Finite Volume Lattice Boltzmann Method Research Articles

Hybrid lattice Boltzmann method using Cartesian and body-fitted grids for turbomachinery aeroacoustic simulations

In this study, we developed a hybrid approach using the lattice Boltzmann method (LBM) and finite-volume lattice Boltzmann method (FVLBM) to perform efficient aeroacoustic simulations with moving boundaries. An entire domain, including the flow and acoustic fields, was computed using the standard LBM with a Cartesian grid. Local domains around the moving objects were computed using the FVLBM with body-fitted grids. These simulations were coupled using the direct forcing method to consider moving boundaries. The hybrid method was validated for several problems including turbulent flows and flow-induced sounds under low-Mach-number conditions. These validation problems covered flows with Reynolds numbers up to 2.0×105 and Mach numbers less than 0.2. In the simulations of the aeolian tone generated from stationary and rotating cylinders, the hybrid method results were consistent with those of the conventional methods. In the simulation of the turbulent flow and broadband sound of the isolated airfoil, the hybrid method was 15 times faster than the standard LBM and produced results consistent with the experimental results. Furthermore, the application of a cross-flow fan demonstrated that the hybrid method successfully simulated the flow and acoustic fields around the rotor.

Application of arbitrary Lagrangian–Eulerian unstructured finite volume lattice Boltzmann method to simulate compressible viscous flows over moving bodies

Application of arbitrary Lagrangian–Eulerian unstructured finite volume lattice Boltzmann method to simulate compressible viscous flows over moving bodies

An improved gas kinetic BGK scheme for finite volume lattice Boltzmann method for nearly incompressible flows

The recently gas kinetic Bhatnagar–Gross–Krook (BGK) scheme proposed by Li et al. for finite volume lattice Boltzmann method (FVLBM) is improved in this work to enhance the numerical stability. Compared with the original scheme, there are two striking features in the present improved method. One is that two auxiliary distribution functions, i.e., f̄α and f̃α, with the inclusion of collision effect are introduced in the present scheme, which eliminates the implicit feature for the update of distribution function. Meanwhile, the update of the distribution function relies on f̄α instead of the original one. Similarly with the original scheme, the fluxes on the cell interfaces are determined by the formal analytical solution of the lattice Boltzmann BGK equation, that is, the collision effects are included in the reconstructed fluxes, which can reduce excessive numerical dissipation. Another different feature is that an auxiliary distribution function f̃α is applied to the reconstructed fluxes on the cell interfaces. These two different features make the distribution function related to the reconstructed fluxes has better positivity-preserving properties, thus Δt/τ in the present method can be much higher than Li et al. scheme, and the numerical stability is enhanced considerably. Additionally, various benchmark studies, including the Kovasznay flow, the lid-driven cavity flow, the laminar boundary layer flow and the unsteady Taylor–Green vortex flow, are conducted to assess the numerical accuracy and stability of the proposed method. The numerical results demonstrate that the present scheme has second-order accuracy and better numerical stability for simulating nearly incompressible flows.

A gas kinetic Lax–Wendroff scheme for low-speed isothermal rarefied gas flows

Previously, a gas kinetic Bhatnagar–Gross–Krook (BGK) scheme was proposed by us for incompressible flows in the continuum limits. [W. Li and W. Li, “A gas-kinetic BGK scheme for the finite volume lattice Boltzmann method for nearly incompressible flows,” Comput. Fluids 162, 126–138 (2018).] In the present work, we extend the gas kinetic BGK scheme to simulate low-speed isothermal rarefied nonequilibrium gas flows. This scheme is a gas kinetic Lax–Wendroff scheme (GKLWS) for the discrete velocity Boltzmann equation in the finite volume discretization framework with second-order accuracy in both time and space. As collision and transport of the molecular particles are coupled in the present GKLWS, the time step of the present method is not limited by the relaxation time, for which the present scheme is efficient for multiscale gas flows. Moreover, the present GKLWS holds the asymptotic preserving (AP) property, which ensures that both the Navier–Stokes solutions in the continuum limits and free-molecular flow solutions in the rarefied limits can be reliably obtained. To validate the accuracy and AP property of the GKLWS, several numerical benchmarks of isothermal low-speed rarefied gas flows are simulated by the present scheme. The numerical results show that the present scheme can be a reliable multiscale method for all Knudsen number low-speed isothermal gas flows.

