ABSTRACT: For flood management modelling of lowland rivers it is important to understand the interaction between river flow and bed forms, specifically dunes. In dune evolution models commonly equilibrium transport formulae like that of Meyer-Peter and Müller (1948) are applied. However, the lag between flow properties and sediment transport is considered a principal cause of bed instability (Nakagawa & Tsujimoto, 1980). Their pick-up and deposition model is used by Shimizu et al. (2009) to model bed load transport in a dune model. Because the properties of step length under dune conditions is highly uncertain they derived a conceptual model for this important parameter.
Sekine & Kikkawa (1992) have made a numerical model of saltation of particles for flat bed conditions and compared it to experimental data. They show that step length strongly correlates with the ratio of friction velocity to settling velocity. Now, we have undertaken experiments to determine step lengths under dune conditions. For a series of dunes the motion of particles has been captured with a high-speed camera. This has been done along the length of the dune to get an idea of the spatial variation. In this paper the first results of the data analysis are presented. The variation of step length distribution along a dune is small, which is against expectation. We hypothesize that this is due to the fact that the relation Sekine & Kikkawa (1992) have found disregards turbulent fluctuation, which is not relevant for a flat bed but is relevant for a dune.
1 INTRODUCTION
Hydraulic roughness values play an important role in correctly determining water levels (Casas et al., 2006; Vidal et al., 2007; Morvan et al., 2008), which is critical for flood management purposes. In rivers with bed sediments ranging in size from silt to grav-el, river dunes are the dominant bed forms (Kostaschuck 2000; Wilbers and Ten Brinke, 2003; Best, 2005; Jerolmack and Mohrig, 2005; Kleinhans et al., 2007). The hydraulic roughness of the main channel is mainly determined by these dunes, which vary greatly in size and shape during a flood wave. To improve flood modelling the development of the bed, and thereby dunes, needs to be understood bet-ter and modelled in computationally cheap ways.
Recently, models have been developed that either are limited to describing only the most important processes correctly (e.g. Shimizu et al., 2001; Nel-son et al., 2005; Tjerry & Fredsøe, 2005; Giri & Shimizu, 2006; Paarlberg et al., 2007, 2009; Shimizu et al., 2009) or try to capture all hydrodynamic and sediment transport details (Nabi et al., 2010). With increasing complexity models become more valua-ble to study detailed hydrodynamic processes, but become too computationally intensive for flood management purposes.
Therefore efforts are made to identify which pro-cesses are most important, and how these can be im-plemented or parameterized in an efficient way. One of the important processes associated with dunes is the transition between various regimes, i.e. the
tran-sition from flat bed to ripples, ripples to dunes and dunes to upper stage plane bed. There are few mod-els that are able to describe all the transitions from a lower-stage plane bed to an upper-stage plane bed. Especially the transition from dunes to an upper stage plane bed is hard to model. During this and other transitions an important driving factor is that the sediment transport and bed shear stress are out of phase (Kennedy, 1963; Nakagawa & Tsujimoto, 1980; Shimizu et al., 2009). Both bed load and sus-pended load contribute to this transition (depending on the conditions simultaneously or one of the two), but here we focus on bed load only.
In the case of the transition to an upper stage plane bed the influence of the phase lag between transport and shear stress is explained as follows. Bed shear stress along a dune reaches a maximum at the crest and decreases on the leeside. If transport follows the same pattern, the crest cannot erode. However, if transport and shear stress are out of phase the maximum transport occurs somewhat downstream of the crest and the crest does erode. This is because the sediment transport rate behind the crest is larger than before the crest.
The model of Shimizu et al. (2009) can describe this transition. Instead of using a straightforward equilibrium transport formula, this model includes the pick-up and deposition formulation of Nakagawa & Tsujimoto (1980) to describe bed load transport. The pick-up is determined from local bed shear stress. The location where the sediment is deposited again is determined using a conceptually derived
Particle step length variation along river dunes
O.J.M. VAN DUIN,
J.S. RIBBERINK, C.M. DOHMEN-JANSSEN, S.J.M.H. HULSCHER
Water Engineering & Management, University of Twente. Enschede, the Netherlands. Contact: o.j.m.vanduin@ctw.utwente.nl
function. This function uses a particle step length (the distance a particle moves from entrainment to deposition) which decreases exponentially with dis-tance from the pick-up point.
