This paper proposes an improved lattice Boltzmann scheme for incompressible axisymmetric flows. The scheme has the following features. First, it is still within the framework of the standard lattice Boltzmann method using the single-particle density distribution function and consistent with the philosophy of the lattice Boltzmann method. Second, the source term of the scheme is simple and contains no velocity gradient terms. Owing to this feature, the scheme is easy to implement. In addition, the singularity problem at the axis can be appropriately handled without affecting an important advantage of the lattice Boltzmann method: the easy treatment of boundary conditions. The scheme is tested by simulating Hagen-Poiseuille flow, three-dimensional Womersley flow, Wheeler benchmark problem in crystal growth, and lid-driven rotational flow in cylindrical cavities. It is found that the numerical results agree well with the analytical solutions and/or the results reported in previous studies.
We present an axisymmetric lattice Boltzmann model based on the Kupershtokh et al. multiphase model that is capable of solving liquid-gas density ratios up to $10^3$. Appropriate source terms are added to the lattice Boltzmann evolution equation to fully recover the axisymmetric multiphase conservation equations. We validate the model by showing that a stationary droplet obeys the Young-Laplace law, comparing the second oscillation mode of a droplet with respect to an analytical solution and showing correct mass conservation of a propagating density wave.
In this paper, a novel lattice Boltzmann (LB) model based on the Allen-Cahn phase-field theory is proposed for simulating axisymmetric multiphase flows. The most striking feature of the model is that it enables to handle multiphase flows with large density ratio, which are unavailable in all previous axisymmetric LB models. The present model utilizes two LB evolution equations, one of which is used to solve fluid interface, and another is adopted to solve hydrodynamic properties. To simulate axisymmetric multiphase flows effectively, the appropriate source term and equilibrium distribution fun…
In this paper, a lattice Boltzmann (LB) model is presented for axisymmetric multiphase flows. Source terms are added to a two-dimensional standard lattice Boltzmann equation (LBE) for multiphase flows such that the emergent dynamics can be transformed into the axisymmetric cylindrical coordinate system. The source terms are temporally and spatially dependent and represent the axisymmetric contribution of the order parameter of fluid phases and inertial, viscous and surface tension forces. A model which is effectively explicit and second order is obtained. This is achieved by taking into accoun…
A lattice Boltzmann method for axisymmetric multiphase flows is presented and validated. The method is capable of accurately modeling flows with variable density. We develop the classic Shan-Chen multiphase model [ Phys. Rev. E 47 1815 (1993)] for axisymmetric flows. The model can be used to efficiently simulate single and multiphase flows. The convergence to the axisymmetric Navier-Stokes equations is demonstrated analytically by means of a Chapmann-Enskog expansion and numerically through several test cases. In particular, the model is benchmarked for its accuracy in reproducing the dynamics…
In this paper, a multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is developed for simulating incompressible axisymmetric thermal flows in porous media at the representative elementary volume (REV) scale. In the model, a D2Q9 MRT-LB equation is proposed to solve the flow field in addition to the D2Q5 LB equation for the temperature field. The source terms of the model are simple and contain no velocity and temperature gradient terms. The generalized axisymmetric Navier-Stokes equations for axisymmetric flows in porous media are correctly recovered from the MRT-LB model through the C…
In this brief report, a thermal lattice-Boltzmann (LB) model is presented for axisymmetric thermal flows in the incompressible limit. The model is based on the double-distribution-function LB method, which has attracted much attention since its emergence for its excellent numerical stability. Compared with the existing axisymmetric thermal LB models, the present model is simpler and retains the inherent features of the standard LB method. Numerical simulations are carried out for the thermally developing laminar flows in circular ducts and the natural convection in an annulus between two coaxi…
We employ a recently formulated axisymmetric version of the multiphase Shan-Chen (SC) lattice Boltzmann method (LBM) [Srivastava et al, in preparation (2013)] to simulate the contraction of a liquid ligament. We compare the axisymmetric LBM simulation against the slender jet (SJ) approximation model [T. Driessen and R. Jeurissen, IJCFD {\bf 25}, 333 (2011)]. We compare the retraction dynamics of the tail-end of the liquid ligament from the LBM simulation, the SJ model, Flow3D simulations and a simple model based on the force balance (FB). We find good agreement between the theoretical predicti…
