PUBLICATIONS. Journal of Geophysical Research: Earth Surface

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1 PUBLICATIONS RESEARCH ARTICLE Key Points: A new method for analysing single point velocity data is presented Flow structures are identified by a sequence of flow states (termed octants) The identified structure exerts high stresses and causes bed-load entrainment Quadrant/octant sequencing and the role of coherent structures in bed load sediment entrainment Christopher J. Keylock 1, Stuart N. Lane 2, and Keith S. Richards 3 1 Sheffield Fluid Mechanics Group and Department of Civil and Structural Engineering, University of Sheffield, Sheffield, UK, 2 Univeristé de Lausanne, Faculté de Geosciences et de l Environnement, Lausanne, Switzerland, 3 Department of Geography, University of Cambridge, Cambridge, UK Correspondence to: C. J. Keylock, c.keylock@sheffield.ac.uk Citation: Keylock, C. J., S. N. Lane, and K. S. Richards (2014), Quadrant/octant sequencing and the role of coherent structures in bed load sediment entrainment, J. Geophys. Res. Earth Surf., 119, , doi: / 2012JF Received 16 DEC 2012 Accepted 9 JAN 2014 Accepted article online 13 JAN 2014 Published online 18 FEB 2014 Abstract To permit the tracking of turbulent flow structures in an Eulerian frame from single-point measurements, we make use of a generalization of conventional two-dimensional quadrant analysis to three-dimensional octants. We characterize flow structures using the sequences of these octants and show how significance may be attached to particular sequences using statistical mull models. We analyze an example experiment and show how a particular dominant flow structure can be identified from the conditional probability of octant sequences. The frequency of this structure corresponds to the dominant peak in the velocity spectra and exerts a high proportion of the total shear stress. We link this structure explicitly to the propensity for sediment entrainment and show that greater insight into sediment entrainment can be obtained by disaggregating those octants that occur within the identified macroturbulence structure from those that do not. Hence, this work goes beyond critiques of Reynolds stress approaches to bed load entrainment that highlight the importance of outward interactions, to identifying and prioritizing the quadrants/octants that define particular flow structures. 1. Introduction Large-scale flow structures in environmental turbulent flows contain a significant part of the turbulent energy. Hence, they are important for momentum exchange as well as mixing, and scalar and sediment transport. Indeed, such coherent structures explain the very nature of near-wall flows [Kline et al., 1967; Adrian et al., 2000; Del Alamo et al., 2006]. These structures are intrinsically three dimensional, but in complex, environmental flows, flow three dimensionality is common at larger scales for a variety of reasons. At the in-channel scale, topographic forcing by three-dimensional dunes [Delgado-Fernandez et al., 2013], and individual clasts [Lacey and Roy, 2007], leads to flow separation in more than one plane, resulting in complex reattachment phenomena. The formation of horseshoe and necklace vortices [Kirkil and Constantinescu, 2010; Paik et al., 2010] about individual clasts or in-channel features highlights the importance of flow three dimensionality. At a larger scale, topographic steering by channel bends and at confluences or diffluences is known to induce motion in the lateral plane, in addition to any vertical motion associated with near-wall bursting processes [Robert, 2003]. In addition, to the complexity of the near-bed flow field, there has been widespread recognition of the need to go beyond Reynolds stress formulations in order to understand the interaction between turbulence and sediment entrainment [Heathershaw and Thorne, 1985; Nelson et al., 1995]. Consequently, for both understanding the flow field and for predicting likely sediment entrainment rates, eddy-resolving numerical methods are being increasingly adopted [Keylock et al., 2005; Hardy et al., 2007; Paik et al., 2010; Constantinescu et al., 2011; Escauriaza and Sotiropoulos, 2011]. For example, in a recent study, Chang et al. [2011] used detached-eddy simulation (DES) to predict that the mean flux of sediment downstream of a cuboid would be 2 to 3 times higher if instantaneous flow fields were used for prediction rather than the mean field.thisisbecauseof the importance of intermittent vortices that are averaged out in a mean flow simulation. Techniques such as acoustic Doppler velocimetry (ADV) [Kraus et al., 1994] allow acquisition of multicomponent (i.e., three orthogonal velocity components) time series at a point. However, they provide no access to the vorticity field or the velocity gradient tensor, meaning that various techniques for identifying flow structures [Dubief and Delcayre, 2000; Chakraborty et al., 2005] are not applicable. Hence, effective techniques for elucidating flow structure from single-point measurements are of significant utility. KEYLOCK ET AL American Geophysical Union. All Rights Reserved. 264

