Standard Lattice Boltzmann Model Research Articles

Lattice Boltzmann model on unstructured grids for radiative transfer in graded-index media

The lattice Boltzmann method (LBM) has already been extended to radiative transfer. Radiative transfer in graded-index media is of great interest in many practical processes and is particularly challenging to solve. Currently, some lattice Boltzmann (LB) models have been developed for radiative transfer in graded-index media. These LB models are limited to uniform Cartesian grids and only applicable to simple geometries, thus limiting its application to practical processes. Here we develop a LB model on unstructured grids for radiative transfer in graded-index media. We adopt the relationship between the equilibrium distribution function and the radiative intensity in the standard LB model to establish this LB model on unstructured grids. In this unstructured LB model, the key is that the equilibrium distribution function is reconstructed on the interface of the unstructured grids. This unstructured LB model can solve the radiative transfer equation of graded-index media directly without using the streaming process on each lattice, thus significantly shortening the computational time compared to the standard LB model. This unstructured LB model allows for a simple and efficient geometrical implementation of complex boundaries for radiative transfer in graded-index media. Numerical results show that this unstructured LB model is well suited for radiative transfer in graded-index media with complex geometries and has good accuracy and stability.

High-order thread-safe lattice Boltzmann model for high performance computing turbulent flow simulations

We present a highly optimized thread-safe lattice Boltzmann model in which the non-equilibrium part of the distribution function is locally reconstructed via recursivity of Hermite polynomials. Such a procedure allows the explicit incorporation of non-equilibrium moments of the distribution up to the order supported by the lattice. Thus, the proposed approach increases accuracy and stability at low viscosities without compromising performance and amenability to parallelization with respect to standard lattice Boltzmann models. The high-order thread-safe lattice Boltzmann is tested on two types of turbulent flows, namely, the turbulent channel flow at Reτ=180 and the axisymmetric turbulent jet at Re = 7000; it delivers results in excellent agreement with reference data [direct numerical simulations (DNS), theory, and experiments] and (a) achieves peak performance [∼5×1012 floating point operations (FLOP) per second and an arithmetic intensity of ∼7 FLOP/byte on a single graphic processing unit] by significantly reducing the memory footprint, (b) retains the algorithmic simplicity of standard lattice Boltzmann computing, and (c) allows to perform stable simulations at vanishingly low viscosities. Our findings open attractive prospects for high-performance simulations of realistic turbulent flows on GPU-based architectures. Such expectations are confirmed by excellent agreement among lattice Boltzmann, experimental, and DNS reference data.

Lattice Boltzmann model for diffusion equation with reduced truncation errors: Applications to Gaussian filtering and image processing

Several image processing approaches require blurring an image using the Gaussian filter. In the present study the filtering method based on a novel four-velocity lattice Boltzmann model with non-local collision kernel is proposed. By applying the Chapman–Enskog expansion up to super-Burnett terms it is shown that the present scheme is equivalent to the diffusion equation with relatively small hyper-viscous defects (truncation errors). Compared to the conventional lattice Boltzmann model the proposed scheme has a small computational overhead but significantly improves accuracy due to the reduced hyper-viscosity. To validate the model the following test problems are considered: diffusion in time of a Gaussian bell, sinusoidal wave, image filtering, feature detection with the application of the modified SIFT method. The numerical experiments indicate that the novel model gives more precise results than the standard lattice Boltzmann model for non-small values of diffusion coefficient and in several cases its computational cost is competitive with OpenCV blurring techniques. In addition, the model can be efficiently used in feature detection algorithms like SIFT.

