Lateral bed load experiments in a flume with strong initial transversal slope, in sub- and supercritical conditions

Transcription

1 WATER RESOURCES RESEARCH, VOL. 45, W01419, doi: /2008wr007246, 2009 Lateral bed load experiments in a flume with strong initial transversal slope, in sub- and supercritical conditions Luigi Fraccarollo 1 and Giorgio Rosatti 1 Received 26 June 2008; revised 17 November 2008; accepted 3 December 2008; published 30 January [1] The role of lateral slope in the generation of bed load transport and bed evolution is investigated. Because of the substantial lack of detailed laboratory or field data, we designed new tests specifically devoted to acquiring accurate measurements in transient flows where the bed surface gradient has a dominant effect on the direction and magnitude of the bed load. The bed was prepared with well-sorted sand and constant nonzero slopes, both longitudinally and transversally over the flume. The flow in each experiment evolved to steady-uniform conditions characterized by a horizontal bed profile in each section and unchanged longitudinal slope. Both sub- and supercritical conditions were investigated. The bounding edges of the bed profile were surveyed on a selected transversal section. Some detailed kinematic description of the grain motion in the bed load layer has been pursued. A shallow-water model for mobile bed, able to include the most recent theories for the treatment of lateral bed load, was developed and numerically applied to the experimental tests. Comparison between measured and calculated data, and between sub- and supercritical conditions, offers several points of interest. Citation: Fraccarollo, L., and G. Rosatti (2009), Lateral bed load experiments in a flume with strong initial transversal slope, in suband supercritical conditions, Water Resour. Res., 45, W01419, doi: /2008wr Introduction [2] The role of slope on the motion of grains over a noncohesive sediment bed takes place in direct and indirect ways. Indirectly, the bed slope is involved in the establishment of the flow dynamics, responsible for grain entrainment and transport in rivers. Directly, grains on a slope are forced by gravity to deviate in the direction of the bed surface elevation gradient. In Figure 1 the core of the physical process focused on here is sketched; it presents the motion of a particle sliding or saltating in the direction of the near-bed liquid velocity distribution when no side slope is present, and the deviation of the particle path from the flow direction when the transversal bed profile is tilted. Morphological effects induced by strong streamwise or lateral slope (the word lateral is used to define the direction in the same plane as the bed surface but orthogonal to the main direction of the flow) take place in bed forms, meanders, banks, bends, bifurcations, confluences, braiding, fans, scours and more. In these situations the local value of the lateral slope may be much larger than the relevant longitudinal component and characterized by high rates of spatial variability. Errors in predicting the lateral bed load produce incorrect representation of the general features (size and shape) of the bed surface. Beyond scientific purposes, there are economical reasons behind modeling river morphology: when hydraulic works (such as abutments, spur dukes or bridge piles) are designed on the basis of wrong information about bed level variations, their 1 Department of Civil and Environmental Engineering, University of Trento, Trento, Italy. Copyright 2009 by the American Geophysical Union /09/2008WR W01419 construction will undergo either failures or unnecessary costs. [3] In spite of the importance of lateral bed load in morphodynamics, there is a clear lack of relevant experimental data to support theoretical assessment. In the works of Ikeda [1982a, 1982b] the incipient motion of noncohesive sands was observed in a tilting wind tunnel; a theoretical analysis that follows previous works of Kikkawa et al. [1976] and Engelund [1974] is there presented. The work of Ikeda [1981] showed that in a straight flume the transversal section evolves toward a stable configuration, almost independent of the initial shape, of the discharge and of the sand diameter. Diplas [1990] and Izumi et al. [1991] performed similar experiments. Talmon et al. [1995] presented an experimental subcritical test in a curved flume with two straight parts, where a laterally sloping bed, with longitudinal dunes, was prepared. Francalanci and Solari [2007] observed paths and velocity of particles on a tilted bed at low shield stress. From this short overview of experimental works it is apparent that different ways to measure the lateral bed load have been pursued. In the case of Francalanci and Solari [2007] a direct measurement of the bed load predictors, such as the grain velocity, is chosen. Talmon et al. [1995] conducted bed-leveling experiments and then, in modeling the experiments, were able to provide an indirect measurement of the lateral bed load. [4] The design of the present research follows Talmon et al. [1995]. The collected data relate to problems where the evolution of a flume bed is influenced by significant lateral bed load. These data include precise measurement of the applied stresses over the bed surface and the time-dependent bed surface elevation at the two edges of the bed profile. The tests consist of a laterally sloping bed surface, subjected to axially directed shear stress, which evolves toward a 1of12