2 Table 1. Definitions of Quadrants and Octants Used in Flow Event Analysis Quadrant Quadrant Name u 1 u 3 τ 13 Octant u 2 1 Outward >0 >0 <0 1 <0 Interaction (OI) +1 >0 2 Ejection <0 >0 >0 2 <0 (E) +2 >0 3 Inward <0 <0 <0 3 <0 Interaction (II) +3 >0 4 Sweep >0 <0 >0 4 <0 (S) +4 >0 A well-known technique for extracting information on flow characteristics, particularly those near the wall, is quadrant analysis [Bogard and Tiederman, 1986] (Table 1). This is a two-dimensional method, meaning it is highly effective for studying nearwall bursting phenomena but will be of more restricted utility when studying fully three-dimensional macroflow structures. It is based on a Reynolds decomposition of an instantaneous velocity measurement, u i,alonga coordinate direction, i: u i ¼ u i þ u i (1) where the overbar indicates a time-averaged quantity and the prime a fluctuating quantity. Quadrant analysis involves consideration of the longitudinal and vertical fluctuating components, u 1 and u 3, respectively. This may be generalized to three dimensions by including the transverse component, u 2, to give octant analysis [Madden, 1997; Keylock, 2007a]. Quadrant analysis shows that consideration of the Reynolds stresses alone, without reference to their distribution between quadrants or their changes through time, provides an insufficient account of the sediment entrainment and transport process [Heathershaw and Thorne, 1985; Nelson et al., 1995]. Octant analysis becomes advantageous when one wishes to analyze turbulent structures with a strong three dimensionality. However, neither traditional quadrant nor octant analysis is sufficient to characterize flow structures. In this paper, we adopt the quadrant/octant approach but we analyze the sequences in which such flow events arise relative to statistical null models. While related ideas have been used in the past for the study of two velocity components [Ferguson et al., 1996], the aim of this paper is to formally characterize such sequences and link them to flow structures. We are then able to show that these structures are responsible for bed load sediment entrainment. We also introduce a statistical testing procedure for placing confidence on our results. We are able to demonstrate the importance of particular flow structures for the flow dynamics and sediment transport and characterize it in terms of a specific octant sequence. 2. Bed Load Sediment Entrainment Shvidchenko and Pender [2001] showed that for similar levels of exerted shear stress, flows with a more dominant presence of turbulent macrostructure were able to entrain bed load more readily. Macroturbulence has been shown to be a common feature of many gravel bed river flows [Roy et al., 2004]. A number of authors have shown the importance of such structures for sediment entrainment [Niño and García, 1996; Hurther and Lemmin, 2003; Chang et al., 2011], and flow and sediment entrainment at confluences (the type of flow geometry considered here) has been of particular interest in recent years [Constantinescu et al., 2012; Ribeiro et al., 2012]. In addition, recent studies have examined the effect of flow structures on near-bed forces or instrumented particles, and the important role of coherent structures on sediment entrainment has been shown [Schmeekle et al., 2007; Detert et al., 2010; Dwivedi et al., 2010, 2011; Wu and Shih, 2012] Bed Load Entrainment and the Navier-Stokes Equations The Navier-Stokes momentum equation for the instantaneous velocity along an orthogonal direction, i, j, k (longitudinal, x 1, transverse, x 2, and vertical, x 3 )is u i t þ u iu j x j ¼ 1 p þ ρ x i x j ν u i þ u j x j x i where p is pressure, the viscosity, ν, is in the kinematic form, and Einstein summation notation is adopted. Application of the Reynolds decomposition, equation (1), gives the Reynolds-averaged Navier-Stokes equations for the time-averaged flow field: u i u j ¼ 1 p þ x j ρ x j x j ν u i x j þ u j x i ρu i u j (2) (3) KEYLOCK ET AL American Geophysical Union. All Rights Reserved. 265

3 Figure 1. The definition of octants and quadrants used in this study. The gray line highlights the identified octant sequence through with the flow moves preferentially. The center of the cube is the mean velocity such that the small cubes define the state of the fluctuations. Hence, the fluctuating velocities enter in the last term via the stress they exert on the mean flow and the covariance of the velocity pairs (proportional to the Reynolds stresses), τ ij ¼ ρu i u j, account for deviation from time-averaged characteristics. This explains why there has been significant interest in coupling force balances for particles on the bed to τ ij. In a classic boundary layer, the dominant Reynolds stress is τ 13, and based on this, the quadrant typology for the flow can be established based on the signs of u 1 and u 3 [Lu and Willmarth, 1973; Bogard and Tiederman, 1986] as shown in Table 1. This provides a means of unpacking the Reynolds stress into particular events: Quadrants two (ejections, E, u 1 < 0, u 3 > 0) and four (sweeps, S, u 1 > 0, u 3 < 0) contribute positively to the Reynolds stresses, while quadrants one (outward interactions, OI, u 1 > 0, u 3 > 0) and three (inward interactions, II, u 1 < 0, u 3 < 0) make a negative contribution. For the case of fine sediment transported into suspension, E events are most important [Niño and García, 1996; Bennett et al., 1998]. However, Heathershaw and Thorne [1985] found that bed load sediment entrainment was correlated with positive u 1 values (S and OI) rather than positive contributors to Reynolds stress (S and E). In addition, although S moved the greatest quantity of sediment, when standardized by their relative frequency, OI were more important. Such effects were highlighted in the study by Nelson et al. [1995] who found that downstream of the point of reattachment for a flow that had separated over a backward-facing step, OI events could be particularly effective. 3. Flow Octant Sequencing 3.1. Flow Octants Following Madden [1997], we define octants according to the fluctuations of all three orthogonal fluctuating velocity components. We use a quadrant type of notation with a sign employed to discriminate between positive and negative contributions in the transverse direction (Table 1 and Figure 1). In this paper we work with time series of the three velocity components directly. This is not optimal for an analysis of turbulence where the fundamental theory [von Kármán and Howarth, 1938; Kolmogorov, 1941] is written in terms of spatial derivatives of velocity components. However, Taylor s hypothesisora moving-average variant [Pinton and Labbé, 1994] is not necessarily applicable given the high turbulence intensity in shallow and complex near-wall flows. In addition, while spatial derivatives in the flow field about a particle at any time will be significant [Schmeeckle et al., 2007], the dominating effect will be the change in the instantaneous forces experienced by a particle on the bed owing to the transport KEYLOCK ET AL American Geophysical Union. All Rights Reserved. 266

