A unified operator splitting approach for multi-scale fluid-particle coupling in the lattice Boltzmann method

A unified framework to derive discrete time-marching schemes for coupling of immersed solid and elastic objects to the lattice Boltzmann method is presented. Based on operator splitting for the discrete Boltzmann equation, second-order time-accurate schemes for the immersed boundary method, viscous force coupling and external boundary force are derived. Furthermore, a modified formulation of the external boundary force is introduced that leads to a more accurate no-slip boundary condition. The derivation also reveals that the coupling methods can be cast into a unified form, and that the immersed boundary method can be interpreted as the limit of force coupling for vanishing particle mass. In practice, the ratio between fluid and particle mass determines the strength of the force transfer in the coupling. The integration schemes formally improve the accuracy of first-order algorithms that are commonly employed when coupling immersed objects to a lattice Boltzmann fluid. It is anticipated that they will also lead to superior long-time stability in simulations of complex fluids with multiple scales

Perfectly Matched Layer Absorbing Boundary Conditions for the Discrete Velocity Boltzmann-BGK Equation

Perfectly Matched Layer (PML) absorbing boundary conditions were first proposed by Berenger in 1994 for the Maxwell\u27s equations of electromagnetics. Since Hu first applied the method to Euler\u27s equations in 1996, progress made in the application of PML to Computational Aeroacoustics (CAA) includes linearized Euler equations with non-uniform mean flow, non-linear Euler equations, flows with an arbitrary mean flow direction, and non-linear clavier-Stokes equations. Although Boltzmann-BGK methods have appeared in the literature and have been shown capable of simulating aeroacoustics phenomena, very little has been done to develop absorbing boundary conditions for these methods. The purpose of this work was to extend the PML methodology to the discrete velocity Boltzmann-BGK equation (DVBE) for the case of a horizontal mean flow in two and three dimensions. The proposed extension of the PML has been accomplished in this dissertation. Both split and unsplit PML absorbing boundary conditions are presented in two and three dimensions. A finite difference and a lattice model are considered for the solution of the PML equations. The linear stability of the PML equations is investigated for both models. The small relaxation time needed for the discrete velocity Boltzmann-BC4K model to solve the Euler equations renders the explicit Runge-Kutta schemes impractical. Alternatively, implicit-explicit Runge-Kutta (IMEX) schemes are used in the finite difference model and are implemented explicitly by exploiting the special structure of the Boltzmann-BGK equation. This yields a numerically stable solution by the finite difference schemes. As the lattice model proves to be unstable, a coupled model consisting of a lattice Boltzmann (LB) method for the Ulterior domain and an IMEX finite difference method for the PML domains is proposed and investigated. Numerical examples of acoustic and vorticity waves are included to support the validity of the PML equations. In each example, accurate solutions are obtained, supporting the conclusion that PML is an effective absorbing boundary condition

Sensitivity analysis and determination of free relaxation parameters for the weakly-compressible MRT-LBM schemes

It is well-known that there exist several free relaxation parameters in the MRT-LBM. Although these parameters have been tuned via linear analysis, the sensitivity analysis of these parameters and other related parameters are still not sufficient for detecting the behaviors of the dispersion and dissipation relations of the MRT-LBM. Previous researches have shown that the bulk dissipation in the MRT-LBM induces a significant over-damping of acoustic disturbances. This indicates that MRT-LBM cannot be used to obtain the correct behavior of pressure fluctuations because of the fixed bulk relaxation parameter. In order to cure this problem, an effective algorithm has been proposed for recovering the linearized Navier-Stokes equations from the linearized MRT-LBM. The recovered L-NSE appear as in matrix form with arbitrary order of the truncation errors with respect to ${\delta}t$. Then, in wave-number space, the first/second-order sensitivity analyses of matrix eigenvalues are used to address the sensitivity of the wavenumber magnitudes to the dispersion-dissipation relations. By the first-order sensitivity analysis, the numerical behaviors of the group velocity of the MRT-LBM are first obtained. Afterwards, the distribution sensitivities of the matrix eigenvalues corresponding to the linearized form of the MRT-LBM are investigated in the complex plane. Based on the sensitivity analysis and the recovered L-NSE, we propose some simplified optimization strategies to determine the free relaxation parameters in the MRT-LBM. Meanwhile, the dispersion and dissipation relations of the optimal MRT-LBM are quantitatively compared with the exact dispersion and dissipation relations. At last, some numerical validations on classical acoustic benchmark problems are shown to assess the new optimal MRT-LBM