Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 1679257, 15 pages

https://doi.org/10.1155/2017/1679257

## Light Particle Tracking Model for Simulating Bed Sediment Transport Load in River Areas

^{1}Department of Agroindustrial Engineering, University of Guanajuato, 38140 Celaya, Salvatierra, GTO, Mexico^{2}Environmental Engineering Department, Universidad de Córdoba, Carrera 6 No. 76-103, 230002 Montería, Colombia^{3}Faculty of Engineering, Autonomous University of San Luis Potosí, 78290 San Luis Potosí, SLP, Mexico^{4}Instituto Politécnico Nacional, Centro de Desarrollo Aeroespacial, Belisario Domínguez 22, 06010 Ciudad de México, Mexico

Correspondence should be addressed to C. Couder-Castañeda

Received 31 October 2016; Revised 16 January 2017; Accepted 21 February 2017; Published 20 March 2017

Academic Editor: Maurizio Brocchini

Copyright © 2017 Israel E. Herrera-Díaz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this work a fast computational particles tracer model is developed based on Particle-In-Cell method to estimate the sediment transport in the access zone of a river port area. To apply the particles tracer method, first it is necessary to calculate the hydrodynamic fields of the study zone to determine the velocity fields in the three directions. The particle transport is governed mainly by the velocity fields and the turbulent dispersion. The mechanisms of dispersion and resuspension of particles are based in stochastic models, which describes the movement through a probability function. The developed code was validated using two well known cases with a discrete transformation obtaining a max relative error around 4.8% in both cases. The simulations were carried out with 350,000 particles allowing us to determine under certain circumstances different hydrodynamic scenarios where the zones are susceptible to present erosion and siltation at the entrance of the port.

#### 1. Introduction

The analysis of sediment transport is extremely important for predicting siltation and contaminant transport in rivers or sea ports. Sediment transport is a crucial process to understand the environmental state of aquatic systems [1], because contaminants in aquatic systems are usually cohesive or hydrophobic. Therefore, it is necessary to estimate accurately the dynamics of the sediment transport to obtain a reliable estimation of the contaminants in the water.

Numerical simulations have shown to be efficient, economic, and practical tools for predicting sediment transport in complex hydrodynamic systems. Nevertheless, to obtain a reliable numerical simulation results, it is necessary to take into account as many parameters as possible, to predict correctly the erosion and deposition. For this work, the number of particles and their size contained in every cell are considered to determine if the concentration, erosion, and deposition are governed by probabilistic functions.

Erosion is the flux of particles from a sediment bed into the overlying water, and deposition is the flux of particles back to the sediment bed. The movement of the water composed by the advection, the turbulent dispersion, and settling are the principal phenomena that affect the sediment transport. For this reason, it is necessary to calculate the velocities field of the water to obtain accurate results.

The main objective of this work is to validate a three-dimensional computational model for the sediment transport, developed specifically to obtain results in a short time period. The model is supported in the Particle-In-Cell method and was implemented in MATLAB; it is very fast, consumes low computer resources, and yields good results. The selected study case is a small port terminal in the Magdalena river in Colombia.

As all the numerical models for the sediment transport, it is necessary to use a hydrodynamic model to estimate the velocities field. The hydrodynamic model used has been proven to be robust and precise, and many research has been carried out with this model and they can be consulted in [2–4].

The numerical computational model developed in this work for the transport of the particles is governed by the Lagrangian approach, where the particles are located following a concentration exponential law or randomly located in the space. The advantage of using Lagrangian models to estimate sediment transport and some temporal changes in the morphology of the bottom lies in the computational speed of using a previously calculated hydrodynamic field for the movement of particles. These allow the approximation of the temporal concentration of sediments contemplating the density of the material and using the PIC method to quantify the sediment transport associated with the displacement of the particles near the bottom.

The Lagrangian approach is widely used in the study of the trajectories of movement of solid particles in fluid environments. This is due to the fact that it is possible to track the movement for each specific particle in more detail, in comparison with determining average concentration for grid cells. However, in practice, the most difficult part to use a Lagrangian based method is the strong dependence on the performance of computational resources, such as the amount of memory required for the particles.

One of the pioneering jobs in the application of the Lagrangian technique regarding particle tracking is introduced by Hunter [5], where the method is described and its superiority with finite-difference and finite-element methods is compared for the prediction of dispersion of contaminant in fluid. The method approximates with greater exactitude the nonlinearity of the advective term in the transport equation. For this reason, it is commonly applied in coastal oceanography [6]. In fact, the Lagrangian method is widely used in the dispersion of cluster of particles, having its origins in atmospheric modelling or thermal engineering, and today is used as a well proven method, which can be reviewed in very recent applications [7, 8].

For particle tracking modelling specifically for the sediment transport research, it is necessary to take into account erosion, transport, settling, deposition, and resuspension. In general, the submodule of tracking for sediments is decoupled of the hydrodynamic model, an overview how to perform particle tracking modelling and setup could be consulted in [9]. To carry out the sediment transport study, it is necessary to first calculate the hydrodynamics and subsequently couple the module of sediments, such as, for example, the module developed for sediments for the Environmental Fluid Dynamics Code provided by EPA [9], and it is common for the modelling of sediment to be done this way, such as DEDPLUME-RW or SEDTRAIL-RW [10].

