Coupling of lattice Boltzmann shallow water model with lattice Boltzmann free-surface model

https://doi.org/10.1016/j.jocs.2019.01.006Get rights and content

Highlights

  • We present a novel scheme for the coupling of a lattice Boltzmann free-surface solver to a lattice Boltzmann shallow water solver. This is of interest for the simulation of multiscale systems in which a large part of the domain can be well approximated by the shallow water model, whereas higher accuracy is needed in a smaller part of the domain which is simulated with the free-surface model.

  • We study the accuracy and performance of the scheme. A quantitative validation of this type of coupling for the lattice Boltzmann method is novel, and opens the door to a range of large-scale simulations of canals and other hydrodynamic systems.

  • The presented model is efficient since the overhead of the coupling is kept small. It saves significant computational effort, in that the computational time of a coupled system simulation is essentially the time taken by the free-surface part of the domain.

  • We present test cases in order to evaluate the capacity of the coupling scheme to simulate flows dominated by current transmission as well as for flows dominated by wave transmission.

Abstract

We present a scheme for coupling a 2D lattice Boltzmann free-surface solver with a 1D lattice Boltzmann shallow water solver, allowing to save computational effort and efficiently realize multiscale systems. The accuracy of the coupling is validated with two tests. First, we compare the numerical and analytical solutions in a setup with fixed inflow current and outflow water level in a canal. Secondly, the physics of wave propagation and reflection in a domain is investigated in a coupled simulation, and compared to the solutions obtained in both a pure free-surface and a pure shallow-water simulation. Finally, a performance test is carried out to demonstrate that the overhead of the coupling is negligible. A quantitative validation of this type of coupling for the lattice Boltzmann method is novel, and opens the door to a range of large-scale simulations of canals and other hydrodynamic systems.

Introduction

The lattice Boltzmann (LB) method is now recognized as a successful numerical solver for the incompressible Navier–Stokes equations (NSE). The method is based on the Boltzmann equation (see [1], [2] for more details on the method), and it is therefore sufficiently general to simulate a wide set of physical systems, including multi-phase and free-surface flows. Free-surface and shallow water models are commonly used to simulate, among others, rivers and coastal flows. The work of [3] demonstrates the ability of shallow water models to simulate erosion with high performance, while [4] uses them to study sediment transport. An application of a shallow water model to marine flow in coastal areas can also be found in [5]. Free-surface flows can be computed with good efficiency in the LB method using a Volume-of-Fluid (VOF) approach [6]. Such an approach is for instance used in [7] for the simulation of sediment transport in rivers. This paper highlights the limits of the VOF approach from a computational perspective, as it shows the inability to simulate a river segment significantly longer than 10 km even using a parallel computer with several hundred CPU cores.

A VOF free-surface model was used in [8] for simulating an Oscillating Water Column wave energy conversion device, and a similar approach using the LB method was studied in our previous work [9]. While a free-surface model allows us to fully simulate a 3D flow, an approach based on the shallow water equation saves computational cost, allowing to simulate much larger domains. Shallow water models may also be considered as a solution to overcome spurious energy dissipation (see [9]) occurring over long distances in underresolved LB VOF simulations. As a downside, the shallow water equation is based on more limiting physical assumptions (see Section 2.1 below) than the free-surface approach. A coupling between a LB shallow water model and a LB free-surface model therefore offers a way to obtain the best of both worlds, using a free-surface model to simulate the fluid on small portions of the domain requiring high accuracy and with complex flow patterns, while pure wave or flow propagation and reflection phenomena are simulated on larger portions with a shallow water model.

The purpose of this article is to present a scheme for coupling a LB free-surface model with a LB shallow water model. To illustrate this coupling and assess its quality, we consider here the 2D-1D case. The coupling accuracy is studied here through two different tests, the former focusing on flow rate transmission between coupled lattices, while the latter focuses on wave propagation. The aim of such a coupling is to save computational time thanks to the shallow water model while preserving accuracy allowed by the free-surface model. As a result of the lower dimensionality of the shallow water model, its computational cost (see Section 5) is one order of magnitude lower than the one of the free-surface model; this particular point is crucial since one can expect this computational scale difference to be reflected in the time and space scale of the corresponding simulated domains, giving the opportunity to simulate multiscale systems.

We present the LB, free-surface LB VOF and LB shallow water methods in the second section. In the third section, we describe the scheme used to couple shallow water and free-surface systems. The coupling model is tested in the fourth section, where different benchmarks are used. Performance considerations are given in the last section.

Section snippets

Lattice Boltzmann method

The fundamental principles of the LB method (see [2], [10], [11] for more details) originate from statistical mechanics. The Boltzmann equation describes the statistical kinetics of gas molecules [12]. As a consequence, the quantity solved for in this method is the particle distribution function, and the macroscopic variables such as velocity or pressure are derived quantities, defined as velocity moments of the distribution function. The method is Cartesian mesh based, allowing fast mesh

Coupling scheme

In a coupled simulation, we use free-surface and shallow water lattices to represent different parts of the physical domain. At the connection between the two domains, an overlap region is used, in which the two lattices are superimposed. At both ends of this overlap region, a coupling between the lattices is performed. In this way, the coupling is not performed at the same location on both lattices, in order to avoid amplification of errors and resulting numerical instabilities. Fig. 1 depicts

Benchmark

Our benchmark case is similar to the one of [20]. It imposes an inflow current Q0 and an outflow water height hL on a canal of given length L, as depicted in Fig. 3, and in this way allows us to test the coupling in both ways (from free-surface to shallow water and vice versa). This benchmark tests the response of both height and velocity to perturbations, and is motivated by its similarity with potential real-life applications to irrigation canals [20], [35]. The simulation domain is

Performance

The free-surface model is computationally much more expensive than the shallow water model, mainly because of the representation of a 2D domain, as opposed to 1D in the shallow water case. Fig. 13 illustrates the dependence of the total execution time on the relative length of the free-surface domain, as a fraction of the total domain. Note that our code has not been optimized and simply follows the description of the coupling model given here. Thus, the performance results depicted below

Conclusion

The proposed coupling scheme allows us to significantly reduce the computational time compared to a pure free-surface approach. Its accuracy is found to be satisfying both in the case of flows dominated by current transmission, as well as for flows dominated by wave transmission, as long as the flow conditions at the coupling interface satisfy the assumptions of the shallow water model.

Time interpolation should be considered in the future as a possible improvement of the current model, as well

Yann Thorimbert is a PhD student in computational physics at the University of Geneva. His main area of interest is the lattice Boltzmann method, including particle suspensions and free-surface systems. He also worked on procedural noise for automatic generation of numerical landscapes.

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