Axisymmetric lattice Boltzmann model for liquid flows with super-hydrophobic cylindrical surfaces

The lattice Boltzmann (LB) method has been extensively applied to the simulation of various fluid flows. Nonetheless, the standard LB model in rectangular coordinates faces some challenges when attempts are made to use it to simulate single-phase liquid flows involving super-hydrophobic cylindrical surfaces with a large slip length. Therefore, in this paper, boundary conditions for an axisymmetric LB model are proposed for the simulation of liquid slip at both convex and concave cylindrical surfaces. Based on the Navier slip boundary conditions, an equilibrium–nonequilibrium boundary scheme is developed to obtain the liquid slip in the azimuthal direction, with a combined bounce-back and specular-reflection boundary condition being developed to obtain the liquid slip in the axial direction. These boundary schemes are verified by simulating several established cases, namely, cylindrical Couette flow, harmonic oscillatory cylindrical Couette flow, Hagen–Poiseuille flow, and annular Poiseuille flow. The results are in excellent agreement with analytical results. Furthermore, some more complex flows involving super-hydrophobic cylindrical surfaces, such as stepwise oscillatory cylindrical Couette flow, are also simulated and studied using the proposed LB model. The axisymmetric LB model presented here overcomes the inability of standard LB models to accurately and efficiently simulate single-phase liquid flows involving super-hydrophobic cylindrical surfaces with a large slip length.

Essentially entropic lattice Boltzmann model: Theory and simulations.

We present a detailed description of the essentially entropic lattice Boltzmann model. The entropic lattice Boltzmann model guarantees unconditional numerical stability by iteratively solving the nonlinear entropy evolution equation. In this paper we explain the construction of closed-form analytic solutions to this equation. We demonstrate that near equilibrium this analytic solution reduces to the standard lattice Boltzmann model. We consider a few test cases to show that the analytic solution does not exhibit any significant deviation from the iterative solution. We also extend the analytical solution for the Ellipsoidal Statistical (ES)-Bhatnagar-Gross-Krook model to remove the limitation on the Prandtl number for heat transfer problems. The simplicity of the analytic solution removes the computational overhead and algorithmic complexity associated with the entropic lattice Boltzmann models.

Hybrid Lattice-Boltzmann-Potential Flow Simulations of Turbulent Flow around Submerged Structures

We report on the development and validation of a 3D hybrid Lattice Boltzmann Model (LBM), with Large Eddy Simulation (LES), to simulate the interactions of incompressible turbulent flows with ocean structures. The LBM is based on a perturbation method, in which the velocity and pressure are expressed as the sum of an inviscid flow and a viscous perturbation. The far- to near-field flow is assumed to be inviscid and represented by potential flow theory, which can be efficiently modeled with a Boundary Element Method (BEM). The near-field perturbation flow around structures is modeled by the Navier–Stokes (NS) equations, based on a Lattice Boltzmann Method (LBM) with a Large Eddy Simulation (LES) of the turbulence. In the paper, we present the hybrid model formulation, in which a modified LBM collision operator is introduced to simulate the viscous perturbation flow, resulting in a novel perturbation LBM (pLBM) approach. The pLBM is then extended for the simulation of turbulence using the LES and a wall model to represent the viscous/turbulent sub-layer near solid boundaries. The hybrid model is first validated by simulating turbulent flows over a flat plate, for moderate to large Reynolds number values, Re ∈[3.7×104;1.2×106]; the plate friction coefficient and near-field turbulence properties computed with the model are found to agree well with both experiments and direct NS simulations. We then simulate the flow past a NACA-0012 foil using a regular LBM-LES and the new hybrid pLBM-LES models with the wall model, for Re = 1.44×106. A good agreement is found for the computed lift and drag forces, and pressure distribution on the foil, with experiments and results of other numerical methods. Results obtained with the pLBM model are either nearly identical or slightly improved, relative to those of the standard LBM, but are obtained in a significantly smaller computational domain and hence at a much reduced computational cost, thus demonstrating the benefits of the new hybrid approach.

Second order accurate boundary conditions for the general propagation lattice Boltzmann method

In the present work, the problem of the development of the second order accurate boundary conditions for the lattice Boltzmann models with the general streaming step is addressed. The proposed boundary conditions are applicable to the flows in domains with solid and curved boundaries. The presented boundary scheme can be considered as the generalization of the Mei-Luo-Shyy conditions, in the case of the standard lattice Boltzmann models both approaches coincide. Several benchmark problems are considered: the Poiseuille flow, the oscillating Womersley flow, the Taylor-Couette flow, and the flow past a circular cylinder. These problems are studied for different spatial grid resolutions, and the convergence rates to the solutions from literature are presented.

