Alps, Figure 2), along a 4-m-wide cascade/step-pool reach with 14% average channel slope. The Rio Cordon was selected as a prototype because of the unique availability of a 20-year record of flow discharge, sediment mobility and transport rates, and channel morphological evolution [Lenzi, 2001; Lenzi et al., 2004a, 2006; Mao and Lenzi, 2007], and recent determination of flow resistance up to quasi-bankfull flows [Comiti et al., 2007]. Following a Froude similitude for flows over fixed beds, this implies that flow rates in the Rio Cordon should be approximately 8 (5/2), i.e., 181, times those in the flume experiments. Given the changing composition (i.e., coarsening) of the streambed during our flume runs, however, such scaling must be viewed as indicative only, leading to a reduced scaling factor. The adoption of such a small scaling factor allows us to neglect scale effects due to viscosity; i.e., experiments feature high flow and grain Reynolds numbers, as will be reported later. Because the present research was aimed at modeling neither bed load rates nor the temporal evolution of the bed, scale effects typical of mobile bed Froude models [Graf, 1971; Chanson, 1999] are not relevant for the present paper. [15] In order to achieve a better resemblance to a natural stepped channel, the vertical flume walls within the study reach were lined with flat large rocks held in place by thin metal wires simulating stable irregular banks which in prototype streams lead to complex three-dimensional flow patterns and stabilizing features for step units. The channel 3of18

4 W04424 COMITI ET AL.: FLOW RESISTANCE IN STEP-POOL CHANNELS W04424 Table 1. Cumulative Grain Size Distributions of Two Sediment Mixtures Used in the Experiments a Sieve Size (mm) Mix A Percentiles Mix B a Percentiles are associated to the passing sieve. width was thus narrowed to an average value of 0.46 m (this translates into 3.7 m at prototype dimensions, very close to the actual Rio Cordon reach width of 4 m), with a range of 0.37 m to 0.55 m and standard deviation of 0.04 m. [16] Nine sediment size components were used in the mixture for the flume, ranging from coarse sand to large cobbles. The three larger classes were painted in different colors (yellow, blue and green) to easily track their displacement. Grain size distribution B derives from mix A by eliminating the largest fraction only; i.e., 14 large cobbles. Two similar series of experiments (hereafter called A and B) were carried out, each on the corresponding sediment mixture. An additional series of tests was then performed on a fixed step configuration, as will be presented later. The sediment size fractions were carefully mixed before being dumped into the flume, and then leveled to create the plane bed configuration used at the beginning of the two series of runs. During each series, the bed was not artificially modified any further. [17] Flow discharges ranged from to 0.09 m 3 s 1, which scales to the interval 1 16 m 3 s 1 for the prototype channel, i.e., from half the bankfull discharge to a flood with >100 years recurrence interval [Lenzi et al., 2004a, 2006] assuming the original scaling factor of 8. However, as mentioned above, the strong coarsening of the bed that occurred with increasing discharge (no sediment feed) prevents use of such scaling reliably for the higher flow rates tested. On the other hand, once the armor layer is removed during formative runs, the flow is once more in contact with a mixture closer to the original one. [18] For the two experimental series, an initial sediment thickness of 0.35 m was set along the 4-m working flume length along which measurements of flow velocity and bed morphology were taken. Between the flume inlet and the study reach, a 0.5-m-long section of large stable cobbles was installed to dissipate excess energy from the head box, followed by a 1.5-m-long section in which the sediment thickness was gradually increased to a depth of 0.35 m. This arrangement allowed a smooth transition between the inlet and the study reach, with the development of a turbulent boundary layer and quasi-uniform flow conditions at the start of the study reach, where the upstream conductivity probe was placed (see section 2.2). A transverse wooden broadcrested weir (6-cm-long streamwise) formed the downstream end of the study reach, functioning as a fixed control section hosting the second conductivity probe at its very upstream end, thus limiting potential downdrop effects on probe readings as a result of flow acceleration. Another wooden weir was