Modified lattice Boltzmann scheme for nonlinear convection diffusion equations

Lattice Boltzmann (LB) methods are usually developed on cubic lattices that discretize the configuration space using uniform grids. For efficient computations of anisotropic and inhomogeneous flows, it would be beneficial to develop LB algorithms involving the collision-and-stream steps based on orthorhombic cuboid lattices. We present a new 3D central moment LB scheme based on a cuboid D3Q27 lattice. This scheme involves two free parameters representing the ratios of the characteristic particle speeds along the two directions with respect to those in the remaining direction, and these parameters are referred to as the grid aspect ratios. Unlike the existing LB schemes for cuboid lattices, which are based on orthogonalized raw moments, we construct the collision step based on the relaxation of central moments and avoid the orthogonalization of moment basis, which leads to a more robust formulation. Moreover, prior cuboid LB algorithms prescribe the mappings between the distribution functions and raw moments before and after collision by using a moment basis designed to separate the trace of the second order moments (related to bulk viscosity) from its other components (related to shear viscosity), which lead to cumbersome relations for the transformations. By contrast, in our approach, the bulk and shear viscosity effects associated with the viscous stress tensor are naturally segregated only within the collision step and not for such mappings, while the grid aspect ratios are introduced via simpler pre- and post-collision diagonal scaling matrices in the above mappings. These lead to a compact approach, which can be interpreted based on special matrices. It also results in a modular 3D LB scheme on the cuboid lattice, which allows the existing cubic lattice implementations to be readily extended to those based on the more general cuboid lattices. To maintain the isotropy of the viscous stress tensor of the 3D Navier–Stokes equations using the cuboid lattice, corrections for eliminating the truncation errors resulting from the grid anisotropy as well as those from the aliasing effects are derived using a Chapman–Enskog analysis. Such local corrections, which involve the diagonal components of the velocity gradient tensor and are parameterized by two grid aspect ratios, augment the second order moment equilibria in the collision step. We present a numerical study validating the accuracy of our approach for various benchmark problems at different grid aspect ratios. In addition, we show that our 3D cuboid central moment LB method is numerically more robust than its corresponding raw moment formulation. Finally, we demonstrate the effectiveness of the 3D cuboid central moment LB scheme for the simulations of anisotropic and inhomogeneous flows and show significant savings in memory storage and computational cost when used in lieu of that based on the cubic lattice.

Peng-Robinson (P-R) equation of state (EOS) has been widely used in the petroleum industry for hydrocarbon fluids. In this work, a three-dimensional diffuse interface model with P-R EOS for two-phase fluid system is solved by the lattice Boltzmann (LB) method. In this diffuse interface model, an Allen-Cahn (A-C) type phase equation with strong nonlinear source term is derived. Using the multiscale Chapman-Enskog analysis, the A-C type phase equation can be recovered from the proposed LB method. Besides, a Lagrange multiplier is introduced based on the mesoscopic character of the LB scheme so that total mass of the hydrocarbon system is preserved. Three-dimensional numerical simulations of realistic hydrocarbon components, such as isobutane and propane, are implemented to illustrate the effectiveness of the proposed mass conservative LB scheme. Numerical results reach a better agreement with laboratory data compared to previous results of two-dimensional numerical simulations.

Local vorticity computation approach in double distribution functions based lattice Boltzmann methods for flow and scalar transport

Lattice Boltzmann simulations of 3D crystal growth: Numerical schemes for a phase-field model with anti-trapping current

The last decade the Lattice Boltzmann method has evolved into a powerfull solver for flow through porous media, where the solid phase is represented fully resolved on the computational grid (Hill, 2001). Packed and stored fresh agricultural produce are frequently modeled as porous media and therefore we envision Lattice Boltzmann to be an important tool for simulating heat and mass transport in these systems via forced or natural convection. However, practical package and storage systems have too large lengthscales for fully resolved simulations. Consequently, simulations on the full system scale have to be performed at the volume-averaged scale. For the volume-averaged scale closure relations are needed such as the drag force, effective diffusivity or conductivity. Also here Lattice Boltzmann can play a vital role. Lattice Boltzmann is also capable of simulations at the volume-averaged scale, though it has not frequently used for this scale. With this capability, we can build a multiscale simulation approach, similar to what is developed for fluidized beds (van der Hoef, 2004). In this contribution we will present the general framework, and applications of Lattice Boltzmann to forced flow through package with spherical produce (fully resolved), natural convection in bulk containers of seed potatoes (volume-averaged) (van der Sman,1997 & 1999), and some benchmark problems like natural convection in cavity (van der Sman, 2006) and flow in a channel with a porous layer (Beavers and Joseph benchmark). References o Hill R.J., Koch D.L., and Ladd A.J.C. Moderate-Reynolds-number flows in ordered and random arrays of spheres JOURNAL OF FLUID MECHANICS 448: 243-278 (2001). o van der Hoef M.A., Annaland M.V., Kuipers J.A.M. Computational fluid dynamics for dense gas-solid fluidized beds: a multi-scale modeling strategy. CHEMICAL ENGINEERING SCIENCE 59 (22-23): 5157-5165 (2004). o van der Sman R.G.M. Solving the vent hole design problem for seed potato packagings, with the Lattice Boltzmann scheme. INTERNATIONAL JOURNAL OF COMPUTATIONAL FLUID DYNAMICS 11 (3-4): 237-248 (1999). o VanderSman R.G.M. Lattice-Boltzmann scheme for natural convection in porous media INTERNATIONAL JOURNAL OF MODERN PHYSICS C 8 (4): 879-888 (1997) o van der Sman R.G.M. Lattice Boltzmann scheme for natural convection on square and rectangular grids. in revision Phys. Rev. E (2006). IUFoST 2006 DOI: 10.1051/IUFoST:20060990

