Standard 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.

A mathematical‐boundary‐recognition domain‐decomposition Lattice Boltzmann method combined with large eddy simulation applied to airfoil aeroacoustics simulation

AbstractBeing a direct computational aeroacoustics method, Lattice Boltzmann method (LBM) has great potential and broad application perspective in the field of numerical simulation of aerodynamic noise due to its low dispersion and low dissipation. A series of numerical algorithms and the related improvements based on the standard LBM method are proposed and developed in this paper to adapt to the airfoil noise calculation with complex grid at middle‐high Reynolds number. First, a new mathematical‐boundary‐recognition algorithm based on Green's formula is proposed to deal with complex curved geometric models, which is validated by three‐element airfoil 30P30N benchmark. Then, in order to reduce grid redundancy and improve computing efficiency, the grid refinement technique of domain decomposition model (DDM) is adopted and also improved, which is verified by calculating the flow and sound fields around 2D and 3D cylinders at Reynolds number equal to 90,000. Finally, three different LES turbulence models are combined with the standard MRT‐LBM method, where different finite difference schemes are used to solve Reynolds stress tensor which is different from the traditional one. Through the direct acoustic numerical simulation of NACA0012 airfoil at Reynolds number equal to 200,000, the effects of Smagorinsky models and Wall‐adapting local eddy‐viscosity (WALE) model on aerodynamic noise prediction are compared and analyzed. Overall, the proposed methodology is shown to be appropriate for predicting the aerodynamic noise at low Mach number and can successfully simulate the generation and propagation of far field acoustics.

An improved axisymmetric interfacial lattice Boltzmann flux solver for large-density-ratio multiphase flows

In this paper, an improved axisymmetric interfacial lattice Boltzmann flux solver abandoning the previous predictor-corrector scheme is proposed. Unlike the previous model starting from the two-dimensional standard lattice Boltzmann method (LBM), the present method is developed using the axisymmetric LBM, which directly incorporates the axisymmetric effects into the distribution functions. As a result, the proposed solver does not need the corrector step involving complicated space derivatives. It makes this method simpler and more computationally efficient. In the present solver, the resultant governing equation is globally resolved by the finite volume method, while the fluxes are reconstructed by local application of the axisymmetric LBM. Therefore, the inconsistency between the local reconstruction and the global governing equation can be eliminated because the global equation can be strictly derived from the axisymmetric LBM, which holds stronger physical basis than the previous method. Numerical experiments about the interface capturing and the multiphase flows are conducted to test the proposed model. Results show that the present method is superior to the fractional step solver in terms of the accuracy, stability, and computational efficiency. In addition, this solver has the capacity of simulating large-density-ratio and complex interfacial change.

High speed flows with particles on demand: Boundary conditions

The particles on demand (PonD) method is a new kinetic theory model that allows for simulation of high speed compressible flows. While the standard lattice-Boltzmann method is limited by a fixed reference frame, significantly reducing the range of Mach numbers, PonD takes advantage of adaptive reference frames to get rid of the restrictions of standard LB and is able to simulate flows at high speeds and with large temperature variation. Previously, PonD has been shown to be a viable alternative for simulation of flows with strong discontinuities and for detonation modelling. However, treatment of flows with complex boundaries has been lacking. Here, we present PonD augmented with a non-equilibrium extrapolation based boundary condition. We present several compressible test cases such as shock-vortex interaction in the Schardin’s problem and supersonic flow over a two-dimensional cylinder at Mach numbers up to 5. We observe that the results agree well with literature, paving the way for a kinetic theory based approach for simulating compressible flows in realistic scenarios.

Study of the convergence of the Meshless Lattice Boltzmann Method in Taylor–Green, annular channel and a porous medium flows

The Meshless Lattice Boltzmann Method (MLBM) is a numerical tool that relieves the standard Lattice Boltzmann Method (LBM) from regular lattices and, at the same time, decouples space and velocity discretizations. In this study, we investigate the numerical convergence of MLBM in three benchmark tests: the Taylor–Green vortex, annular (bent) channel and a porous medium flow. We compare our MLBM results to LBM and to the analytical solution of the Navier–Stokes equation. We investigate the method’s convergence in terms of the discretization parameter, the interpolation order, and the LBM streaming distance refinement. We observe that MLBM outperforms LBM in terms of the error value for the same number of nodes discretizing the domain. We find that LBM errors at a given streaming distance δx and timestep length δt are the asymptotic lower bounds of MLBM errors with the same streaming distance and timestep length. Finally, we suggest an expression for the MLBM error that consists of the LBM error and other terms related to the semi-Lagrangian nature of the discussed method itself.

