The Lattice Boltzmann Method

Krüger, Timm; Kusumaatmaja, Halim; Kuzmin, Alexandr; Shardt, Orest; Silva, Gonçalo; Viggen, Erlend Magnus

doi:10.1007/978-3-319-44649-3

Cited by 1,110 publications

(1,014 citation statements)

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“…The Boltzmann equation describes the evolution of a collection of particles via their associated probability density functions in space and velocity . Interactions between these particles are accounted by a collision operator, which we take here to have the simple BGK form, such that,

((), \frac{\partial}{\partial t} + 𝛏 \cdot \nabla_{x} + Φ_{B} \cdot \nabla_{ξ}) f = \frac{1}{τ} ((), \overset{f}{true} - f) .

…”

Section: Hydro‐micromechanical Modelmentioning

confidence: 99%

“…A boundary node (blue square) is located in midway between the fluid and solid nodes at x w . Fluid particles advecting along the fluid‐solid links

{bold-italicc}_{d_{f - s}}

are bounced back at x w (figure adapted from Krüger et al) [Colour figure can be viewed at wileyonlinelibrary.com]…”

Section: Hydro‐micromechanical Modelmentioning

confidence: 99%

“…The predicted drag force is seemingly independent of the relaxation time, especially in the intermediate Reynolds number regime. Interested readers should refer to Krüger et al and Rettinger and Rüde for the detailed verification and comparison of the fluid‐solid coupling schemes.…”

Section: Verification Of the Hydro‐micromechanical Modelmentioning

confidence: 99%

“…Depending on the spatial and temporal resolution of hydrodynamic interactions in the pore fluid, various fluid‐solid coupling methods have been proposed for dense or loose granular materials submerged in viscous fluids. While conventional computational fluid dynamics (CFD) methods couple the drag forces on solid particles locally into the Navier‐Stokes equations, direct numerical simulation (DNS) techniques such as the lattice Boltzmann method (LBM) resolve hydrodynamic interactions at the pore scale and maintain the momentum balance at no‐slip fluid‐solid interfaces . Conventional CFD methods are sufficient for modeling dilute suspensions of particles in which local pore‐scale fluid flow is not essential, whereas DNS techniques are better suited for fully/partially saturated dense granular media.…”

Section: Introductionmentioning

confidence: 99%

“…The LBM resolves the pore‐scale fluid flow because of the propagation of elastic waves, while DEM describes the motion and interparticle forces of solid particles. The fluid‐solid coupling is handled with the momentum exchange method (MEM), which computes the hydrodynamic forces on solid particles and updates local flow at the no‐slip fluid‐solid interfaces. In order to propagate waves in the fluid‐solid system, an oscillating pressure boundary condition that emits acoustic waves coming from the fluid is implemented.…”

Section: Introductionmentioning

confidence: 99%

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Hydro‐micromechanical modeling of wave propagation in saturated granular crystals

Cheng

Luding

Rivas

et al. 2019

Num Anal Meth Geomechanics

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Summary Biot theory predicts wave velocities in a saturated granular medium using the pore geometry, viscosity, densities, and elastic moduli of the solid skeleton and pore fluid, neglecting the interaction between constituent particles and local flow, which becomes essential as the wavelength decreases. Here, a hydro‐micromechanical model, for direct numerical simulations of wave propagation in saturated granular media, is implemented by two‐way coupling the lattice Boltzmann method (LBM) and the discrete element method (DEM), which resolve the pore‐scale hydrodynamics and intergranular behavior, respectively. The coupling scheme is benchmarked with the terminal velocity of a single sphere settling in a fluid. In order to mimic a small amplitude pressure wave entering a saturated granular medium, an oscillating pressure boundary on the fluid is implemented and benchmarked with the one‐dimensional wave equation. The effects of input waveforms and frequencies on the dispersion relations in 3D saturated poroelastic media are investigated with granular face‐centered‐cubic crystals. Finally, the pressure and shear wave velocities predicted by the numerical model at various effective confining pressures are found to be in excellent agreement with Biot analytical solutions, including his prediction for slow compressional waves.

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