A hybrid scheme coupling lattice Boltzmann method and finite-volume lattice Boltzmann method for steady incompressible flows

We present a new hybrid method coupling the adaptive mesh refinement lattice Boltzmann method (AMRLBM) and the finite-volume lattice Boltzmann method (FVLBM) to improve both the simulation efficiency and adaptivity for steady incompressible flows with complex geometries. The present method makes use of the domain decomposition, in which the FVLBM sub-domain is applied to the region adjacent to the walls, and is coupled to the lattice Boltzmann method (LBM) sub-domain in the rest of the flow field to enhance the ability of the LBM to deal with irregular geometries without sacrificing the high efficiency and accuracy property of the LBM. In the LBM sub-domain, a cell-centered lattice structure-based AMRLBM is used and, in the FVLBM sub-domain, the gas-kinetic Bhatnagar–Gross–Krook (BGK) scheme-based FVLBM is adopted to reduce the numerical dissipation and enhance the efficiency of FVLBM. Moreover, not like the conventional LBM and Navier–Stokes equation solver-based hybrid schemes, the present hybrid scheme combines two kinds of lattice Boltzmann equation solvers, that is, AMRLBM and FVLBM, which makes the present scheme much simpler and better consistency than the conventional hybrid schemes. To assess the accuracy and efficacy of the proposed method, various benchmark studies, including the Kovasznay flow, the lid-driven cavity flow with Reynolds number Re=100, 400, and 1000, and the steady flow past a cylinder with Re=20 and 40, are also conducted. The numerical results show that the present scheme can be an efficient and reliable method for steady incompressible flows.

Parallel unstructured finite volume lattice Boltzmann method for high-speed viscid compressible flows

Based on the double distribution function Boltzmann-BGK equations, a cell-centered finite volume lattice Boltzmann method on unstructured grids for high-speed viscid compressible flows is presented. In the equations, the particle distribution function is introduced on the basis of the D2Q17 circular function, and its corresponding total energy distribution function is adopted. In the proposed method, the advective term is evaluated by Roe’s flux-difference splitting scheme, and a limiter is used to prevent the generation of oscillations. The distribution functions on the interface are calculated by piecewise linear reconstruction, in which the gradient is computed by the least-squares approach. In order to do large-scale simulations, a parallel algorithm is illustrated. The present method is validated by a flow around the NACA0012 airfoil and a flow past a circular cylinder at high Mach numbers. The results agree well with the published results, which demonstrate that the present method is an efficient numerical method for high-speed viscid compressible flows. The parallel performance results show that the proposed parallel algorithm achieves 90% parallel efficiency on 4800 cores for a problem with [Formula: see text] unstructured triangle cells, which shows the potential to perform fast and high-fidelity simulations of large-scale high-speed viscid compressible flows in complicated computational domains.

A scalable parallel unstructured finite volume lattice Boltzmann method for three‐dimensional incompressible flow simulations

AbstractThe standard lattice Boltzmann method, which employs certain regular lattices coupled with discrete velocities as the computational grid, is limited in its flexibility to simulate flows in irregular geometries. To simulate large‐scale complex flows, we present a cell‐centered finite volume lattice Boltzmann method for incompressible flows on three‐dimensional (3D) unstructured grids and its corresponding parallel algorithm. The advective fluxes are calculated by the low‐diffusion Roe scheme, and the gradients of the particle distribution functions are computed with a least squares method. The presented scheme is validated by three benchmark flows: (a) a 3D Poiseuille flow, (b) cubic cavity flows with Reynolds numbers Re = 100 and 400, and (c) flows past a sphere with Re = 50, 100, 150, 200, and 250. Some parallel performance results are presented to show the scalability of the method, which reveal that the proposed parallel algorithm has considerable scalability and that the parallel efficiency is higher than 87% on 3840 processor cores. It can be seen that the presented parallel solver has significant potential for the accurate simulation of flows in complex 3D geometries.

Numerical analyses of liquid slosh by Finite volume and Lattice Boltzmann methods

Liquid slosh phenomenon has been widely observed in aerospace and aeronautic engineering. In this paper, Finite volume method (FVM) and Lattice Boltzmann method (LBM) are proposed to perform the quick evaluation of liquid slosh behaviour under different container designs. The effectiveness of the numerical models is validated by the existing experimental data. The validated models are then used to study the fuel slosh behaviour with different baffle ring designs. By inserting the baffle to divide the liquid region into multiple sections, the slosh motion of the liquid is suppressed. At the same time, the effectiveness and time efficiency of LBM model have been demonstrated by a parallel comparison with the results from the FVM model. It is found that the computational time cost of LBM model is only around 7.0% of the time cost for FVM model without compromise of the numerical accuracy. This supports the future utilisation of LBM model in quick design evaluation of slosh-like problem.

