Modelling and Simulation in Engineering

Volume 2018, Article ID 4245658, 11 pages

https://doi.org/10.1155/2018/4245658

## Numerical Simulation of Dam Break Flows Using a Radial Basis Function Meshless Method with Artificial Viscosity

^{1}National School of Applied Sciences, First Mohammed University, Al-Hociema, Morocco^{2}LME, Faculty of Sciences, First Mohammed University, Oujda, Morocco

Correspondence should be addressed to Elmiloud Chaabelasri; moc.liamg@irsalebaahc

Received 29 December 2017; Revised 1 March 2018; Accepted 13 March 2018; Published 15 May 2018

Academic Editor: Dimitrios E. Manolakos

Copyright © 2018 Elmiloud Chaabelasri. 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

A simple radial basis function (RBF) meshless method is used to solve the two-dimensional shallow water equations (SWEs) for simulation of dam break flows over irregular, frictional topography involving wetting and drying. At first, we construct the RBF interpolation corresponding to space derivative operators. Next, we obtain numerical schemes to solve the SWEs, by using the gradient of the interpolant to approximate the spatial derivative of the differential equation and a third-order explicit Runge–Kutta scheme to approximate the temporal derivative of the differential equation. For the problems involving shock or discontinuity solutions, we use an artificial viscosity for shock capturing. Then, we apply our scheme for several theoretical two-dimensional numerical experiments involving dam break flows over nonuniform beds and moving wet-dry fronts over irregular bed topography. Promising results are obtained.

#### 1. Introduction

Development of robust meshless methods for the numerical solution of partial differential equations has attracted considerable interest over the past twenty years, for example, [1–3]. There are three types of meshless techniques: meshless techniques based on weak forms such as the element-free Galerkin method [4], meshless techniques based on collocation techniques such as the meshless collocation technique based on radial basis functions (RBFs) [5], and meshless techniques based on the combination of weak forms and collocation technique. In the literature, several meshless weak form techniques have been reported; among others, the smooth particle hydrodynamic method [6] and boundary point interpolation method are worth noticing [7]. The weak forms are used to derive a set of algebraic equations through a numerical integration process using a set of quadrature domain that might be constructed globally or locally in the domain of the problem. In this topic, Liu et al. [8] applied the notion of meshless local Petrov–Galerkin and developed the meshless local radial point interpolation method. This method was studied and used on a class of three-dimensional wave equations later by Shivanian [9]. A combination of the natural neighbour finite element method with the radial point interpolation method, using the multiquadric radial basis function, is developed by Dinis et al. [10] and Belinha et al. [11] to analyse a three-dimensional solid.

Initially, the radial basis function meshless method was developed for data surface fitting, and later, with the work developed by Kansa [5], the radial basis function was used for solving partial differential equations. Fedoseyev et al. [12] and Cheng et al. [13] have shown that meshless radial basis functions (RBFs) are attractive options because of the exponential convergence of certain RBFs. Various RBFs have been successfully applied to obtain accurate, efficient solutions of partial differential equations of engineering interest. This method has been applied to solve inviscid compressible flows [14], natural convection [15], heat conduction [16], three-dimensional incompressible viscous flows [17], and long waves in shallow water [18].

The Saint-Venant equations, also called shallow water equations, have often been preferred for the propagation of the flow in the open channel, and they exhibit a simplified mathematical structure, with an ability to take into account the smoothly varying flow conditions and flow discontinuities such as hydraulic jumps, moving bores, and propagation over dry beds, despite it still poses many theoretical and practical problems [19]. The dam break flow problem is an ideal theoretical example that involves these hydraulic challenges. In this paper, we examine the application of the radial basis function meshless method to the numerical solution of the shallow equations for dam break flow problems involving wetting/drying over complicated, frictional bed topography. In the formulation of RBFs, the radial basis functions represented by the multiquadric functions are employed as basis functions to compute weighting coefficients for space differential operators, over a global set of computational collocation points [20]. The friction term is included in the momentum equations and discretized by a splitting implicit scheme [21]. A third-order Runge–Kutta algorithm is used for time integration [22].

It is known that the system of shallow water equations admits nonsmooth solutions that may contain shocks and rarefaction waves and, in the case of nonsmooth bottom topography, also contain contact discontinuities. To perform properly, a numerical method must be nonlinearly stable because linearly stable methods may develop large spurious oscillations and even blow up. To stabilize the proposed RBF model for slowly varying flows, as well as rapidly varying flows involving shocks or discontinuities such as dam breaks and hydraulic jumps, an artificial viscosity technique is used, following [23]. Furthermore, to avoid numerical instability caused by negative water depth near wet/dry fronts, the local flow variables are modified, imposing zero discharge when the water height became very small. As a consequence, the present numerical scheme ensures preservation of nonnegative water depth, and there is no need to limit the fluxes during simulation.