A cascaded lattice Boltzmann (LB) approach based on central moments and multiple relaxation times to simulate thermal convective flows, which are driven by buoyancy forces and/or swirling effects, in the cylindrical coordinate system with axial symmetry is presented. In this regard, the dynamics of the axial and radial momentum components along with the pressure are represented by means of the 2D Navier-Stokes equations with geometric mass and momentum source terms in the pseudo Cartesian form, while the evolutions of the azimuthal momentum and the temperature field are each modeled by an adve…
In this paper, an improved three-dimensional color-gradient lattice Boltzmann (LB) model is proposed for simulating immiscible multiphase flows. Compared with the previous three-dimensional color-gradient LB models, which suffer from the lack of Galilean invariance and considerable numerical errors in many cases owing to the error terms in the recovered macroscopic equations, the present model eliminates the error terms and therefore improves the numerical accuracy and enhances the Galilean invariance. To validate the proposed model, numerical simulation are performed. First, the test of a mov…
In this paper we show that standard implementations of fluctuating Lattice Boltzmann methods do not obey Galilean invariance at a fundamental level. In trying to remedy this we are led to a novel kind of multi-relaxation time lattice Boltzmann methods where the collision matrix depends on the local velocity. This new method is conceptually elegant but numerically inefficient. With a small numerical trick, however, this method recovers nearly the original efficiency and allows the practical implementation of fluctuating lattice Boltzmann methods with significantly improved Galilean invariance. …
In this paper, an improved thermal lattice Boltzmann (LB) model is proposed for simulating liquid-vapor phase change, which is aimed at improving an existing thermal LB model for liquid-vapor phase change [S. Gong and P. Cheng, Int. J. Heat Mass Transfer 55, 4923 (2012)]. First, we emphasize that the replacement of \[{\left( {ρ{c_V}} \right)^{ - 1}}\nabla \cdot \left( {λ\nabla T} \right)\] with \[\nabla \cdot \left( {χ\nabla T} \right)\] is an inappropriate treatment for diffuse interface modeling of liquid-vapor phase change. Furthermore, the error terms ${\partial_{t0}}\left( {Tv} \right) + …
In this study, a phase-field lattice Boltzmann model based on the Allen-Cahn equation with a filtered collision operator and high-order corrections in the equilibrium distribution functions is presented. Here we show that in addition to producing numerical results consistent with prior numerical methods, analytic solutions, and experiments with the density ratio of 1000, previous numerical deficiencies are resolved. Specifically, the new model is characterized by robustness at low viscosity, accurate prediction of shear stress at interfaces, and removal of artificial dense bubbles and rarefied…
Modeling liquid-vapor phase change using the lattice Boltzmann (LB) method has attracted significant attention in recent years. In this paper, we propose an improved three-dimensional (3D) thermal multiphase LB model for simulating liquid-vapor phase change. The proposed model has the following features. First, it is still within the framework of the thermal LB method using a temperature distribution function and therefore retains the fundamental advantages of the thermal LB method. Second, in the existing thermal LB models for liquid-vapor phase change, the finite-difference computations of t…
Lattice Boltzmann simulations of three-dimensional, isothermal hydrodynamics often use either the D3Q19 or the D3Q27 velocity sets. While both models correctly approximate Navier-Stokes in the continuum limit, the D3Q19 model is computationally less expensive but has some known deficiencies regarding Galilean invariance, especially for high Reynolds number flows. In this work we present a novel methodology to construct lattice Boltzmann equilibria for hydrodynamics directly from the continuous Maxwellian equilibrium. While our new approach reproduces the well known LBM equilibrium for D2Q9 and…
Cascaded or central-moment-based lattice Boltzmann method (CLBM) proposed in [Geier \textit{et al.}, Phys. Rev. E \textbf{63}, 066705 (2006)] possesses very good numerical stability. However, two constraints exist in three-dimensional (3D) CLBM simulations. Firstly, the conventional implementation for 3D CLBM involves cumbersome operations and requires much higher computational cost compared to the single-relaxation-time (SRT) LBM. Secondly, it is a challenge to accurately incorporate a general force field into the 3D CLBM. In this paper, we present an improved method to implement CLBM in 3D. …
