Downslope Transport (Transverse Sediment Transport)

Transcription

1 Third Chinese-German Joint Symposium on Coastal and Ocean Engineering National Cheng Kung University, Tainan November 8-16, 2006 Downslope Transport (Transverse Sediment Transport) Abstract Jens-Uwe Wiesemann*, Peter Mewis and Ulrich C.E. Zanke Institute of Hydraulic and Water Resources Engineering, Darmstadt University of Technology, Darmstadt The sediment transport in transverse direction to the main flow is a very important factor in morphodynamic-numeric computations. The transport direction differs from the direction of the main flow affected by an inclined bed. Following the gravity force the sediment particles move downwards the bed slope. The quantitative knowledge of this effect is important for morphological simulations with numerical models. The morphological evolution in coastal and in fluvial areas depends on the same physical processes. So the transverse transport process, a phenomenon which has to be investigated further on, is fundamental in all processes with moving sediments. The knowledge of the downslope transport, the gravity driven transverse component of the general, total sediment transport, and the development of a useful mathematical formulation are very important for testing, improving and applying morphodynamic-numeric models. With respect to the relevance of this process there is a need for more detailed information supported by experimental data. The presented work shows some new experimental results for the description of the transverse transport process. Investigations in sediment transport on a transversely sloped bed were conducted for different flow conditions. The studies took place in the laboratory of the Institute of Hydraulic and Water Resources Engineering at Darmstadt University of Technology. Measurements were carried out in a 60m long and 1m wide tilting flume under different hydraulic and sedimentologic conditions to succeed the state of ripples and dunes. The time dependant decay of the prepared cosine-shaped bed was determined by measuring multiple longitudinal bed profiles. Changing sedimentologic parameters led to different downslope transport approaches. 1 Introduction In investigations of hydraulic engineering topics the morphological evolution of the channel bed is often one important component. The river morphology and the flume bed morphology (laboratory) depend on the behaviour of the moving bed material. Therefore it is very important to know exactly how much material is transported and in which paths (direction) the sediments travel.

2 A transversely inclined bed affects the sediment transport: Following the gravity force the sediment particles move downwards the bed slope. A transversely sloped bed influences the beginning of sediment transport and also the general transport process over a wide range of hydraulic and sedimentologic parameters. The direction of bed-load transport over a transverse sloping bed plays an important role in calculations of the morphology of rivers (Struiksma et al. 1985). On a sloping bed the bed-load transport includes a gravity component parallel to the bed surface (van Bendegom 1947). The sediment transport transverse to the main flow, affected by an inclined bed, is a phenomenon which is not sufficiently investigated and described yet. Quantitative knowledge of this effect, often described as transverse slope effect, is important for simulations of scouring and of river and coastal morphology with numerical models. If this transverse transport (transverse bed slope effect) is modelled too strong the calculated bed topographies will be flattened. If modelled too weak an unrealistic steep variation of bed topography will result. Therefore calculated depths of 3D-scour holes, channel systems and hypsometry of estuaries will turn out to be wrong. The transverse transport has become a tuning parameter in the art of conducting numerical computations. This bad habit has to be erased as quickly as possible. Developers and users of mathematical models are provided an alternative by the release of new data (Talmon & Wiesemann 2006). There is still a lack of experimental data which validates an approach describing the sediment transport in downslope direction transverse to the main flow. Therefore detailed investigations are necessary to develop and to modify approaches which are suitable for numerical investigations. In this paper experiments on sediment transport on a transversely sloped bed are presented. The results are compared to those reported in the technical literature. For analysing the downslope transport and the sediment transport direction an arrangement like that used by Talmon, van Mierlo & Struiksma 1995 was applied (see also Wiesemann, Mewis & Zanke 2004). The relevance for the implementation of a downslope transport approach in a morphodynamic numeric model was recently shown e.g. in Mewis 2002 or Rüther & Olsen 2005 among others. In the present study different grain materials were used and the experiments were carried out with various hydraulic and morphodynamic conditions, like bed shear stress and given bed forms. 2 Applications and Relevance The downslope transport is an important factor for many applications. In topical works at the Institute of Hydraulic and Water Resources Engineering (Darmstadt University of Technology) the gravitational component of the sediment transport direction is included in simulations with the morphodynamic numerical models TIMOR and SMOR3D.