Non-orthogonal multiple-relaxation-time lattice Boltzmann method for vorticity-streamfunction formulation

In this work, we propose a non-orthogonal multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) for the two-dimensional (2D) incompressible Navier-Stokes equations in vorticity-stream function formulation. In this method, the vorticity and the streamfunction equations are calculated by two separate LB equations, which are constructed by using the non-orthogonal transformation matrix. Most importantly, to eliminate the residual term that may appear in the recovered vorticity equation, the convection term is handled as a source term in the LB equation rather than being directly recovered by the evolution equation. In addition, due to the convection term in our method is computed with a local scheme, it retains the parallel feature of the standard LBM. We then test the accuracy and the stability of the present LBM by considering various benchmark examples. The results show that the present method has a second-order convergence rate in space. Our numerical results indicate that the present LBM is a good candidate for solving streamfunction-vorticity formulation.

A hybrid lattice Boltzmann - Navier-Stokes method for unsteady aerodynamic and aeroacoustic computations

A hybrid numerical method coupling the standard lattice Boltzmann method and a compressible finite-volume Navier-Stokes solver is proposed in the context of unsteady aerodynamic and aeroacoustic simulations. The trend being towards more realistic and detailed simulations in a reasonable amount of CPU time, lattice Boltzmann and Navier-Sokes methods can be combined to solve the same problem. The present hybrid method relies on a zonal decomposition of the computational domain thus allowing to exploit the numerical features of both methods to their optimal extent in specific flow regions.The key issue when combining the lattice Boltzmann method with a Navier-Stokes solver is to ensure a smooth transition of the flow variables at the two-way coupling interface. While existing approaches consider overlapping meshes, a direct grid coupling is here derived. The mapping from the macroscopic flow variables to the set of lattice Boltzmann distribution functions is performed analytically thanks to a Chapman-Enskog expansion and draws a direct link to advanced regularised collision operators. Unsteady computations are enabled by coupling the lattice Boltzmann stream and collide algorithm with explicit and implicit Navier-Stokes time-stepping schemes. The hybrid method is then assessed on four time-dependent test cases representative of aerodynamic and aeroacoustic problems. The proposed approach is proven to yield very accurate results while keeping the numerical advantages of both methods and reducing the overall computational cost of direct noise computations.

Acoustic multipole source–simplified lattice Boltzmann method for simulating acoustic propagation problems

AbstractCombining the advantages of the acoustic multipole source (AMS) method and the simplified lattice Boltzmann method (SLBM), a method called AMS–SLBM is proposed for simulating the propagation of acoustic multipole sources in fluids. A particle source term is introduced to the right‐hand side of the lattice Boltzmann equation using the AMS method, and the macroscopic equations with source terms are derived via the Chapman–Enskog expansion analysis. Employing the fractional step technique, the solving process of the macroscopic equations can be divided into three steps: the predictor, corrector, and supplement steps. In the predictor and corrector steps, macroscopic equations without source terms are solved by SLBM, and in the supplement step, the time advancement of the source terms is solved using the finite difference method. AMS–SLBM uses SLBM to simulate the propagation of sound waves by directly evolving the macroscopic variables, which evades the evolution and storage of the distribution function, and the computational process is simpler and memory can be reduced compared to the standard LBM. Moreover, since the acoustic source term is introduced to the right‐hand side of the lattice Boltzmann equation by the AMS method, AMS–SLBM avoids the disadvantage that the traditional forced equilibrium distribution function method will interfere and cover the original flow field during the calculations. Several cases including the propagation of a plane wave, a Gaussian pulse and acoustic monopole, dipole, and quadrupole sources are simulated to validate the robustness and accuracy of the present method. The results show that AMS–SLBM can well simulate acoustic multipole sources propagation, and it affords second‐order accuracy.

Prediction and Validation of Flow Properties in Porous Lattice Structures

Abstract High-porosity metal foams have been extensively studied as an attractive candidate for efficient and compact heat exchanger design. With the advancements in additive manufacturing, such foams can be manufactured with controlled topology to yield highly tailorable mechanical and transport properties. In this study, a lattice Boltzmann method (LBM)-based pore-scale model is implemented to simulate the fluid flow in additively manufactured (AM) metal foams with unit cell topologies of Cube, Face Diagonal (FD)-Cube, Tetrakaidecahedron (TKD), and Octet lattices. The pressure gradient versus average velocity profiles predicted by the LBM model were validated against in-house measurements on the AM lattice samples with the same unit cell topologies. Based on the simulation results, a novel hybrid model is proposed to accurately predict the volume averaged flow properties (permeability and inertial coefficients) of the four structures. Specifically, the linear LBM (neglecting inertial forces) is first implemented to obtain the intrinsic permeability, and then the standard LBM is applied to obtain the inertial coefficient. Convenient correlations for those flow properties as a function of porosity and fiber diameter are constructed. The effects of the AM print qualities on the flow properties are also discussed. The advantages of the hybrid model compared to the polynomial fitting approach for determining flow properties are discussed and compared quantitatively. The hybrid model and presented results are valuable for flow and thermal transport evaluation when designing new metal foams for specific applications and with different materials and topologies. The presented correlations based on pore-scale simulations can also be conveniently used in volume-averaged models to predict the macroscale flow behavior in such complex structures.