This new approach leads to good results. Howev-er, the properties of step lengths under dune condi-tions are highly uncertain (as opposed to a flat bed situation) and the authors have to assume a concep-tual relation between the Shields parameter and mean step length (Shimizu et al., 2009). To better understand which parameters and processes are im-portant in determining step lengths of particles transported as bed load along a dune surface we have undertaken experiments, of which the first re-sults are presented in this paper.
2 STEP LENGTH
Assuming equilibrium between shear stress and transport, the formula devised by Meyer-Peter and Müller (1948) can be directly applied. As Nakagawa & Tsujimoto (1980) argue, bed instability is princi-pally caused by three separate lags, i.e. 1) the phase lags between bed shear stress and bed form, 2) the phase lag between bed load transport rate and eleva-tion change and 3) the probability distribueleva-tion of sediment particle step length, which causes a lag dis-tance between bed shear stress and bed load transport rate. We focus on the sediment particle step length, which is the distance travelled from en-trainment to deposition as defined by Einstein (1950).
Francis (1973), Fernandez Luque & Van Beek (1976) and Sekine & Kikkawa (1984) have done ex-periments to determine the dependence of particle step length on various parameters under flat bed conditions. The data shows a range of values for the particle step length of approximately 40 to 240 times the particle diameter. Sekine & Kikkawa (1992) have used this data to make a numerical model of saltation of particles which reproduces the experi-mental values well. Their model shows that the mean step length varies between near zero and about 350 times the particle diameter. They also relate the step length to certain hydrodynamic parameters. Most importantly it correlates positively with the ra-tio of fricra-tion velocity to particle settling velocity (which is a measure for the relative importance of suspended load), and negatively with the ratio of critical friction velocity to particle settling velocity.
It is likely that observations of step lengths for dunes will differ from existing observations (for flat beds) as the effects of the non-uniformity of flow (higher turbulence, transport occurs more in pulses), upwards slope, avalanching at the crest and (possi-ble) flow separation after the crest must have influ-ence on the distribution of step length. Along a dune the depth averaged velocity increases from the trough towards the crest, due to the increasing bed
level and the (slight) decrease of water level at the crest. This means that near bed velocities, the veloci-ty gradient at the bed and friction velocities increase as well. It may therefore be expected that this will lead to greater step lengths at the crest than at the trough.
3 EXPERIMENTAL SET-UP
For these experiments we are interested in the steps particles make when they are transported as bed load along a dune. For this purpose we have used a high-speed camera which shoots grayscale images at 200 Hz. The camera was installed in a Perspex tube, so that it could be placed in the water, with the camera parallel to the flume bottom. It was placed above several locations along a dune, cover-ing an area of about 15 cm by 15 cm each time. To ensure sharp images the height of the camera was adjusted (and noted), so that it would always be ap-proximately 30 cm above the area in the frame. At each location the camera was first used to take a pic-ture of a reference object (with known size) and then took pictures for 2 seconds (filling the internal stor-age), and was moved to the next location. The imag-es are 1280 pixels in the mean direction of flow, and 1024 pixels perpendicular to the flow. A schematiza-tion of the measurement set-up can be seen below.
Camera
Figure 1: schematization of the experimental set-up
The experiments were conducted at the Leicht-weiß-Institute (LWI) for Hydraulic Engineering and Water Resources at the TU Braunschweig in Ger-many. A recirculating flume with a length of 30 me-ters was used. It has a width of 2 meme-ters, but the width was reduced to 1 meter to limit three-dimensional behaviour and reduce the amount of sand needed to cover the bottom. At the end of the flume a settling basin and sediment trap was present. All the sediment and a small part of the water dis-charge was transported back to the beginning of the flume to ensure a constant supply of sediment from upstream. The remaining discharge flowed over a weir into a basin, to be pumped back up. The charac-teristic grain sizes of the used sand can be found in Table 1.
