Author(s): Sara Venturi; Jessica Padrone; Silvia Di Francesco

Linked Author(s): Silvia Di Francesco

Keywords: Lattice Boltzmann method; Shallow water; Cascaded and cumulant; Multilayer; Multi relaxation time approach

Abstract: In this work the multi relaxation time (MRT) cascaded (CaLB) and cumulant (CLB) lattice Boltzmann model [4] is extended and improved to simulate multilayer shallow water flows. Cascaded and cumulant LB models are based on the use of collision operators (CO) different from BGK CO. Standard BGK models are based on a linear single relaxation time (SRT) CO, while cascaded and cumulant models use a multiple relaxation times approach (MRT). In cascaded model, the use of central moments as basis allows to overcome the defects in Galilean invariance of the original MRT method allowing to improve stability and accuracy; an alternative approach is based on a cumulant collision operator, which relaxes, in the collision step, quantities that are Galilean invariant by construction. The original LBM c-code is adapted and developed to reproduce the dynamics of a multi-layered liquid, made of immiscible shallow-layers of different density, to obtain stratified horizontal flow velocities at various depths, avoiding the expensive Navier–Stokes equations. The multilayer shallow water equations under the hydrostatic assumption present an alternative solution to the free surface Navier–Stokes system and lead to a precise description of the vertical profile of the horizontal velocity while preserving the robustness and computational efficiency of the shallow water equations [1]. The main advantage of using the LBM is that after selecting the appropriate equilibrium, the LB algorithm is only slightly modified for each layer and retains all the simplicities of the LBM within the computing environment [3]. To validate the model, the attention will be first limited to 2D, two-layered, shallow-water flows on a given topography. The interaction is considered by means of forces terms at the interfaces [2] allowing to transfer momentum between the two layers. Equilibrium central moments/cumulants depend on the thickness of the corresponding layer. Results obtained from the LBM developed in this work are compared with solutions published in literature, referred to an unsteady state, verifying the accordance between the numerical solution and the benchmark test case. [1] Luca Bonaventura et al. “Multilayer shallow water models with locally variable number of layers and semi-implicit time discretization”. In: Journal of Computational Physics 364 (2018), pp. 209–234. [2] Michele La Rocca et al. “Development of a lattice Boltzmann method for two-layered shallow water flow”. In: International Journal for Numerical Methods in Fluids 70.8 (2012), pp. 1048–1072. [3] Kevin R Tubbs and Frank T-C Tsai. “Mrt-lattice Boltzmann model for multilayer shallow water flow”. In: Water 11.8 (2019), p. 1623. [4] Sara Venturi et al. “A new collision operator for lattice Boltzmann shallow water model: a convergence and stability study”. In: Advances in Water Resources 135 (2020), p. 103474.

DOI: https://doi.org/10.3850/IAHR-39WC2521711920221319

Year: 2022