4 of flow structures with high momentum and/or vorticity relative to mean flow conditions. This is why sediment movement can be correlated to changes in u i [Heathershaw and Thorne, 1985; Nelson et al., 1995; Niño and García, 1996]. Given an array of discretely sampled, multicomponent velocity data 2 3 u 1 ðt ¼ 1Þ u 2 ðt ¼ 1Þ u 3 ðt ¼ 1Þ u 1 ðt ¼ 2Þ u 2 ðt ¼ 2Þ u 3 ðt ¼ 2Þ u ¼ u 1 ðt ¼ NÞ u 2 ðt ¼ NÞ u 3 ðt ¼ NÞ (4) the column means, u, maybesubtractedfromu to give u.basedontheclassificationgivenintable1,an octant value may be assigned to all N rows in u. We then group consecutive octants of the same value into octant flow events. It is the sequence of these octant events, ω { 4, 3, 2, 1,+1,+2,+3,+4}, that are used in this study to analyze coherent structures. It should be noted that in quadrant analysis it is common to adopt a threshold condition (termed a hole size ) and exclude from consideration fluctuations that fail to exceed this [Lu and Willmarth, 1973]. However, the choice of threshold is generally rather arbitrary and we do not use one here (i.e., we set the hole size to 0.0). This issue is discussed further in section 7.1. Hence, we define a new discrete time scale, ξ, where a change from ξ to ξ + 1 occurs with a change in an octant event. Hence, the minimum duration of an octant event is the inverse of the sampling frequency. We may then calculate the empirical conditional probabilities of the flow moving in to an octant, ω ξ, given that it is currently in ω ξ 1, p(ω ξ ω ξ 1 ), where ω ξ ω ξ 1 from our definition of ξ. We may generalize this to look at longer term dependencies; i.e., we do not make the assumption that this process is Markovian [Ash, 1970]. Thus, we study p(ω ξ ω ξ 1, ω ξ 2,..., ω ξ Λ ) where the maximum number of octant events considered, Λ, is controlled by the nature of the data and the nature of the flow. The former constraint depends on the length of the original record, N, and the structure of the data. If the duration of each octant event is relatively high or N is small, then the maximum for Λ is small. The latter constraint concerns the types of structure seen in the flow field. There will be some value for Λ beyond which the observed conditional probabilities become statistically insignificant. In this case, conditional probabilities defined until Λ contain the detectable structure Significance Determination We define the significance of the observed flow structures in two different ways. In the first, we generate surrogate series for the matrix u and compare the properties of the observed data set to these statistical null models. The surrogate data generation method is explained in the Appendix and is a recently proposed generalization of a well-known technique [Schreiber and Schmitz, 1996]. The latter method is based on the calculation of transition probabilities assuming that the octant event sequences indexed by our time scale, ξ, obeyed the Markovian property. With an infinitely high data acquisition frequency, the case would never arise of a change in more than one velocity component from ξ 1toξ. Hence, our null model is p ω ξ ω ξ 1 Þ ¼ 1 (5) null 3 irrespective of the value for ω, and because this distribution is independent of any particular data set, it may be used to assess the significance of first-order, Markovian transition probabilities of the form given in equation (5) by comparing observed probabilities to those derived from Bernoulli trials. However, for a finite sampling frequency, the assumption that there will never be more than one change in a velocity component will be violated. We handle this in the simplest way, by only examining the cases where a single change in sign arises, with our observed probabilities renormalized accordingly. The null model for higher-order transition probabilities follows as p ωξ ωξ 1; ;ωξ ΛÞ ¼ 1 (6) null 3 Λ KEYLOCK ET AL American Geophysical Union. All Rights Reserved. 267

5 Figure 2. The confluence geometry and coordinate system used for laboratory experiments in this study. In the postconfluence region, either 0.02 m thickness of mobile sediment (preliminary experiments), or a board roughened with immobile grains are used such the bed is flush with the smaller step. Thus, the null hypothesis for the total probability of a sequence of Λ + 1 octant events {ω ξ,ω ξ 1,,ω ξ Λ }is Λ 4. Experimental Design 4.1. Experiment Configuration The experiments reported in this study were undertaken using the recirculating flume at the Department of Geography, University of Cambridge, which was constructed by Armfield Engineering Ltd. It has a working section m wide, m deep and 7.5 m long. The total length of the tank is m, and the slope is adjustable, although zero gradient was used in this study. The flow geometry adopted to induce a complex flow was based on the interaction between two perpendicular shear layers, one from a backward-facing step [Driver et al., 1987; Simpson, 1989; Wee et al., 2004] and the other due to a vertical splitter plate in the center of the flume (Figure 2). The backward-facing step was introduced by placing two steps either side of the splitter plate, one 0.02 m in height and the other 0.07 m in height. In the postconfluence region, a m thick board was placed with its surface roughened by unimodal, immobilized grains with a median diameter of D 50 = 2.36 mm and 95% of grains sized between 2 mm and 3.36 mm. The exception to this is the precursor experiments in 4.2 where the postconfluence region consisted of a 20 mm depth of fully mobile sediment instead. Hence, in effect, one channel underwent a 0.05 m change in bed elevation, while the other was concordant with the postconfluence bed elevation. This geometry is similar to that used for studying turbulence at parallelchannel confluences [Best and Roy, 1991; Biron et al., 1996; Bradbrook et al., 1998]. Table 2 lists the flow properties used in this experiment where Table 2. Flow Characteristics for the two Input Channels for the Confluence Experiment the Weber number was calculated using the formula of Peakall and Warburton [1996]: Flow Property Raised Channel Unraised Channel Flow depth (m) Mean velocity (m s 1 ) Discharge (m 3 s 1 ) Reynolds number Froude number Weber number We ¼ U 1 2 ρd (7) σ where d is the flow depth, U 1 is the cross-sectional mean velocity, and σ is the surface tension. Discharges KEYLOCK ET AL American Geophysical Union. All Rights Reserved. 268