Table 1. Grain size distribution of the used sand
Nth percentile Grain size mm 10 0.6
50 0.8
90 1.2
As we are interested in particle behaviour along dunes, conditions were selected that fall in the dune regime. For this initial set of experiments we aimed for the lower dune regime, because we wanted to avoid large amounts of suspended sediment. And we are primarily interested in the step length of sedi-ment that is transported as bed load. Discharges of 0.150 m3/s and 0.165 m3/s were selected, and we aimed for a water depth of 30 cm (making the aver-age flow velocity around 0.50 m/s).
Starting from a flat bed we let the bed develop towards equilibrium. During this stage we monitored bed and water levels and adjusted the slope of the flume and the level of the downstream weir with two goals. The first goal was to reach the desired water depth of 30 cm and the second to get a uniform sec-tion in the streamwise direcsec-tion of the flume from about 13 meters to 23 meters (with 0 meters being the most upstream). When the slope of the water level was parallel to the slope of the bed we saw the section as uniform. When the dune height and lengths were in a (dynamic) equilibrium as well, the camera measurements started. For the two experi-ments, the resulting slopes and the number of dunes ‘measured’ with the camera as described before can be found in Table 2.
Table 2. Details of the experiments
Experiment Discharge
m3/s Slope 10-4 m/m Dunes measured
1 0.150 0.19 10
2 0.165 2.03 8
For now, we will only present results of one dune in the second experiment.
4 RESULTS
By analysing series of successive images the move-ment of individual particles was determined. This was done manually, as attempts to do this automati-cally were until now not successful. For a moving particle the moment and position of entrainment was determined, as was the moment and position of dep-osition. From this we can directly determine particle velocities and step lengths. In the following the first results of this analysis for two locations along a sin-gle dune are shown.
Along this dune we have defined two places of interest: just before the crest and just after the flow reattachment point in the trough. A histogram of the particle step lengths Λ divided by the median grain size D50 at these two locations is shown in Figure 2.
Figure 2: Histogram of step length for the crest and trough. Total number of observed particle steps per location is 30.
As can be seen the results differ, but not strongly. The particles at the trough seem to tend to higher step lengths, which is not what we expected. With the number of particle steps per location (30) the dis-tribution seems to be fairly well represented. While it’s not completely smooth, it is very unlikely that with a larger number of observations the picture will change strongly. In Table 3 the mean value and standard deviation of the previously presented re-sults per location can be found. The mean value at the trough is slightly higher than at the crest. How-ever, this might not be significant, as this difference could be caused by statistical uncertainty. The standard deviations are rather high: between half and three quarters of the mean values.
Table 3. mean value and standard deviation (SD) of step length
Location Mean Λ/D50SD Λ/D50
Crest 18.9 10.7
Trough 20.2 16.6
Sekine & Kikkawa (1992) have derived a relation between the observations of particle step lengths for bed load over a flat bed and certain flow and sedi-ment properties (see paragraph two). Most im-portantly, step length increases with ratio of friction velocity to particle settling velocity. Along a dune, friction velocity increases towards the crest and higher step lengths are therefore expected. In the Sekine & Kikkawa (1992) relation friction velocity is a turbulence-averaged value so (turbulent) fluctua-tions are disregarded. This does not matter for the uniform flow over a flat bed, as the fluctuations are the same everywhere. However, this is not the case
along dunes. With this in mind we propose the fol-lowing hypothetical explanation for the differences between our expectations and our observations.
While the friction velocity at the trough of a dune (after the flow reattachment point) is lower than at the crest, the turbulent fluctuations are much higher here. It seems these two differences between flow at the crest and trough are fairly well balanced, leading to similar average step lengths. It should also be not-ed that the settling of a particle in the more turbulent flow at the trough (with larger recirculating eddies) will also be different than at the crest. This is not re-flected in the settling velocity which only depends on particle properties and viscosity.
Other average results are presented in Table 4. The average step length in streamwise direction (Λx), perpendicular to the streamwise direction (Λy)
and (again) the total (Λ) are given as well as the av-erage particle velocity in streamwise direction (u), perpendicular to the streamwise direction (v) and the total (ū). Results are both given in real dimensions, and divided by the median grain size.
