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In this paper, the mesoscopic representation of non-uniform forcing is investigated by an inverse design approach, for several MRT (multiple-relaxation-time) lattice Boltzmann models. First, the mesoscopic forcing representation of a standard LBM-MRT model on a square lattice is re-visited. By the multiscale Chapman–Enskog expansion, we derive the most general form of the representation by taking advantage of the MRT formulation. In particular, we show that there are three free components in the mesoscopic representation of forcing. Second, by the same methodology, the forcing scheme of two new rectangular MRT lattice Boltzmann models are derived based on the requirements of the Navier–Stokes equations. These theoretical results are then validated by numerical simulations of a forced Taylor–Green vortex flow, with several different forms of non-uniform forcing to alter the kinetic-energy evolution of the system. The numerical results are in excellent agreement with the corresponding time-dependent analytical solution of flow.

The standard lattice-Boltzmann method (LBM) for fluid flow simulation is based on a square (in 2D) or a cubic (in 3D) lattice grids. Recently, two new lattice Boltzmann schemes have been developed on a 2D rectangular grid using the MRT (multiple-relaxation-time) collision model, either by adding a free parameter in the definition of moments or by extending the equilibrium moments. These models satisfy all isotropy conditions and are fully consistent to the Navier–Stokes equations. Here we developed a lattice Boltzmann model on a 3D cuboid lattice, namely, a lattice grid with different grid lengths in different spatial directions. We designed the moment equations, derived from our MRT–LBM model through the Chapman–Enskog analysis, to be fully consistent with the Navier–Stokes equations. A second-order term is added to the equilibrium moments in order to not only satisfy all isotropy conditions but also to better accommodate different values of shear and bulk viscosities. The form of the second-order term and the coefficients of the extended equilibrium moments are determined through an inverse design process. An additional benefit of the model is that the shear viscosity can be adjusted, independent of the stress-moment relaxation parameter, thus potentially improving the numerical stability of the model. The resulting cuboid MRT–LBM model is then validated through benchmark simulations using the laminar channel flow, the turbulent channel flow, and the 3D time-dependent, energy-cascading Taylor–Green vortex flow. The second-order accuracy of the proposed model is also demonstrated. The aspect ratios for numerical stability appear to be constrained at high flow Reynolds numbers, especially for turbulent flow simulations, an aspect requiring further investigation.

In this paper, starting from the standard lattice BGK (LBGK) equation with an extended equilibrium distribution and applying the multiscale Chapman–Enskog (CE) analysis, we show theoretically that the correct Navier–Stokes (N–S) equations can be reproduced on a non-standard rectangular grid, using only the BGK collision model. The parameters in the extended equilibrium distribution are determined through an inverse design process. This new LBGK model is then validated using two benchmark cases, i.e., the 2D decaying Taylor–Green vortex flow and the lid-driven cavity flow. The accuracy and stability of the new model are discussed. The new model clearly extends the standard LBGK model as it was previously thought to be impossible to construct an LBGK model for the N–S equations on a rectangular grid.

In this work, we revisit implementation issues in the lattice Boltzmann method (LBM) concerning moving rigid solid particles suspended a viscous fluid. Three aspects relevant to the interaction between flow of a viscous fluid and moving solid boundaries are considered. First, the popular interpolated bounce back scheme is examined both theoretically and numerically. It is important to recognize that even though significant efforts had previously been devoted to the performance, especially the accuracy, of different interpolated bounce back schemes for a fixed boundary, there were relatively few studies focusing on moving solid surfaces. In this study, different interpolated bounce back schemes are compared theoretically for a moving boundary. Then, several benchmark cases are presented to show their actual performance in numerical simulations. Second, we examine different implementations of the momentum exchange method to calculate hydrodynamic force and torque acting on a moving surface. The momentum exchange method is well established for fixed solid boundaries, however, for moving solid boundaries there are still open issues such as unphysical force fluctuations and Galilean invariance errors. Recent progress in this direction is discussed, along with our own interpretations and modifications. Several benchmark cases, including a particle-laden turbulent channel flow, are used to demonstrate the effects of different modifications on the accuracy and physical results under different physical configurations. The third aspect is the refilling scheme for constructing the unknown distribution functions for the new fluid nodes that emerge from the previous solid region as a particle moves relative to a fixed lattice grid. We examine and compare the performance of the refilling schemes introduced by Fang etźal. (2002), Lallemand and Luo (2003), and Caiazzo (2008). We demonstrate that improvements can be made to suppress force fluctuations resulting from refilling.