2 W01419 FRACCAROLLO AND ROSATTI: EXPERIMENTS ON LATERAL BED LOAD W01419 an in-depth assessment of the state of the art theories, with different outcomes in sub- and supercritical conditions. [7] We present first our experimental setup, along with a complete description of the test conditions, which have been already sketched. The presentation of experimental results is the core of this work. In this section kinematic details of the grains moving within the bed load are provided. A short presentation of existing theories to predict the bed load movement over arbitrarily sloping bed follows, along with our shallow-water model to consider the gravitational effects on bed load. Application of our model to test cases allows us to compare computed data with the measurements and to assess the theory of lateral bed load transport. Figure 1. Motion of a particle when (left) no side slope is present and when (right) the bed surface is tilted. steady-uniform state with no lateral slope. The selected laboratory test is quite schematic. This choice allows a simple analysis of the fluid dynamics and of the consequent stresses generated at the bed level. In such a way all the measurements we need are straightforward and consist of the time-dependent bed elevation at the two edges of a flow section. Consequently, modeling analyses are also straightforward (as much as possible, at least); this permits a reliable assessment of theories for the lateral bed load flux and, when computed data agree with the measurements, a good evaluation of the actual lateral bed load. In our tests the bed surface evolves quickly; on the contrary, the physical problems laying at the horizon of this research, and above listed, are not as transient (e.g., bed forms) and are even almost steady (e.g., scours around piers and abutments). Therefore our unsteady tests were not selected to mimic some specific real problems; instead they allow us to study a phenomenon where the lateral bed load flux is the key process and, at the same time, the flow is essentially uniform. [5] A couple of features of our experiments deserves to be highlighted. First, we have considered both super- and subcritical conditions, whereas in other works just the subcritical is examined. Second, steep lateral slopes are tested (about 20% at the beginning of the tests), much more than in the experimental tests of Talmon et al. [1995]. These values of the lateral slope allow for a considerable lateral bed load transport and are consistent with data in real cases of interest in river morphology. As an example, we refer here to the case of scour around abutments, reported by Kwan and Melville [1995]; in that case the grains move over a bed with steep slope values and high rates of spatial changes [e.g., see Kwan and Melville, 1995, Figure 5a or 5b]. [6] The work we are presenting also deals with the assessment of recent theories on bed load transport over an arbitrarily sloping bed. To accomplish this we developed a shallow-water mathematical model able to embed bed load transport theories, and a numerical tool for applying the mathematical model to the test cases. For the specific issue of bed load movement over an arbitrarily sloping bed, among some recent contributions, the theory presented by Kovacs and Parker [1994] has been considered. The comparison between measured and calculated data has permitted 2. Experimental Setup and Test Conditions 2.1. Flume, Sediments, Circulating System, and Instrumentation [8] The facility is a 7 m long, 25 cm wide and 50 cm high flume, with the bare inner walls forming a regular, rectangular transversal section. Parts of the sidewalls are transparent, to allow the application of the imaging technique. The flume has a horizontal bottom and is supported on the floor by a steel frame. The flume is equipped with upstream and downstream tanks. At the downstream tank the water is separated and most of it is returned to the water tank at the upstream end of the flume by means of a submerged pump. All the sediments are trapped in the bottom of the downstream tank, from which a cyclone pump conveys a con- Figure 2. View inside the flume with the sieving net placed over the tilted sandy bed. 2of12

3 W01419 FRACCAROLLO AND ROSATTI: EXPERIMENTS ON LATERAL BED LOAD W01419 Table 1. Physical Parameters of the Supercritical Uniform Flow (Final Condition) Long. Slope Q liq /b [m 2 s 1 ] h [m] Froude Shield Grain-Reynolds Q s /b [m 2 s 1 ] j~t* j=t* c centrated mixture of solids and water to a hopper placed over the upstream reach of the flume. The amount of water in this line is small compared to the total discharge in the flume. The hopper has a rectangular opening that releases sediments into the flume. Regulation of the solid discharge is obtained by tuning the rotation velocity of a cylinder equipped with radial short rubber wings, placed horizontally just beneath the hopper opening. In this way an open system for the return sediments is created. Kinetic energy associated with water discharge at the upstream end of the flume is dissipated by both a weir and a porous steel plate. Immediately downstream, the cyclone pump system delivers the concentrated flow-sediment mixture over a diffuser, which provides the mixture some momentum in the longitudinal flume direction. In this upstream location where water and sediments mix, the flume bed is paved with cobbles (size is about 2 3 cm), in order to avoid local erosive processes. [9] The sediment is quartz sand obtained from industrial grinding; particles are angular and sharp edged with a grainsize range between mm and d 50 = 1.2 mm. [10] All along the flume the mobile bed is shaped with the help of a rigid plane template, kept orthogonal to the flume axis and laid over two straight rods stuck over the inner sidewalls. It is manually moved along the length of the flume. In this way the lateral slope of the bed corresponds to that of the bottom blade of the template, while the longitudinal slope is given by the rod inclination. [11] The water discharge is measured by means of an ultrasound flowmeter placed in the pipeline of the liquid circuit and by bulk samples at the exit of the hopper, in the cyclone pump system. The sum of the two gives the total liquid discharge, which is kept constant. The bed load is measured from bulk samples taken at the exit of the hopper; the bed load is measured only toward the end of the run, when the flow is steady and uniform. A common imaging technique is used to measure, in a selected section at about 4 m from the upstream end, the time-dependent elevation of the two edges of the transversal bed profile. The freesurface position is also detected via the images. [12] The bed load layer, in steady-uniform conditions, is also analyzed by means of a powerful imaging technique, where a grey-scale high-speed digital camera acquires digital footage of the flow through the flume sidewall. At the chosen video rate of 250 frames per second, the digital images have a resolution of pixels. To prevent motion-induced blur, the camera shutter is set at 1/4000 s. The lighting system is such to allow flickering-induced artificial displacements of the order of 0.1 pixels. In a first low-level