3 In morphodynamic numerical simulations of bed forms like alternate bars an accounted transverse transport leads to a wavelength of bed forms which is comparable to real observed ones. The instability of the bed which leads to alternate bars depends on the downslope transport process. In the two-dimensional case the instability occurs on the shortest wavelengths. The shorter the bar is the faster it grows. This was shown in Mewis Additionally the transverse transport is a crucial factor for the evolution of scours in river bends. The interaction between spiral water flow and transverse transport was shown in several publications (e.g. Zimmermann, Kennedy 1978, Odgaard 1988). Spiral water flow and transverse transport are in equilibrium when the scour is fully developed. Exemplified applications according to the accounted downslope transport are shown in Mewis 2002 or Rüther, Olsen In figure 1 the results of a numerical simulation of Mewis 2002 are given. The effect of neglecting and of considering the downslope transport is shown. Figure 1 Morphodynamic instability for the case of neglected downslope transport. Simulated section of the river Elbe (flow direction is from right to left). Left: Initial state from measurements; Centre: Development of diagonal structures in the morphological numerical simulation (neglected downslope transport); Right: Simulation including downslope transport. (Numerical simulation Mewis 2002) In pier scours additionally to the longitudinal slope the transverse slope affects the local transport rates. So the shape and the dimensions of the scour are affected by this downslope process. Recent works on three-dimensional simulations of scours incorporate the transverse transport component. Especially on locally steep slopes the downslope component of the transport direction is a determining element. Numerical Simulations of Scouring show according to Link 2006 some typical scour forms which are in transverse direction a little wider than in nature (see e.g. Weilbeer). The correct shape of the scour could be achieved by consideration of the slope inclination, on one hand by the threshold of motion and on the other hand by the transverse transport process (downslope transport). In figure 2 the numerical simulation of a pier scour is given (Link & Zanke 2006). The left depiction shows the scour hole without a downslope transport approach

4 and the right depiction shows the scour simulation with considered downslope transport. The shape of the scour with the approach of the gravitational effect fits the measured shape from laboratory measurements well. With neglecting the downslope transport the scour shape is a little distorted. Figure 2 Consideration of downslope transport on a numerical simulated scour (Link & Zanke 2006), Left: neglected downslope transport, Right: included downslope transport. 3 Downslope Transport Approaches 3-1 Sediment Transport Direction The sediment transport direction (including transverse and longitudinal component) can be described by focusing on the force balance of a grain. The forces affecting a grain on a transverse bed slope are depicted in figure 3. In addition to the fluid drag force, the gravity force and the inclined bed turn the resulting force in downslope direction. According to van Bendegom 1947, Struiksma et al and others the transport rate in transverse direction on transversely sloped beds is modelled by: tan (α) = sin( δ ) cos( δ ) 1 zr f ( Θ) y 1 zr f ( Θ) x (1) With: α = deviation angle of transport direction, δ = deviation angle of flow direction, f(θ) = downslope transport function

5 F D F L R F G sin Main flow F G Figure 3 Forces acting on grain on a transversely sloped bed: F G = gravity, F D = fluid drag, F L = fluid lift, R = resulting force (transport direction), β = angle of transverse inclination, α = angle of transport direction. Neglecting the spiral water flow (δ=0) and assuming a horizontal bed in stream-wise direction yields the following formulation: q q y x = tan(α) = 1 f z r ( Θ) y (2) With: 1/f(Θ) = transverse bed slope coefficient, q x = bed load transport rate in longitudinal direction per unit width and q y = bed load transport rate in transverse direction per unit width, z r = bed surface level (referred to reference level). The approaches for the transverse transport direction consist of formulations which include the function of the Shields parameter and the critical shear stress. The approaches differ in the exponent n of the Shields arguments and in the quantifying factor a (see eq. 3). The general formulation for the downslope transport function f(θ) has to be determined. For the analysis of the experiments in this investigation the following general expression is used: f ( Θ ) = a Θ n (3) With: τ u Θ = τ = Fr* = = (4) ( ρ W ) g d Sediment ρ m ρ g dm in which τ = shear stress on the bed, ρ Sediment = density of bed material, ρ W = density of water, d m = representative grain diameter, ρ = specific density, u* = shear stress velocity. 2