This paper is organized as follows: Section 2 outlines the shallow water equations. Section 3 presents the general formulation of partial differential problem interpolation using RBFs. Section 4 describes the application of RBFs to generate the discrete form of shallow water equations. Details are given of the numerical methods used to represent the friction term and carry out time integration. Section 4 presents the way to use the artificial viscosity for shock capturing. Section 6 discusses validation and application of the method; several numerical experiments are carried out for previously published benchmark cases in order to confirm the potential of the proposed scheme. Section 7 summarizes the main findings.

#### 2. Two-Dimensional Shallow Water Equations

In conservation form, the two-dimensional nonlinear shallow water equations are given by (1) below [24]. The depth-averaged continuity and momentum equations in the *x*- and *y*-directions arewhere is the total depth from the sea bed to the free surface, and are the depth-averaged velocity components in the Cartesian and directions, is the bed elevation above a fixed horizontal datum, is the acceleration due to gravity, and and are the bed shear stress components, which are defined aswhere is the water density and is the bed friction coefficient, which may be estimated from , where is the Manning coefficient.where is the vector of dependent variables, and are the inviscid flux vectors, and is the vector of source terms. In full, the vectors are

#### 3. The Radial Basis Function Meshless Method

Let be an open domain of . Suppose is a function to be approximated in a set of *N* pairwise distinct nodes . In the RBF meshless scheme, the approximation of at the node can be written as a linear combination of *N* RBFs:where are the function values at node , is a RBF centered at , denotes the Euclidean norm between nodes and , and are the coefficients to be determined.

One of the most commonly used RBFs is the multiquadric (MQ) RBFs [25]. Here, we use the infinitely smooth multiquadric radial basis function defined aswhere is a shape parameter that controls the fit of a smooth surface to the data. In the present work, we used the following selection [26]:where denotes the smallest nodal distance. The expansion coefficients in (5) are obtained by solving the following linear system of algebraic equations of :

The expansion coefficients are then calculated bywhere is the vector of approximate solutions and is an matrix of RBFs given as

Space derivatives of the interpolant (5) may be readily calculated, due to its linearity. Generally, the first- and second-order spatial derivatives at the point can be calculated aswhere = [1, 2] denotes the first- and second-order derivatives. In a compact matrix form, using (9), (11) can be written aswhere

#### 4. Discretized Form of Shallow Water Equations by the RBF Meshless Method

##### 4. 1. Convective Flux and Bottom Topography Term Approx-imations

Let us assume that are known convective fluxes along the axes and at time , where . Using (5), it can be approximated by

In the matrix form, this equation becomes

However, the expansion coefficients are then calculated by

From (14), the partial flux derivative can be written as

Then, the compact matrix form of the partial derivative of the flux vector is

Applying the same procedure described above for the bottom topography function , it is easy to obtain its discrete form. The topography function can be approximated by

Its partial derivative is then expressed as

In the same way, the compact matrix form of the bottom topography term along the axis is given as

##### 4.2. Addition of Friction Effects

To incorporate friction into the present numerical scheme, the friction term is discretized using an operator splitting procedure described by Boushaba et al. [27], which splits the shallow water equations (2) into two equations:where . In the first step of the calculation, the upper ordinary differential equation in (23) is approximated by an implicit method as described in [21]:where the friction term may be expressed using a Taylor series as

Rearranging the above equation leads to the following formula for updating water discharge at the new time step:

In the second step, the value is taken to be the initial condition when solving the second equation in (23).

##### 4.3. Time Integration and Stability Condition

To date, the forward Euler method has been mainly used as the preferred time-stepping scheme for RBF methods. However, the forward Euler method is only the first-order accurate in time and so may introduce excessive numerical dissipation in the computed RBF solutions. To achieve a higher order of accuracy, we use the explicit Runge–Kutta method recommended by [22]. The procedure to advance the solution from the time to the next time is carried out aswhere represents the time level and is the time step. To achieve stability, for this explicit scheme, the time step must meet the following criterion:where is the Courant number, such that 0 < CFL < 1, and denotes the smallest nodal distance between collocation points.

##### 4.4. Boundary Conditions

In this work, transmissive boundary conditions for open inflow/outflow and reflective boundary conditions for solid walls are applied in the simulations. At a transmissive boundary, the flow variables at a collocation point on the boundary are set to the same values as at the interior point normal to the boundary. At a reflective boundary, the value at a collocation point is simply the mirror image of that at the associated boundary point so that the normal velocity component is zero at the boundary. However, the representation of boundary conditions within an RBF method is less trivial.

##### 4.5. Implementation Algorithms

For a given initial condition , the time integration procedure is described in Algorithm 1. The right-hand side (RHS) used in the algorithm represents the convective flux and bottom topography that are calculated in Algorithm 2.