A kinetic theory-based axisymmetric lattice Boltzmann flux solver for isothermal and thermal swirling flows

In this study, an alternative axisymmetric lattice Boltzmann flux solver (LBFS) is proposed within the framework of the kinetic theory-based axisymmetric lattice Boltzmann (LB) model. The macroscopic equations recovered from the axisymmetric LB model are globally resolved by the finite volume method, while numerical fluxes on the cell interface are locally reconstructed from solutions to the axisymmetric LB equation with the second-order of accuracy. Unlike the previous fractional step – axisymmetric LBFS which is based on the standard LB model and contains complicated source terms of various space derivatives, the present scheme originates from the kinetic theory-based lattice Boltzmann equation defined on the cylindrical coordinates which naturally incorporates the axisymmetric effects into the distribution functions and greatly simplifies the source terms involved. As a result, no space derivatives emerge in the source terms, and the corrector step in the previous LBFS model is removed, which yields more compact formulations. Moreover, previous inconsistency between the local reconstruction and the global governing equations is fixed in the present solver, which helps to nurture the computational results. By consolidating the evolutions of the azimuthal velocity and the temperature into the present axisymmetric LBFS, it is particularly suitable for simulations of isothermal and thermal swirling flows. The accuracy and robustness of the proposed scheme are validated through various benchmark tests. Comprehensive comparisons with the results from the previous solver demonstrate that the present solver is quantitatively more accurate than the previous solver, particularly in predicting the pressure distributions.

A Method for Simulating the Dynamics of Rarefied Gas Based on Lattice Boltzmann Equations and the BGK Equation

A hybrid method for solving boundary value problems for rarefied gas using the Bhatnagar–Gross–Krook (BGK) model and the lattice Boltzmann equation is studied. One-dimensional boundary value problems subject to membrane-type boundary conditions are considered. In strongly nonequilibrium regions, the BGK model should be used, and in the regions in which the distribution function is close to Maxwell’s one, the lattice Boltzmann equations can be used. On the region boundaries, a matching procedure should be performed; such a procedure is proposed in this paper. Note that the standard lattice Boltzmann models distort the distribution function on the region boundaries, but this distortion has no physical meaning. It is shown that, in order to correctly join the solutions on the region boundaries, the semi-moments of Maxwell’s distribution must be exactly reproduced. For this purpose, novel lattice models of the Boltzmann equation are constructed using the entropy method. Results of numerical computations of the temperature and density profiles for the Knudsen number equal to $$0.1$$ are presented, and the numerically obtained distribution function at the matching point is compared with the theoretical distribution function. Computation of the matching point is discussed.

A multi-component discrete Boltzmann model for nonequilibrium reactive flows

We propose a multi-component discrete Boltzmann model (DBM) for premixed, nonpremixed, or partially premixed nonequilibrium reactive flows. This model is suitable for both subsonic and supersonic flows with or without chemical reaction and/or external force. A two-dimensional sixteen-velocity model is constructed for the DBM. In the hydrodynamic limit, the DBM recovers the modified Navier-Stokes equations for reacting species in a force field. Compared to standard lattice Boltzmann models, the DBM presents not only more accurate hydrodynamic quantities, but also detailed nonequilibrium effects that are essential yet long-neglected by traditional fluid dynamics. Apart from nonequilibrium terms (viscous stress and heat flux) in conventional models, specific hydrodynamic and thermodynamic nonequilibrium quantities (high order kinetic moments and their departure from equilibrium) are dynamically obtained from the DBM in a straightforward way. Due to its generality, the developed methodology is applicable to a wide range of phenomena across many energy technologies, emissions reduction, environmental protection, mining accident prevention, chemical and process industry.