A coupled gas-kinetic BGK scheme for the finite volume lattice Boltzmann method for nearly incompressible thermal flows

This work presents a coupled gas-kinetic Bhatnagar-Gross-Krook (BGK) scheme based finite volume lattice Boltzmann method (FVLBM) for nearly incompressible thermal flows. In the present formulation, a thermal lattice BGK model with double distribution functions coupled through the Boussinesq approximation is adopted. Compared with the conventional thermal finite volume/finite difference lattice Boltzmann schemes, the present method has two striking advantages. One is that the fluxes on cell interfaces are determined by the formal analytical solution of the lattice BGK equation rather than interpolation or solving the Riemann problem, which reduces numerical dissipation considerably. Another is that, an implicit collision of the lattice Boltzmann equations (LBE) is adopted to remove the time-step constrain from the relaxation time, which improves the efficiency for high Rayleigh number thermal flows. Additionally, the accuracy of the proposed scheme are validated by the numerical simulations of four test cases in two dimensions: (a) the thermal Couette flow with wall injection; (b) the natural convection in a square cavity with differently heated vertical walls; (c) the Rayleigh-Bénard convection in a rectangle heated from bottom; and (d) the turbulent Rayleigh-Bénard convection in a square cavity heated from bottom. The numerical results show that the present scheme has second-order accuracy and can be a reliable tool for incompressible thermal flows.

A novel median dual finite volume lattice Boltzmann method for incompressible flows on unstructured grids

A novel median dual finite volume lattice Boltzmann method (FV-LBM) for the accurate simulation of incompressible flows on unstructured grids is presented in this paper. The finite volume method is adopted to discretize the discrete velocity Boltzmann equation (DVBE) on median dual control volumes (CVs). In the previous studies on median dual FV-LBMs, the fluxes for each partial face have to be computed separately. In the present second-order scheme, we assume the particle distribution functions (PDFs) to be constant for all faces grouped around a particular edge. The fluxes are then evaluated using the low-diffusion Roe scheme at the midpoint of the edge, and the PDFs at the faces of the CV are obtained through piecewise linear reconstruction of the left and right states. The gradients of the PDFs are computed with the Green–Gauss approach. The presented scheme is validated on four benchmark flows: (a) pressure driven Poiseuille flow; (b) the backward-facing step flow with [Formula: see text], 100, 200 and 300; (c) the lid-driven flow with [Formula: see text] and 1000; and (d) the steady viscous flow past a circular cylinder with [Formula: see text], 20 and 40.

Scalable parallel finite volume lattice Boltzmann method for thermal incompressible flows on unstructured grids

A parallel cell-centered finite volume thermal lattice Boltzmann method (FV-TLBM) for two-dimensional unstructured grids is proposed to simulate thermal incompressible flows. The governing equations for the velocity and temperature fields are solved using the D2Q9 lattice model in the lattice Boltzmann method and the cell-centered finite volume method. In this paper, the advective fluxes in the governing equations are evaluated by a low-diffusion Roe scheme, and the gradients of the particle distribution functions are computed with the least-square approach. To simulate large-scale complex flows within a reasonable time, a parallel algorithm for the FV-TLBM on unstructured grids is devised. The present method is validated by numerical simulations of the natural convection in a square cavity and natural convection in a concentric annulus at different Rayleigh numbers. The results obtained by the proposed method agree well with previous studies. The parallel performance results show that the proposed algorithm has considerable scalability and the parallel efficiency is as high as 96.50% for a case with over 5 billion grid cells using 6000 processor cores.

A simplified finite volume lattice Boltzmann method for simulations of fluid flows from laminar to turbulent regime, Part II: Extension towards turbulent flow simulation

