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The lattice Boltzmann method: Fundamentals and acoustics

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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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... 38 Recent applications to acoustics have demonstrated the capability of the LBM to capture weak pressure fluctuations in pure fluids with high accuracy. In Viggen, 39 the acoustic behavior was reproduced with a monopole source, which was later extended to multipole in both 2D 40 and 3D. 41 In the present work, for the first time, two-way coupled LBM-DEM modeling is exploited to study the fluid-solid interaction at the pore scale associated with the propagation of pressure and shear waves in saturated poroelastic media. ...
... The propagation of a plane wave in a viscous fluid is chosen as the first benchmark for the verification of the acoustic source, following Viggen. 39,40 The 2D cross section at x 2 = 6 of the cuboidal fluid domain (12 × 12 × 620) is shown in Figure 4A. The acoustic source is located on the left-hand side (x 3 = 1), where a P-wave is agitated by changing the local density according to a sinusoidal waveform (see Equation 25). ...
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... The waves are emitted from a circular sound source placed at the center of the left boundary of a 3D cavity using the acoustic point source (APS) technique. In 2D, this technique has been widely addressed by many researchers [23,37,49] using the single relaxation time model. However, according to [37], the investigation of the wave behavior with the MRT model is more stable and accurate than with the SRT model. ...
... However, carrying out physical simulations can impose some constraints on the real units. It is then important to refer to the adequate conversion between LBM and physical units, as it is done in references [37,49]. In LBM units, the dimensions of the cavity and the source diameter ( = 2 = /3) are defined by the numbers of points of the LBM lattice (points from 0 to ). ...
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This paper implements the lattice Boltzmann method to simulate the propagation of sound waves in three dimensions. The numerical model is exercised on the lid-driven cavity flow. Tests are then proposed on acoustic situations. The results are first confronted with analytical solutions of the spherical waves emitted by a single point source inside a cubic cavity. Then, we studied the case where the waves are emitted from a circular sound source placed at the center of the left boundary of a parallelepipedic cavity filled with water. With the circular source discretized as a set of point sources, we were able to simulate the wave propagation in 3D and calculate the sound pressure amplitude. Tests using different emission conditions and LBM relaxation times finally allowed us to get good comparisons with analytical expressions of the pressure amplitude along the source axis, highlighting the performance of the lattice Boltzmann simulations in acoustics.
... 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. ...
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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.
... [11] , [22] Recently, the increase in computational power and the development of the so-called Lattice Boltzmann Method (LBM) made possible to describe this process meticulously. [22] , [23] , [24] 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. ...
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... The lattice Boltzmann method based on Multi-Relaxation Time (MRT) is used to simulate the coupling between heat transfer and wave propagation. This MRT model is preferred because it presents high precision and stability compared to Single Relaxation Time (SRT) model [22]. D2Q9-MRT scheme is applied to determine the macroscopic quantities such as the fluid density, the velocities and the pressure. ...
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In this paper, the lattice Boltzmann method is used to study the acoustic waves propagation inside a differentially heated square enclosure filled with air. The waves are generated by a point sound source located at the center of this cavity. The main aim of this simulation is to simulate the interaction between the thermal convection and the propagation of these acoustic waves. The results have been validated with those obtained in the literature and show that the effect of natural convection on the acoustic waves propagation is almost negligible for low Rayleigh numbers (Ra ≤ 10⁴), which begins to appear when the Rayleigh number begins to become important (Ra ≥ 10⁵) and it becomes considerable for large Rayleigh numbers (Ra ≥ 10⁶) where the thermal convection is important.
... c s is the isothermal (or Newtonian) speed of sound, and it corresponds to the lattice constant when the lattice Boltzmann unit system is adopted [121]. ...