placed 1.5 m farther downstream, creating a storage basin where the transported sediment was deposited for collection at fixed time intervals during each run Experimental Procedure and Measurement Methods [19] Starting from an initial plane bed tilted at a slope of 0.14 for each of the two sediment mixtures, flow rates were progressively increased, creating 8 different bed structures (A1 A8 for mix A, B1 B8 for mix B; see Table 2) characterized by varying equilibrium slopes, morphologies and macroroughness geometry. After each bed morphology was stable with its forming discharge Q formative (see Table 2), several lower flows were run to determine a sort of at-astation hydraulic geometry (Figure 3), where reach-averaged values for velocity, hydraulic radius and flow resistance are used instead of cross-section values (see also Comiti et al. [2007] for the same approach applied in the field). The experimental procedure and the reach-scale analysis of flow velocity and flow resistance are thus similar to that adopted by Aberle and Smart [2003]. [20] It must be pointed out the most of the morphological evolution took place within the first min of each Figure 3. (top) Starting from an initial plane bed, flow rates were progressively increased creating different beds, characterized by varying morphologies and macroroughness geometry (runs for series A are shown). (bottom) After each bed morphology was stable with its forming discharge, lower-than-forming flows were run (A7 is depicted, with numbers referring to the Q/Q formative ratio for each flow rate). 4of18

5 W04424 COMITI ET AL.: FLOW RESISTANCE IN STEP-POOL CHANNELS W04424 Table 2. Main Characteristics of the Bed-Forming Runs a Series Run Q formative (m 3 s 1 ) S (m m 1 ) D 84 (m) D 84tra (m) D 100tra (m) s prof (m) H (m) (H/L)/S V (m s 1 ) Fr A A A A A A A A A A A A A A A A B B B B B B B B B B B B B B B B C C a Q formative, discharge forming each bed morphology; S, equilibrium bed slope for each formative discharge; D 84, 84th percentile of sediment distribution of the surface layer and of the transported material (D 84tra ) collected at the exit of the flume; D 100tra, largest sediment size class transported by the flow; s prof, profile standard deviation; H, step height; (H/L)/S, steepness factor; V, flow velocity; Fr, reach-averaged Froude number. formative run, but test durations lasted 2 3 h until no sediment at all was observed to deposit in the downstream storage area (i.e., equilibrium corresponding to absence of sediment transport). Also, formative discharges were not chosen a priori but adjusted to achieve a different bed morphology for each run, and this led to uneven increments of flow rates between series A and B. Once equilibrium was achieved, measurements of flow velocity and water elevation were taken (see later). Subsequently, the pump was shut off to allow dry bed measurements of bed topography and sediment size. [21] The pump was then reactivated to perform measurements of flow velocity on the same bed but with lower flows. We started from the lowest flow able to occupy the entire cross section, then increased the discharge at roughly even intervals which varied from to m 3 s 1, until again reaching the formative discharge. Therefore, the number of at-a-station velocity measurements is larger for higher formative discharges. For each lower discharge, characteristics and location of hydraulic jumps were noted in order to discern possible thresholds in flow regime type (i.e., nappe versus skimming). Eventually, when the complete set of velocities for the lower-than-forming flows was obtained, a higher formative discharge was run, and the process repeated as described. [22] The very heterogeneous characteristic of the bed sediments (i.e., the presence of randomly located large clasts which were immobile during the initial runs) very likely makes each bed configuration unique in the sense that the same flow conditions repeated with the same grain size distribution would lead to a different bed topography at the unit scale (i.e., step and pool location), depending on the location of the immobile elements. This mimics the conditions encountered in real step-pool streams [Church and Zimmermann, 2007]. However, the overall reach geometry would presumably be similar. [23] Flow discharge, Q, was measured though a digital Venturi meter and an orifice plate located on the pipelines conveying water into the head box, which then transferred water to the flume inlet though a vertical honeycomb-like screen which dampened flow oscillations. [24] Reach-averaged