6 Figure 3. The postconfluence region for the parallel-channel confluence flow after one hour of erosion of a mobile bed. The superimposed crosses are the sampling locations where velocity data were collected over a fixed bed. Flow direction is from left to right, H is the step height and w f is the flume width. Redrawn from Keylock [2009] with kind permission of Springer Science and Business Media. were determined by integrating mean velocity profiles obtained in the channels using the ADV, and the discharge ratio Q r between the raised and unraised channel discharges had a value of when calculated following Ashmore and Parker [1983]: Q r ¼ Q raised Q unraised 1 Q 2ð raised þ Q unraised Þ (8) The Shields parameter, θ 50, for the median grain size has a value of This value is the average of estimates from the raised and unraised channels and is given by [Shields, 1936] 2 ρu θ 50 ¼ (9) ðρ s ρþgd 50 where u * is the shear velocity obtained from the law of the wall and g is the gravitational acceleration. The grain Reynolds number is Hence, the flow has a subcritical Froude number and Shields number, with a turbulent grain Reynolds number, meaning that sediment entrainment is induced by the action of particularly energetic flow structures. The flow depth ratio (f H ) was calculated by dividing the flow depth in the raised channel by the flow depth in the deeper channel [Biron et al., 1993]. The value used in this study is 0.07 / 0.12 = 0.583, which is within the range of 0.26 to 0.66 observed for the Bayonne-Berthier confluence and is close to the mean of 0.52 for that river [Biron et al., 1993]. Hence, although we consider an idealized case, our results are potentially relevant in field studies Precursor Experiments The fixed bed in the postconfluence region was replaced by a 0.02 m thick sediment layer, which was smoothed flat to simulate the conditions for the sediment entrainment experiments described in section 4.4 The surface grains were painted in colored patches to help identify where erosion was occurring (removal of colored grains to reveal the underlying natural colors) and where they were transported to (different color with respect to that beneath). With the flow conditions described in section 4.1, an experiment was run for 1 h to identify regions of erosion and deposition. Figure 3 shows that the majority of the bed was undisturbed. However, as shown by Best and Roy [1991], our analysis of the near-bed intermittency structure for these data [Keylock, 2009] and our LES results [Keylock et al., 2005], vortices shed from the step, interact with those generated by the central splitter plate to produce a curved reattachment zone and it is this region (between 3 < x/h < 5for0.2< y/w f < 0.4) where sediment transport has occurred, where x is the longitudinal distance from the postconfluenceorigin(showninfigure2),y is the distance across the flume in the transverse distance (Figures 2 and 3), H is the step height, and w f is the flume width. Thus, we have achieved the marginal transport regime predicted from the flow characteristics in section 4.1. As will be shown below, particular coherent structures are responsible for incipient motion and where the flow is KEYLOCK ET AL American Geophysical Union. All Rights Reserved. 269

7 Figure 4. Some of the properties of the postconfluence flow field. The number of sediment movement events observed in a 10 min period under the flow conditions in Table 2 are given in (a). Contours begin at 1 andincreaseto25instepsof4.therelativefrequencyof the { 1, 4, +4} octant sequence as a proportion of the frequency of all possible three octant sequences at (x/h =5,y/w f =0.233)isin (b). Contours begin at 0.02 and increase to 0.05 in steps of The mean velocity vectors near the bed in the x-y plane are in (c). Vectors are scaled such that a length of one step height equates to 0.20 m s 1. less perturbed (e.g., y/w f > 0.5), stresses are insufficient to mobilize sediment. The map of bed erosion shown in Figure 3 is supported by the contour map in Figure 4a. This is an experiment with the same flow conditions as described in section 4.1 where entrainment was observed from one of forty two locations in the postconfluence region during a 10 min period (sufficient to result in a number of entrainment events, without significantly deforming the bed and altering flow structure). These two initial experiments were used to select the sites for velocity acquisition based on the potential for sediment entrainment, and these are overlain on Figure Flow Measurement Experiments Velocity time series were obtained at these sites, close to the bed, at a frequency of 25 Hz for 300 s duration using an acoustic Doppler velocimeter (ADV) [Kraus et al., 1994; Lohrmann et al., 1995]. In this study, the probe was aligned in the downstream direction by mounting the instrument in a housing that set it parallel to the walls of the flume. There is a close correspondence between our ADV data and near-bed time series extracted from a large-eddy simulation for this flow configuration at locations affected by the vortex shedding processes [Keylock et al., 2005]. This is shown in Figure 5 and provides evidence that our measurement procedure was robust. For experiments without mobile grains, we worked carefully to produce data where the probe head correlations remained above R = 0.98 which meant that the base of the measuring volume was 7 mm above the bed. For experiments with mobile grains, this same height meant that the probe was ~ 5 mm above the height of the 3.3 mm diameter mobile grains (as they sit in pockets formed by the fixed layer below). In addition, it is likely that the discrete values for u 1 (t), u 2 (t), and u 3 (t) (rather than merely the bulk statistics) are robust because these data yield the correct slopes for the joint velocity-intermittency structure [Keylock et al., 2012c] when compared to those obtained at 5 khz sampling frequencies in wind tunnels using hot wire anemometry [Keylock et al., 2012b]. To test whether the 300 s sampling time interval is sufficient for robust statistics to be derived, we have undertaken a convergence analysis. Figure 6 shows the first four moments of the three velocity components as a function of the time period of data acquisition (in terms of the number of samples all measurements at 25 Hz). The data are from (x/h = 5.0; y/w f = 0.233) where sediment entrainment peaked. Further downstream of the step, convergence is more rapid. A record of 20,000 samples was analyzed. The values for the first four moments for this record were calculated and are plotted as a dotted line in each panel. Thereafter, the data set was subdivided into sample periods with a duration given on the x axis. The bars show the range of values KEYLOCK ET AL American Geophysical Union. All Rights Reserved. 270

