Comparative Study of Lattice Boltzmann Models for Complex Fractal Geometry

A standard model, one of the lattice Boltzmann models for incompressible flow, is broadly applied in mesoscopic fluid with obvious compressible error. To eliminate the compressible effect and the limits in 2D problems, three different models (He-Luo model, Guo’s model, and Zhang’s model) have been proposed and tested by some benchmark questions. However, the numerical accuracy of models adopted in complex geometry and the effect of structural complexity are rarely studied. In this paper, a 2D dimensionless steady flow model is proposed and constructed by fractal geometry with different structural complexity. Poiseuille flow is first simulated to verify the code and shows good agreements with the theoretical solution, supporting further the comparative study on four models to investigate the effect of structural complexity and grid resolution, with reference results obtained by the finite element method (FEM). The work confirms the latter proposed models and effectively reduces compressible error in contrast to the standard model; however, the compressible effect still cannot be ignored in Zhang’s model. The results show that structural error has an approximately negative exponential relationship with grid resolution but an approximately linear relationship with structural complexity. The comparison also demonstrates that the He-Luo model and Guo’s model have a good performance in accuracy and stability, but the convergence rate is lower, while Zhang’s model has an advantage in the convergence rate but the computational stability is poor. The study is significant as it provides guidance and suggestions for adopting LBM to simulate incompressible flow in a complex structure.

A virtual force method to rectify the equation of state of the lattice Boltzmann models

In this work, to rectify the equation of state (EOS) of a recently introduced constant speed entropic kinetic model (CSKM), a virtual force method is proposed. The CSKM, as shown in Zadehgol and Ashrafizaadeh [J. Comp. Phys. 274, 803 (2014)] and Zadehgol [Phys. Rev. E 91, 063311 (2015)], is an entropic kinetic model with unconventional entropies of Burg and Tsallis. The dependence of the pressure on the velocity, in the CSKM, was addressed and it was shown that it can be rectified by inserting rest particles into the model. This work shows that this dependence can also be removed by treating the pressure gradient as a pseudo force term, expanding the source term using the Fourier series, and applying the modified method of Khazaeli et al. [Phys. Rev. E 98, 053303 (2018)]. The proposed method can potentially be used to remove other pseudo-force error terms of the CSKM, e.g. the residual error terms which become significant at high Mach numbers, ensuring thermodynamic consistency of the entropic model, at the compressible flow regimes. The accuracy of the method is verified by simulating benchmark flows.

Lattice Boltzmann simulations on the tumbling to tank-treading transition: effects of membrane viscosity

The tumbling to tank-treading (TB-TT) transition for red blood cells (RBCs) has been widely investigated, with a main focus on the effects of the viscosity ratio λ (i.e., the ratio between the viscosities of the fluids inside and outside the membrane) and the shear rate γ ˙ applied to the RBC. However, the membrane viscosity μ m plays a major role in a realistic description of RBC dynamics, and only a few works have systematically focused on its effects on the TB-TT transition. In this work, we provide a parametric investigation on the effect of membrane viscosity μ m on the TB-TT transition for a single RBC. It is found that, at fixed viscosity ratios λ , larger values of μ m lead to an increased range of values of capillary number at which the TB-TT transition occurs; moreover, we found that increasing λ or increasing μ m results in a qualitatively but not quantitatively similar behaviour. All results are obtained by means of mesoscale numerical simulations based on the lattice Boltzmann models. This article is part of the theme issue ‘Progress in mesoscale methods for fluid dynamics simulation’.

Volumetric Lattice Boltzmann Models in General Curvilinear Coordinates: Theoretical Formulation

A theoretical formulation of lattice Boltzmann models on a general curvilinear coordinate system is presented. It is based on a volumetric representation so that mass and momentum are exactly conserved as in the conventional lattice Boltzmann on a Cartesian lattice. In contrast to some previously existing approaches for arbitrary meshes involving interpolation approximations among multiple neighboring cells, the current formulation preserves the fundamental one-to-one advection feature of a standard lattice Boltzmann method on a uniform Cartesian lattice. The new approach is built on the concept that a particle is moving along a curved path. A discrete space-time inertial force is derived so that the momentum conservation is exactly ensured for the underlying Euclidean space. We theoretically show that the new scheme recovers the Navier-Stokes equation in general curvilinear coordinates in the hydrodynamic limit, along with the correct mass continuity equation.

Extended Lattice Boltzmann Model

Conventional lattice Boltzmann models for the simulation of fluid dynamics are restricted by an error in the stress tensor that is negligible only for small flow velocity and at a singular value of the temperature. To that end, we propose a unified formulation that restores Galilean invariance and the isotropy of the stress tensor by introducing an extended equilibrium. This modification extends lattice Boltzmann models to simulations with higher values of the flow velocity and can be used at temperatures that are higher than the lattice reference temperature, which enhances computational efficiency by decreasing the number of required time steps. Furthermore, the extended model also remains valid for stretched lattices, which are useful when flow gradients are predominant in one direction. The model is validated by simulations of two- and three-dimensional benchmark problems, including the double shear layer flow, the decay of homogeneous isotropic turbulence, the laminar boundary layer over a flat plate and the turbulent channel flow.