6 With regard to a comparison of a bed level evolution describing mathematical formulation and measured bed level data from laboratory experiments, the function f(θ) could be found. 3-1 Previously Released Approaches and Formulations Bagnold 1956, 1966 considers in his transport formula (energy-equation) the inclination of the bed. A deviation in the sediment transport direction was not given. Engelund 1974, 1981 investigated the deviation of the transport direction on basis of a force balance on the grain. His work yields a direct proportionality to the bed inclination and to the bed friction factor μ d. In the 1981 released publication the modification of the angle of repose was shown, which changes its value on a moving bed due to the bed shear stresses. Ikeda 1982 presented a formulation to evaluate transverse transport rates on basis of own experiments. This formulation was modified by Parker in 1984 and transformed to a formulation describing the deviation of the transport direction. Struiskma, Olesen, Flokstra und de Vriend 1985 use in their analysis of investigations of river bend experiments a formulation due to van Bendegom The weight function was replaced by the Shields parameter Θ and the factor f s. The factor f s, which is described as a form factor of sand grains, has values from 1 to 2. Sekine and Parker 1992 investigated the saltation of grains. In the presented particle model an influence of the inclined bed on the saltating particle was implemented. The saltating grain is bouncing on the inclined surface and gets a deviating saltation, oriented in downslope direction. This leads to a transverse transport. Talmon, van Mierlo and Struiksma 1995 publicised a formulation similar to Engelunds formulation. Here the downslope transport depends on the square root of the Shields parameter and on the transverse slope. Parker, Seminara and Solari 2003 presented an alternative entrainment formulation on basis of the by Seminara 2002 found failure of the Bagnold hypothesis. The failure of this approach was observed even for nearly horizontal beds. The presented entrainment formulation yields consistent results on finite longitudinal and transverse bed slopes. 3-2 Experimental Works The number of true data-points for the transverse slope effect is small. Only dedicated well executed bed levelling experiments in straight flumes classify as true data points. Under the experimental conditions in a straight flume one can focus on sediment transport processes without complications by 3D flow effects (secondary flow). The force balance of transport in longitudinal and transverse direction specifies the actual sediment transport direction, Talmon & Wiesemann As a kind of an indirect evaluation downslope transport approaches could be

7 formulated from investigations of scouring. Here the works from Zimmermann and Kennedy 1978 and from Olesen 1987 could be mentioned for example. One important disadvantage in such an analysis is the fact that the secondary flow in the bed near region increases strongly. Conducted measurements especially over a ripple bed are therefore not accurate. For the evaluation of the downslope transport the influences of the secondary flow must be calculated with the help of additional simplified formulations. Direct measurements of the downslope transport were presented only in few works. One of the earliest works is by Fredsøe Here the deformation of an initial trapezoidal channel in a laboratory flume was investigated. Such a filling of the channel is observable in dredged channels in nature. Two experiments with 1.1mm sand and with 0.55mm sand were conducted. The results were comparable to the formulation of Engelund Ikeda 1981, 1982 analysed the transverse transport with experimental data of wind tunnel experiments. Three different grain materials were used under changing conditions with varying transverse bed inclinations and bed shear stresses. The transverse transport rate was measured by gathering the downslope moving material with laterally located holes in the side wall of the flume. The total amount of sediments was measured periodically. In the experiments with water in 1981 the erosion process of the side banks of a longitudinal depression was measured. These experiments were conducted by the use of 1.3mm sand. One essential result from these investigations is that the downslope transport is directly proportional to the transverse bed inclination. Sekine and Parker 1992 refer to experimental data of several Japanese publications: Hirano 1973, Hasegawa 1981, Ikeda 1982 (see before) and Yamasaka et al The work from Yamasaka et al. investigated the transverse transport in a wind tunnel. Hirano and Ikeda did not measure the longitudinal transport rate, so another calculated, on other formulations based, longitudinal transport rate is a fundamental parameter. This seems not to be useful for the analysis of the transverse transport. The transverse transport rate is determined by observing the bed deformation. Hasegawa and Yamasaka et al. measured the transverse transport rate in a direct way. In these experiments according to Sekine and Parker no bed forms (ripple or dunes) were present. Nakagawa, Tsujimoto and Murakami 1986 presented an investigation on a bank. The inclined bank evolved out of the water. The secondary flow is the reason for the evolution of this longitudinal structure. This longitudinal deformation of the channel leads to a different stress on the bed. The varying stress distribution was measured too. The measurements were carried out by using a video-camera and sediment traps. The sediment transport rates and the angle of the transport direction referred to the longitudinal direction were measured. The investigated bank slope angles had values of 27 and 34. In their experiment transport direction angles from 20 to 50 were observed. Talmon, van Mierlo and Struiksma 1995 investigated the transport direction in a straight flume section with a transversely inclined bed. Grain sizes of 0.09mm, 0.16mm and 0.78mm were used. Dunes and ripples were present in all of the

