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The lattice Boltzmann method: Fundamentals and acoustics
Abstract and Figures
The lattice Boltzmann method has been widely used as a solver for incompressible flow, though it is not restricted to this application. More generally, it can be used as a compressible Navier-Stokes solver, albeit
with a restriction that the Mach number is low. While that restriction may seem strict, it does not hinder the application of the method to the simulation of sound waves, for which the Mach numbers are generally
very low. Even sound waves with strong nonlinear effects can be captured well. Despite this, the method has not been as widely used for problems where acoustic phenomena are involved as it has been for incompressible problems.
The research presented this thesis goes into three different aspects of lattice Boltzmann acoustics. Firstly, linearisation analyses are used to derive and compare the sound propagation properties of the lattice
Boltzmann equation and comparable fluid models for both free and forced waves. The propagation properties of the fully discrete lattice Boltzmann equation are shown to converge at second order towards those of the discrete-velocity Boltzmann equation, which itself predicts the same lowest-order absorption but different dispersion to the other fluid models.
Secondly, it is shown how multipole sound sources can be created mesoscopically by adding a particle source term to the Boltzmann equation. This method is straightforwardly extended to the lattice Boltzmann
method by discretisation. The results of lattice Boltzmann simulations of monopole, dipole, and quadrupole point sources are shown to agree very well with the combined predictions of this multipole method and the linearisation analysis. The exception to this agreement is the immediate vicinity of the point source, where the singularity in the analytical solution cannot be reproduced numerically.
Thirdly, an extended lattice Boltzmann model is described. This model alters the equilibrium distribution to reproduce variable equations of state while remaining simple to implement and efficient to run. To
compensate for an unphysical bulk viscosity, the extended model contains a bulk viscosity correction term. It is shown that all equilibrium distributions that allow variable equations of state must be identical for the one-dimensional D1Q3 velocity set. Using such a D1Q3 velocity set and an isentropic equation of state, both mechanisms of nonlinear acoustics are captured successfully in a simulation, improving on previous isothermal simulations where only one mechanism could be captured. In addition, the effect of molecular relaxation on sound propagation is simulated using a model equation of state. Though the particular implementation used is not completely stable, the results agree well with theory.
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... The values of the weight coefficients and lattice sound speed directly influence the stability of simulations. For the discrete vectors of the LBM model, certain assumptions must be satisfied to ensure the isotropic nature of the discrete lattice, which is essential for the accurate derivation of the macroscopic Navier-Stokes equations [22]. This subsection will address four commonly used 2D velocity models employed in the square lattices, i.e., D2Q4, D2Q5, D2Q9, and D2Q13. ...
... Diverging from the LBM-SRT model, the collision process in the LBM-MRT model necessitates operation within moment space, wherein the particle distribution function vector f is mapped into the moment space vector m. Equation (1) can be transformed into the following form [22,48]: ...
... Before conducting numerical simulations of the seismic wavefields using the LBM schemes, it is crucial to perform numerical dispersion and computational stability analysis [22,48]. This analysis serves a fundamental purpose and holds significance for ensuring the accuracy, reliability, and credibility of the subsequent simulations. ...
The simulation of seismic wavefields holds paramount significance in understanding subsurface structures and seismic events. The lattice Boltzmann method (LBM) provides a computational framework adept at capturing detailed wave interactions, offering a new approach to improve seismic wavefield simulations. Our study involves a novel comparative analysis of wavefields using different lattice Boltzmann models, focusing on how relaxation times, discrete velocity models, and collision operators affect simulation accuracy and efficiency. We explore the impacts of distinct relaxation times and evaluate their effects on wave propagation speed and fidelity. By incorporating four discrete velocity models of LBM, we innovatively investigate the trade-off between spatial resolution and computational complexity. Additionally, we delve into the implications of employing three collision operators—single relaxation time (SRT), two relaxation times (TRT), and multiple relaxation times (MRT). By comparing their accuracy and stability, we provide insights into selecting the most suitable collision operator for capturing complex wave interactions. Our research provides a comprehensive framework to optimize the LBM parameters, enhancing both accuracy and efficiency in seismic wave simulations, and offers valuable insights to benefit wave simulation across diverse disciplines.