8 (a) (b) Figure 5. Boxplots showing the instantaneous velocity values recorded for the three velocity components at two locations in the postconfluence region: (x/h =6,y/w f = 0.233) in (a) and (x/h =7,y/w f = 0.233) in (b). The boxplot center line and limits indicate the median and upper and lower quartiles, while the whiskers extend for up to 1.5 times the interquartile range. for each statistic, the tick on each bar is the mean value from all these samples, and the dashed lines indicate ± 1 standard error on these means. It is clear for all cases that by 5000 samples the mean of the samples has converged on the mean for the full record and that the range of values obtained for each statistic is comparable to the standard error. Hence, our sampling period of 7500 samples is sufficient. Furthermore, as is established in section 5, our dominant flow structure has a frequency of ~0.5 Hz. Consequently, we sample this structure some 150 times in a typical flow record, which is the appropriate order for sampling recommendations (e.g., Luchik and Tiederman [1987] suggest 200 flow structures). Figure 6. Convergence statistics for an 800 s time series (20,000 samples) at (x/h =5.0; y/w f = 0.233). Each vertical bar indicates the range of observed values, with the central tick, the mean value. The horizontal dotted line is the value estimated over all samples. The dashed lines (barely visible) are upper and lower confidence intervals on the mean value and represent ± 1 standard error for the mean of the samples. KEYLOCK ET AL American Geophysical Union. All Rights Reserved. 271

9 4.4. Sediment Entrainment Experiments To link flow structures to sediment movement, five colored test grains were placed on the fixed bed of immobilized grains within a 10 mm 10 mm area. This was similar to the projected measuring area of the ADV, which was positioned such that the base of the measuring volume was ~ 1.5 diameters above the mobile patch of grains. These five grains had a median intermediate axis of b=3.3 mm and a maximum projection sphericity [Sneed and Folk, 1958] of (c 2 /ab) 1/3 = 0.718, where a is the long axis and c the short axis. Approximating the maximum projected area for each grain by π ab/4where a and b are the long and intermediate axes of the particles, respectively, the total area of these five grains is some 65 mm 2.After allowances for void spaces, this results in compact, but not imbricated, packing. Five hundred and seventy-six individual sediment entrainment events were observed in the flume at eleven different locations in the postconfluence region for x/h < 7 and y/w f < 0.5. The sample size was sufficient to account for variability in grain packing and protrusion between experiments. On the flat bed used in this study, more complex form resistance effects are eliminated and the effects of turbulent structure can be isolated. The input flow conditions were held to the values given in Table 2. Entrainment was defined as at least one of the mobile grains moving out of its pocket on the bed. Typically, it would roll some five to ten diameters along the bed before momentum gained from the passage of a particular flow structure was lost and it settled once more. Entrainment was detected manually and stamped on the velocity time series. The flume was stopped, and the grains were repositioned and held in place. The flume was restarted, and when the velocity had attained the conditions in Table 2, the grains were released once more. 5. Overview of Flow Characteristics Figure 4c shows the mean velocity vectors close to the bed in the longitudinal-transverse plane. Unlike a classical backward-facing step flow [Simpson, 1989], the reattachment zone is markedly curved [Best and Roy, 1991; Bradbrook et al., 1998] and it is also much shorter (approximately half the length) owing to the stronger pressure gradient. In terms of the coherent structures generated by this flow configuration, there are both similarities and differences with respect to a backward-facing step flow. This can be seen from a consideration of the frequency of occurrence of a flow structure, f, nondimensionalized using a Strouhal number, N str = fh/u 1. Concerning first the similarities to a backward-facing step, variability in the reattachment length due to the advection of pairs of spanwise Kelvin-Helmholtz vortices occurs at N str = [Eaton and Johnston, 1980; Driver et al., 1987; Wee et al., 2004], which corresponds to an observed peak at frequencies of Hz for this study (Figure 7) and is in agreement with the frequencies for such vortices observed at the Bayonne- Berthier confluence [Biron et al., 1993] and LES simulations of such a confluence (N str = ) [Bradbrook et al., 2000]. Driver et al. [1987] noted that the shear layer forming over a backward-facing step has a characteristic flapping frequency of N str = This is in agreement with the periodicity of eddy shedding for a parallel-channel confluence noted by Bradbrook et al. [2000] who performed their LES with a Smagorinsky closure scheme, a time step of 0.1 s and a mesh of cells for the 1.0 m 0.3 m 0.1 m domain. For a comparison with the study by Bradbrook et al. [2000], who extracted Fourier amplitude spectra at (x/h =6,y/w f =0.25)and(x/H =10,y/w f = 0.1), amplitude spectra from this study obtained at (x/h =6,y/w f = 0.233) and (x/h = 10, y/w f = 0.167) were examined. Note that the former is close to the region of maximum sediment entrainment in Figure 3, while the latter is someway downstream, where a reduction in energy and a decay of peak structures are anticipated. Power spectra for each velocity component at these two locations, for the flow conditions given in Table 2, were studied. In addition, an additional experiment at lower velocity but the same flow depths (mean velocity in the raised channel of 0.44 m s 1 and in the unraised channel of 0.39 m s 1 )was also examined. Six spectra (those for the flow conditions in Table 2) are shown in Figure 7. Of the 12 spectra examined altogether, nine had one of the five greatest spectral peaks for N str = , while eight showed a peak at the Kelvin-Helmholtz frequency range (N str = ). However, the dominant peak frequency in the region of sediment entrainment occurred at N str = (Figure 7) and was a structure that emerged from the coupling of the two shear layers. Keylock et al. [2005] showed using LES that this spectral peak disappears in a wider confluence, explaining why it did not occur in KEYLOCK ET AL American Geophysical Union. All Rights Reserved. 272