Thermal analysis of multi-finned plate employing lattice Boltzmann method based on Taylor-series/least-squares

A comprehensive experimental/numerical investigation is carried for the purpose of heat transfer and hydrodynamic analysis of natural convection within a cavity equipped with active rectangular fins. Numbers of hot and cold fins are fitted at the surface of square cavity which have remarkable effect on the hydrothermal performance of cavity. In this work, some parameters with significant impact on hydrothermal performance are selected to be studied such as value of Rayleigh number (Ra), value of nanoparticle volume concentration (φV) and arrangement of active fins (Ther.Arr A, B, C and D). To reach this purpose, numerical approach and empirical observations are coupled to provide an accurate data. In order to make the standard Lattice Boltzmann Method (LBM) compatible with the non-uniform grid distribution, the Taylor series expansion and least square-based lattice Boltzmann method (TLLBM) is used. Furthermore, firstly, the SiC-EG nanofluid samples are prepared using modern experimental, and then its rheological and thermal properties are measured using modern devices. It is concluded that using of nanofluid and the thermal arrangement of fins, have significant impact on the thermal performance of cavity.

Parallel Scheme for Multi-Layer Refinement Non-Uniform Grid Lattice Boltzmann Method Based on Load Balancing

The large-scale numerical simulation of complex flows has been an important research area in scientific and engineering computing. The lattice Boltzmann method (LBM) as a mesoscopic method for solving flow field problems has become a relatively new research direction in computational fluid dynamics. The multi-layer grid-refinement strategy deals with different-level of computing complexity through multi-scale grids, which can be used to solve the complex flow field of the non-uniform grid LBM without destroying the parallelism of the standard LBM. It also avoids the inefficiencies and waste of computational resources associated with standard LBMs using uniform and homogeneous Cartesian grids. This paper proposed a multi-layer grid-refinement strategy for LBM and implemented the corresponding parallel algorithm with load balancing. Taking a parallel scheme for two-dimensional non-uniform meshes as an example, this method presented the implementation details of the proposed parallel algorithm, including a partitioning scheme for evaluating the load in a one-dimensional direction and an interpolation scheme based on buffer optimization. Simply by expanding the necessary data transfer of distribution functions and macroscopic quantities for non-uniform grids in different parallel domains, our method could be used to conduct numerical simulations of the flow field problems with complex geometry and achieved good load-balancing results. Among them, the weak scalability performance could be as high as 88.90% in a 16-threaded environment, while the numerical simulation with a specific grid structure still had a parallel efficiency of 77.4% when the parallel domain was expanded to 16 threads.

Investigation on reactive flow through porous media by quadtree Lattice Boltzmann

In this study, in order to investigate the effect of the underlying pore-scale processes on continuum scale simulations of porous media dissolution, we improve the standard Lattice Boltzmann method using Quadtree grid refinement approach to simulate fluid flow and reactive transport through large domain sizes. Our results have shown considerable computational improvements up to 80% in simulation time together with increased numerical accuracy. The results and the added value of the new approach are discussed using comparison of our model with the conventional LBM. Moreover, we have applied a systematic analysis by increasing complexity levels and starting from fluid flow and continuing with tracer transport and reactive transport in single pores, and ultimately, reactive transport in dissolving porous structures. For each case, the accuracy and computational benefits of the Quadtree models are discussed and dimensionless numbers are used to characterize regimes of flow and reaction in each step of comparison. Porosity-permeability variation in a 2D pore structure and mass transfer coefficient in a closed-end fracture with porous walls are then evaluated under different flow conditions.

A curvilinear lattice Boltzmann scheme for thermal flows

Lattice Boltzmann schemes are known for their efficiency and low dissipation properties. However, the standard lattice Boltzmann method (LBM) is limited to Cartesian grids, and this approach can be troublesome when approximating thermal flows over curved walls. The present work proposes to solve the two-dimensional lattice Boltzmann equation under curvilinear coordinate transformation to simulate thermal flows with body-fitted grids. Several test cases are discussed, and the results are extensively compared with a very favourable outcome to the available numerical and experimental data, confirming the importance of the implemented methodology. Natural convection in a square cavity and a wavy cavity, as well as flow in a concentric cylindrical annulus are used for validation. Forced convection over a heated two-dimensional cylinder is also included.