... Li and Shan (2011) conduct initial numerical evidence that adiabatic sound can be adequately simulated by the LBM provided sufficient hydrodynamic moments are retained in velocity-space discretization and the sound frequency is much lower than the molecular collision frequency. Viggen (2014) provides a detailed account of the application of the LBM scheme for acoustic waves and finds that the LBM is relevant as a simple compressible Navier-Stokes solver for cases where there is an interaction between the flow field and the acoustic field. Dhuri et al. (2017) offer a linear lattice Boltzmann formulation for the simulation of linear acoustic wave propagation, and they derive the dispersion relation for 2D LBM schemes and offer the optimal parameters to implement LBM by comparing with that of the finite-difference method (FDM). ...
... As we are using LBE to simulate acoustic waves in the LBM scheme, one would wonder whether the wave equation characterizing the wave problem can be derived from the LBE? As LBM is an efficient solving algorithm for the Navier-Stokes equation, we try to recover the viscoacoustic wave equation from the following Navier-Stokes-Fourier equations (Feireisl et al., 2012;Viggen, 2014): ...
The lattice Boltzmann method (LBM), widely employed in computational fluid mechanics, is introduced as a novel mesoscopic numerical scheme for viscoacoustic wavefield simulation. Through mathematical derivation, a mapping model between the relaxation time of LBM and the quality factor based on the Kelvin-Voigt model is established, which provides a theoretical background for the comparison of the viscoacoustic wavefields obtained respectively by LBM and finite difference method (FDM) formulated on the traditional wave equation. By defining the transmission and reflection coefficients and adopting a Newton interpolation algorithm to modify the streaming process of the LBM, we extend the conventional LBM to simulate the wavefields in complex media with an acceptable accuracy. A two-dimensional (2D) homogeneous model, two layered models and the modified Marmousi model are tested in the numerical simulation experiments. The simulation results of LBM are comparable to those of FDM, and the relative errors are all within reasonable range, which can verify the effectiveness of the proposed forward modeling kernel. The modified LBM offers a new numerical scheme in seismology to simulate viscoacoustic wave propagation in complex media and even in porous media considering its flexible boundary condition and high discrete characteristic.
... The multipole approach adds a source term directly in the LBE, acting as a particle source [45,46]. The MRT regularized BGK operator was implemented to allow convergence at a low viscosity. ...
The Lattice Boltzmann Method (LBM) can be applied to several fluid dynamic problems in the time domain. This numerical method indirectly solves the Navier-Stokes equations in a weakly compressible limit that allows acoustic wave propagation. This work presents a systematic literature review concerning the application of the LBM in acoustics. Applications found in the literature are classified and presented in different categories, including wave theory, boundary conditions, sound absorption materials, aeroacoustics, and musical acoustics. The increasing amount of research in recent years about aeroacoustics is remarkable, thanks to the intrinsically coupled treatment of the acoustical field and the mean flow, the potential of studying different wave phenomena such as diffraction and scattering, the easy way to model complex geometric boundaries in 2D and 3D, and finally thanks to the increasing available computational power. Some examples were included to illustrate the LBM capabilities to simulate sound wave phenomena, including point source modeling, diffraction and interference of sound waves, jet noise, and edge noise. This work will give a retrospective of the research developed in the past and a perspective on how this numerical method might evolve in the acoustical field.
... However, for acoustic problems, which involve wave propagation, other conversion techniques have been proposed. The most well-known procedure [36,[58][59][60] is that which allows direct relationships between the LBM and physical acoustic quantities by means of the space step x (m) and the time step t (s), which represent the physical distance between the LBM lattice nodes and the physical time needed by a particle to pass from one node to another, respectively. ...
The paper presents a three-dimensional numerical study of the acoustic streaming induced by the dissipation of ultrasounds during their propagation in the air. The waves are generated by a circular acoustic source positioned at the center of the left wall of a parallelepipedic cavity. The simulations are performed with the lattice Boltzmann method associated with the D3Q19 multiple relaxation time model. A validation of this model is first performed by comparing the numerical and analytical acoustic intensities along the central axis of the acoustic source. The main objective of this study is to use two different methods to calculate the acoustic streaming flow. The first method is the direct calculation of the mean velocity fields as the mean values of the instantaneous velocities. The second method is an indirect technique, which first calculates the acoustic streaming force and then injects this force into the numerical code to produce the streaming. A comparison between the results obtained by the two methods was carried out and a good agreement was found between them. These different investigations, rather new in three-dimensional configurations, have allowed us to discuss the advantages and limitations of the lattice Boltzmann approach to simulate real situations of wave propagation and acoustic streaming.