8 different runs. The transverse slope angle was about 3 at the beginning of the experiments. The cross section was preformed in a sine-function. This form was stable through the run and the transverse transport rate could be determined by striking the balance inside the flume. Francalanci, Solari and Vignoli 2006 presented an experimental study on the gravitational effect of transversal slopes on the bed load transport. The bottom topography of a laterally tilted bed was measured at the beginning and during the experiments until the final laterally flat equilibrium configuration was reached. In the experiments disc shaped steel particles with an approximately three times higher density than sand grains were used. The measured data was compared with the results of a three-dimensional model simulation which includes a linear formulation (Ikeda 1982) and a non linear formulation (Parker et al. 2003) for the downslope transport. In this study the simulation including the linear formulation underestimates the measured data. Based on the experimental setup presented by Talmon, van Mierlo and Struiksma 1995 extended experiments in a straight flume were carried out in the hydraulic laboratory of the Darmstadt University of Technology, which is described in the following chapter. 3-3 Laboratory Investigations 1 The experiments of this work were conducted in a 60m long tilting flume. The investigations include experiments with prevailing bed-load transport and experiments with a noticeable fraction of suspended load. The laboratory tilting flume is 1m wide and 0.5m deep. In figure 4 the sediment characteristics and the prevailing conditions and parameters are given. screen underflow [%] series R0 series R1 series R grain size [mm] Figure 4 Sediment characteristics (left), Prevailing conditions and parameters (right). To quantify the transported material it is inevitable to measure transport rates under various conditions to find representative approaches of sediment transport prognoses. In bed levelling experiments the transverse slope parameter, describing the 1 This work was partly funded the by the German Research Foundation (DFG).

9 downslope transport, is determined from measured transverse transport (bed levelling rate) and measured longitudinal bed-load transport. The most accurate method to get the quantity of the transported material is an integrated weight measurement. The weighing of the total transported material which was transported out of the flume with a movable bed arranged over the total length of the flume up to the outflow region yield an accurate total transport rate. If the bed morphology and its changing with time are known, then the different volumes of the bed forms and its propagation gives a good estimate of the total transported volume. By knowledge of the porosity the sediment transport rate can be calculated. In the conducted bed levelling experiments the grain mixture of the mobile bed was preformed previous to every experiment with a cosine in cross-section like it is described in Talmon, van Mierlo and Struiksma 1995 and Wiesemann, Mewis and Zanke The bed s shape could be described by a cosine-function throughout the deformation process. Only the amplitude decreases with time according to a mathematical formulation. This provides an improved accuracy in the measurements of the decaying amplitude, especially if bed forms are present. A sketch of the laboratory flume with varying bed forms is given in fig. 5. q streamwise q normal q streamwise q normal main flow main flow Figure 5 Section of the laboratory flume with a ripple bed (left) and with a dune bed (right). The bed levels at different positions in the cross section, the water levels and the total sediment transport rate were measured. The location of the bed level measurements was chosen according to the requirements of the bed morphology. Five bed profile values yield the cross sectional shape of the flume bed. For data acquisition five Laser-Distance-Sensors (LDS) to detect the bed levels and two Ultrasonic-Distance-Sensors (UDS) to detect the water levels were used. The sediment transport rates were measured by gathering the material at the end of the flume and for the experiments with prevailing dunes by analysing the movement of bed forms. The experimental setup and the used devices enable a high resolved simulation of the movable flume bed. The experiment was controlled by adjusting the tilt of the flume, adjusting the flow discharge and positioning the sliding panel at the outlet of the flume. In figure 6 the positioning of the Laser-Distance-Sensors (LDS) and the measuring carriage are shown. The LDS were positioned at the transversal coordinates 0.96m, 0.79m, 0.5m, 0.21m and 0.06m with placing the origin at the left side of the flume.