analysis, particle capture and tracking algorithms are used to extract particle positions and velocities from the digital images. These raw data are then processed using high-level statistical tools to obtain mean and fluctuation velocities. In particular, an error analysis and correction, based on the correlations of the particle displacements along trajectories [Larcher et al., 2007], has been applied. For both the low-level and high-level processing, the techniques used are based on the Voronoï diagram [see Okabe et al., 1992], and were introduced by Capart et al. [2002]. Using this technique we measured the local mean velocity and the kinetic energy the grain velocity fluctuations. Measurements of the local solid concentration were extremely overestimated, as expected because the applied image technique is not stereoscopic [Spinewine et al., 2003]. The concentration data are not presented Test Conditions [13] A movable bed is prepared above the fixed horizontalbottom of the flume. As an initial condition the bed surface is given a constant longitudinal slope and an independent constant transversal slope. Two different cases have been studied, characterized by a different value of the longitudinal slope, although the transversal slope is the same. In the test with streamwise higher slope the flow is supercritical, in the other case (with milder slope) the flow is subcritical. The free surface is, initially and over all the runs, well above the movable bed. On this point the performed runs are different from those ones made by Ikeda [1981], later chosen as case study by Kovacs and Parker [1994]. In our experiments the mobile bed is fully submerged, with some important consequences. Firstly, the asymptotic solution is a priori known and quite simple, consisting of a plane bed surface keeping the initial longitudinal slope and with no lateral slope. Secondly, the erosional front advancing toward the bank in the experiments of Ikeda [1981] is not present. Thirdly, the edges of the transversal bed profile are easily detected from visual inspection. [14] Given the initial conditions for the test, with a bed surface gradient that is constant throughout the flow domain but not aligned with the mainstream longitudinal direction, a transient process of transversal motion of the sediments takes place; the lateral grain migration fades out when the transversal slope vanishes everywhere. The bed evolution has a stronger rate of change in the beginning and becomes weaker and weaker as time passes, such that the final equilibrium configuration is reached asymptotically. [15] Although initial and final configurations of the bed surface are simple, its evolution is not simple to be described. Table 2. Physical Parameters of the Subcritical Uniform Flow Test (Final Condition) Long. Slope Q liq /b [m 2 s 1 ] h [m] Froude Shield Grain-Reynolds Q s /b [m 2 s 1 ] j~t* j=t* c of12

4 W01419 FRACCAROLLO AND ROSATTI: EXPERIMENTS ON LATERAL BED LOAD W01419 The transversal bed profile is not straight but is a smooth curved line, changing as time passes. The movement of the sediment toward the depressed corner of the bed profile causes an average movement of the water column in the opposite direction. This also triggers, as a secondary effect, mild wave activity of the transversal free surface. Furthermore, when bed forms develop and travel, as in the subcritical test, they strongly affect the transport mode and, as a consequence, the morphological evolution. [16] The uniformity of the flow could be hampered, to some extent, by secondary circulation and by boundary conditions at both ends of the flume. As far as secondary circulations are concerned, we consider their contributions to the bed surface stresses very small and negligible. As far as the physical boundary conditions are concerned, we did not achieve perfect transmissive (nonreflecting) conditions and strove to reduce the boundary biases as much as possible. At the upstream end, we shaped the tilted plane bed almost up to the location where the grains are introduced. At the tail of the flume, a transversal bottom threshold was inserted which rotates around its midpoint; in such a way it became possible to regulate the profile of the bed in the downstream section to mimic transmissive boundary conditions. The movement of the threshold is manually operated following the information gathered during preliminary tests. [17] The situation we wished to simulate requires a sharp physical trigger. At first, a clear water flow in steadyuniform conditions is set up, with cross sections having a straight tilted bed line (Figure 3) and no sediment motion, as if the bed were frozen. The bed is imaged to be switched on, becoming mobile, instantaneously. This initial condition introduces a problem in its practical realization. The way we realized the initial operations made use of a sieving net, bordered by a thin steel frame, which was designed to cover and protect all the bed surface. The sieve has a mesh size which is just smaller the sediment d 50, and is tight on the frame. A secondary array of steel rods is applied to the frame, to support the sieve and to adhere to the bed surface. A few rigid steel handles, made as small as possible to reduce their disturbance to the flow, allow us to grasp and remove the bed shield system: this operation represents the starting time (t = 0 s) of the test. The high porosity of the sieve surface makes it possible to remove it rather quickly and without appreciable disturbances to the bed. Figure 2 shows the described device. Given the described bed preparation and given a certain natural dispersion of data in this kind of experiments, five repetitions of the supercritical test and four of the subcritical one were performed. 