Convergence and Optimal Parameters of the Inertial Relaxed Lattice Boltzmann Method in Fluid and Solid Simulations

Abstract

This paper introduces an accelerated lattice Boltzmann method (LBM) tailored for both fluid and solid simulations, utilizing the inertial relaxed Bhatnagar-Gross-Krook (IR-BGK) operator, referred to as IR-LBM. The Navier-Stokes equations and the elastic lamina deformation equation are derived from the Boltzmann equation using the Chapman-Enskog expansion. It is demonstrated that the Boltzmann equation based the IR-BGK operator exhibits a second-order error related to both the time step and the inertia term. Numerical tests employing the D2Q9 and D3Q15 lattice velocity models are conducted to simulate the streamline distribution of cavity flow across various Reynolds numbers and the deformation of elastic lamina under different load patterns using IR-LBM. Results indicate that IR-LBM achieves acceptable accuracy and superior convergence rates compared to the original LBM. Optimal configurations for the inertia term in fluid and solid simulations are provided. This work potentially offers a new approach for enhancing LBM-based algorithms for fluid-structure interaction in future applications. This paper proposes an algorithm based on the lattice Boltzmann method, referred to as the inertial relaxed lattice Boltzmann method, which incorporates an inertial term into the Boltzmann equation to achieve a faster convergence rate compared to the traditional lattice Boltzmann method with the Bhatnagar-Gross-Krook operator. With a suitable choice of equilibrium distribution function, this method is applicable to both fluid and solid simulations. The results of the numerical experiments demonstrate that, by selecting a proper range for the inertia term, the method accelerates convergence rate in algorithms based on the lattice Boltzmann method involving fluid-structure interaction, enhancing computational efficiency while maintaining the relative error with respect to the benchmark solution within an acceptable range. This model is well suited for fluid-structure coupling calculations in fields such as engineering structures and biomechanics, with potential applications including the response of bridges to wind forces and interactions between cardiovascular structures and blood flow.