10 Figure 6 Measurement platform with laser distance sensors (LDS) and ultrasonic distance sensors (UDS) (left), Three-dimensional ripple bed at the end of a run (centre), submerged LDS in a dip tank measuring a bed profile, moving with average flow velocity in stream wise direction (right)., By locating the measurement section in sufficiently long distance from the boundaries interferences could be regarded as negligible. Every run includes several measurements of longitudinal bed profiles of 30m length, which were measured by moving the devices on a measuring carriage with mean flow velocity to minimise the disturbances to the flow conditions. Each run is short because the main goal was the bed levelling. No feeding of sediment at the inflow was necessary. The determination of experimental series yields bed level data which gives information about the transverse transport. The initial bed slope inside the flume decays with time. This decay was checked by measuring longitudinal bed profiles with laser distance sensors (non-touch method) and compared with the results of a mathematical formulation (eq. 5,6). Based on this comparison the general formulation for downslope transport (eq. 3) could be modified and adapted to the measured data. h( y) = hˆ a T = i e t / T B (1 p) π 2 y cos π B f ( Θ) q S (5) (6) With: T = time scale of the decay, B = width of the flume, h y = water depth, a i = amplitude of initial bed slope (cosine shape), p = porosity of bed layer, ĥ = mean water depth, q s = longitudinal transport rate and f(θ) = function for the transverse transport.

11 3-4 Main Results of the Experimental Investigation The experimental data from the investigation described before could be compared to approaches from the technical literature. A selection of known approaches was used to illustrate the measurements. In figure 7 the measured data of the experimental series R0 and literature data are shown. Plotted is the Shields parameter Θ versus 1/f(Θ). The Shields parameter for this measurement series is in the range from 0.1 to 0.6. A good correlation is observed Downslope Transport Measurements d m =0.25mm R0 fit Struiksma et al Measurements 1/f(θ) [-] 10 0 Talmon et al Engelund 1974 Engelund 1981 Sekine&Parker 1992 bed levelling measurement Talmon et al Struiksma et al Engelund 1974 Engelund 1981 Sekine&Parker θ [-] Figure 7 Downslope transport: Fine sand Two experiments which were conducted with increasing Shields parameters yield different values for the existing downslope transport approach. Dunes were present at these two runs whereas in all other runs a three-dimensional ripple bed evolved. The fairly good correlation for a three-dimensional ripple bed is no longer existing for a condition where dunes evolved from the bed. This leads to the conclusion that the bed forms have an effect on the transverse transport. In the region with smaller shear stresses, where some two dimensional ripple formations were observable, one measurement deviates from the fit as well. This has to be investigated in further experiments to confirm this first impression. An adaption of the general downslope transport approach (eq. 3) in the region of Shields parameters from 0.16 to 0.5 according to the measurement data leads to the following downslope transport approach (Wiesemann, Mewis, Zanke 2004): fine sand, ripples 1 2 ( Θ) =1.0 Θ f (7) In the series R1 with coarse sand (d m =0.96mm) dunes were present at the bed. Figure 8 shows particularly the deviating gradient of the fit, which leads to a differing exponent in the downslope transport approach compared to the first