10 Figure 7. Example velocity power spectral density functions at (x/h =6,y/w f = 0.233), shown in black and (x/h = 10, y/w f = 0.167), shown in gray plotted against Strouhal number. The results in panels (a), (b) and (c) correspond to spectra for u 1, u 2, and u 3, respectively. Vertical dotted lines delimit the frequency ranges discussed in the text. the field data of Biron et al. [1993]. This structure is an emergent property of a narrow-confluence configuration and represents an evacuation of fluid from the recirculation region. It is a key difference between the confluence case and the backward-facing step. As shown below, this flow structure is crucial for understanding entrainment of sediment from the bed downstream from the confluence geometry. Figure 7 highlights the reduced energy of the spectral peaks at x/h = 10 and the dominance of the u 1 and u 2 components at x/h =6, compared to u 1 at x/h = 10 where the boundary layer is beginning to recover. It can also be seen that at x/h = 10, the dominant peak occurs for u 1 and at a lower frequency (St < 0.01) owing to amalgamation of structures generated near reattachment as they are advected downstream. 6. Results 6.1. Dominant Octant Sequence Determination The off-diagonal stresses, and the turbulence intensity, I turb ¼ 3 i¼1 1 ðu i Þ 2 2 all peaked just downstream of the average reattachment length and at y/w f < 0.5, with the maximum close to (x/h =5,y/w f = 0.233). This is shown in Figure 8, where in addition to I turb, a summary measure of the instantaneous stress magnitude, hu i u j i u i u j, is also plotted. It is clear that the mean instantaneous stress exerted at y/wf = (squares) is much less than at y/w f = and y/w f = 0.167, at 3 x/h 5, indicating that the reattachment structure is highly three dimensional. Furthermore, the highest values are seen for the components that involve the transverse velocities, indicating that the flow is fully three dimensional in this region. Table 3 gives the Markovian octant transition probabilities at (x/h = 5.0, y/w f = 0.233). A similar pattern was detected at all sites downstream of reattachment for x/h < 10 and y/w f < 0.5. In general, there is a clockwise movement through the conventional quadrants [Ferguson et al., 1996]. However, the transition probabilities indicate three dimensionality to the flow. For example, ω = +2 and ω = 2 show opposite behavior in terms of the ranking of particular sign changes and although ω = 1 and ω = +1 both prioritize changes in u 3 followed by u 1 and then u 2, the probability of these changes varies dramatically (e.g., p( 4 1) = 0.63, compared to p(+4 + 1) = 0.37). With the exception of ω = +1 and ω = 3,themostprobablechangeisapproximately twice as likely as any other. This suggests that the passage of a structure is strongly influencing the flow behavior, and Table 3 implies an octant sequence of { 1, 4, +4, +3, +2, 2, 1}, explaining the lack of occurrences for ω = 3 andω = +1 in the final column. KEYLOCK ET AL American Geophysical Union. All Rights Reserved. 273

11 Figure 8. Longitudinal profiles of the turbulence intensity, I turb as well as the mean, absolute, kinematic stresses, u i u j ui u j. The circles are for data from y/w f = 0.167, the diamonds are for y/w f = and the squares are for y/w f = Renormalizing the probabilities in Table 3 to eliminate changes between octants of more than one sign, the { 1, 4, +4, +3, +2, 2, 1} structure has an overall probability of , while a null hypothesis for six independent changes each with a probability of 0.33 is = Hence, this structure is highly significant. Note that the sequence of changes to velocity components represented by this structure is vertical, lateral, downstream, vertical, lateral, and downstream. This analysis is not on its own sufficient because the six octant changes are characterized by Markovian transition probabilities. However, a direct examination of the probability of occurrence of any six octant sequences does give { 1, 4, +4, +3, +2, 2} as the most probable combination, with a probability of based on all observed sequences. This is followed by related sequences such as {+3, +2, 2, 1, 4, +4} (p = ) and { 2, 1, 4, +4, +3, +2} (p = ), indicating that the patterns deduced from single transition probabilities are robust. As the actual probability of this structure is greater than expected from the single transition probabilities (p = ), it may be deduced that this is a highly significant coherent structure in the flow. In terms of three consecutive octants, the { 1, 4, +4}, { 2, 1, 4} and {+3, +2, 2} elements of the identified sequence were the most probable at (x/h =5,y/w f = 0.233) with probabilities (from all observed transitions) of 0.055, 0.047, and 0.046, respectively, and occurring 147, 126,and 123 times from 2687 octants observed in 300 s. Note that 147/300 equates to a frequency of 0.49 Hz and N str of This corresponds to the dominant spectral peak in Figure 7 in this region discussed in section 5. Hence, a combination of octant sequencing and spectral analysis provides a physical explanation for a dominant frequency. The mean duration for the { 1, 4, +4} sequence was 0.33 s at (x/h =5,y/w f = 0.233), meaning that the flow was in this state approximately one sixth of the time. Table 3. Octant Switching Probabilities at (x/h = 5.0, y/w f = 0.233) a Octant u 1 Transition u 2 Transition u 3 Transition Number of Octant Occurrences 1 2 (0.12) +1 (0.13) 4 (0.63) (0.30) 1 (0.16) +4 (0.37) (0.42) +2 (0.18) 3 (0.21) (0.07) 2 (0.70) +3 (0.16) (0.29) +3 (0.18) 2 (0.35) (0.13) 3 (0.21) +2 (0.54) (0.15) +4 (0.52) 1 (0.21) (0.39) 4 (0.20) +1 (0.19) 378 a The most probable transitions for each octant are highlighted in bold. The probabilities do not sum to 1.0 because of transitions involving more than one sign within the 0.04 s sampling interval. Figure 9 employs gradual wavelet reconstruction and the surrogate data algorithms reviewed in the appendix to determine the significance of the number of occurrences of { 1, 4, +4}. It is clear that when the control parameter, η, explained in the Appendix equals 0.0, and the cross spectrum between velocity components is only preserved by chance, the maximum number of occurrences of this sequence is more than 3 times smaller than the number observed. From η = 0.90, there is no KEYLOCK ET AL American Geophysical Union. All Rights Reserved. 274