flow velocity, V, was measured at equilibrium (here defined as the final, stable bed configuration with no sediment transport) through the salt tracer technique along the 4-m-long study reach of the flume, by using two conductivity meters with data loggers (acquisition time 0.5 s) installed on the downstream weir and on a vertically adjustable support upstream, both located on the flume centerline. The salt-water mixture was injected into the flow using a 0.4-m-wide tipping bucket at the flume inlet, thus allowing sufficient length (2 m) for salt mixing before the upstream probe. Peak travel time (i.e., time between the maximum conductivity values for the two probes) was used to calculate mean flow velocity V (then used to derive flow resistance by equation (4)) over the 4-m flume length, a method commonly used in the field [Curran and Wohl, 2003; Comiti et al., 2007]. The large dimensions of the laboratory sump allowed for proper dilution of the salt plugs, and therefore effects from previous injections were not detectable. Potential errors would have been possible if flow discharge had been measured by a salt dilution technique, but that was not the case here. [25] A point gage mounted on a gantry was used to measure bed elevation. The free surface of the flow was also measured with the point gage for step-forming runs, but not for lower flows (see below). Measurements were ordinarily taken every 5 cm along the flume centerline, but additional points were taken at particular morphological locations such as step crests and maximum pool depths. Equilibrium bed slope (S, used in equation (4)) was assessed along the study reach using a linear regression of bed elevations, which also served to calculate the standard deviation of the bed profile. Flow depth was not measured directly, but calculated by continuity, i.e., h = q/v, where q is the unit discharge (i.e., q = Q/w, with w = 0.46 m as reported above), as also done by Lee and Ferguson [2002] and Aberle and Smart [2003] in similar experiments. The reach-integrated flow depth h was then converted to a hydraulic radius R h by using the rectangular cross-section approximation, R h =(wh)/(w +2h). Errors in flow velocity and therefore in flow depth were estimated on the basis of measured travel times compared to the probe 5of18

6 W04424 COMITI ET AL.: FLOW RESISTANCE IN STEP-POOL CHANNELS W04424 acquisition intervals. These errors averaged 10 15%, although values up to 20% occurred during the faster flows. [26] The sediment grain size distribution of the surface layer was obtained by taking 8 vertical digital photographs of the bed within the study reach after each step-forming run and analyzing the photographs through a dedicated commercial image analysis software(digital Gravelometer#) [Graham et al., 2005]. A rectangular wooden frame was used in each picture as a scale for the software. The program determines diameters of all grains within the frame, but a lower bound truncation at 5 mm was applied because of limited photo resolution. The original area-by-number distribution is converted to a grid-by-number sampling by the software so that percentiles are comparable to field-derived grain size distributions. [27] Sediment transport volumes and rates during stepforming runs were derived by collecting the sediment in the retention basin at the downstream end of the flume into burlap bags at fixed intervals from the beginning of each run (5, 15, 30, 60, 120 min). Once dry, the content of each bag was weighed and sieved in the lab to obtain its grain size distribution (volume-by-weight method). [28] Finally, morphological units such as steps and pools were visually identified at low flows and located by their longitudinal distance along the reach. Their individual characteristics (step height H, drop height z, pool depth s = H z, pool length l p ) were then measured from the profiles (see Figure 1). Averaged step-pool geometrical variables were subsequently used to characterize each bed structure. Step-tostep distance L was calculated on the basis of the number ns of steps counted within the study reach length (4 m) for each bed morphology; i.e., L = 4/ns. For each bed, step wavelength H/L was calculated on the basis of the average H and on step distance L, and the so-called steepness factor [Abrahams et al., 1995] was derived by dividing average step wavelength by the equilibrium bed slope (H/L/S). [29] In order to characterize the bed geometry in an integrated rather than average manner, drop heights z of each step-pool unit were summed