3. Experimental Results [18] The results are split into data sets relevant to the supercritical and to the subcritical tests. The two data sets share in common the experimental setup, the type of sediment, the liquid discharge (almost), the initial transversal slope and the position of the monitored flume section. They differ, therefore, mainly in the longitudinal slope. Depending on this change, the liquid velocity, the flow section area, the sediment transport (in both directions) and the timescale of the bed evolution are different. [19] With reference to the final steady-uniform conditions, Tables 1 and 2 report a summary of data, including Figure 3. Sketch of the prepared transversal bed profile. The asymptotic bed profile is identified with the dotted horizontal line. Froude, grain-reynolds and Shield numbers, for the supercritical and subcritical flows, respectively. Data in these tables all refer to the final steady-uniform condition. Q liq is the total liquid discharge, b is the flume width, j~t*j/t* c is the transport stage, that is the ratio between the applied bottom shear stress and the critical stress. The initial lateral slope is tanq = 0.20 for both tests, being q reported in Figure 3. The ratio between lateral and longitudinal slopes evolves from about 5 to 0 in the supercritical case, and from 40 to 0 in the subcritical one. [20] After the steel network is removed, the bed load process adapts to the flow conditions. Accordingly, since the flowing grains are picked up from the bed, a small and fast decrease of the bed elevation takes place; after this adaptation stage the process is considered started and position t = 0 s assigned. [21] The results consist of the time-dependent elevation, on both flume sides, of the margins of the transversal bottom profile in the selected section, located at about 4 m from the upstream end and 2 m from the downstream one. For sake of clarity, we stress that the initial bed profile, as depicted in Figure 3, has its lowest point conventionally on the left (y = 0), and the maximum on the right (y = b). The time-dependent variation of the bed elevation at the two edges of the transversal bed profile is made dimensionless by division with half of the initial elevation difference W (see Figure 3), equal to 5 cm, between the left and the right margins; precisely, these quantities are and ~z b;l ðþ¼ t z bðy ¼ 0; tþ z b ðy ¼ 0; t ¼ 0Þ 1 2 W ~z b;r ðþ¼ t z bðy ¼ b; tþ z b ðy ¼ b; t ¼ 0Þ 1 2 W ~z b;l ranges from 0, at t = 0 s, to 1, at the asymptote. Vice versa, ~z b;r ranges from 0, at t = 0 s, to 1, at the asymptote. [22] For the supercritical case we will provide a detailed analysis of the bed load layer under steady-uniform conditions (section 3.1.2), useful to calibrate the bed load 4of12

5 W01419 FRACCAROLLO AND ROSATTI: EXPERIMENTS ON LATERAL BED LOAD W01419 Figure 4. Evolution of the bed edges in supercritical conditions. formulation later employed in mathematical model. For the subcritical case a supplementary discussion (section 3.2.2) concerns the observed bed forms, which interact heavily with the evolution of the lateral slope Supercritical Conditions Evolution Stage [23] The experimental results are presented in Figure 4, illustrating the behavior of the bed elevation on left (~z b;l ) and right (~z b;r ) edges of the bed profile in the measuring section. Results from five tests are graphed. In the supercritical case the run is quite fast. The phenomenon, although its theoretical behavior is asymptotic, can be considered complete after about 65 s. [24] After the protective sieve is removed, the formation of a strong sideward bed load movement, according to the direction of the descent lateral slope, causes aggradation of the bed elevation on the left side and degradation of the bed elevation on the right side. On both left and right sides the bed variation is almost monotone, with a rate which is clearly highest at the beginning of the process. Some effects due to the boundary conditions show up after 15 seconds on the left side. Across the flume, the lateral slope of the bed progressively lowers until the bed is flat. From the continuity balance of the solid phase it is possible to argue that the reduction of the lateral slope takes place sooner in positions close by the sidewalls and later around the center of the flume. This conjecture cannot be directly checked through data, but will be illustrated by the mathematical model. By comparing the rate of variation of the bed elevation at the two margins, the data show that the changes are slightly faster on the left margin (the depressed corner) than on the right one for the first 20 seconds, and vice versa later Detailed Measurements in the Bed Load Layer at the Asymptotic Stage [25] In the supercritical flow regime bed forms are absent. By exploiting the imaging technique described in section 2.1, the following quantities for the solid phase in the bed load can be obtained: the thickness of the bed load layer, the profiles of the mean velocity and of the kinetic energy of the fluctuations. Some of these data will be used in the modeling analysis of the present work (section 5) to assess the bed load formulation. Figure 5 is a snapshot of the bed load layer with the distributions of the time-average (or mean) streamwise granular velocity U s and of U s ± hu 0 si superimposed (where U s + u 0 s is the instantaneous fluctuating velocity, and hi is the r.m.s operator). hu 0 si gives a measure closely related to the concept of granular temperature T s [Chapman and Cowling, 1971]. Given the high number of frames (about 6000) and of particles in each frame, the errors on the mean velocities are null and those on mean velocity fluctuation negligible. The profiles in Figure 5 allow us to evaluate the thickness of the bed load layer compared with the average grain size. Both U s and hu 0 si are null in the bed and their profiles reach the bed with null gradients. The mean velocity is almost linear away from the bed; the granular temperature grows from the bed, till it reaches a maximum, which is in the central part of the Figure 5. Detailed velocity U s and granular temperature hu 0 si profiles in the bed load layer (U s [solid line], U s ± hu 0 si [dashed line]) placed over a photograph of the bed load layer, in the supercritical case, taken from the transparent sidewall. 5of12

