Highlyresolved numerical Simulations of bedload Transport in a turbulent open Channel Flow
Prediction of turbulenceinduced erosion and nearbed transport of sediment particles in turbulent flow is important for many processes in environmental engineering. Beyond its relation to sediment transport, the results of the present study are relevant as well for numerous industrial applications, particularly in the field of process technology, where solid particles are conveyed by a carrier flow. Traditional methods for the prediction of sediment transport are empirical and based on averaged bulk quantities. The predictive power of these formulae is low, because homogeneity of the sediment is postulated. A detailed understanding of sediment stability and the physical mechanisms involved in sediment transport is still missing due to the lack of highlyresolved data under controlled flow conditions. The presented study employs Direct Numerical Simulations (DNS) of a flow laden with a large number of particles with parameters of the disperse phase chosen similar to laboratory experiments [1, 2]. The highlyresolved simulations provide detailed and physically reliable instantaneous information on bedload transport at medium Reynolds number, covering parameter ranges so far not reached.

Figure 1: Particle position, background Eulerian grid and fluid velocity in the streamwise direction in a plane through the particle centre with 22 grid points per diameter. 
Numerical Method
The study was conducted with the inhouse code PRIME (Phase Resolving sIMulation Environment) using an EulerLagrange approach with individual particles being geometrically resolved. The fluid is described by the unsteady threedimensional NavierStokes equations discretized on a regular Cartesian grid. For each particle, the equations of motion are solved in terms of translational and angular velocity. The coupling between both phases is accomplished by an immersed boundary method [3]. For particle contact, the Adaptive Collision Model (ACM) was used [4]. It accounts for all relevant mechanisms that have to be modelled during the collision process and was validated in great detail for single and multiple simultaneous collisions.
Computational Setup
An openchannel flow is considered with a computational domain 24 H× (H+Hsed) × 12H in streamwise, vertical, and spanwise direction, respectively, with H the channel height and Hsed the sediment height at the top of two layers of Np,fixed = 27000 monodisperse spheres, which are arranged in a hexagonal pattern [5]. Periodic boundary conditions are applied in streamwise and spanwise direction and a freeslip condition at the top. The bulk Reynolds number based on the channel height and the bulk velocity of the fluid is 2941, i.e. slightly higher than the threshold for turbulent flow of an unladen channel. The resulting friction Reynolds number for an unladen flow (Case Fix) is Reτ = 193 and the particle Reynolds number in wall units is D+ = 21. The resolution of the equidistant, Cartesian grid is set to D/∆x = 22.2 thus guaranteeing proper resolution of the viscous effects (Fig. 1). This results in a total amount of 1.4·109 grid cells, which is to the knowledge of the authors the largest grid employed so far for this kind of problem. In the reference run (case Ref) the upper layer of the sediment bed was released on top of the remaining sediment bed held fixed, which gives a total of Np,mobile = 13500 particles. The mobility is slightly above the threshold of initiation of motion [6]. Subsequently, additional simulations with lower mass loading (case FewPart) and lower mobility (case LowSh) were carried out to elucidate the effect of these two key parameters on bedload transport. After initialization, the simulations were run until an equilibrium between erosion and deposition was obtained [5]. For each simulation, statistical data was gathered for more than 240 bulk units.

Figure 2: Instantaneous particle distribution of the reference run (Ref). Isosurfaces of fluid fluctuations blue: u’ /Ub = −0.3, particles in yellow: fixed, white: up  < 1.5 uτ , black: up  ≥ 1.5 uτ 

Figure 3: Same as Fig. 2, but for case FewPart. 