up for each run and normalized by the respective total elevation drop, to give the relative total step drop (RTSD). Similarly, a proxy for pool volume (per unit width) was calculated as residual pool depth s times pool length l p, thus assuming a constant pool shape [Comiti, 2003], and the sum of all pools divided by the total elevation drop times reach length, to give the relative total pool volume (RTPV). [30] In addition to the self-formed stepped morphologies (series A and B), an additional series of experiments (series C) was carried out with 6 fixed steps composed of the largest rocks used for mix A and built to be as stable as possible, in order to test a partly alluvial channel more closely resembling a stepped spillway. These steps were constructed at 0.67-m intervals (corresponding to 5.4 m if scaled to the Rio Cordon, in accordance with the prototype value of the modeled reach) along the study reach, with a 10% slope between their crests. The arrangement (Figure 4) may thus be viewed as an ideal step-pool geometry for mix A, where all the largest clasts are grouped into transverse steps, or as a sequence of boulder check dams for stream restoration following geomorphic criteria [Lenzi and Comiti, 2003]. Only one pool-forming discharge (0.056 m 3 s 1 ) was released, with the intent of shaping large pools below the artificial steps, which remained Figure 4. The artificial step-pool sequence (series C) at (a) low and (b) high flows. Apparently, a nappe flow regime persisted up to highest flow rates tested. stable. This flow rate was selected because when scaled up to the prototype, it approximates a high-magnitude flood (recurrence interval about years [see Lenzi et al., 2006]). Notwithstanding the large discharge value, steps remained in place thanks to the careful assemblage and their large (about 15 cm) foundation depth or structure height below the bed surface, which is crucial in limiting undermining by scour [Lenzi et al., 2004b]. Flows lower than those that created the pools below the fixed steps were subsequently run to measure flow velocity at different discharges. Flow velocity, bed topography, and surface grain size distribution were measured applying the same methodology used for the selfformed stepped morphologies described above. [31] For all series of experiments (A, B, and C), the Reynolds numbers of the flow (i.e., Re = Vh/n, with n kinematic viscosity of the flow) are >12,000, and grain Reynolds number (i.e., Re* =u*d 50 /n, with u* shear flow velocity) >7000, thus indicating fully developed turbulent characteristics of the flow which allow us to consider the present experiments as being indicative of prototype conditions [Graf, 1971]. [32] Flow Froude number is calculated as Fr = V/(gh) 0.5, where V is the reach-averaged flow velocity, g is acceleration due to gravity, and h is the continuity-derived flow depth. [33] Total flow resistance f tot (i.e., including grain, form and spill sources) is calculated by equation 4 using the measured average flow velocity V, bed slope S, and the calculated hydraulic radius R h, as described above. In order to establish a tentative partitioning among the different flow resistance terms, skin friction is estimated following Millar and Quick s [1994] adaptation of the Keulegan equation: f grain ¼ 2:03 log 12:2R 2 h ð7þ D 50 as done by Curran and Wohl [2003], MacFarlane and Wohl [2003], and Comiti et al. [2008]. Millar and Quick [1994] recommended the median grain size rather than a larger measure, such as D 84, to avoid incorporating form resistance. In contrast, spill resistance is assessed by subtracting values of total resistance for the same bed geometry with and without hydraulic jumps at steps; i.e., at nappe and 6of18

7 W04424 COMITI ET AL.: FLOW RESISTANCE IN STEP-POOL CHANNELS W04424 Figure 5. Bed profiles from series A and series B for runs of comparable discharge, with increasing flow rates: (a) Q = m 3 s 1 ;(b)q = m 3 s 1 ;(c)q = m 3 s 1 ;(d)q = m 3 s 1. See Table 2 for further details. skimming flow regimes. Form resistance is assumed to compose the remaining fraction. [34] Finally, all statistical analyses were carried out using the software STATISTICA version 7.1 (StatSoft, Tulsa, Oklahoma). Linear multiple regression analyses were performed on log-transformed variables using the ordinary least squares method. 