12 experimental series. The aforementioned adaption to measurements leads for this experimental series with a 0.96mm-sand to the following downslope transport approach (Wiesemann, Mewis, Zanke 2006a): coarse sand, dunes ( Θ) 0.9 f (8) 10 1 Downslope Transport Measurements d m =0.96mm Struiksma et al /f(θ) [-] 10 0 Talmon et al R1 fit Measurements Engelund 1974 Engelund 1981 Sekine&Parker 1992 bed levelling measurement Talmon et al Struiksma et al Engelund 1974 Engelund 1981 Sekine&Parker θ [-] Figure 8 Downslope transport: Coarse sand 10 1 Downslope Transport Measurements d m =3.00mm Talmon et al Struiksma et al Engelund /f(θ) [-] 10 0 Measurements Engelund 1974 Sekine&Parker 1992 bed levelling measurement Talmon et al Struiksma et al Engelund 1974 Engelund 1981 Sekine&Parker θ [-] Figure 9 Downslope transport: Gravel

13 Recent results from experiments with a gravel flume bed (experimental series R2, fig. 9) yield downslope transport values which are in the same order of magnitude as the coarse sand experiments of series R1. Dunes were present at the bed. A reliable statement concerning the gradient of a fit of all measurement points, i.e. the exponent n of the downslope transport approach f(θ), could not be given yet. A more complete investigation with varying hydraulic parameters should be continued. Particularly some more experiments with the same Θ-values should be conducted to get comparable parameters to the experimental series R0 and R1. To succeed the higher dimensionless shear stresses in additional runs the applied measurement technique and the experimental procedure if necessary have to be adapted to the required parameter values. 4 Conclusions The results show that with prevailing of dunes on the bed the ratio of transverse to longitudinal transport q y /q x decreases, so the downslope transport decreases at the same quantity of the longitudinal transport rate. These results yield a clear dependence of the downslope transport on the prevailing bed forms. This was not observed in any other experimental investigation before. In figure 10 the measured downslope transport values of the above mentioned experimental series are given. The different grain-mixtures yield varying downslope transport values. Particularly the deviating exponent of the transverse transport function f(θ) is remarkable. Results -Downslope Transport- different bed materials 10 dm=0,25 mm 1/f( ) 1 dm=0,96 mm dm=3,00 mm Θ Figure 10 Comparison of downslope transport measurements of three experimental series with fine sand, coarse sand and gravel. The present study shows good results. These results point at a dependence of downslope transport on prevailing bed forms. An important goal is to get more experimental data which supports this first attempts. Continuing experimental work

14 should give evidence of the previous results. Based on these results a confirmation of validity for other sediments with deviating properties has to be found. For this a variation of the bed material, including a changing density and a changing form factor, is necessary. From experiments with varying density some essential information about the influence of the density could be given. The changing gravity component due to the deviating density of the grains leads to differing transport rates and to differing gravity force components. Resulting from this extension of investigation a more general formulation for the downslope transport should be found. 5 References Bagnold, R.A.. The Flow of Cohesionless Grains in Fluids, Proc. Royal Soc. Philos. Trans., London, Vol. 249, Bagnold, R.A.. An Approach to the Sediment Transport Problem, General Physics Geological Survey, Prof. Paper 422-I, Washington, Bendegom, L. van. Eenige beschouwingen over Riviermorphologie en rivierverbetering, De Ingenieur, Vol.59, No. 4, pp. B1-B11, in Dutch, 1947, English translation: Some Considerations on River Morphology and River Improvement, Nat. Res. Council of Canada, Tech. Transl. 1054, Engelund, F.. Flow and Bed Topography in Channel Bends, Journal of the Hydraulics Division, ASCE, 100(HY11), Proc. Paper 10963, , Engelund, F.. The Motion of Sediment on an Inclined Bed, Prog. Rep. Inst. Hydrodyn. and Hydraulic Engrg., Tech. Univ. Denmark, No. 53, pp , Francalanci, S, L. Solari and G. Vignoli, Gravitational Effects on River Morphodynamics, International Conference of Fluvial Hydraulics - Riverflow 2006, Lisboa, Portugal, Proceedings, Volume 2, pp , Fredsøe, J.. Sedimentation of River Navigation Channels, Journal of the Hydraulic Division, ASCE, vol. 104, No. HY2, Hasegawa, K.. Bank-erosion Discharge Based on a Non-equilibrium Theory, Trans. Japan Soc. Civ. Engrg., 316, 37-52, (in Japanese) Ikeda, S.. Self-formed Straight Channels in Sandy Beds, Journal of Hydraulic Division, ASCE, 107(HY4), , Ikeda, S.. Lateral Bed Load Transport on Side Slopes, Journal of Hydraulic Division, ASCE, 108(11), , Link, O. and U. Zanke. Messung und Modellierung der Kolkbildung an einem schlanken zylindrischen Pfeiler in sandigem Boden, Österreichische Wasser- und Abwasserwirtschaft, in print, 2006.