A expansion to fourth-order for lattice Boltzmann methods is presented. This expansion provides an easy model for finding fourth-order corrections to lattice Boltzmann methods for various physical systems. The fourth-order terms can give rise to improved results over traditional second-order lattice Boltzmann implementations. Although, this manuscript solely deals with fourth-order expansions, this expansion is easily extended to arbitrary order. We present examples of how this expansion is utilized and provide basic analysis to show how the fourth-order methods differ from lower order models …
We generalize to three dimensions (3D) a recently developed improved multi-component pseudopotential lattice Boltzmann method and analyze its applicability to simulate flows through realistic porous media. The model is validated and characterized via benchmarks, and we investigate its performance by simulating the displacement of immiscible fluids in 3D geometries. Two samples are considered, namely, a pack of spheres obtained numerically, and a Bentheimer sandstone rock sample obtained experimentally. We show that, with this model it is possible to simulate realistic viscosity ratios, to tune…
The free-energy lattice Boltzmann (LB) model is one of the major multiphase models in the LB community. The present study is focused on a class of free-energy LB models in which the divergence of thermodynamic pressure tensor or its equivalent form expressed by the chemical potential is incorporated into the LB equation via a forcing term. Although this class of free-energy LB models may be thermodynamically consistent at the continuum level, it suffers from thermodynamic inconsistency at the discrete lattice level owing to numerical errors [Guo et al., Physical Review E 83, 036707 (2011)]. Th…
Chemical potential is an effective way to drive phase transition or express wettability. In this letter, we present a chemical-potential-based lattice Boltzmann model to simulate multiphase flows. The nonideal force is directly evaluated by a chemical potential. The model theoretically satisfies thermodynamics and Galilean invariance. The computational efficiency is improved owing to avoiding the calculation of pressure tensor. We have derived several chemical potentials of the popular equations of state from the free-energy density function. An effective chemical-potential boundary condition …
A D2Q9 Hybrid Lattice Boltzmann Method (HLBM) is proposed for the simulation of both compressible subsonic and supersonic flows. This HLBM is an extension of the model of Feng et al: [12], which has been found, via different test cases, to be unstable for supersonic regimes. The improvements consist of: (1) a new discretization of the lattice closure correction term making possible to properly simulate supersonic flows, (2) a corrected viscous stress tensor that takes into account polyatomic gases, and (3) a novel discretization of the viscous heat production term fitting with the regularized …
In this work, we improve the accuracy and stability of the lattice Boltzmann model for the Kuramoto-Sivashinsky equation proposed in \cite{2017_Otomo}. This improvement is achieved by controlling the relaxation time, modifying the equilibrium state, and employing more and higher lattice speeds, in a manner suggested by our analysis of the Taylor-series expansion method. The model's enhanced stability enables us to use larger time increments, thereby more than compensating for the extra computation required by the high lattice speeds. Furthermore, even though the time increments are larger than…
Owing to its conceptual simplicity and computational efficiency, the pseudopotential multiphase lattice Boltzmann (LB) model has attracted significant attention since its emergence. In this work, we aim to extend the pseudopotential LB model to simulate multiphase flows at large density ratio and relatively high Reynolds number. First, based on our recent work [Li et al., Phys. Rev. E. 86, 016709 (2012)], an improved forcing scheme is proposed for the multiple-relaxation-time pseudopotential LB model in order to achieve thermodynamic consistency and large density ratio in the model. Next, thro…
Immiscible fluid displacement in porous media occurs in several natural and industrial processes. For example, during petroleum extraction from porous rock reservoirs, water is used to displace oil. In this paper, we investigate primary drainage and imbibition in a heterogeneous porous medium using an improved numerical model based on the multicomponent pseudopotential lattice Boltzmann method. We apply recent developments from the literature and develop new pressure boundary conditions. We show that the proposed model is able to simulate realistic viscosity ratios, and it allows independent t…
