Hybrid boundary condition combined with data assimilation for simulations of free surface flows using lattice Boltzmann method Yoshiki Nishi ⇑ , Phan Viet Doan Department of Systems Design for Ocean-Space, Faculty of Engineering, Yokohama National University, 79-5 Tokiwadai, Hodogaya, Yokohama, Kanagawa 2408501, Japan article info Article history: Received 16 May 2012 Received in revised form 3 August 2013 Accepted 29 August 2013 Available online 7 September 2013 Keywords: Lattice Boltzmann method Free surface flows Hybrid boundary condition Data assimilation abstract In this paper, we combine the single-phase lattice Boltzmann method (LBM) including the hybrid bound- ary condition with the assimilation of experimental data into a simulation. This combination improves the accuracy of numerical simulations of violent free surface flows performed using the LBM. An unknown wall effect parameter used in the hybrid boundary condition is determined through iterative calculations to minimize the objective function that measures the discrepancy between the simulation and measurement. The present method is applied to a dam failure problem involving a very large defor- mation of a free surface. We confirm that the method proposed in this study can optimally incorporate the hybrid boundary condition into the LBM without ad hoc specification of the wall effect parameter. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction The lattice Boltzmann Method (LBM) has recently developed into a promising numerical scheme for modeling physics in fluids and simulating fluid flows. The scheme is particularly successful in flow simulations involving interfacial dynamics and complex boundaries. The theory and algorithm of LBM are based on microscopic and mesoscopic kinetic models for fluid particles. The LBM does not re- quire complex techniques for discretizing macroscopic variables, unlike conventional methods based on macroscopic continuum equations which include spatial and temporal differentiations with regard to the macroscopic variables. The kinetic models in LBM are described in terms of the velocity distribution function of fluid particles. The macroscopic variables are calculated as the moments of the velocity distribution func- tions and satisfy the macroscopic equations [1]. There are several LBM models to simulate the interfacial dynamics developed in the past decade (e.g., Xing et al. [2].A LBM model, called as single-phase LBM model, for gas–liquid inter- face simulations was proposed by Korner et al. [3], which was developed for the simulation of metal foams to optimize and en- hance the production process. This LBM model assumes that the gas phase has negligible influence on the liquid phase and thus is not simulated as a fluid. The distribution functions streaming from gas cells into interface cells are reconstructed quite simply. Only the distribution functions of liquid phase are updated, enabling this LBM model to offer an efficient computation and to request low memory. We use this model to simulate a violent free surface flow. Many previous works on the simulation of free surface flows using LBM have adopted slip or no-slip boundary conditions (BCs) on solid walls, which can be implemented simply in the LBM. However, computations with these BCs often give inaccurate results espe- cially when the free surface involves violent deformations. We present a method for resolving this issue by adopting a hy- brid BC, which is a linear combination of slip and no-slip BCs. To apply the hybrid BC, a wall effect parameter needs to be specified. To date, any methods for logically determining this parameter is yet to be developed; we have inevitably substituted a seemingly appropriate value and manually modified the value after observing results. This trial and error parameterization lowers the practical- ity of the simulations. The assimilation of measured data into a simulation is expected to provide a legitimate method for determining the wall effect parameter. We define the objective function for measuring the discrepancy between a simulation and measurement, and employ an optimization theory to minimize the objective function and consequently obtain the optimized value of the wall effect param- eter. To demonstrate to what extent the method proposed in this study works, the method is applied to a dam failure simulation. This paper is organized as follows: in Section 2, the LBM algorithm, free surface tracking scheme, hybrid boundary condi- tion, and optimization algorithm are described; in Section 3, the performance of the present method are evaluated; conclusions are presented in Section 4. 0045-7930/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compfluid.2013.08.010 ⇑ Corresponding author. Tel.: +81 45 339 4087; fax: +81 45 339 4099. E-mail address: ynishi@ynu.ac.jp (Y. Nishi). Computers & Fluids 88 (2013) 108–114 Contents lists available at ScienceDirect Computers & Fluids journal homepage: www.elsevier.com/locate/compfluid