3. Results [35] This section first presents changes in bed and flow characteristics associated with increasing formative discharge. Subsequently it moves to the description of how flow velocity and resistance vary with discharge for each stable bed geometry in order to discern possible dissimilar flow regimes existing at different flow stages, establishing a comparison with studies on stepped spillways. The determination of the most adequate model for velocity and resistance in step-pool channels is then addressed, followed by an attempt to partition flow resistance for two bed morphologies at different stages Bed Evolution and Associated Flow Characteristics [36] Table 2 reports a summary of the main bed and flow characteristics measured for the runs with self-formed steps (series A and B). Figure 5 shows how the channel bed response for similar flow rates differed between series A and B as a consequence of the larger clasts present in the former mix. A much higher flow rate could be tested with mix A, because the faster bed degradation (see equilibrium slope S in Table 2) occurring with mix B eventually reached the flume bottom. Unfortunately, the comparison between series A and B cannot rely on exactly similar discharges for all of the runs because flow rates used in the two series differ because they were selected to have a different morphology for each run, as already explained. [37] The presence of a few larger clasts in mix A (i.e., the larger sieve size in Table 1) stabilized the entire bed such that the average channel slope did not substantially lower from the initial value (0.14) up to Q = m 3 s 1, a discharge for which S < 0.10 in series B runs (Table 2). In mix A, the largest sediment fraction became mobile only at the highest discharge tested (A8), in contrast to mix B where it was already mobile at B3. This also led to higher degrees of armoring, expressed by the ratio D 50sur /D 50bulk (ratio of median diameters relative to the surface layer and the initial sediment bulk still present in the subsurface layer), which was in excess of 4 for mix A (Figure 6a). In contrast, the transport ratio (i.e., the ratio between the median diameters of the transported and bulk sediment D 50tra /D 50bulk ) attains unity (Figure 6a) at much lower flow rates in mix B (run B6) than in mix A (run A8). [38] Parallel to these trends in sediment transport and bed armoring, bed morphology exhibits important adjustments. No antidunes were observed during the tests. Taller steps formed at random locations adjacent to immobile clasts and smaller steps formed downstream from immobile clasts as a result of local scouring processes of berm deposition. [39] The following analysis on bed evolution will use several variables relative to step-pool geometry: average steppool height H; average step-pool steepness H/L; average 7of18

8 W04424 COMITI ET AL.: FLOW RESISTANCE IN STEP-POOL CHANNELS W04424 Figure 6. Variation in bed sediments and morphology with increasing discharge (normalized to its max tested value) for both experimental series: (a) bed armoring and transport conditions; (b) step-pool geometry (error bars indicate standard deviation of the values); and (c) relative total pool volume (RTPV) and step drop (RTSD). relative steepness (H/L)/S; average ratio between step height and the 90th percentile of surface sediment, H/D 90; plus the relative total pool volume RTPVand relative step drop RTSD. See section 2 for a description of the calculation procedure for these variables. [40] Step height and step-pool steepness (both absolute and relative to bed slope) increase with discharge up to A7 during mix A experiments, and up to B6 for mix B (Table 2 and Figure 6b). After the peak, step-pool steepness abruptly drops, then rises again slightly in mix B. The peak value of (H/L)/S is considerably higher (2.1 versus 1.2) for the finer grain size tests (Table 2). A7 and B6 also produce the peak values of the ratio H/D 90, which are 1.0 and 1.2 for series A and B, respectively (Figure 6b). Differences 8of18

9 W04424 COMITI ET AL.: FLOW RESISTANCE IN STEP-POOL CHANNELS W04424 Figure 7. Flow resistance and Froude number measured during bed forming events as a function of flow discharge relative to the maximum value tested. Both sediment mixtures are plotted (series A and B). between series A and B with regard to H/L, (H/L)/S, and H/D 90 are statistically significant (p <0.01)basedonan analysis of covariance including flow discharge as a predictive variable. Figure 6c shows how the cumulative measures of step-pool development (i.e., relative total pool volume RTPV and relative total step drop RTSD) peaked during A7 for mix A. For mix B, however, B6 had the maximum fraction of elevation drop due to steps (90%), but not the largest