Mathematical Problems in Engineering

Volume 2017, Article ID 8917360, 5 pages

https://doi.org/10.1155/2017/8917360

## Multispeed Lattice Boltzmann Model with Space-Filling Lattice for Transcritical Shallow Water Flows

^{1}State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China^{2}Scientific Computing Department, STFC Daresbury Laboratory, Warrington WA4 4AD, UK

Correspondence should be addressed to J. M. Zhang; nc.ude.ucs@nimnaijgnahz

Received 21 February 2017; Revised 7 June 2017; Accepted 5 July 2017; Published 24 August 2017

Academic Editor: Ling Qian

Copyright © 2017 Y. Peng 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

Inspired by the recent success of applying multispeed lattice Boltzmann models with a non-space-filling lattice for simulating transcritical shallow water flows, the capabilities of their space-filling counterpart are investigated in this work. Firstly, two lattice models with five integer discrete velocities are derived by using the method of matching hydrodynamics moments and then tested with two typical 1D problems including the dam-break flow over flat bed and the steady flow over bump. In simulations, the derived space-filling multispeed models, together with the stream-collision scheme, demonstrate better capability in simulating flows with finite Froude number. However, the performance is worse than the non-space-filling model solved by finite difference scheme. The stream-collision scheme with second-order accuracy may be the reason since a numerical scheme with second-order accuracy is prone to numerical oscillations at discontinuities, which is worthwhile for further study.

#### 1. Introduction

The shallow water equations (SWEs) have been used to model free surface flows in rivers and coastal areas under the assumption of the hydrostatic pressure [1]. Numerically, the SWEs can be solved by using conventional numerical method such as finite difference methods [2], finite element methods [3], and finite volume methods [4]. Alternatively, the lattice Boltzmann method (LBM) [5–8] can be used to model SWEs at mesoscopic level [1, 9–18]. In particular, one-dimensional shallow water flows have been studied by Frandsen [19], Thang et al. [20], and Chopard et al. [21].

A major limitation of LBM is its inability to model supercritical flows. For this reason, Chopard et al. [21] developed an asymmetric lattice Boltzmann model for one-dimensional flow flows which can simulate flows with Froude numbers larger than 1. La Rocca et al. [22] proposed a multispeed model with a non-space-filling lattice which is solved by finite difference scheme and successfully simulated supercritical flows. Here, by using the word “multispeed,” it means lattices that have more than one nonzero speed in one dimension; see, for example, La Rocca et al. [22] and Brownlee et al. [23]. For instance, the commonly used D1Q3, D2Q9, and D3Q19 lattices are not classified as multispeed lattice. Also, “non-space-filling”/“space-filling” means a lattice model that cannot/can fit into the standard steam-collision scheme.

Inspired by the success of the non-space-filling multispeed model, the capability of their space-filling counterpart will be investigated on simulating flows with finite Froude number. Specifically, two space-filling models will be derived by matching hydrodynamic moments (see, e.g., [1, 22, 24]) and then tested by using two typical shallow water problems.

#### 2. Multispeed Lattice Boltzmann Models for 1D Shallow Water Equation

The one-dimensional SWEs readwhich describe the evolution of water depth and depth-averaged velocity . The force term is used to model various interesting effects such as wind-induced surface stress and the bed gradient.

SWEs can be modelled by using a mesoscopic evolution rulewhich describes the fluid motion using a distribution function . Important factors, such as wind-induced surface stress, are also modelled by a force term at the right hand side; that is, and . The weight factor is denoted by for a discrete velocity , and the sound speed can be calculated by using , where is a reference quantity varying with lattice.

To successfully simulate shallow water flows, the key is to define an appropriate local equilibrium function and the associated lattice. Here, the equilibrium distribution function in Zhou [1] is generalized towhere the coefficients and can be adjusted to satisfy the conservation of mass and momentum, i.e.,By substituting (3) into (4), the following equations can be obtained:The weights in (3) will depend on the choice of lattice. Here two sets of velocity lattices will be adopted, that is, discrete velocities (0, ±1, ±2) derived by Qian and Zhou [25] and (0, ±1, ±3) derived by Chikatamarla and Karlin [26]. They are named as D1Q5A and D1Q5B, respectively, and the relevant parameters are listed in Table 1. For convenience, the standard (0, ±1) (D1Q3) lattice also is listed and it will be used for comparison in the following simulations.