6 W01419 FRACCAROLLO AND ROSATTI: EXPERIMENTS ON LATERAL BED LOAD W01419 Table 3. Comparison Between Experimental and Modeled Values Relevant to the Bed Load in the Supercritical Uniform Flow (Final Condition) ~v p [m s 1 ] h s /D j~q s j [m 2 s 1 ] Measured Calculated bed load layer; on the top of the bed load layer the strong increase of the temperature is a measurement artifact due to a reduction of the statistical quality of the data. [26] The attention here given to T s could create confusion, since it is true that existing bed load theories (as the adopted approach of Kovacs and Parker [1994]) do not even consider the granular temperature; in spite of that, there is a significant amount of work [Jenkins and Hanes, 1998; Armanini et al., 2005; Aragon, 1995] that, dealing with sheet flows, have presented robust rheological interpretation based on the kinetic theory. We deem that, at least when travelling bed forms are absent, bed load flows and sheet flows are not different phenomenon in any respect, apart from the thickness of the transport layer compared to the flow depth. Profiles of Figure 5 remarkably match those observed in recent flume experiments devoted to massive sediment-laden flows [Armanini et al., 2005; Larcher et al., 2007]. This shows that there is a clear analogy between the physics of bed load transport and of debris flows. The theoretical advances in debris flows may therefore open new ways for a better understanding of the bed load transport and the measurement of the granular temperature is crucial in this concern. [27] Data useful to calibrate the mathematical model, later presented, are the bed-load-averaged grain velocity j~v p j and the bed load thickness h s for the steady-uniform flow; these values are easily obtained by applying the image analysis technique (section 2.1) and are reported in Table 3. [28] As already explained, measurements of the concentration profile are not available. The concentration ranges from the bed concentration (0.67 [-] in our flume bed) to zero above the top of the bed layer. An average value of about 0.40 [-], as reported in other papers dealing with experiments for sediment-laden flows [Larcher et al., 2007], may been assumed Subcritical Conditions Evolution Stage [29] In Table 2 and at the beginning of section 3 the experimental conditions of the subcritical test are reported. The experimental results from four runs are graphed in Figure 6. [30] The major variations from the previous supercritical case consist of the duration of the phenomenon, now lasting for about 30 minutes, and of the appearing of bed forms which have an increasing effect, as they form, on the behavior of the flow and on the asymptotic conditions of bed profile adjustments. As a result the data are much more scattered, from the beginning of the test, than in the supercritical case. Similar experiments, but with differences in the experimental conditions, are those reported by Talmon et al. [1995]; they prepared the bed with a transverse slope milder than in the present tests and with manually built dunes. We preferred to let dunes form spontaneously, since in our case their formation requires much less time than the duration of the whole phenomenon. [31] The side view of the experiment shows, during the first few tens of seconds, almost no variations. Later, subsequent bed form passages determine the stepwise approach to a final equilibrium. The final conditions do not consist of a transversal flat bed profile, as it is in the supercritical case, but are marked by bed forms across the flume. As a consequence the elevation of the edges of the bed profiles, in the monitored transversal section, is not strictly asymptotic as the time passes, but wavy, jumping over and below the average values ~z b;l = 0 at the left margin and ~z b;r = 1 at the right one Steady-Uniform Asymptotic Stage [32] From the side view observations, either during the evolution of the bed or after the steady conditions are reached, the propagating bed forms appear as shown in Figure 7. From a top view of the bed, taken just after the end of the run, a morphological alternate pattern is apparent (Figure 8), with a wavelength scaled to the flume width. Relevant wave number is about unity and the aspect ratio is about 5; alternate bars, from literature, are reported to have a wave number smaller than 0.5 and an aspect ratio about double of what was found here. Flow and geometric parameters, instead, are in a range where dunes Figure 6. Evolution of the cross-section bed edges in subcritical conditions. 6of12

7 W01419 FRACCAROLLO AND ROSATTI: EXPERIMENTS ON LATERAL BED LOAD W01419 boundaries of the flume, which have been actually minimized in the experiments (section 2.1). By doing so, we designed a problem which is unsteady in time, spatially evolving along the y transversal direction, but uniform in the longitudinal x direction. Similar assumptions were adopted in the modeling analysis developed by Talmon et al. [1995]; in subcritical conditions they were able to obtain relevant model predictions based on a simplified description of the lateral bed load. [35] Because of the uniform conditions, the downstream momentum and mass (liquid and solid) fluxes disappear. With these assumptions, the flow evolution in the cross section is modeled. The mass balance for the bulk mixture ðh þ z @y vh þ xbv p v ¼ 0 ð1þ the mass balance for the solid phase only ðc b z @y bxv p ¼ 0 ð2þ Figure 7. Subsequent snapshots illustrating the downstream propagation of a single bed form and grain trajectories. Bed form propagation celerity is L D /(t 3 t 1 ). are reported to exist. A single bed form passage, as shown in Figure 7, marks a sharp variation of the local bed elevation of about 1 cm. This explains the stepwise behavior of the time series data and the presence of strong oscillations in the asymptotic stage. The bed form profile of Figure 7, taken through the lateral wall, propagated at a speed L D /(t 3 t 1 ) of about ms 1. The bed forms, beyond their making the bed surface more spatially irregular in all directions, participate in the process of grain movements along both the streamwise and the lateral directions. This aspect makes clear that the morphological evolution in the sub- and supercritical tests (where no bed forms formed) is governed by different mechanisms and that specific theoretical approaches need to be devised. This point is important in the theoretical analysis of these experimental results. 4. Modeling 4.1. Mathematical Model [33] Given the quasi-uniformity of the flow, the compact shape of the flow section and the smoothness of the sidewalls versus the erodible bed, no significant three-dimensional flow features develop. Hence the shallow-water model appears a suitable tool for the description of the phenomenon observed in the flume. In this depth-integrated model the hypotheses concern the vertical profiles of the involved physical quantities (velocity, concentration) and the hydrostatic pressure distribution. The model consists of a system of equations expressing mass and momentum balances, endowed with proper phenomenological closures which reduce the number of unknowns and make the system of equations solvable in general terms. The closure relations the model needs concern the shear stress on the bed and the bed load. [34] To model the investigated experiments, we have neglected the influences of the upstream and downstream and the momentum equations, in the streamwise and transversal directions x and huv ¼ ghi sin a t hv 2 þ 1 2 gh2 ¼ t y r Figure 8. View of the bed taken just after the end of the run in the subcritical case. The morphological pattern may be devised. ð3þ ð4þ 7of12