Graphical abstract
... It is easy to implement [28] and contains many local operations, which allows it to scale well in parallel computing, e.g. on compute clusters and GPUs, [28,46] and thus offer high performance [31]. At the same time, it is also able to handle complex or even moving geometries well [15,41,46] and can simulate particle movement, sound generation, and sound waves interacting with surrounding flow [49], thus offering many interesting features for the simulation of urban flows. ...
The study of low-rise buildings' interaction with wind is gravitating toward large eddy simulation (LES) for simulating airflow. As it is straightforward to implement and also offers the option to simulate particles, sound, and heat flow, the lattice Boltzmann method (LBM) may be an interesting option among the LES methods but has so far barely been explored for this application. In this work, the LBM is investigated and assessed as a tool for the analysis of wind flow around low-rise buildings. The hardware resources used are limited to one GPU to mimic the limited resources an industrial setting or a preliminary study might allow for. We suggest an efficient LBM simulation setup and compare its results from the wind flow around nine exemplary gable roofed low-rise buildings to findings of previous studies, both in detail and in terms of found behaviors. All comparisons show good agreement between our findings and the previous results obtained via other simulation methods or wind tunnel measurements. These results confirm that the LBM may be used to investigate low-rise buildings, even with limited hardware. It is thus a further contender among the available LES methods and our setup may serve as starting point for its application.
... [156] , [161] Recently, the increase in computational power and the development of the so-called Lattice Boltzmann Method (LBM) made possible to describe this process meticulously. [162] , [163] , [164] Compared to other techniques, the strength of the LBM is its mesoscopic nature based on the discrete kinetic theory. At the mesoscopic level, the LBM models combine microscopic dynamics, such as fluid-fluid and fluid-solid boundary interactions, and the macroscopic kinetic theory of fluids, like the Navier-Stokes equation in the bulk flow. ...
The aim of the thesis was to create the models which describes electrolyte infiltration and characterize its performance. The electrochemical impedance spectroscopy model should be developed based on Newman model. The Lattice Boltzmann Method must be used to describe the electrolyte infiltration process. All the models were combined with experimental results. The thesis was part of the ERC-Artistic project. The first stage of the thesis was devoted to develop the Electrochemical impedance Spectroscopy model. EIS constitutes an experimental technique used for the characterization of LIB porous electrodes tortuosities. For the first time, a 4D (3D in space + time) physical model is proposed to simulate EIS carried out on NMC porous cathodes, derived from the simulation of their manufacturing process, in symmetric cells. This methodology allows to understand the limitations of using EIS, electric circuit models and homogenized physical models for the determination of the tortuosity of NMC-based cathodes, revealing a complex interplay between the conductivity of the solid phases, the electrolyte properties and the cathode meso/microstructure. The second stage of the thesis was devoted to development of the electrolyte infiltration process. This step is crucial as it is directly linked to LIB quality and affects the subsequent time consuming electrolyte wetting process. It was reported here for the first time a 3D-resolved Lattice Boltzmann Method model able to simulate electrolyte filling upon applied pressure of LIB porous electrodes obtained both from experiments (micro X-ray tomography) and computations (stochastic generation, simulation of the manufacturing process using Coarse Grained Molecular Dynamics and Discrete Element Method). The model allows obtaining advanced insights about the impact of the electrode mesostructures on the speed of electrolyte impregnation and wetting, highlighting the important of porosity, pore size distribution and pores interconnectivity on the filling dynamics. Furthermore, we identify scenarios where volumes with trapped air (dead zones) appear and evaluate the impact of those on the electrochemical behavior of the electrodes. Further an innovative machine learning model, based on deep neural networks, to fast and accurately predict fluid flow in three dimensions, as well as wetting degree and time for LIB electrodes. The ML model is trained on a database generated using a 3D-resolved physical model based on the Lattice Boltzmann Method. We demonstrate the ML model with a NMC electrode mesostructure obtained by X-ray micro-computer tomography. The extracted pore