Table 4. mean values of various parameters in dimension and non-dimensional units
Parameter Crest Trough Parameter Crest Trough Λ [cm] 1.51 1.62 Λ/D50 18.90 20.21 Λx [cm] 1.36 1.43 Λx/D50 17.01 17.84 Λy [cm] 0.22 0.17 Λy/D50 2.78 2.09 ū [cm/s] 5.36 5.88 ū/D50 [s-1] 67.03 73.46 u [cm/s] 4.79 5.21 u/D50 [s-1] 59.92 65.10 v [cm/s] 0.96 0.71 v/D50 [s-1] 12.01 8.83
The particle velocities in the streamwise direction are in the order of 10% of the dune-averaged streamwise flow velocity, both at the crest and trough. The velocities and step lengths of the parti-cles are dominated by streamwise movement: the to-tal step lengths and velocities are only somewhat higher than the step lengths and velocities in streamwise direction.
There is still some movement perpendicular to the streamwise direction. The velocities and step lengths in this direction are around a factor 7 smaller than in the streamwise direction. That the particle move-ment over the dune is not solely in the streamwise direction is caused by the local topography of the dune.
Some qualitative properties of the particle move-ment at the crest and trough were also observed. In general, there was much more particle movement at the crest. This is to be expected, because when dunes migrate in equilibrium condition with a fairly con-stant dune shape, the sand transport at the crest must be considerably higher than at the trough.
Further-more it is observed that to a large extent sediment at the trough is transported in bursts. Often, a group of particles would start moving simultaneously. At the crest there was also some movement that seemed to be triggered by bursts, but most of it seems to be set in motion continuously likely due to the higher aver-age bed shear stress. Another observation is that at the crest and trough particles can sometimes be seen to start moving due to collision with another particle and that their paths are also influenced by collisions. 5 CONCLUSIONS
A series of experiments have been undertaken to de-termine the statistical properties of step lengths of particles transported as bed load in the dune regime (as opposed to the flat bed situation). While the analysis of the data is in its early stages, some inter-esting characteristics have been found.
From the results presented here it seems that there is little difference between the step lengths found at the crest and trough of a dune, while there is a con-siderable variation of step length at a single location.
A qualitative difference between particle move-ment at the crest and trough was also observed. There is much more particle movement at the crest, while particle movement in the trough seems to be mostly triggered by turbulent bursts.
Based on the new data we hypothesize that the average step length at the crest is mainly determined by a high fairly constant bed shear stress, while the average step length in the trough is mainly deter-mined by the high turbulent fluctuations. We further hypothesize that these two aspects of local flow are of similar importance, making the average step lengths at both locations roughly the same.
6 FUTURE WORK
This analysis will be repeated for more dunes (the images are already collected) and for more locations along the dune. Also, per location more observations will be gathered. With this the results presented here can be improved further, including their statistics.
Later on the step length models derived for flat bed situations will be verified and the results imple-mented in a dune evolution model. Possibly further experiments under different conditions will be done. 7 ACKNOWLEDGMENTS
This study is carried out as part of the project ‘Bed-FormFlood’, supported by the Technology Founda-tion STW, the applied science division of NWO and the technology programme of the Ministry of Eco-nomic Affairs. During the experiments described in this paper the primary author was helped by Sul-eyman Naqshband my fellow PhD-candidate in this project.
The experiments were carried out at the Leicht-weiß-Institute (LWI) for Hydraulic Engineering and Water Resources at the TU Braunschweig in Ger-many. We are grateful for all the help and support the staff of LWI has given.
REFERENCES
Best, J. (2005). The fluid dynamics of river dunes: A review and some future research directions. Journal of
Geophysi-cal Research, 110.
Casas, A., G. Benito, V.R. Thorndycraft, M. Rico (2006). The topographic data source of digital terrain models as a key element in the accuracy of hydraulic flood modelling. Earth
Surface Processes and Land Forms, 31, pp. 444-456.
Einstein, H.A. (1950). The bed load function for sediment transportation in open channel flows. Technical bulletin,
No. 1026, U.S. Department of Agriculture, Soil
Conserva-tion Service.