The lattice Boltzmann method (LBM) has gained increasing popularity in incompressible viscous flow simulations, but it uses many more variables than necessary. This defect was overcome by a recent approach that solves the more actual macroscopic equations obtained through Taylor series expansion analysis of the lattice Boltzmann equations [Lu et al., J. Comp. Phys., 415, 109546 (2020)]. The key is to keep some small additional terms (SATs) to stabilize the numerical solution of the weakly compressible Navier-Stokes equations. However, there are many SATs that complicate the implementation of t…
During the last decade, lattice-Boltzmann (LB) simulations have been improved to become an efficient tool for determining the permeability of porous media samples. However, well known improvements of the original algorithm are often not implemented. These include for example multirelaxation time schemes or improved boundary conditions, as well as different possibilities to impose a pressure gradient. This paper shows that a significant difference of the calculated permeabilities can be found unless one uses a carefully selected setup. We present a detailed discussion of possible simulation set…
We develop a relativistic lattice Boltzmann (LB) model, providing a more accurate description of dissipative phenomena in relativistic hydrodynamics than previously available with existing LB schemes. The procedure applies to the ultra-relativistic regime, in which the kinetic energy (temperature) far exceeds the rest mass energy, although the extension to massive particles and/or low temperatures is conceptually straightforward. In order to improve the description of dissipative effects, the Maxwell-Juettner distribution is expanded in a basis of orthonormal polynomials, so as to correctly re…
Generalized Onsager-Regularized Lattice Boltzmann Method for error-free Navier-Stokes models on standard lattices
Istituto Italiano di Tecnologia（意大利理工大学）；Fondazione Istituto Italiano di Tecnologia（意大利理工大学基金会）；Consiglio Nazionale delle Ricerche（意大利国家研究委员会）；Harvard University（哈佛大学）
We show that, when a single relaxation time lattice Boltzmann algorithm is used to solve the hydrodynamic equations of a binary fluid for which the two components have different viscosities, strong spurious velocities in the steady state lead to incorrect results for the equilibrium contact angle. We identify the origins of these spurious currents, and demonstrate how the results can be greatly improved by using a lattice Boltzmann method based on a multiple-relaxation-time algorithm. By considering capillary filling we describe the dependence of the advancing contact angle on the interface ve…
Several applications exist in which lattice Boltzmann methods (LBM) are used to compute stationary states of fluid motions, particularly those driven or modulated by external forces. Standard LBM, being explicit time-marching in nature, requires a long time to attain steady state convergence, particularly at low Mach numbers due to the disparity in characteristic speeds of propagation of different quantities. In this paper, we present a preconditioned generalized lattice Boltzmann equation (GLBE) with forcing term to accelerate steady state convergence to flows driven by external forces. The u…
We propose a single-step simplified lattice Boltzmann algorithm capable of performing magnetohydrodynamic (MHD) flow simulations in pipes for very small values of magnetic Reynolds numbers $R_m$. In some previous works, most lattice Boltzmann simulations are performed with values of $R_m$ close to the Reynolds numbers for flows in simplified rectangular geometries. One of the reasons is the limitation of some traditional lattice Boltzmann algorithms in dealing with situations involving very small magnetic diffusion time scales associated with most industrial applications in MHD, which require …
Phonon Boltzmann transport equation (BTE) is an important tool for studying the nanoscale thermal transport. Because phonons have a large spread in their properties, the non-gray (i.e. considering different phonon bands) phonon BTE is needed to accurately capture the nanoscale transport phenomena. However, BTE solvers generally require large computational cost. Non-gray modeling imposes significant additional complexity to the numerical simulations, which hinders the modeling of real nanoscale systems. In this work, we address this issue by a systematic investigation on the phonon band discret…
Immersed boundary-lattice Boltzmann method (IB-LBM) has been widely used for simulation of particle-laden flows recently. However, it was limited to small-scale simulations with no more than O(103) particles. Here, we expand IB-LBM for massive particle-laden flows with more than O(104) particles by two sequential works. First is the Part I: serial computing on a single CPU core and following the Part II: parallel computing on many CPU cores. In this Part I paper, a highly efficient and localized implementation of IB-LBM is proposed for serial computing. We optimize in three main aspects: swap …