RTPV (Figure 6c). [41] The changes in flow resistance f tot associated with the morphological adjustments just described display a complex pattern, slightly out-of-phase with trends in bed geometry. In particular, flow resistance steadily diminished (Figure 7) up to A4 and B4, during which the bed is characterized by few, poorly developed step-pool units. Flow resistance then rises and peaks at A6 for mix A and B5 for mix B, to suddenly drop thereafter. Therefore, both test series show local maxima in flow resistance at discharges just before those that maximize the step-pool morphological parameters (see section 2). A similar complex pattern is evident by looking at Froude numbers (Figure 7), which overall tend to increase with discharge for both series, eventually attaining values around unity (mix A) or slightly larger (mix B). However, it should be kept in mind that uncertainties associated with such formative Froude numbers (i.e., the higher for each bed morphology), which result mostly from measuring errors in determining average velocity, are expected to range from 9% (A1) to 30% (A8) Variation of Flow Resistance With Discharge Over Stable Beds [42] Examining how flow resistance varies with discharge for each stable bed morphology (i.e., a sort of at-a-station variation but at the reach scale [see Comiti et al., 2007]), Figures 8a and 8b report results from series A and B, respectively. In Figure 8a, data from the artificial step-pool sequence (see section 2) are also reported, because the original bulk grain size distribution is equivalent to Mix A. Flow discharges Q and friction factors f tot for each run are normalized to the respective values of formative discharge and formative friction factor. [43] For both series A and B, as well as for the artificial step-pool series C, it is apparent how friction factors for the lower flows tested ( the formative discharges) are about times higher than at formative conditions. However, it is evident how the relationships change in slope and shape moving toward higher formative discharges for both initial sediment mixes. Furthermore, the slopes of the curves approaching such thresholds, which occur when Q/Q formative is about , are steeper than after the threshold is crossed, when the slopes of the curves even become negative (A6 and A8). It is interesting to note that the runs before A5, which are characterized by a chaotic tumbling flow over a cascade-like morphology, have steeper curves than runs with more defined step-pool sequences. [44] The artificial step-pool case exhibits a regular trend up to Q/Q formative of about 0.4, above which several small abrupt drops and plateaus emerge in the curve. For series B (Figure 8b), sudden decreases in the friction factor are very evident for B6 and B7. Such abrupt drops in flow resistance are smaller than in series A experiments, but similar to series A, thresholds seem to mostly occur for Q/Q formative of In contrast, B8 includes two abrupt changes for moderate flow rates (i.e., Q/Q form < 0.4), then a regular trend up to a final decrease associated with the formative discharge (as in A5). [45] Froude numbers (Figure 9) indicate that supercritical flow conditions were established (but with Fr < 1.2) only for a few runs at flow rates close to the formative stage. A8 displays a striking pattern of low Froude numbers (about 0.5) up to the flow threshold discussed above, when suddenly the Froude number increases to slightly above unity, only to steadily decrease with higher discharges. A pattern parallel to A8 is shown by B7. Overall, critical (Fr 1) flow conditions seem to form an upper limit for both sediment mixtures. In the artificial step geometry (series C in Figure 9), subcritical conditions were measured even at the highest discharge tested. [46] The abrupt changes (thresholds) in flow resistance and Froude numbers observed in the runs with developed, self-formed step-pool morphology were visually correlated with the disappearance of rollers in the hydraulic jump region below steps, stronger flow aeration and the onset of 9of18