network from tomography data was also used to train our ML neural network. The results show that the ML model is able to predict the electrode filling process, with ultralow computational cost (few seconds) and with high accuracy when compared with the original data generated with the physical model. Also, systematic sensitivity analysis was carried out to unravel the spatial relationship between electrode mesostructure parameters and predicted infiltration process characteristics, such as saturation dynamics, filling time among others. Finally, the EIS, LBM and Machine learning models will be integrated into the ARTISTIC platform. The platform can be used to simulate, understand, and optimize battery manufacturing. The platform will be free of charge and all the data and codes will be available
... An alternative and promising scheme for simulating seismic wave propagation is the so-called lattice Boltzmann method (LBM) (Guangwu, 2000), which is a mesoscopic numerical method that does not require to solve macro equations. LBM originated from lattice gas automata (LGA) (Frisch et al., 1986;Treurniet et al., 2000), and was first introduced in computational fluid dynamics by McNamara and Zanetti (1988) to overcome some of LGA's main shortcomings, such as statistical noise, limited range of physical parameters, and non-Galilean invariance (D'Humieres, 2002;Viggen, 2014. Since it is simple to implement and efficiently parallelize, and the processing of boundary conditions is flexible, LBM thereby has successful applications in the fields of acoustic simulation (Li and Shan, 2011;Xia et al., 2017;Zhuo and Sagaut, 2017, computational fluid dynamics (Tiribocchi et al., 2009;Zhao and Wang, 2019), computational aeroacoustics (Kim et al., 2007;Popescu and Johansen, 2009), and fluid multiphase flow , and so on. ...
In general, the wave equation can be taken to characterize the seismic wavefield propagation from a macroscopic perspective. The lattice Boltzmann method (LBM) is an alternative strategy to model seismic wavefields on mesoscopic scale. Due to its fully discrete nature and flexible boundary processing, LBM therefore has attracted increasing attention in seismology. The stability problem of LBM is a critical aspect in its seismic wavefield simulation. We give the steps of employing LBM for seismic wavefield modeling and some comparisons between LBM and the wave equation. Further, we derive theoretically the stability conditions of Bhatnagar-Groos-Krook (BGK) LBM and multiple-relaxation-time (MRT) LBM. Although the stability adjustment of MRT-LBM is more flexible, it is rather difficult to obtain an analytical solution concerning its relaxation parameters and other factors. In view of this, we first demonstrate that the stability and accuracy of MRT-LBM is superior to BGK-LBM to a certain extent by means of seismic wavefield modeling. Then, we investigate the individual effect of each relaxation parameter of MRT-LBM on the seismic waveforms. Most significantly, we construct the stability models for MRT-LBM only related to the relaxation parameters, based on the stability results of a large number of wavefield simulations in the homogeneous medium. Our stability models can well guide MRT-LBM for stable seismic wavefield modeling without considering other factors. Finally, we verify the correctness as well as the wide applicability of the proposed stability models by some simple layered media and modified Marmousi and BP models. The stability models may play a nice guiding role for further wavefield simulations based on MRT-LBM, especially for low-viscosity media.
... The simplest way is to perform this conversion between the two sets of units by using reference physical quantities for length, time, and density. We will use the spacing of the nodes Δ (m), the physical time step Δ (s) and the mean density of the fluid (Kg/ m 3 ) [19,49]. With this choice, Δ = = 1 , Δ = = 1 and = 1. ...
This paper presents a numerical investigation of the propagation of acoustic waves generated by a linear acoustic source using the lattice Boltzmann method (LBM). The main objective of this study is to compute the sound pressure and acoustic force produced by a rectangular sound source located at the center of the west wall of a rectangular cavity, filled with water. The sound source is discretized into a set of point sources emitting waves according to the acoustic point source method. The interference between the generated cylindrical waves creates an acoustic beam in the cavity. An analytical study is carried out to validate these numerical results. The error between the numerical and analytical calculations of the wave propagation is also discussed to confirm the validity of the numerical approach. In a second step, the acoustic streaming is calculated by introducing the acoustic force into the LBM code. A characteristic flow structure with two recirculating cells is thus obtained.