8 W01419 FRACCAROLLO AND ROSATTI: EXPERIMENTS ON LATERAL BED LOAD W01419 Figure 9. where g is the gravity acceleration, c b is the static bed concentration, z b is the local bed elevation, {u, v} are the x and y components of the liquid phase velocity ~u l, averaged over h, laying in the flow domain basal plane. h s is the thickness of the bed load layer, u p and v p are the velocity components of ~v p, the particle velocity, averaged over h s ; x is the areal concentration, defined as x = c s h s, where c s is the averaged solid concentration in h s. Liquid and solid velocities, ~u l and ~v p respectively, are generally different in both magnitude and direction. In equations (1) and (2) the bed load per unit of width in the x and y directions is given by q sx ¼ bu p x; q sy ¼ bv p x: Let us focus on the coefficient b. We have seen, in Figure 5, that the mean velocity profile is nonuniform in the bed load layer; the concentration profile, although not measured, is expected to decrease from the bottom to the top of the bed load layer. This explains the expression for the bed load vector reported in equation (5), where b is a compensation coefficient defined as b ¼ 1 xj~v p j Z hs 0 U p C s dy ¼ 1 xj~v p j q s being U p and C s the time-averaged local values for the streamwise particle velocity and concentration. To derive this coefficient we used the measured value of the bed load transport per unit of width q s and of ~v p ;asfarx(= c s h s ), is concerned, we obtained h s from image analysis (see Figure 5) and assumed c s equal to 0.40, as reported in section The calculation gives b = [36] In the model the four unknown quantities are the flow depth h, the bed elevation z b and the two components of the flow velocity ~u l. All the other quantities are described by supplementary closure relations. In particular the bed load vector ~q s, described by equation (5), is managed as described in the specific section 4.2. Some of the included physical quantities are reported in Figure 9. The closure equations for the shear stress components t x and t y in the momentum equations are obtained by means of the following Manning-type expression ~t r ¼ g K 2 S h1=3 Flow scheme. ð5þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 þ v 2 ~u l ð6þ being K S is the Gauckler-Strickler dimensional parameter characterizing the bed roughness, available from the experiments when the uniform flow reaches the steady state Closure Relation for the Bed Load Arbitrarily Directed Over a Slope [37] In the present paper the data interpretation is performed through a classical physically based model of bed load, which comes from the product of two independent predictors, one for the average particle velocity and one for the areal concentration [Bagnold, 1980; Ashida and Michiue, 1972; Engelund and Fredsoe, 1976; Wiberg and Smith, 1985, 1989; Sekine and Kikkawa, 1992; Niño and Garcia, 1994; Niño et al., 1994]. The areal concentration, in turn, requires an assumption for the value of the fluid stress at the bed. Commonly, it is assumed that the fluid stress at the bed is kept about equal to the critical shear stress by the action of the grains entrained into the flow (Bagnold hypothesis [Bagnold, 1956]). This view has been positively assessed by Fernandez Luque and van Beek [1974] in the case of large transport stage, which is when the applied boundary shear stress is at least three times the conventional critical shear stress value; the same authors were the first to detect the inadequacy of Bagnold assumption in cases of low transport stages. Recent works offering new insights into the validity of Bagnold assumption are by Whiting and Dietrich [1990], McEwan et al. [1999], Niño and Garcia [1994], Niño et al. [1994], Drake et al. [1988], Kirchner et al. [1990], Schmeeckle and Nelson [2003], and McEwan and Heald [2001]. [38] For the specific issue of bed load over an arbitrarily sloping bed, a linear modification of the transport capacity formula, applicable when the bed slope is gentle, is presented by Chiew and Parker [1994] and Christensen [1995]. Sekine and Parker [1992] extended a saltating model to the case of transverse bed slope. Theoretical light was shed by Kovacs and Parker [1994], who proposed a vectorial bed load formulation based on an extended version of the Bagnold s hypothesis. Later, Parker et al. [2003] abandoned the Bagnold s hypothesis in favor of a novel one, extrapolated from the field data set of Fernandez Luque and van Beek [1974]. [39] In particular, the works by Kovacs and Parker [1994], Parker et al. [2003], and Seminara et al. [2002] show how to evaluate the three quantities involved in the expression of the local bed load, which are (i) the magnitude t c of the critical shear stress, (ii) the average particle velocity ~v p in the bed load layer, and (iii) the areal concentration x. [40] With ~v p and x available, the vectorial bed load transport is readily obtained (equation (5)). The relations involved in (i) (iii) contain coefficients which must be assigned. They are: z p D, a dimensionless distance from the bed where the fluid drag on a moving grain is taken to act, being D the representative grain size of the sediment; c D,a drag force coefficient; c L, a lift force coefficient; r L, a constant given by expression (51b) in the work of Seminara et al. [2002]; m, the tangent of the angle of repose; l, a coefficient to take into account that a particle, once dislodged, typically must undergo a number of jumps before it settles on the bed (see expression (61) in the work of Seminara et al. [2002]); f ¼ u b u, the ratio between the liquid 8of12