10 W04424 COMITI ET AL.: FLOW RESISTANCE IN STEP-POOL CHANNELS W04424 Figure 8. Variation of friction factor f tot relative to the formative value ( f tot formative ) for each stable bed geometry as a function of flow rate (Q) relative to bed forming discharge (Q formative ): (a) series A and (b) series B tests. V-shaped shock waves. All this suggests a transition toward a flow regime (skimming) different from that characterizing the lower flows (nappe). [47] In order to determine if the existence of different flow regimes affects flow resistance, the present data on selfformed steps are analyzed as if they were stepped spillways (see section 1), and therefore, the ratio between critical flow depth (i.e., h c =(q 2 /g) 1/3 ) and drop height z becomes relevant (see section 1). Figure 10 shows the friction factor f tot plotted against the h c /z ratio only for runs selected among those which display a knee-shaped curve in the relationship (A7, A8, B6, B7, B8). [48] These runs present bed geometries featuring the most developed step-pool structure. A precise quantitative definition could not be found, even though they seem to be characterized by a relative total step drop > 50% and by relative step-pool steepness (H/L)/S > 1. However, a remarkable exception is run A8, characterized by only two tall steps with deep pools, whose relative step drop and steepness are very low (see Table 2 and Figure 6c). Also run B5, which has a relative drop ratio of about 50% but a relative steepness slightly <1, shows a small drop in flow resistance for h c /z = 1.22 (run not shown in Figure 10 for sake of clarity). Several small drops in flow resistance can be observed in series C, but a major decrease is lacking. [49] In general, a sharp reduction in flow resistance appears to occur when the critical flow depth h c is approximately times the average elevation drop between steps z (Figure 10). Before this point, the curves show similar negative slopes; afterward they display a much flatter trend, which in the case of A8 even reverses. However, as already noted, the higher flows for run A8 include the largest measurement uncertainties, so the last reversing trend should be treated with caution. Nonetheless, data [Chanson et al., 2002] from stepped spillways of slopes ranging from 0.10 to 0.32, which overlaps S values for the runs analyzed here, indicate that a constant friction factor holds for any step submergence once skimming flow is established. [50] It is worth highlighting the much lower flow resistance measured in the smoother stepped spillways ( f tot between 0.10 and 0.20) compared to rougher alluvial steppools (minimum f tot between 0.40 and 0.80). Indeed, other data collected in stepped spillways at lower slopes [e.g., Chanson and Toombes, 2002] (S = 0.06) indicate even lower ( f tot down to 0.02) friction factors at skimming flows as well as at nappe flows (Figure 10). 10 of 18

11 W04424 COMITI ET AL.: FLOW RESISTANCE IN STEP-POOL CHANNELS W04424 Figure 9. Relationship between reach-averaged Froude number and flow rate relative to bed forming discharge for selected runs, from both series A and B, plus the artificial step-pool sequence. [51] In order to determine whether our data do reflect a transition between nappe and skimming flow regimes, threshold equations developed for stepped spillway (equations (1) (3)) are tested in Figure 11a. In the plot, the threshold lines above which a definite drop in flow resistance was observed are marked. In general, runs with flow rates above these thresholds plot in the skimming region. Among the tested models, equation (2) seems to better predict the transition for the self-formed steps. However, runs A8 and B8 display a transition in flow resistance well above equation (2), and the artificial step-pool geometry (series C), as mentioned above, does not show any clear transition despite its high ratios h c /z. [52] Skimming flows in stepped spillways are nonetheless characterized by supercritical (Fr > 1) conditions [Chanson, 1994]. Figure 11b shows that flows above the thresholds marked in Figure 11a (apart from those belonging to B5) have Fr > 0.9, reflecting quasi-critical to supercritical conditions. This apparently confirms that these flows include skimming characteristics, or at least a transitional status. Flows characterized by large h c /z ratios (>1.3) and subcritical flows virtually indicate a nappe flow but with submerged jets. This might be favored by very dissipative morphologies (i.e., deep pools as in series C) as well as by lower bed slopes (as in run B8), which may prevent the onset of supercritical conditions even though steps become submerged. Finally, thecaseofsupercriticalflowswithlowh c /z ratios (i.e., nappe flow without hydraulic jumps at steps (Figure 11b)) was not observed in our tests and it is unlikely to occur in natural step-pool channels due to the presence of deep pools, in contrast to the quasi-horizontal step treads of concrete spillways where this kind of flow can form [Chanson and Toombes, 2002]. Figure 10. Variation of friction factor as a function of step submergence (ratio h c /z) for selected runs from series A and B. Data from stepped spillways models for skimming flow only [Chanson et al., 2002] (slope 0.10 < S < 0.32) and for both nappe and skimming regimes [Chanson and Toombes, 2002] (slope of 0.06) are also plotted. The range within which the abrupt decrease in flow resistance occurs for selfformed steps is 1.2 < h c /z < of 18

