Force and torque model sensitivity and coarse graining for bedload-dominated sediment transport
Graphical abstract
Introduction
Sediment particles are set in motion when an erodible particle bed is placed below a turbulent shearing flow. Such flows are ubiquitous in nature and examples include turbidity currents (e.g. [1]), rivers (e.g. [2]), and coastal sea waves (e.g. [3]), to name a few. It is important to estimate the amount of sediment that is transported by such shearing flows (e.g. [4], [5]). Transport can occur as bedload or as suspended load depending on how particles are transported. When transported particles remain at close proximity to the bed and their motion consists primarily of rolling, sliding, and saltation, sediment transport is labeled as bedload (e.g. [6]). Bedload transport occurs when the bed shear stress at the sediment–fluid interface is marginally in excess of the critical shear stress necessary for incipient motion of particles. The bedload transport rate is a statistical quantity and is usually obtained from empirical relations as a function of the non-dimensional shear stress at the bed (e.g. [7], [8]). On the other hand, if the overlying turbulent flow is relatively vigorous such that the bed shear stress is well above the critical shear stress, particles may be entrained into and remain suspended by the flow. This mode of transport is labeled as suspended load (e.g [9]). In the present study, we will restrict our attention to bedload-dominated transport.
Predicting sediment transport at the grain level is a daunting task because of the large variability in the flow velocity around a grain due to turbulent fluctuations in addition to the stochastic arrangement of neighboring particles around that grain within the bed (e.g. [10], [11]). However, upon averaging over a large enough area or over long enough times, the variability in both the flow field and the bed arrangement become less important. Under such conditions, simple empirical expressions for the sediment flux can be obtained as a function of the bed shear stress. In fact, many such expressions have been proposed. Meyer-Peter and Müller [5] were among the first to propose an expression for the sediment flux as a function of the excess Shields stress. Their expression came to be known as the ”Swiss formula”. Wong and Parker [4] (WP) suggested a correction to the Swiss formula, namely , where is the sediment flux per unit width, is the average Shields stress at the sediment bed, and is the critical Shields stress for incipient motion.
The Swiss formula, and consequently the WP expression, were derived to estimate the sediment flux under temporally- and spatially-averaged conditions. Turbulence-resolved simulations of the flow above a sediment bed is becoming increasingly possible with the application of bedload transport models such as the Swiss formula or the WP expression as the bottom boundary conditions (e.g. [12], [13]). Care however must be taken with such an application. Due to turbulence resolution, the near-bed grid can be of the order of sediment size and correspondingly the time step is small enough for temporal resolution of turbulence. It therefore becomes important to average the turbulent shear stress over space (on the order of a hundred particle diameters) and over time in order to apply the bedload transport models as appropriate boundary conditions [12], [13], [14], [15]. In this study, we start with Euler–Lagrange simulations, where the dynamics of every sediment grain within the computational domain is accurately tracked. The simulations results thus yield highly accurate space and time-dependent bedload transport information that has been obtained with sediment-level resolution. This accurate information is then space–time averaged as a post-processing step for different levels of spatial and temporal filter widths. With such a coarse-graining post-processing operation, we plan to compare the average bedload transport obtained from the simulations against the WP expression. We propose to identify an optimal length scale of averaging below which local fluctuations of bedload transport will substantially deviate from that predicted by the model and above which the averaging will match closely the model prediction.
For the sake of clarity, We want to stress that the present simulations do not use the Swiss formula or any other bedload transport correlation. We solve the motion of each and every particle within the bed. The coarse graining operation is only a post-processing step. We also would like to stress the difference between the coarse graining and the filtering process represented by the anisotropic Gaussian filter in (1). The latter is on the order of a few particle diameters, while the former is on the order of 100 particle diameters. The coarse graining filter will not modify the flow nor the sediment transport. The coarse graining operation reveals that only when averaged over lengths of the order of 100 particle diameters the averaged shear stress and the averaged bedload transport are correlated in accordance with the Swiss formula and other such correlations. When averaged over smaller lengths, the departure between average bedload calculated from post-processing the simulations results and that estimated with the Swiss formula increased. Thus the optimal coarse graining filter widths are an order of magnitude larger than the length scale of the feedback filter in (1).
Euler–Lagrange (EL) simulations of particle-laden flows have been used in many physical contexts, from the study of fluidized beds (e.g. [16]) to particle dispersal in sprays (e.g. [17]). Particle-resolved simulations are the best, if they can be afforded, since they resolve all the relevant physics of particle–fluid interaction accurately and only the inter-particle collisional physics is modeled (e.g. [18], [19], [20], [21], [22], [23], [24]). The high computational cost of particle-resolved simulations of particle-laden flows makes EL simulations a useful alternative when modeling flows with large number of particles (). However, adequate selection of closure models for hydrodynamic forces and torques acting on particles is fundamental for accurate modeling of the physics of a specific problem. Dilute flows where volume fraction is small allow for the use of “one way” coupled simulations where particle–particle collisions as well as back-coupling from particles to the surrounding fluid is ignored. On the other hand, flows with high concentration of particles, as in the present work, must use “four-way” coupled simulations that account for the back-coupling from the particles to the fluid and particle–particle collisions. In the present context of bedload transport, the high gradient of volume fraction at the particle bed surface requires models that accurately capture the forces on particles saltating above the surface as well as those rolling and sliding at or below the surface.
Another purpose of this paper is to test the influence of different force models on the accuracy of bedload transport in EL simulations by comparing the resulting mean sediment flux and excess Shields to the Wong and Parker formula [4]. We also test the influence of different force and torque components within the model, specifically the role of such force and torque contributions for accurate representation of sediment flux. These components include the lift force, the tangential collision force, the particle rotation and the associated torque on the particle. Additionally, we examine the effect spatial coarse graining (or spatial averaging) and time averaging on the relation between sediment flux and the excess Shields stress. Existence of a unique relation between the coarse-grained excess Shields stress and sediment flux is important in the implementation of Euler–Euler (EE) simulations where individual sediment grains are not tracked. Furthermore, improved empirical sediment flux correlations such as in [4], [5] can be advanced for use in EE simulations. We quantify the effect of coarse graining and time averaging by computing the standard deviation of the scatter for the sediment flux, the excess Shields stress, and the fluid velocity components at the sediment bed surface.
Section snippets
Numerical model
In the present setup, a large number of monodispersed particles are placed at the bottom of the domain to form a random close-packed bed. The process of generating the bed consists of first placing a layer of hexagonally arranged particles at the bottom of the numerical domain, in which every second particle is lifted by a random vertical distance ranging between one particle radius to one particle diameter. Moreover, these particles, which form the bottommost layer are kept fixed during the
Results and discussion
The simulations consist of a particle bed composed of nearly 1.3 million particles of dimensionless diameter below a unidirectional open-channel turbulent flow. Ten simulations are considered, the details of which are shown in Tables 2, 3, and 4. The Galileo number is defined as . In the present study, the particle and fluid densities are chosen to be and , and consequently, is fixed for all simulations. Furthermore, we use and
Conclusions
We presented results from Euler–Lagrange simulations of a turbulent flow over a monodisperse erodible particle bed at a shear Reynolds number of and over a range of between 1.32 and 5.98. Two drag models were investigated along with the influence of lift force, particle rotation, and tangential collision force for each model. We find the temporally and spatially-averaged particle flux and excess Shields stress to vary little between the two drag models and to compare well with WP
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors gratefully acknowledge support from ExxonMobil Upstream Research Company, USA through grant number EM09296. The simulations discussed in this work where performed with the help of the High Performance Computing Center at the University of Florida.
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