In this paper, the original finite volume lattice Boltzmann method (FVLBM) on an unstructured grid (Part I of these twin papers) is extended to simulate turbulent flows. To model the turbulent effect, the k−ω SST turbulence model is incorporated into the present FVLBM framework and is also solved by the finite volume method. Based on the eddy viscosity hypothesis, the eddy viscosity is computed from the solution of the k−ω SST model, and the total viscosity is modified by adding this eddy viscosity to the laminar (kinematic) viscosity given in the Bhatnagar–Gross–Krook collision term. Because of solving for the collision term with the explicit method in the original FVLBM scheme, the computational efficiency is much lower for simulating high Reynolds number flow. This is due to the fact that the largest time step decided by the stability condition of the collision term, which is less than twice the relaxation time, is much smaller than that decided by the CFL condition. In order to enhance the computational efficiency, the three-stage second-order implicit–explicit (IMEX) Runge–Kutta method is used for temporal discretization, and the time step can be one or two orders of magnitude larger as compared with the explicit Euler forward scheme. Although the computational cost is increased, the final computational efficiency is enhanced by about one-order of magnitude and good results can also be obtained at a large time step through the test case of a lid-driven cavity flow. Two turbulent flow cases are carried out to validate the present method, including flow over a backward-facing step and flow around a NACA0012 airfoil. The numerical results are found to be in agreement with experimental data and numerical solutions, demonstrating the applicability of the present FVLBM coupled with the k−ω SST model to accurately predict the incompressible turbulent flows.

A simplified finite volume lattice Boltzmann method for simulations of fluid flows from laminar to turbulent regime, Part I: Numerical framework and its application to laminar flow simulation

In this paper, an unstructured grid based finite volume lattice Boltzmann method (FVLBM) that can be used for the simulation of incompressible laminar flows is presented and studied in detail. This method is derived from a simple modification of the cell-vertex unstructured grid based FVLBM proposed by Stiebler et al. (2006). Compared with other complex flux reconstruction methods, the present scheme has a low computational cost and can also achieve second-order spatial accuracy. Furthermore, depending on the use of the different temporal discretization schemes, the temporal accuracy can be adjusted for both steady and unsteady flows. Besides, some comparisons of the computational cost and accuracy with another FVLBM scheme are also presented. Meanwhile, different boundary conditions are illustrated that are easy to implement on complex geometries. To validate the present method, four cases are carried out, including a Couette flow driven by one plate for an accuracy test, flow in a square cavity, flow around a single circular cylinder and a more complex flow around double circular cylinders. Numerical experiments show that the present scheme can simulate steady and unsteady flows at relatively high Reynolds number with relatively few grid cells, thus demonstrating the good capability of the present method.

A conservation-moment-based implicit finite volume lattice Boltzmann method for steady nearly incompressible flows

This paper presents an efficient, low memory cost, implicit finite volume lattice Boltzmann method (FVLBM) based on conservation moments acceleration for steady nearly incompressible flows. In the proposed scheme, not as the conventional implicit schemes, both the micro lattice Boltzmann equations (LBE) and the associated conservation moment equations are solved by the matrix-free, lower-upper symmetric Gauss-Seidel scheme (LUSGS) and the conservation moment equations are used to predict equilibrium distribution functions at the new time, which eliminates the storage of the Jacobian matrix of the collision term in the implicit LBE system and provides a driving force for the fast convergence of the LBE. Moreover, by utilizing the projection matrix and the collision invariant, we can construct the fluxes of the moment equations efficiently from the fluxes of the LBE and avoid the time-consuming reconstruction procedure for obtaining the fluxes of the moment equations. To demonstrate the accuracy and high efficiency of the proposed scheme, comparison studies of simulation results of several two-dimensional testing cases by the present scheme and an explicit FVLBM are conducted and numerical results show that the proposed implicit scheme can be as accurate as its explicit counterpart with 1 or 2 orders times speedup.

A gas-kinetic BGK scheme for the finite volume lattice Boltzmann method for nearly incompressible flows

This work presents a gas kinetic BGK scheme for the finite volume lattice Boltzmann method (FVLBM). In the present method, the formal analytical solution of the lattice Boltzmann BGK equation is used to determine the fluxes on cell interfaces. With such formal analytical solution, there are two remarkable features in the present FVLBM. One is that the effects of the collision term in the lattice Boltzmann equation (LBE) is included in the reconstructed fluxes on cell interfaces, which reduces numerical dissipation, and another is that, not as conventional flux schemes of FVLBM, the present flux scheme is a function of both space and time, which gives second order accuracy of both time and space in one step for unsteady flows. Furthermore, thanks to the collision invariant, by introducing a prediction step for the macro variables, the implicit feature of the equilibrium distribution function is removed and thus, the time-step in the present scheme is only limited by the Courant–Friedrichs–Lewy (CFL) condition. Additionally, some numerical implementations of boundary conditions are discussed and, to validate accuracy of the present scheme, comparison studies of several two dimensional incompressible unsteady and steady benchmark flows are also provided. The accuracy and effectiveness of the present scheme are clearly verified.