A novel coupled lattice Boltzmann model for low Mach number combustion simulation

A family of analytical solutions of a nonlinear diffusion–convection equation

A three-dimensional model of wave interactions with permeable structures using the lattice Boltzmann method

Lattice Boltzmann modeling of phonon transport

Lattice BGK direct numerical simulation of fully developed turbulence in incompressible plane channel flow

Spurious interface and boundary behaviour beyond physical solutions in lattice Boltzmann schemes

In this paper, an alternative multiple-relaxation-time lattice Boltzmann (LB) scheme for conjugate heat transfer is proposed. We have derived the correct energy governing equationfrom the first law of thermodynamics and clarified that the thermal boundary for conjugate heat transfer should be the continuity of the conductive heat flux, which is essential to guarantee Galilean invariance. The development of the present model has two distinct features. First, the temperature evolution equationis modified to include the effect of the specific heat capacity. Second, the convection term is considered as a source term and is calculated in moment space, thus preserving the inherent advantages of the LB method without finite difference calculations. To verify the thermal governing equations, we conduct numerical simulations of heat conduction in a moving rod configuration with varying thermal capacities. Furthermore, to validate the present model's in accuracy, we examine several benchmark problems, including the effective thermal conductivity of parallel and series dual-component materials, steady and unsteady coupled heat transfer in horizontal interface channels, stable heat conduction in a two-layer annulus, natural convection in a square cavity with 16 solid blocks, and thermocapillary flow with two superimposed planar fluids. In the realm of application research, we apply our model to simulate the cooling process in random porous media. The results demonstrate that our model achieves satisfactory accuracy and exhibits a second-order convergence rate in space. The simulation results show the potential of the current LB method to simulate complex applications, highlighting its versatility and accuracy.

Tempered fractional calculus, as an extension of fractional calculus, has been successfully applied in numerous scientific and engineering fields. Although several traditional numerical methods have been improved for solving a variety of tempered fractional partial differential equations, solving these equations by the lattice Boltzmann (LB) method is an unresolved issue. This paper is dedicated to presenting a novel LB method for the tempered time-fractional diffusion equation. The tempered time-fractional diffusion equation is first transformed into an integer-order partial differential equation by approximating the tempered fractional derivative term. Then the LB method is proposed to solve the transformed objective equation. The Chapman-Enskog procedure is conducted to confirm that the present LB method can accurately recover the objective equation. Some numerical examples with an analytical solution are employed to validate the present LB method, and a strong consistency is observed between the numerical and analytical solutions. The numerical simulations indicate that the LB method is a second-order accurate scheme. The proposed LB method presents a new approach to solving the tempered time-fractional diffusion equation, which is beneficial for the widespread application of the tempered time-fractional diffusion equation in addressing complex transport problems.

In this paper, an incompressible two-dimensional (2-D) and three- dimensional (3-D) thermohydrodynamics for the lattice Boltzmann scheme are de- veloped. The basic idea is to solve the velocity fleld and the temperature fleld using two difierent distribution functions. A derivation of the lattice Boltzmann scheme from the continuous Boltzmann equation for 2-D is discussed in detail. By using the same procedure as in the derivation of the discretised density distribution function, it is found that new lattice of four-velocity (2-D) and eight-velocity (3-D) models for internal energy density distribution function can be developed where the viscous and compressive heating efiects are negligible. These models are validated by the numerical simulation of the 2-D porous plate Couette ∞ow problem where the analytical solution exists and the natural convection ∞ows in a cubic cavity.

In this article, two Hybrid Thermal Lattice Boltzmann (HTLB) methods are proposed to simulate thermally driven flows. In these methods, the velocity field is solved by a Fractional Volumetric Multi Relaxation Time Lattice Boltzmann (FV-MRT-LB) scheme and an En-tropic Lattice Boltzmann (ELB) method, while the advection-diffusion equation for temperature is solved separately by a finite difference technique. In order to demonstrate the ability of the FV-MRT-LB and ELB methods for simulation of isothermal flows, a two-dimensional cavity-driven flow has been simulated at different Reynolds numbers. Then two classical cases of buoyancy-induced flows, i.e., a differentially heated cavity flow and a Rayleigh-Be′nard convective flow, have been studied numerically at different Rayleigh numbers with a Prandtl number of 0.71. The numerical results show that both HT-FV-MRT-LB and HT-ELB methods can produce accurate results.