The paper describes application of the mesoscopic lattice Boltzmann (LB) method to the simulations of both single-phase turbulence and particle-laden turbulence which are maintained by a large-scale forcing. The disturbance flows around finite-size solid particles are resolved, providing the opportunity to study the detailed interactions between fluid turbulence and solid particles at the particle-fluid interfaces. Specifically, a nonuniform time-dependent stochastic forcing scheme is implemented within the mesoscopic multiple-relaxation-time LB approach. The statistics of single-phase forced turbulence obtained from the LB approach are found to be in excellent agreement with those from the pseudo-spectral simulations, provided that the grid resolution in the LB simulation is doubled. It is shown that the flow statistics is not sensitive to the velocity scale used for the LB simulation. Preliminary results on forced turbulence laden with non-sedimenting solid particles at a particle-to-fluid density ratio of 5, solid volume fraction of 0.102, and particle diameter to Kolmogorov length ratio of 8.05 are interpreted, using a systematic analysis conducted at three levels: whole-field, phase-partitioned, and profiles as a function of distance from the surface of solid particles. It is found that the particle-laden turbulence is much more dissipative in terms of the non-dimensional dissipation rate, due to both reduction of the effective flow Reynolds number and the viscous boundary layer on the surfaces of solid particles. The thickness of the boundary layer is found to be about 0.4r"p. While this boundary layer region accounts for 19.5% of the space within the fluid, it contributes to 57.5% of total viscous dissipation. The vorticity magnitude exhibits a maximum inside the boundary layer and a minimum outside the boundary layer, showing detachment of the vorticity structure from the solid surface. The sharp gradients near the particle surface contribute dominantly to the value of velocity derivative flatness, making the flatness in particle-laden flow much larger than that of single-phase turbulence. In the spectral space, presence of solid particles attenuates energy at large scales including the forcing shells and augments energy at the small scales. The pivot wavenumber is found to be very similar to the value previously found in decaying particle-laden turbulence under the similar parameter setting.

The paper describes a particle-resolved simulation method for turbulent flow laden with finite size particles. The method is based on the multiple-relaxation-time lattice Boltzmann equation. The no-slip boundary condition on the moving particle boundaries is handled by a second-order interpolated bounce-back scheme. The populations at a newly converted fluid lattice node are constructed by the equilibrium distribution with non-equilibrium corrections. MPI implementation details are described and the resulting code is found to be computationally efficient with a good scalability. The method is first validated using unsteady sedimentation of a single particle and sedimentation of a random suspension. It is then applied to a decaying isotropic turbulence laden with particles of Kolmogorov to Taylor microscale sizes. At a given particle volume fraction, the dynamics of the particle-laden flow is found to depend mainly on the effective particle surface area and particle Stokes number. The presence of finite-size inertial particles enhances dissipation at small scales while reducing kinetic energy at large scales. This is in accordance with related studies. The normalized pivot wavenumber is found to not only depend on the particle size, but also on the ratio of particle size to flow scales and particle-to-fluid density ratio.

Transport of sub-micron colloid particles in soil porous media has been mostly studied numerically with unit-cell-based grain-scale geometries. In this study, we develop a more general approach by combining a multiple-grain pore-scale flow simulation with Lagrangian tracking of individual colloids. First, two numerical methods are applied simultaneously to solve viscous flows in a channel partially or fully packed with spherical grain particles, this allows cross-validation of the numerical methods for considered model geometries. It is demonstrated that the mesoscopic lattice Boltzmann approach can more accurately simulate three-dimensional pore-scale flows with multiple grain–grain and grain–wall contact points. Colloid transport is simulated under the combined influence of hydrodynamic forces, Brownian force, and physicochemical forces. Preliminary results demonstrate the capture of colloids by the secondary energy minimum (SEM) well. The local hydrodynamic retardation is shown to reduce the ability for colloids to move into the SEM well, but does not prevent this. Trajectories before and after the capture are also discussed.

In order to understand the transport behavior of colloids near an air–water interface (AWI), two computational methods are applied to simulate the local water flow field near a moving AWI in a 2D microfluidic channel. The first method is a mesoscopic multicomponent and multiphase lattice Boltzmann (LBM) model and the second is the macroscopic, Navier–Stokes based, volume-of-fluid interface tracking method. In the LBM, it is possible to predict the dynamic contact angles after the static contact angle is correctly set, and the predicted dynamic contact angles are in good agreement with previous observations. It is demonstrated that the two methods can yield a similar flow velocity field if they are applied properly. The flow field relative to AWI depends on the direction of the flow, and exhibits curved streamlines that transport fluid between the center of the channel and the wall region. Using the obtained flow, the motion of sub-micron colloids in a de-ionized water solution is then studied by a Lagrangian approach. The observed colloid trajectories are in qualitative agreement with our visualizations using a confocal microscope.