Finite-difference lattice Boltzmann model for nonlinear convection-diffusion equations

In this paper, a finite-difference lattice Boltzmann (LB) model for nonlinear isotropic and anisotropic convection-diffusion equations is proposed. In this model, the equilibrium distribution function is delicately designed in order to recover the convection-diffusion equation exactly. Different from the standard LB model, the temporal and spatial steps in this model are decoupled such that it is convenient to study convection-diffusion problem with the non-uniform grid. In addition, it also preserves the advantage of standard LB model that the complex-valued convection-diffusion equation can be solved directly. The von Neumann stability analysis is conducted to discuss the stability region which can be used to determine the free parameters appeared in the model. To test the performance of the model, a series of numerical simulations of some classical problems, including the diffusion equation, the nonlinear heat conduction equation, the Sine-Gordon equation, the Gaussian hill problem, the Burgers–Fisher equation, and the nonlinear Schrödinger equation, have also been carried out. The results show that the present model has a second-order convergence rate in space, and generally it is also more accurate than the standard LB model.

A Finite-Difference Lattice Boltzmann Approach for Gas Microflows

AbstractFinite-difference Lattice Boltzmann (LB) models are proposed for simulating gas flows in devices with microscale geometries. The models employ the roots of half-range Gauss-Hermite polynomials as discrete velocities. Unlike the standard LB velocity-space discretizations based on the roots of full-range Hermite polynomials, using the nodes of a quadrature defined in the half-space permits a consistent treatment of kinetic boundary conditions. The possibilities of the proposed LB models are illustrated by studying the one-dimensional Couette flow and the two-dimensional square driven cavity flow. Numerical and analytical results show an improved accuracy in finite Knudsen flows as compared with standard LB models.

An alternative lattice Boltzmann model for three-dimensional incompressible flow

In this work, an alternative lattice Boltzmann (LB) model for three-dimensional (3D) incompressible flow is proposed. The equilibrium distribution function (EDF) of the present model is directly derived in accordance with the incompressibility conditions by applying the Hermite expansion. Moreover, an alternative formula for pressure computation is designed from the second order moment of the distribution function. The present 3D LB model inherits the advantageous features of Guo’s LB model: the density is a constant, the fluid pressure is independent of density and the Navier–Stokes (N–S) equations for incompressible flow can be derived. Two benchmark tests, flow over a backward-facing step and the lid-driven cavity flow, are applied to validate the present model. Accurate results for these tests are obtained with the present model, and further comparisons with the previous LB models (the standard LB model, the He–Luo model and Guo’s LB model) demonstrate that the present model provides better accuracy in the region of high deviatoric stress and such advantage is further enhanced by using the D3Q27 lattice.

Regional scale transient groundwater flow modeling using Lattice Boltzmann methods

Lattice Boltzmann (LB) models can be used to simulate flow in porous media at scales much larger than pore size. LB-based models for such macroscopic scale porous media flow simulations are an extension of standard LB models. There are at least two alternative approaches for implementing such models. In the first approach the local velocity is altered during the collision step by incorporating an external force, F , equivalent to the damping effect of solid particles in porous media. The porous media can be permeable or impermeable depending upon the external forcing term. A sink term is introduced in the LB model to simulate a pumping well and this model is further applied to solve transient ground water well problems for confined aquifers. Directly solving the ground water flow equation with an LB model by exploiting its ability to solve the diffusion equation is another strategy. The second order transient ground water flow equation is analogous to the diffusion equation and mass diffusivity is analogous to hydraulic diffusivity. This diffusion model is used to solve transient ground water problems. Simulated results accurately match analytical solutions of the transient ground water flow equation.

Preconditioned lattice-Boltzmann method for steady flows

In this paper we propose a preconditioned lattice Boltzmann (LB) method for steady incompressible flows. For steady flows, the macroscopic equations derived from this LB model are equivalent to those from the standard LB model, but with an improved eigenvalue system. The proposed model can be viewed as an explicit solver for preconditioned compressible Navier-Stokes equations. Linear stability analysis is performed and the results show that the stability of the model is the same as that of the standard LB model for low Mach numbers. The proposed model retains the structure of the standard LB model and, hence, possesses all the advantages. Numerical tests show that the convergence rate can be enhanced as much as an order of magnitude compared to the standard lattice Boltzmann method. The accuracy of the solutions is improved as well.