12 Figure 9. Boxplots illustrating the gradual wavelet reconstruction of the number of occurrences of the { 1, 4, +4} octant sequence at (x/h =5, y/w f = 0.233). The dashed line equals the actual number of occurrences and the boxplots show the median, upper and lower quartiles, with whiskers extending to the maximum and minimum values for the particular choice of η. Multi indicates surrogates built using a surrogate variable algorithm that preserves the cross spectra. significant difference and a comparison with the multivariate algorithm indicates that preservation of the cross spectrum equates to η ~ Hence, it is necessary to preserve some of the nonlinear properties of the original data in order to attain 147 { 1, 4, +4} octant sequences as preservation of the spectral and cross-spectral properties is not on its own sufficient to replicate the observed occurrence frequency. Figure 4b shows the relative frequency of the { 1, 4, +4} sequence compared to the occurrences of all observed sequences of three octants. The peak in frequency of this structure coincides with the sediment entrainment peak in Figure 4a. This is strongly suggestive that it is this structure that is responsible for sediment entrainment. For (x/h =4,y/w f = 0.233), the leading two sequences reversed ranking with { 2, 1, +4} having an occurrence probability of and { 1, 4, +4} a probability of 0.050, while downstream of (x/h =5,y/w f = 0.233), { 1, 4, +4} was ranked first, but the probability of occurrence decayed with distance from flow reattachment from at (x/h =6,y/w f = 0.233) to at (x/h =7,y/w f = 0.233), to at (x/h =10,y/w f = 0.233). That the { 2, 1, 4} tends to rank second suggests that the ejection of fluid ( 2) and the subsequent { 1, 4, +4} sequence is more closely coupled than the link between { 1, 4, +4} and the following +3 octant. This is supported by the result that while the { 1, 4, +4, +3, +2} structure occurred 22% of the time that a { 1, 4, +4} structure developed, sequences of { 1, 4, +4, +2} and { 1, 4, +4, +1, +2} also occurred 10% and 9% of the time, respectively. Because the former of these two sequences has a simultaneous change in two velocity components, it suggests a rapid movement through the +3 octant. An equivalent analysis based on quadrants at (x/h =5, y/w f = 0.233) yields some differences. While the {1, 4} sequence, the quadrant equivalent of { 1, 4, +4}, dominates single transitions with a probability of 0.17 (318 occurrences), the {1, 4, 3, 2} structure (the equivalent to { 1, 4, +4, +3, +2, 2}) is as probable as {3, 2, 1, 4} (116 compared to 117 occurrences, respectively). Significance testing at η =0 gives a maximum value for the number of occurrences of {1, 4} of 259. While 318 occurrences are significantly greater than this, it is so only by 22%, rather than ~3 times for octants (the first box in Figure 9). This indicates that while a quadrant analysis reveals important and significant structure, it subsumes a more keenly expressed three-dimensional structure Contribution of the Identified Octant Sequence to Exerted Stresses Figure 10 shows duration histograms for each octant at (x/h =5.0, y/w f = 0.233). It is clear from Figure 10 that ω = +2 is particularly persistent, while ω = 1, ω = +1, and ω = 3 rarely last more than 0.3 s. The relatively low duration of the latter two octants is not too surprising as they are not part of the six octant sequence identified in section 6.1. The short duration of 1 suggests a rapid transition from ejection to sweep in the { 2, 1, 4} part of the identified octant sequence similar to the rapid transition through +3 identified in the previous paragraph. Figures 11a, 11c, and 11e show cumulative histograms for the KEYLOCK ET AL American Geophysical Union. All Rights Reserved. 275

13 Figure 10. Histograms indicating the duration (in seconds) of particular octant events at (x/h =5, y/w f = 0.233). Each octant is identified toward the top of each histogram panel. peak instantaneous values for ρ u i u j in each octant. In each case there are four octants exerting higher stresses than the others and these form consistent groupings. For ρ u 1 u 2 in Figure 11a, the high stresses in the { 1, 4} part of the flow structure are accompanied by the return motion through +3 and +2, although the stresses in quadrants 1 and 4 are higher than those for 2 and 3 in both the high- and low-stress groups. As expected, ρ u 1 u 3 in Figure 11c reveals a bias toward octants ±2 and ±4 for the peak stresses, while Figure 11e shows that the ρ u 2 u 3 peak stresses are dominated by 1, 2, +3, and +4 octants, which is consistent with the results in Figure 11c given the 90 rotation of the relevant reference frame. These patterns P P P i = 1, j = i = 1, j = i = 2, j = / / Peak values for ρ u uj i (Pa) (a) (c) (e) i = 1, j = i = 1, j = i = 2, j = / / Peak values for ρ u uj i (Pa) (b) (d) (f) Figure 11. Cumulative probability plots of the peak fluctuating stresses exerted during particular octant events at (x/h =5, y/w f = 0.233), truncated at a lower cumulative probability limit of 0.2. The left-hand plots give results for each octant, with solid lines indicating positive octants, and dotted lines, negative. Quadrants 1, 2, 3, and 4 are colored red, blue, green and black, respectively. The right-hand plots separate results for octants 1, 4 and +4 by their occurrence within the { 1, 4, +4} structure (magenta) or not (gray). The dotted line is for 4, the solid line for +4 and the dashed line is 1. KEYLOCK ET AL American Geophysical Union. All Rights Reserved. 276