9 W01419 FRACCAROLLO AND ROSATTI: EXPERIMENTS ON LATERAL BED LOAD W01419 Figure 10. Numerical evolution of the transversal bed profile in the supercritical case. velocity u b, taken at a distance z p from the bed, and the shear velocity u * (u * = t r ); t c0, the critical shear stress on a nearly horizontal slope. [41] Points (i) and (ii) require independent calculations, whereas point (iii) employs results from the former two points. The above mentioned works of Kovacs and Parker [1994] and Parker et al. [2003] are quite similar as far as parts (i) and (ii) are concerned, while they differ in the calculation of x (part (iii)). In their approach, Kovacs and Parker [1994] made use of the new proposition ~t fb ^s ¼ t c, which represents an extension to the arbitrarily sloping bed of the original Bagnold hypothesis (~t fb ¼ t c [Bagnold, 1956]), where ~t fb is the residual shear stress vector acting in the liquid phase at the bottom basis of the bed load layer, t c is the critical shear stress at a given slope, ^s is the unit vector relevant to the local flow-depth-averaged liquid velocity, D ¼ r s r r is the submerged relative density of the grains (having density r s ) in water (density r). For Kovacs and Parker [1994], therefore, at the level of the bed surface the fluid stress drops to the critical value relevant to the given slope and type of sediments, but only in the flow direction ^s. Parker et al. [2003] abandoned the Bagnold hypothesis in favor of one developed by matching entrainment and deposition rates in an equilibrium state over the river data reported by Fernandez Luque and van Beek [1974]. Parker et al. [2003] admitted, anyway, that their approach would require a theoretical revision of the parameters involved. The transport stage j~tj/t c in both the super- and subcritical flow data presented here (see Tables 1 and 2) is well above the critical value (about three) for the application of Bagnold hypothesis, as determined by many authors [Fernandez Luque and van Beek, 1974; McEwan et al., 1999; Whiting and Dietrich, 1990]. Given these considerations, we have accepted to use the Bagnold hypothesis, in the form proposed by Kovacs and Parker [1994], in our model. 5. Comparison Between Measured and Computed Data 5.1. The Supercritical Case [42] The shallow-water model (1) (4) and the attendant closure relations are used to analyze the experimental data. The values for the coefficients presented above are extracted from the paper of Seminara et al. [2002] and are herein listed: z p D =0.5,c D = 0.59, c L =0.5,r L = 0.74, m =0.7, l = t 0.7, f = 6.81, p ffiffiffiffiffiffiffi c0 = 0.036; sand grains have D = gdd Changes in any of them would alter the solution but not the theoretical path. We may exploit the experimental data referring to the steady-uniform flow condition to make some assessment of these coefficients. First, the measured values of the longitudinal slope, of the water discharge and of the flow depth (reported in Table 1), the Gauckler- Strickler coefficient, in equation (6), is calculated to be 50 m 1/3 s 1. This value is consistent with the friction factor f already discussed. Secondly, we have considered the agreement between the following measured and computed quantities: j~v p j, h s (they are obtained by the image analysis technique, section 2.1) and the bed load per unit of width q s. As shown in Table 3, all calculated quantities agree sufficiently well with the corresponding measurements. Deviations may partly be attributed to the fact that the measurements are relevant to the sidewall position, whereas the flow and the grain transport in the bed load layer are probably slightly nonuniform in the transversal direction. [43] Figure 10 shows the calculated transversal bed profile at different times. As argued in the presentation of the test data (section 3.1.1), the central part of the section is the last one to undergo morphological variations. Figure 10 also shows that the elevation of the bed in the middle of the flow section is not steady. In a first period it moves down and, later, it recovers the initial position, as expected. This behavior is due to the greater shear stress applied on the grains over the left side half flume, where the flow depths are higher. [44] Figure 11 compares experimental and modeling results. Some delay in reaching the asymptotic limit is detected, that is the predicted timescale of the process is slightly too high. At t = 60 s, when the physical test is almost complete, the calculated morphological variation is about 80% of the measured one. Playing with the model we have noticed that an increment (just in the model) of about 20% of the flow depth would produce a result well comparing (i.e., with no delay) with the measured data. This means that the lateral bed load is slightly underestimated by the employed expression. [45] It may be concluded that the one-dimensional shallow-water model employed, along with the Bagnold-type evaluation of the bed load proposed by Kovacs and Parker 9of12

10 W01419 FRACCAROLLO AND ROSATTI: EXPERIMENTS ON LATERAL BED LOAD W01419 Figure 11. Numerical (solid line) and average experimental results (dotted line) for the elevation of the cross-section bed edges in the supercritical test. [1994], is able to predict, although not with great accuracy, the morphological experiments, which are quite demanding for the wide range of longitudinal to lateral slope ratios and the strong unsteadiness Subcritical Case [46] Numerical results have been calculated using the same parameters as in the supercritical case. Let us check how these coefficients work in uniform conditions for the subcritical case. The calculated water discharge, using a Gauckler-Strickler coefficient equal to 50 m 1/3 s 1,asinthe supercritical case, matches quite well the measured value reported in Table 2 (Q liq /b = m 2 s 1 ). On the contrary, the predicted bed load is almost 50% higher than the experimental one (Table 4). [47] The evolution stage for the model is illustrated in Figure 12. It shows that the asymptotic steady state is approached too quickly. The predicted evolution lasts about one fifth the time of the observed one. Figure 12 shows that after three hundreds seconds the modeled evolution is complete whereas the experimental data (relevant dots are also reported in Figure 12) are far from the final stage. The