Multi-level lattice Boltzmann model on square lattice for compressible flows

A compressible lattice Boltzmann model is established on a square lattice. The model allows large variations in the mean velocity by introducing a large particle-velocity set. To maintain tractability, the support set of the equilibrium distribution is chosen to include only four directions and three particle-velocity levels in which the third level is introduced to improve the stability of the model. This simple structure of the equilibrium distribution makes the model efficient for the simulation of flows over a wide range of Mach numbers and gives it the capability of capturing shock jumps. Unlike the standard lattice Boltzmann model, the formulation eliminated the fourth-order velocity tensors, which were the source of concerns over the homogeneity of square lattices. A modified collision invariant eliminates the second-order discretization error of the fluctuation velocity in the macroscopic conservation equation from which the Navier–Stokes equation and energy equation are recovered. The model is suitable for both viscous and inviscid compressible flows with or without shocks. Two-dimensional shock-wave propagations and boundary layer flows were successfully simulated. The model can be easily extended to three-dimensional cubic lattices.

Three-dimensional lattice Boltzmann model for compressible flows.

A three-dimensional compressible lattice Boltzmann model is formulated on a cubic lattice. A very large particle-velocity set is incorporated in order to enable a greater variation in the mean velocity. Meanwhile, the support set of the equilibrium distribution has only six directions. Therefore, this model can efficiently handle flows over a wide range of Mach numbers and capture shock waves. Due to the simple form of the equilibrium distribution, the fourth-order velocity tensors are not involved in the formulation. Unlike the standard lattice Boltzmann model, no special treatment is required for the homogeneity of fourth-order velocity tensors on square lattices. The Navier-Stokes equations were recovered, using the Chapman-Enskog method from the Bhatnagar-Gross-Krook (BGK) lattice Boltzmann equation. The second-order discretization error of the fluctuation velocity in the macroscopic conservation equation was eliminated by means of a modified collision invariant. The model is suitable for both viscous and inviscid compressible flows with or without shocks. Since the present scheme deals only with the equilibrium distribution that depends only on fluid density, velocity, and internal energy, boundary conditions on curved wall are easily implemented by an extrapolation of macroscopic variables. To verify the scheme for inviscid flows, we have successfully simulated a three-dimensional shock-wave propagation in a box and a normal shock of Mach number 10 over a wedge. As an application to viscous flows, we have simulated a flat plate boundary layer flow, flow over a cylinder, and a transonic flow over a NACA0012 airfoil cascade.

Lattice Bhatnagar–Gross–Krook model for the Lorenz attractor

In this paper, we propose a new lattice Bhatnagar–Gross–Krook model for simulation of the Lorenz attractor. The new model is based on a nonlinear diffusion-reaction system. We formulate a lattice Bhatnagar–Gross–Krook model for the nonlinear diffusion-reaction system by using a method of higher moments of the lattice Boltzmann equation. As a special case where the diffusion effect disappears, we get the Lorenz equation. In the model, we obtain a series of lattice Boltzmann equations at different time scales, and the conservation law at time scale t 0. The equilibrium distribution functions are simpler than those for the standard lattice Boltzmann model. The numerical examples show that the method can be used to simulate the Lorenz equation.

Lattice Boltzmann Model for Compressible Fluid on a Square Lattice

A two-level four-direction lattice Boltzmann model is formulated on a square lattice to simulate compressible flows with a high Mach number. The particle velocities are adaptive to the mean velocity and internal energy. Therefore, the mean flow can have a high Mach number. Due to the simple form of the equilibrium distribution, the 4th order velocity tensors are not involved in the calculations. Unlike the standard lattice Boltzmann model, no special treatment is need for the homogeneity of 4th order velocity tensors on square lattices. The Navier-Stokes equations were derived by the Chapman-Enskog method from the BGK Boltzmann equation. The model can be easily extended to three-dimensional cubic lattices. Two-dimensional shock-wave propagation was simulated.

A lattice gas model for erosion and particles transport in a fluid

We consider a simple lattice gas model to simulate erosion, deposition and particle transport in a streaming fluid. In our approach, the fluid is described by a standard lattice Boltzmann model and the granular suspension by a multiparticle cellular automata. A good agreement is obtained between the predictions of the model and field measurements, as observed by analyzing the deposition patterns resulting from various snow and sand transport phenomena. In particular we study the case of ripples formation and simulate the scour appearing around a submarine pipe.