15 Mewis, P.. Morphodynamisch-numerische Modellierung von Flußkurven, Dissertation, Mitteilungen des Institutes für Wasserbau, Heft 126, TU Darmstadt, Nakagawa, H., Tsujimoto, T. und Murakami S., Non-equilibrium Bed Load Transport along Side Slope of an Alluvial Stream, Third international Symposium on River Sedimentation, The University of Mississippi, March 31 April 4, Olesen, K.W.. Bed Topography in Shallow River Bends, Communications on Hydraulic and Geotechnical Engineering, Faculty of Civil Engineering, Delft University of Technology, Rep. No. 87-1, Parker, G.. Discussion of Lateral Bed Load Transport on Side Slopes, Journal of Hydraulic Engineering, ASCE, 110(2), , Parker, G., G. Seminara, and L. Solari. Bedload at Low Shields Stress on Arbitrarily Sloping Beds, Alternative Entrainment Formulation. Water Resour. Res., 39(7), 1183, Rüther N. and N.R.B. Olsen. Three Dimensional Modeling of Sediment Transport in a Narrow 90 Channel Bend, Journal of Hydraulic Engineering, Vol. 131, No. 10, , Sekine, M. and G. Parker. Bed load Transport on a Transverse Slope, Journal of Hydr. Engrg., ASCE, Vol.108, No.4, , Struiksma, N., K.W. Olesen, C. Flokstra and H.J. de Vriend. Bed Deformation in Curved Alluvial Channels, Journal of Hydraulic Research, 23(1), 57-79,1985. Talmon, van Mierlo and Struiksma. Laboratory Measurements of the Direction of Sediment Transport on Transverse Alluvial-bed Slopes, Journal of Hydraulic Research, Vol. 33, No. 4, Talmon, A.M. and J.-U. Wiesemann. Influence of Grain Size on the Direction of Bed-Load Transport on Transverse Sloping Beds, International Conference on Scour and Erosion - ICSE-3, Netherlands, in print, Weilbeer, H.. Numerische Simulation von Strömung und Kolkung an Wasserbauwerken, Dissertation, Institut für Stömungsmechanik, Universität Hannover, Bericht Nr. 65/2001. Wiesemann J.-U., P. Mewis and U. Zanke. Laboratory Measurements of Sediment Transport on Transverse Sloped Beds, Second International Conference on Scour and Erosion, Singapore, Proceedings, Volume 2, , Wiesemann, J.-U., P. Mewis and U. Zanke. Bed Levelling Investigations on Transverse Sloped Beds, International Conference of Fluvial Hydraulics - Riverflow 2006, Lisboa, Portugal, Proceedings, Volume 2, p , Wiesemann, J.-U., P. Mewis and U. Zanke. Sediment Transport Rate in the Dune

16 Regime in Bed Levelling Experiments, International Conference on Scour and Erosion ICSE3 2006, Netherlands, in print, Zanke, U.. Grundlagen der Sedimentbewegung, Berlin, Heidelberg, New York, Springer-Verlag, Zanke, Ulrich C.E.. Zur Physik von strömungsgetriebenem Sediment (Geschiebetrieb), Mitteilungen des Instituts für Wasserbau und Wasserwirtschaft, Technische Universität Darmstadt, Heft 106, Zanke, U.. Sedimenttransport auf geneigter Sohle (Hangabtrieb), Wasserbauliche Mitteilungen der Technischen Universität Dresden, Heft 23, S , 2002.