Francis, J.R.D. (1973). Experiment on the motion of solitary grains along the bed of a water stream. Proceedings of the
Royal Society of London, A332, pp. 443-471.
Fernandez Luque, R. and R. Van Beek (1976). Erosion and transport of bed sediment. Journal of hydraulic research, 14 (2), pp. 127-144.
Giri, S. & Y. Shimizu (2006). Computation of sand dune mi-gration with free surface flow. Water Resources Research, 42.
Jerolmack, D.J. and D. Mohrig (2005). A unified model for subaqueous bedform dynamics. Water Resources Research, 41.
Kennedy, J.F. (1963). The mechanics of dunes and antidunes in erodible-bed channels. Journal of Fluid Mechanics, 16, pp. 521-544.
Kleinhans, M.G., A.W.E. Wilbers and W.B.M. Ten Brinke (2007). Opposite hysteresis of sand and gravel transport up-stream and downup-stream of a bifurcation during a flood in the River Rhine, the Netherlands. Netherlands Journal of
Geosciences - Geologie en Mijnbouw, 86 (3), pp. 273–285.
Kostaschuk, R. (2000). A field study of turbulence and sedi-ment dynamics over subaqueous dunes with flow separa-tion. Sedimentology, 47, pp. 519–531.
Meyer-Peter, E. and R. Müller (1948). Formulas for bed-load transport. Proceedings of the 2nd IAHR congress, Vol. 2, pp. 39–64.
Morvan, H., D. Knight, N. Wright, X. Tang, A. Crossley (2008). The concept of roughness in fluvial hydraulics and its formulation in 1D, 2D and 3D numerical simulation models. Journal of Hydraulic Research, 46(2), pp. 191-208. Nabi, M., H.J. De Vriend, E. Mosselman, C.J. Sloff, Y.
Shimi-zu (2010) – Simulation of subaqueous dunes using detailed hydrodynamics. River, Coastal and Estuarine
Morphody-namics: RCEM 2009.
Nakagawa, H. & T. Tsujimoto (1980). Sand bed instability due to bed load motion. Journal of the Hydraulics Division, 106(12), pp. 2029-2051.
Paarlberg, A.J., C.M. Dohmen-Janssen, S.J.M.H. Hulscher, P. Termes (2007). A parameterization of flow separation over subaqueous dunes. Water Resource Research, 43.
Paarlberg, A.J., C.M. Dohmen-Janssen, S.J.M.H. Hulscher, and A.P.P. Termes (2009). Modelling river dune evolution us-ing a parameterization of flow separation. Journal of
Geo-physical Research. Pt. F: Earth surface, 114.
Sekine, M., and H. Kikkawa (1984). Transportation mechanism of bed-load in an open channel. Proceedings of the
nese Society of Civil Engineering., 351, pp. 69-75 (in
Japa-nese).
Sekine, M., and H. Kikkawa (1992). Mechanics of Saltating Grains II. Journal of Hydraulic Engineering, ASCE, 118 (4), pp. 536-558.
Shimizu, Y., M.W. Schmeeckle, J.M. Nelson (2001). Direct numerical simulations of turbulence over two-dimensional dunes using CIP methods. Journal of Hydroscience and
Hydraulic Engineering, 19(2), pp. 85-92.
Shimizu, Y., S. Giri, I. Yamaguchi, J. Nelson (2009). Numeri-cal simulation of dune-flat bed transition and stage-discharge relationship with hysteresis effect. Water
Re-sources Research, 45.
Tjerry, S. & J. Fredsøe (2005). Calculation of dune morpholo-gy. Journal of Geophysical Research - Earth Surface, 110. Vidal, J.-P., S. Moisan, J.-B. Faure, D. Dartus (2007). River
model calibration, from guidelines to operational support tools. Environmental Modelling & Software, 22, pp. 1628– 1640.
Wilbers A.W.E., W.B.M. Ten Brinke (2003). The response of subaqueous dunes to floods in sand and gravel bed reaches of the Dutch Rhine. Sedimentology, 50, pp. 1013–1034.