Figure 4: Same as Fig. 2, but for case LowSh. 
Results
The reference run (Fig. 2) produces two dunelike particle clusters with their major axis in spanwise direction. The clusters have an average distance of about 12 H and travel on the surface of a layer of inactive particles. The latter are particles, which are free to move but have come to rest temporarily. The layer of inactive particles shows an increased porosity, because more than 26 % of the mobile particles are moving. This allows a significant flow at this level. On top of the inactive layer, the dunes form a bedload layer with a thickness of more than 2 D. This enhances the turbulent fluctuations in this regions substantially.
Reducing the mass loading results in inactive particle structures oriented in streamwise direction, socalled ridges (case FewPart Fig. 3). These enhance coherent vortex structures in the fluid oriented in streamwise direction. Furthermore, they generate a mean secondary flow in the troughs with component perpendicular to the main flow direction in turn transporting particles towards the ridges. On the other hand, fast travelling particles move in the troughs and occasionally erode the ridges at different locations. As the fixed bed is not completely covered by inactive particles, spatial heterogeneity in the spanwise direction develops that allows a significant flow with a high turbulent intensity in the sediment region. The relative amount of moving particles is reduced in comparison to Ref, as the inactive ridges are selfstabilising through hiding and shading mechanisms induced by the spatial heterogeneity. The ridges extend 12 H on average in the streamwise direction and move very slowly resulting in the separation of the fluid time scales and of the morphological time scales of the particle clusters.
In case LowSh, almost all mobile particles settle onto the fixed particles due to their increased submerged density and form an almost closed plane bed (Fig. 4). Only a small percentage of particles is eroded. The lack of hiding and shading mechanisms leads to higher translational particle velocities. In this dilute regime, the particles only have small impact on the fluid flow. Hence, the turbulent structures resemble the largescale behaviour observed for unladen flows over rough walls with high submergence [7].

Figure 5: Twopoint correlation function of the fluid in streamwise direction at y = 0.5 D. The thin horizontal line represents the value of decorrelated fluid. 

Figure 6: CDF in the streamwise direction of moving particles. The thin horizontal line represent the value of a random distribution. 
The interaction of the disperse and the continuous phase was analyzed by suitable statistical measures, such as the twopoint correlation for the streamwise velocity component of the fluid in the nearwall region Ruu(rx), with rx the correlation length, and the Cartesian Distribution Function (CDF) of particle pairs G(ξx, ξz) [8,9] depicted in Fig. 5. Particle structures were assessed using the probability of a particle to find another particle at a given distance ξx, ξz in x and zdirection respectively. A value of one reflects a random distribution, while values larger than one reveal particle clustering. The CDF was conditioned by the particle velocity with Gm only accounting for pairs of moving particles (Fig. 6) and Gr only pairs of two resting particles. This measure allows to distinguish between the dunelike and small scale clusters on the one hand, and the closed plane bed and the ridges on the other. To compare the twopoint correlation of the fluid to the CDF of particles pairs, the values the interval 0.5 D < ξx < 12 H and ξz = 0 are shown here only. This allows to determine the relevant length scales for the two phases and to correlate them. Note that both functions Ruu(rx) and G(ξx, ξz) are symmetric for negative values of rx and ξx for large datasets.
The characteristic distance of the dunelike structures of case Ref becomes obvious for both the particle and the fluid statistics in Fig. 5 and 6. This suggests a strong interaction between the two phases. The phaseresolving approach of the present numerical scheme forces the fluid to flow around particle clusters and to adopt to their length scales. The same can be observed for the inactive ridges reported in case FewPart. Since the characteristic length of a ridge is up to 12 H, only one ridge fits into the computational domain driving Ruu below zero for rx > 6 H. The regime of case LowSh is very dilute as only a small percentage of the particles are actually moving across the closed bed. The moving particles are randomly distributed in space and time and thus do not have the potential to alter the flow field substantially.
Conclusions
Highlyresolved simulations of three particleladen flows across an idealized sediment bed elucidate the impact of mass loading and mobility on bedload transport. The resulting interaction of the two phases was analyzed by suitable statistical tools to distinguish between the length scales of eroded and inactive clusters. The strong similarity of the statistical measures extracted from the two phases reveal a strong interaction. This suggest practical implications for the structural design of sedimentladen open channels, as the mobile bed ultimately alters the hydraulic resistance of the channel by introducing spatial heterogeneity that can occur in streamwise direction (dunelike clusters) or spanwise direction (ridges). It also illustrates the limits of the pointparticle approach, where particles are treated as mass points without spatial extension and their motion is modelled by empirical correlations.
References
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Phaseresolving simulations of bed load transport. In THESIS2013 symposium, Chatou, France, 2013
• Bernhard Vowinckel
• Tobias Kempe
• Jochen Fröhlich
Institut für Strömungsmechanik, TU Dresden
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