A Fully Implicit Finite Volume Lattice Boltzmann Method for Turbulent Flows

AbstractAlmost all schemes existed in the literature to solve the Lattice Boltzmann Equation likestream & collide, finite difference, finite element, finite volumeschemes are explicit. However, it is known fact that implicit methods utilizes better stability and faster convergence compared to the explicit methods. In this paper, a method named herein as Implicit Finite Volume Lattice Boltzmann Method (IFVLBM) for incompressible laminar and turbulent flows is proposed and it is applied to some 2D benchmark test cases given in the literature.Alternating Direction Implicit, an approximate factorization method is used to solve the obtained algebraic system. The proposed method presents a very good agreement for all the validation cases with the literature data. The proposed method shows good stability characteristics, the CFL number is eased. IFVLBM has about 2 times faster convergence rate compared with Implicit-Explicit Runge Kutta method even though it possesses a computational burden from the solution of algebraic systems of equations.

A matrix-free, implicit finite volume lattice Boltzmann method for steady flows

Abstract In the present paper, a matrix-free, implicit finite volume lattice Boltzmann method for steady flow on unstructured mesh is proposed. The approximate linear system arising from the implicit finite volume discretization of lattice Boltzmann equation (LBE) is solved by a novel algorithm, which combines the lower-upper symmetric Gauss–Seidel (LU-SGS) and the Jacobi iteration schemes. A remarkable feature of the present implicit method is that the storage of the Jacobian matrixes of the convection and collision terms can be completely eliminated by approximating the Jacobian matrix-solution incremental vector product with appropriate numerical flux incremental vector and the numerical increment of collision term, resulting a matrix-free implicit scheme. The present method is validated by several two-dimensional testing cases.

Finite Volume Lattice Boltzmann Method for Nearly Incompressible Flows on Arbitrary Unstructured Meshes

AbstractA genuine finite volume method based on the lattice Boltzmann equation (LBE) for nearly incompressible flows is developed. The proposed finite volume lattice Boltzmann method (FV-LBM) is grid-transparent,i.e., it requires no knowledge of cell topology, thus it can be implemented on arbitrary unstructured meshes for effective and efficient treatment of complex geometries. Due to the linear advection term in the LBE, it is easy to construct multi-dimensional schemes. In addition, inviscid and viscous fluxes are computed in one step in the LBE, as opposed to in two separate steps for the traditional finite-volume discretization of the Navier-Stokes equations. Because of its conservation constraints, the collision term of the kinetic equation can be treated implicitly without linearization or any other approximation, thus the computational efficiency is enhanced. The collision with multiple-relaxation-time (MRT) model is used in the LBE. The developed FV-LBM is of second-order convergence. The proposed FV-LBM is validated with three test cases in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the Blasius boundary layer; and (c) the steady flow past a cylinder at the Reynolds numbers Re=10, 20, and 40. The results verify the designed accuracy and efficacy of the proposed FV-LBM.

Modified momentum exchange method for fluid-particle interactions in the lattice Boltzmann method.

In this paper, a modified momentum exchange method for fluid-particle interactions is proposed based on the finite-volume lattice Boltzmann method. The idea of the improvement is to remove the restriction that the boundary points must be set as the midpoints of the grid lines or the intersection of the grid lines with the solid boundaries. The particle surface is represented by a set of arc (area) elements, and the interior fluid is used which the geometric conservation law is naturally satisfied. The interactions between fluid and arc (area) elements of particle boundary are considered using the momentum exchange method, and the mass of the fluid particles which collide with an arc (area) element is obtained by means of numerical integration in the control volume. The fluid field is corrected with the help of the smooth kernel function. Moreover, a generalized explicit time marching scheme is introduced to resolve the motion of particle in the problems with the ratio of particle density to fluid density is close to or less than 1. Finally, some numerical case studies of particle sedimentation are carried out to validate the present method. The corresponding results have a good agreement with the previous literature, which strongly demonstrates the capability of the improved method.

Transient conduction simulation of a nano-scale hotspot using finite volume lattice Boltzmann method

This paper uses the finite volume lattice Boltzmann method (FVLBM) to simulate the transient heat conduction from macro- to nano-scales corresponding to Kn = 0.01–10. This model is used for two dimensional (2D) transient hotspot modeling. The results of the diffusive regime are compared with those of the Fourier law as a model of continuum mechanics and an excellent agreement is found in this regime. After the validation of model for the case of Kn = 0.01, it has been used for high-Knudsen number simulations and a test case with Kn = 10 is studied. By increasing the Knudsen number from 0.01 up to 10, the transition from totally diffusive to totally ballistic behavior has been discussed and the wave-feature of heat transport through the solid material has been investigated.