overestimate of the bed load is not enough to explain the fast rate with which asymptotic conditions are achieved in the model. To achieve a match with the experimental data, unacceptable values have to be imposed on some of the coefficients (listed in section 5.1). Therefore it has been found that the prediction of the bed load, and of its lateral component in particular, is inaccurate when bed forms are present and these are not properly reproduced by the model. This point was also highlighted by Nelson et al. [1995]. Shallow-water models, because of the hydrostatic pressure assumption, do not capture flow separation and therefore cannot generate dunes. The interaction and participation of the bed forms in the processes of longitudinal and lateral transport requires a different approach, which presents a goal for future theoretical work. 6. Conclusions [48] New experimental data for the problem of the lateral bed load transport of noncohesive grains, deviated by gravitational forces from the streamwise direction, have been presented. There has been a lack of data on this issue and these new data can be used to improve understanding and modeling. A special condition of unsteady-uniform flow with initially high values of lateral bed slope has been tested in a flume. Both supercritical and subcritical conditions have been investigated. The evolution of both edges of the bed profile in a transversal section has been measured. [49] A recent theoretical approach to bed load over arbitrarily sloping bed has been evaluated, and a way to lump it into a morphological shallow-water model has been found and used. [50] The experimental data were then compared with the model results. Since the predicted streamwise bed load is calibrated with the experimental datum, the prediction of the lateral bed load is crucial for the determination of the timescale of the selected evolving tests. The supercritical test presents a good match between experimental and modeled results. In the subcritical case the transport mode is dominated by propagating bed forms, and there is strong interaction between them and the lateral bed load transport process. As a result, the timescale of the phenomena is much longer than predicted, indicating new research issues in the theoretical modeling. In natural geomorphic problems (e.g., in meanders or in local scours, etc) model inaccuracies in evaluating the lateral bed load, such as those detected for the experiments of this work (and in particular for the subcritical case), would have a different consequence, that is an incorrect representation of the general features (size and shape) of the bed surface. Table 4. Comparison Between Experimental and Modeled Values Relevant to the Bed Load in the Subcritical Test Uniform Flow (Final Condition) ~v p [m s 1 ] h s /D j~q s j Measured Calculated of 12

11 W01419 FRACCAROLLO AND ROSATTI: EXPERIMENTS ON LATERAL BED LOAD W01419 Figure 12. Numerical (line) and experimental results (dots) for the elevation of the cross-section bed edges in the subcritical test. [51] The original data (excel files and movies) are available for use by researchers interested to assess their theories for the lateral bed load problems. Require data by to the corresponding author. Notation g gravity acceleration (m/s 2 ) D representative grain size (m) h flow depth (m) b width of the flow section (m) z b local bed elevation (m) x,y Cartesian spatial coordinates; x is the longitudinal flume axis, y is transversal (m) a slope angle of the longitudinal bed profile J angle of the transversal bed profile at the beginning of the tests ~u l ={u, v} liquid velocity vector, averaged over h, and components (m/s) U p time-averaged streamwise granular velocity (m/s) u 0 p instantaneous deviation of the fluctuating streamwise velocity from the local mean value (m/s) ~v p ={u p, v p } particle velocity vector, averaged over h s, and components (m/s) h s thickness of the bed load layer (m) x areal concentration C s local solid concentration c s bed load layer averaged solid concentration c b bed concentration b compensation coefficient (equation (5)) c D particle drag coefficient c L particle lift coefficient c L parameter describing the effect of lift z p /D dimensionless distance from the bed of the grain prone to move ~t ={t x, t x } average shear stress vector and components at the bed level (N/m 2 ) ~q s ={q sx, q sy } volumetric solid discharge per unit of width vector and components (m 2 /s) liquid discharge (m 3 /s) Strickler parameter (m 1/3 /s) r density of water (kg/m 3 ) r s density of grains (kg/m 3 ) D submerged relative density of the grains t c0 critical Shields stress on a nearly horizontal bed (N/m 2 ) t c critical Shields stress on an arbitrarily sloping bed (N/m 2 ) m tangent of the angle of repose f friction factor ~m b representative near-bed liquid velocity (m/s) m * shear velocity (m/s) l coefficient relating the Shields stress necessary for bed load transport to stop to the critical Shields stress for the initiation of bed load transport ~t b fluid shear stress at the upper surface of Q liq K S W the bed load layer (N/m 2 ) bed elevation difference between the left and right margins in a section at t = 0 s ~Z b;l ; ~Z b;r dimensionless variation of the bed elevation at the left and right edges of the transversal bed profile L D, H D a longitudinal length and a vertical height in Figure 7 t 1, t 2, t 3 time values in Figure 7 [52] Acknowledgments. The work was funded by DIATA, a project supported by MIUR, the Italian Ministry of University and Research. Special thanks to our colleague Michele Larcher for his generous help in the assistance and use of imaging techniques during experiments, to Angela Ferrari for her time spent in the laboratory and precious analyses of the raw data, and to Emanuele Sebastiani for having spent some months in the laboratory for preparing tests and for tidily arranging data. References Aragon, J. A. G. (1995), Granular-fluid chute flow Experimental and numerical observations, J. Hydrol. Eng., 121, Armanini, A., H. Capart, L. Fraccarollo, and M. Larcher (2005), Rheological stratification of liquid-granular debris flows down loose slopes, J. Fluid Mech., 532, Ashida, K., and M. Michiue (1972), Study on hydraulic resistance and bedload transport rate in alluvial streams, Trans. Jpn. Soc. Civ. Eng., 206, of 12