We describe an algorithm for the Rosenbluth Monte Carlo enumeration of clusters and lattice animals. The method may also be used to calculate associated properties such as moments or perimeter multiplicities of the clusters. The new scheme is an extension of the Rosenbluth method for growing polymer chains and is a simplification of a scheme reported earlier by one of the authors. The algorithm may be used to estimate the number of distinct lattice animals on any lattice topology. The method is validated against exact and Monte Carlo enumerations for clusters up to size 50, on a two dimensiona…
First of all, this paper presents some improvements of DSMC method in the form of new schemes and approaches, that, for a wide class of problems, increase performance and reduce the demands on computer resources. The most important improvement is the scheme of temporal factors, allowing the use of different time step for different sorts of particles, thus reducing the complexity and/or resource usage in simulation of stationary problems with very different collisional cross-sections between components of a mixture. Other improvements include the similarity parameter for efficient estimation of…
Pool boiling characteristics in two computational domains with and without considering conjugate heat transfer (CHT) were numerically simulated by an improved hybrid pseudopotential phase-change lattice Boltzmann method (LBM). The effects of constant temperature boundary condition (BC) with nucleate spots and fluctuant temperature on the boiling process were investigated in detail. It was found that for the computational domain without CHT, the treatment of constant temperature BC with nucleate spots is quite easier to produce film boiling than the temperature BC with small fluctuation. Howeve…
We assess the performance of the hybrid Open Accelerator (OpenACC) and Message Passing Interface (MPI) approach for multi-graphics processing units (GPUs) accelerated thermal lattice Boltzmann (LB) simulation. The OpenACC accelerates computation on a single GPU, and the MPI synchronizes the information between multiple GPUs. With a single GPU, the two-dimension (2D) simulation achieved 1.93 billion lattice updates per second (GLUPS) with a grid number of $8193^{2}$, and the three-dimension (3D) simulation achieved 1.04 GLUPS with a grid number of $385^{3}$, which is more than 76% of the theore…
A lattice Boltzmann method (LBM) with enhanced stability and accuracy is presented for various Hermite tensor-based lattice structures. The collision operator relies on a regularization step, which is here improved through a recursive computation of non-equilibrium Hermite polynomial coefficients. In addition to the reduced computational cost of this procedure with respect to the standard one, the recursive step allows to considerably enhance the stability and accuracy of the numerical scheme by properly filtering out second (and higher) order non-hydrodynamic contributions in under-resolved c…
A hybrid thermal lattice Boltzmann (LB) model is presented to simulate thermal multiphase flows with phase change based on an improved pseudopotential LB approach [Q. Li, K. H. Luo, and X. J. Li, Phys. Rev. E 87, 053301 (2013)]. The present model does not suffer from the spurious term caused by the forcing-term effect, which was encountered in some previous thermal LB models for liquid-vapor phase change. Using the model, the liquid-vapor boiling process is simulated. The boiling curve together with the three boiling stages (nucleate boiling, transition boiling, and film boiling) is numericall…
The pseudopotential multiphase lattice Boltzmann (LB) model is a very popular model in the LB community for simulating multiphase flows. When the multiphase modeling involves a solid boundary, a numerical scheme is required to simulate the contact angle at the solid boundary. In this work, we aim at investigating the implementation of contact angles in the pseudopotential LB simulations with curved boundaries. In the pseudopotential LB model, the contact angle is usually realized by employing a solid-fluid interaction or specifying a constant virtual wall density. However, it is shown that the…
Single component pseudo-potential lattice Boltzmann models have been widely studied due to their simplicity and stability in multiphase simulations. While numerous model have been proposed, comparative analysis and advantages and disadvantages of different force schemes is often lacking. A pseudo-potential model to simulate large density ratios proposed by Kupershtokh et al. is analyzed in detail in this work. Several common used force schemes are utilized and results compared. Based on the numerical results, the relatively most accurate force scheme proposed by Guo et al. is selected and appl…
