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