An efficient implementation of nodal discontinuous Galerkin lattice Boltzmann method and validation for direct numerical simulation of turbulent flows

The application of finite element method in solving the lattice Boltzmann equation has been successfully demonstrated by many researchers. However, the computation cost of these methods is much higher than the standard lattice Boltzmann method. In this paper, an efficient implementation of nodal discontinuous Galerkin lattice Boltzmann method is presented. It relies on a simplified nodal discontinuous Galerkin formulation, which makes full use of the possible sparse property of elemental matrices in quadrangles and hexahedrons. Some key points of its implementation in the program are also discussed. Performance tests confirm the superior efficiency of the current program, which exceeds the record of previous researchers. The accuracy of the current method is investigated and compared with the standard lattice Boltzmann method. The high-order discretization can give similar accuracy with lower number of grid points and computational time and the efficiency of the present method is sometimes higher than standard lattice Boltzmann method. Direct numerical simulations of several incompressible turbulence benchmark cases are performed. The present method also shows better accuracy and stability especially in the turbulent pipe flow simulations.

A high-order implicit-explicit flux reconstruction lattice Boltzmann method for viscous incompressible flows

In this paper, a high-order implicit-explicit flux reconstruction lattice Boltzmann method (FRLBM) is proposed for simulating viscous incompressible flows. The discrete velocity Boltzmann equation (DVBE) is solved by using a high-order flux reconstruction scheme (FR) for spatial discretization and a high-order implicit-explicit Runge-Kutta scheme (IMEX RK) for time discretization in the generalized curvilinear coordinates with uniform and non-uniform grids to obtain high simulation efficiency. Numerical validations of the proposed method are implemented by simulating (a) Taylor-Green vortex problem, (b) doubly periodic shear layer flow, (c) lid-driven cavity flow, (d) cylindrical Couette flow and (e) flow over a circular cylinder. Numerical results demonstrate that the present method is stable and accurate even at high Reynolds number. Compared with the existing high-order off-lattice Boltzmann methods, the present method is more efficient. In addition, the present method holds the compactness of the standard LBM for parallel computing, but greatly improves the adaptability of complicated geometries.

Family of parametric second-order boundary schemes for the vectorial finite-difference-based lattice Boltzmann method.

In this paper, we propose a family of parametric second-order boundary schemes for the vectorial finite-difference-based lattice Boltzmann method (FD-LBM), which consist of convex combinations. The FD-LBM unifies several different numerical schemes for the Navier-Stokes equations, and thereby these boundary schemes are naturally applicable for the standard LBM. The accuracy of the boundary schemes is independent of the boundary location, and it is validated by several numerical experiments, two- and three-dimensional flow problems, with straight and curved boundaries.

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.

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.

Development of an immersed boundary-multiphase lattice Boltzmann flux solver with high density ratio for contact line dynamics

Interaction between a two-phase fluid and a structure involving contact line dynamics is a common phenomenon. In this paper, we aim to develop a fluid–solid coupling model that can study contact line dynamics in the case of a high density ratio between the two fluids. The fluids are treated using a multiphase lattice Boltzmann flux solver (MLBFS) that uses the cell-centered finite volume method to obtain macroscopic flow variables, and the interface fluxes are reconstructed locally by the standard lattice Boltzmann method (LBM) solutions. This approach retains the advantages of the original LBM while being more flexible in handling nonuniform grids and external force terms. The immersed boundary method (IBM) is an effective method for processing structural information, and here, the implicit boundary-condition-enforced IBM is used to accurately satisfy the Dirichlet boundary condition (no-slip boundary). Moreover, the Neumann boundary condition is deemed to represent the contribution from the structure boundary flux and is incorporated into the IB-MLBFS. The developed IB-MLBFS is verified by several test cases, including contact line motion of a two-phase fluid along a circular cylinder and droplet spreading on a flat plate, where both equilibrium results and dynamic process are correctly reproduced for different density ratios and wettability conditions. Furthermore, based on the IB-MLBFS established here, the contact line dynamics of a two-phase fluid between two square cylinders or two circular cylinders is studied. The effects of distance, structure size, and wettability on the interface state and the contact angle are studied in detail. The robustness of the proposed model is verified.

Regularization and the particles-on-demand method for the solution of the discrete Boltzmann equation

In its classical formulation, the lattice Boltzmann method (LBM) is applicable in the range of subsonic velocities and small temperature ratios. The novel Particle-on-demand method (PonD) allows to numerically solve the discrete Boltzmann equation for high Mach numbers. In comparison with the standard LBM, the collision step is simple, but the streaming step is implicit, not mass conserving and computationally heavy. A large part of the computational cost comes from matrix inversions during the rescaling of the discrete distribution functions (DF) from one gauge to another. To avoid matrix inversions, we propose another method of discrete DF rescaling, where the discrete DF are restored from the moments, while the conversion to a different reference frame is in the moment space. Results obtained by this improved method were compared to results, received by standard PonD for a number of problems. This improvement is validated to produce similar results to the original PonD, and is computationally cheaper in comparison with PonD.