12 W04424 COMITI ET AL.: FLOW RESISTANCE IN STEP-POOL CHANNELS W04424 Figure 11. Thresholds in step submergence (h c /z) for nappe/skimming flow regimes as a function of channel slope, compared to equations developed for stepped spillways (see section 1). (a) V-shaped segments mark the threshold evident from drops in flow resistance, and lie within the skimming flow region. (b) Flows above these thresholds (boxes) feature near-critical to slightly supercritical (0.9 < Fr < 1.2) conditions (except for B5), thus confirming their skimming characteristics. Field data (Cordon) relative to flows up to near-bankfull stage are also plotted Modeling Flow Resistance and Flow Velocity in Step-Pool Channels [53] Figure 12 shows the relationship between the friction factor and the relative submergence R h /D 84 for all the flows tested on the self-formed bed geometries. On the basis of the analysis presented above, runs are separated between skimming and nappe regimes. The upper flows on the B5 bed, being characterized by subcritical conditions, are included in the latter group despite the drop in flow resistance they showed (see above). [54] Skimming flows form a distinct group with friction factor in the range , whereas nappe flow data display values ranging from about 1 up to 30. The scatter in the graph is very large, and best fit power equations showed very low R 2, ranging from 0.16 (nappe flow data) to 0.25 (skimming flow data). These equations are, therefore, not reported because of their poor performance. [55] Overall, nappe flow data show a very steep variation of flow resistance with the submergence ratio, which becomes near vertical for series A. Such a steep relationship is somewhat similar to Aberle and Smart s [2003, equation (9)] (recast for f ) equation, whereas Lee and Ferguson s [2002, equation (7)] (recast for f ) formula features a less steep curvature. Both curves intersect the nappe flow data in their lower range but would overestimate flow resistance for the skimming flow data, 12 of 18

13 W04424 COMITI ET AL.: FLOW RESISTANCE IN STEP-POOL CHANNELS W04424 Figure 12. Relationship between friction factor f tot and relative submergence R h /D 84, showing series A and B runs separated between the hypothesized nappe and skimming flows. Field data from the Rio Cordon are also plotted, as well as data from Lee and Ferguson [2002] and Aberle and Smart [2003] formula (see text). which are also characterized by a flatter negative trend with relative submergence. [56] Figure 12 also suggests that friction factors for similar relative submergence differ between the two series of runs. Indeed, an analysis of covariance on the entire data set (nappe and skimming flows, 139 data points), with the relative submergence as a predictive continuous variable and series A B as the categorical factor, indicates that differences due the original sediment mix and thus possibly to the associated bed morphologies are statistically significant (degrees of freedom = 1; F = 13.28; p < 0.001). [57] When data are analyzed in terms of dimensionless flow velocity V* =V/(gD 84 ) 0.5 and unit discharge q* = (gd 3 84 ) 0.5, which is a more robust approach for very rough channels as argued by Aberle and Smart [2003], Comiti et al. [2007], and Ferguson [2007], skimming flows (the upper discharge for runs A7, A8, B6, B7 and B8) still tend to separate or plot higher from nappe flow data (Figure 13). The best fit power equation for nappe flow data only (n = 127, R 2 = 0.87, p < 0.001) is V* ¼ 1:18q* 0:82 ð8þ whereas for skimming flow data the equation reads (n = 12, R 2 = 0.58, p < 0.01) V* ¼ 1:10q* 0:38 ð9þ Figure 13. Relationship between dimensionless flow discharge q* =q/(gd 3 84 ) 0.5 and dimensionless flow velocity V* =V/(gD 84 ) 0.5, showing series A and B runs separated between the hypothesized nappe and skimming flows. Rio Cordon field data are also plotted, as well as data from Comiti et al. [2007] equation (i.e., V* = 0.92q* 0.66 ) developed from field measurements including Cordon data. Best fit equations for nappe and skimming flow data separately and together are also shown. 13 of 18