When gas flows through corrugated pipes, pressure waves interacting with vortex shedding can produce distinct tonal noise and structural vibration. Based on established observations, a model is proposed which couples an acoustic pipe and self-excited oscillations with vortex shedding over the corrugation cavities. In the model, the acoustic response of the corrugated pipe is simulated by connecting the lossless medium moving with a constant velocity with a source based on a discrete distribution of van der Pol oscillators arranged along the pipe. Our time accurate solutions exhibit dynamic behavior consistent with that experimentally observed, including the lock-in frequency of vortex shedding, standing waves and the onset fluid velocity capable of generating the lock-in.
This thesis describes the development of a Lattice Boltzmann (LB) model for a binary gas mixture. Specifically, channel flow driven by a density gradient with diffusion slip occurring at the wall is studied in depth.
The first part of this thesis sets the foundation for the multi-component model used in the subsequent chapters. Commonly used single component LB methods use a non-physical equation of state, in which the relationship between pressure and density varies according to the scaling used. This is fundamentally unsuitable for extension to multi-component systems containing gases of differing molecular masses that are modelled with the ideal gas equation of state. Also, existing methods for implementing boundary conditions are unsuitable for extending to novel boundary conditions, such as diffusion slip. Therefore, a new single component LB derivation and a new method for implementing boundary conditions are developed, and validated against Poiseuille flow. However, including a physical equation of state reduces stability and time accuracy, leading to longer computational times, compared with 'incompressible' LB methods. The new method of analysing LB boundary conditions successfully explains observations from other commonly used schemes, such as the slip velocity associated with 'bounce-back'.
The new model developed for multi-component gases avoids the pitfalls of some other LB models, a single computational grid is shared by all the species and the diffusivity is independent of the viscosity. The Navier-Stokes equation for the mixture and the Stefan-Maxwell diffusion equation are both recovered by the model. However, the species momentum equations are not recovered correctly and this can lead to instability. Diffusion slip, the non-zero velocity of a gas mixture at a wall parallel to a concentration gradient, is successfully modelled and validated against a simple one-dimensional model for channel flow. To increase the accuracy of the scheme a second order numerical implementation is needed. This can be achieved using a variable transformation method which does not result in an increase in computational time.
Simulations were carried out on hydrogen and water diffusion through a narrow channel, with varying total pressure and concentration gradients. For a given value of the species mass flux ratio, the total pressure gradient was dependent on the species concentration gradients. These results may be applicable to fuel cells where the species mass flux ratio is determined by a chemical reaction and the species have opposing velocities. In this case the total pressure gradient is low and the cross-channel average mass flux of hydrogen is independent of the channel width.
Finally, solutions for a binary Stefan tube problem were investigated, in which the boundary at one end of a channel is permeable to hydrogen but not water. The water has no total mass flux along the channel but circulates due to the slip velocity at the wall. The cross-channel average mass flux of the hydrogen along the channel increases with larger channel widths. A fuel cell using a mixture of gases, one being inert, will experience similar circulation phenomena and, importantly, the width of the pores will affect performance.
This thesis essentially proves the viability of LB models to simulate multi-component gases with diffusion slip boundaries, and identifies the many areas in which improvements could be made.
To simulate various underwater warfare situations, a virtual undersea environment with reasonable acoustic communication among the platforms is needed. This paper describes a lattice Boltzmann approach to simulate undersea acoustic propagation for underwater warfare. The lattice Boltzmann methods were developed from the lattice gas cellular automata of Frisch, Hasslacher, et al. This method has also been used to demonstrate various engineering phenomena, and one of its most promising application fields is the flow simulation of viscous fluid, the reaction-diffusion system, acoustics, etc. In this study, the lattice Boltzmann method was used to solve the acoustic propagation problem within the framework of the undersea environment. The undersea space is represented by the lattices, and each node in this lattice can be evolved with a certain acoustic evolution equation by itself. A self-noise from each platform is considered an acoustic point source. The proposed space model is based on the discrete event system specification (DEVS) formalism. To confirm the functions of the proposed model, simple anti-surface ship warfare was simulated.