14 Figure 12. Cumulative probability plots of the impulse exerted during particular octant events at (x/h =5,y/w f = 0.233) truncated at a lower cumulative probability limit of 0.2. The upper figures are for I(u R ) and the lower figures are for I(u 1 ). The line styles and coloration are as given in Figure 11. contextualize the results shown in Figures 11b, 11d, and 11f where the histograms for the 1, 4, and +4 quadrants are decomposed based on their occurrence within the { 1, 4, +4} sequence. If a particular octant belongs to the group of four low-stress cases in either Figures 11a, 11c, or 11e, there is little difference between the two histograms. However, if they are one of the four high-stress octants, then presence within the { 1, 4, +4} structure (magenta lines) is clearly indicative of higher exerted stresses. Thus, not only is there a spatial congruence between the { 1, 4, +4} structure and zones of sediment entrainment (Figures 4a and 4b) but the structure is shown to exert preferentially greater peak stresses. A critical Shields parameter value of 0.05 equates to a critical shear stress of ~3 Pa, for the median intermediate axis of the grains used in the sediment entrainment experiments. Although not of direct relevance to the characterization of sediment entrainment by macroturbulence (the shear velocity in equation (9) is derived from a mean velocity profile), this provides a guide as to the necessary stress to entrain sediment. According to Figure 11c, quadrants 1 and 3 rarely attain this value, while it is exceeded 20% to 40% of the time for quadrants 2 and 4. Similarly, Figure 11e shows that the +4 octant attains a peak for ρ u 2 u 3 that exceeds 3 Pa 40% of the time. Figures 11c and 11e suggest that ω = +4 would be the most effective octant for mobilizing sediment. Figure 11d shows that this effectiveness increases greatly if ω = +4 occurs in the { 1, 4, +4} sequence, with 3 Pa attained close to 70% of the time. The increase seen in Figure 11f is less dramatic, but still important, with peak values for ρ u 2 u 3 exceeding 3 Pa 50% of the time. Consideration of the stress components individually almost certainly underestimates the efficacy of the { 1, 4, +4} structure as rotation in the x-y plane induced by the change from 4 to +4 will rotate the particle, increasing the chance of it rolling over a saddle between underlying grains Contribution of This Sequence to Exerted Impulse Because recent work demonstrates the importance of impulse for the entrainment of bed load sediment [Diplas et al., 2008; Valyrakis et al., 2010], we also present results in Figure 12 for impulse exerted per octant event. Here we integrate a force-related variable over limits corresponding to the octant event duration. Note that Diplas and coworkers were working with a well-developed boundary layer where the magnitude and variation associated with u 1 is much greater than for the other velocity components. Hence, changes in u 2 or u 3 are unlikely to be sufficient to prevent a force being sustained against a grain and the appropriate time KEYLOCK ET AL American Geophysical Union. All Rights Reserved. 277

15 scale is fluctuations in u 1. In contrast, close to reattachment, and under conditions of full three dimensionality as studied here, the means and variances of all the components are similar. Hence, while the traditional definition of impulse is likely to perform well as it would correspond to the integrated effect of the { 1. 4, +4} structure along x 1, variability in the other components is significant and means that forces are unlikely to be sustained on a given grain for the whole duration of such an impulse event. We simplify the expression for the drag force to F D ðu R Þ ¼ C D Aρu 2 R (10) where C D is the drag coefficient, A is the exposed area of the particle, and the square of the resultant velocity in the bed parallel plane is u 2 R ¼ u2 1 þ u2 2. We also consider u2 1 instead of u2 R in equation (10), to provide an alternative: F D ðu 1 Þ ¼ C D Aρu 2 1. While the value for C D for our grain Reynolds number of 1600 is ~ 0.55 [Fenton and Abbott, 1977], and our flat bed conditions minimize variation in particle exposure and protrusion, there will still be some variation in C D A. Consequently, we simplify to F D ðu 1 Þ u 2 1 and F Dðu R Þ u 2 R [Diplas et al., 2008]. Hence, Iu ð R Iu ð 1 t 2 Þ ¼ u 2 R dt t 1 t 2 Þ ¼ t 1 u 2 1 dt (11a) (11b) where the limits t 1 and t 2 are the times for the beginning and end of a particular octant event. As it will be needed in section 6.5, we also introduce the more traditional definition of impulse, I(trad), which is given by equation (11b) but with limits defined in terms of sign changes in u 1 rather than in any component. As a consequence of our definition of impulse, the results in Figure 12 have units of meters squared per second. The differences between the upper and lower rows of Figure 12 are minor. The patterns resemble those in Figure 11, although greater emphasis is placed on the role of ω = 4 in Figures 12a and 12c. Again, there is a clear difference between impulses exerted by individual octants when conditioned on the occurrence of the { 1, 4, +4} structure. The largest separation between the gray and magenta lines is for ω = +4 as was the case in Figures 11d and 11f. Therefore, this analysis also supports the earlier conclusion that ω = +4 is a critical part of the identified structure Statistical Testing of the Significance of the Observed Stress Distributions We compared the cumulative histograms in Figure 11d to those from 19 surrogate series obtained using the Iterated, Amplitude Adjusted Fourier Transform (IAAFT) algorithm (Figure 13) and its cross-spectrum preserving variant (Figure 14) for τ 13. The IAAFT algorithm is described in the Appendix and provides data of a type that corresponds to η = 0 in Figure 9. Figure 11d shows that the stress histogram for ω = 1 had low values and did not exhibit a dependence on the occurrence of the { 1, 4, +4} structure. Figure 13 shows that the synthetic data yield values for the peak stresses that are at least as large as those observed. There is a clear contrast with ω = 4, which shows much higher stresses than the synthetic data when part of { 1, 4, +4}, and for ω = +4, which exhibits a significant difference irrespective of the occurrence of { 1, 4, +4}, but this is much stronger when it is part of the { 1, 4, +4} sequence. The results in Figure 14 indicate that preservation of the original values, Fourier spectra, and cross spectra is sufficient to replicate the stress distributions. This is in contrast to the number of occurrences of the { 1, 4, +4} sequence (Figure 9). Thus, in order for total event effectiveness [Nelson et al., 1995] (the product of frequency and exerted forces) to be preserved, nonlinear interactions are significant, but they control the frequency of the generated structures (Figure 9). The stress distribution given that { 1, 4, +4} occurs can be replicated with linear, randomized data (Figure 14), but cross correlation is crucial to this (comparison of Figures 13 and 14) Sediment Entrainment Experiments For 322 cases from the 576 total entrainment experiments (56%), one of octant peak stress, octant impulse, or impulse defined traditionally attained its maximum value in the time series in the 0.6 s before entrainment occurred. For 486 cases (84%), one of these variables attained one of its highest five ranked values in the 0.6 s interval. Note that the ADV sampling volume is slightly above the grains to maximize the quality of the signal KEYLOCK ET AL American Geophysical Union. All Rights Reserved. 278