Basically, to the modelling of particle tracking hydrodynamics, the field used corresponds to a stable field and with convergence in time; then the velocity and the turbulent parameters can be considered constant or periodic. The temporal variation could be bigger than the used for the hydrodynamic module. Thus, the particles transport simulation could be carried out for bigger simulation times than the obtained in the hydrodynamic. Therefore, the simulation of the particle transport could use the hydrodynamic field repeatedly until the desired simulation time period is reached. The velocities that move the particles are located in the sides of a three-dimensional cell that build a mesh. It is possible to approximate the velocities in any point inside the cell using a linear interpolation. In fact, this modelling process is applied for any pollutant discharge governed by the transport equation [11] as thermal plume discharges [4, 12].

The Lagrangian mathematical approximations, based on methods of random movement, are well established tools for the calculus of pollutant discharges into aquatic environments. The discharges are treated as a finite number of particle; these particles move under the influence of the previously established flow field. The amount of contaminant represented by each particle in the model can be decremented through the time to represent processes of disappearance. With this method, it is relatively easy to estimate the concentration that is required at a given volume. It is necessary to just carry out a simple calculation to relate the number of particles contained in a fixed space with the number of particles included in a mesh of known concentration [13].

Basically in a Lagrangian approach consisting in determining the next accurate location for each particle in the computational domain, a particle located at the position is moved to the position at time , where , , and , are expressed as follows:where the drift terms are governed for river flows by the advection-diffusion and the random displacements by some probabilistic functions. For this reason, in this research the advection of a diffusion is tested under controlled cases to determine the precision with different numbers of particles and a probabilistic function is proposed.

Additionally, among the benefits of the Lagrangian models over Eulerian Advection-Diffusion models is the computational efficiency [14]. However, it requires the use of mobile coordinate system that significantly increases the computational resources [15]. In addition, it is necessary to have a good solution in order to have a large number of particles that generally demand large memory resources and introduce an additional calculation time for the determination of the trajectories at every time step. A Lagrangian based particle tracking method, versus a Eulerian method, has the advantage of realistic subgrid scale motion and best approximates the movement of particles [16]. However, there are still currently some studies which are carried out by solving the transport equation using a Eulerian approach [17].

Sensibility and accuracy of the Lagrangian Particle Tracking for sediment transport have been well proven. To exemplify, in this research [18] the research first by resolving the hydrodynamic using the nonconservative form of the shallow water equations is carried out. Nonetheless, for the particle movement, a fourth-order Runge-Kutta is used which is computationally heavy. For this reason, OpenMP is used to parallelize computation over memory shared systems [19]. For this reason, some studies are performed avoiding the resolution for ordinary differential equations to lighten the use of computing resources [20]; this is similar to what is proposed in this paper.

For the particle movement, it is considered the specific weight of each particle, as well as the drop velocity [21]. This analysis is very important to calculate the number of particles in suspension. The Lagrangian model has the capacity to be applied after the hydrodynamic fields were calculated or at the same time that the hydrodynamics is simulated. Nevertheless, it is preferable to apply the particle model once the hydrodynamics is estimated for computational resources saving reasons.

Separating the hydrodynamic calculation of the motion simulation of particles allows developing independent transport simulations. Thus, we can simulate particle movement with different locations and sources, various lengths simulations, different transport parameters, and different physical properties of particles (specific weight and diameter), all this supported by the hydrodynamic flow simulation previously calculated. In comparison with other models this is convenient because the hydrodynamic module and the transport module are separated and the erosion and deposition are calculated with particles instead of estimation formulae as part of the hydrodynamic module [1].

Finally, according to some current review papers regarding the Lagrangian modelling of Saltating Sediment Transport [22, 23], a model for the transport of sediment has to include mainly the motion of saltating grains, diffusion of particles, and calculation of bedload transport rate and to improve the motion of particles representing more natural shapes. And, for rivers, it has to consider the nonuniform character of sand, distribution of particle saltation lengths, and excursion lengths may be more important in determining the morphodynamic behavior of the channel bed than the average particle motion. Therefore, future research should focus on relating particle travel distances of individual particles to their shape and physical parameters.

This paper is organized as follows. Section 1 focuses on the introduction. In Section 2 are presented the materials and methods where the hydrodynamic model is introduced, and the description of the particle motion model proposed for the sediment transport. Section 3 shows validation tests of the code proving the advection and the diffusion with different number of particles and are subsequently described; then simulations are carried out in the study area under two dominant scenarios, dry and rainy seasons, and finally the discussion is presented.

#### 2. Materials and Methods

##### 2.1. Hydrodynamic Model

The governing equations for the velocity field are the shallow waters equations, where the model is simplified with the assumption that the vertical scale is smaller than the horizontal scales with the hypothesis of the hydrostatic pressure and considering the Reynolds approximation [2]:where is the coefficient of effective viscosity obtained adding the turbulent velocity coefficient and the molecular velocity coefficient . The coefficient could be approximated as follows [24]:where the vertical longitude is defined as for and for ; is the von Karman constant with typical value of 0.41, is the distance from the river bottom, is the thickness of the boundary layer, and is a constant with a value of 0.09.

##### 2.2. Sediment Transport Model

The numerical model for the particle transport developed in this research is based on a Lagrangian approach (Particle-In-Cell). For this model, the particles are located or placed following an exponential law of concentrations. In addition, it is possible to place the particles randomly in the domain [25].

For the particles movement, it is considered a three-dimensional stochastic model (see Figure 1). For an accuracy simulation, it is necessary to consider the specific weight of each particle, as well as the fall velocity. The movement of the particles in the next time step is calculated from the previous step as follows:where , , and represent the position of the particle in the time () and are the average velocities located in . is the coefficient of turbulent viscosity, is the differential time, and is the sediment fall velocity. The movement equations of the particles 6 are taken from [26] but are modified adding the random movement (Brownian motion).