Article
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Over the last decades, several types of collision models have been proposed to extend the validity domain of the lattice Boltzmann method (LBM), each of them being introduced in its own formalism. This article proposes a formalism that describes all these methods within a common mathematical framework, and in this way allows us to draw direct links between them. Here, the focus is put on single and multirelaxation time collision models in either their raw moment, central moment, cumulant, or regularized form. In parallel with that, several bases (nonorthogonal, orthogonal, Hermite) are considered for the polynomial expansion of populations. General relationships between moments are first derived to understand how moment spaces are related to each other. In addition, a review of collision models further sheds light on collision models that can be rewritten in a linear matrix form. More quantitative mathematical studies are then carried out by comparing explicit expressions for the post-collision populations. Thanks to this, it is possible to deduce the impact of both the polynomial basis (raw, Hermite, central, central Hermite, cumulant) and the inclusion of regularization steps on isothermal LBMs. Extensive results are provided for the D1Q3, D2Q9, and D3Q27 lattices, the latter being further extended to the D3Q19 velocity discretization. Links with the most common two and multirelaxation time collision models are also provided for the sake of completeness. This work ends by emphasizing the importance of an accurate representation of the equilibrium state, independently of the choice of moment space. As an addition to the theoretical purpose of this article, general instructions are provided to help the reader with the implementation of the most complicated collision models.
... LBM was a new and promising method and widely used in computational fluid dynamics and acoustic wave. Viggen (2014) provided a detailed account of the application of lattice Boltzmann scheme for acoustic waves, which were usually simulated by finite difference method (FDM) in seismology. Dhuri (2017) analysed a linear lattice Boltzmann (LB) formulation for the simulation of linear acoustic wave propagation in heterogeneous media, which proved that the LB scheme performance was comparable to the classical second-order finite-difference schemes. ...
... Introducing the integration rule for a spherically symmetric integrand [Viggen, 2014] ...
Thesis
Despite the inherent efficiency and low dissipative behaviour of the standard lattice Boltzmann method (LBM) relying on a two step stream and collide algorithm, a major drawback of this approach is the restriction to uniform Cartesian grids. The adaptation of the discretization step to varying fluid dynamic scales is usually achieved by multi-scale lattice Boltzmann schemes, in which the computational domain is decomposed into multiple uniform subdomains with different spatial resolutions. For the sake of connectivity, the resolution factor of adjacent subdomains has to be a multiple of two, introducing an abrupt change of the space-time discretization step at the interface that is prone to trigger instabilites and generate spurious noise sources that contaminate the expected physical pressure signal.In the present PhD thesis, we first elucidate the subject of mesh refinement in the standard lattice Boltzmann method and point out challenges and potential sources of error. Subsequently, we propose a novel hybrid lattice Boltzmann method (HLBM) that combines the stream and collide algorithm with an Eulerian flux-balance algorithm that is obtained from a finite-volume discretization of the discrete velocity Boltzmann equations. The interest of a hybrid lattice Boltzmann method is the pairing of efficiency and low numerical dissipation with an increase in geometrical flexibility. The HLBM allows for non-uniform grids. In the scope of 2D periodic test cases, it is shown that such an approach constitutes a valuable alternative to multi-scale lattice Boltzmann schemes by allowing local mesh refinement of type H. The HLBM properly resolves aerodynamics and aeroacoustics in the interface regions. A further part of the presented work examines the coupling of the stream and collide algorithm with a finite-volume formulation of the isothermal Navier-Stokes equations. Such an attempt bears the advantages that the number of equations of the finite-volume solver is reduced. In addition, the stability is increased due to a more favorable CFL condition. A major difference to the pairing of two kinetic schemes is the coupling in moment space. Here, a novel technique is presented to inject the macroscopic solution of the Navier-Stokes solver into the stream and collide algorithm using a central moment collision. First results on 2D tests cases show that such an algorithm is stable and feasible. Numerical results are compared with those of the previous HLBM.
... Introducing the integration rule for a spherically symmetric integrand [Viggen, 2014] ...
Thesis
Full-text available
Despite the inherent efficiency and low dissipative behaviour of the standard lattice Boltzmann method (LBM) relying on a two step stream and collide algorithm, a major drawback of this approach is the restriction to uniform Cartesian grids. The adaptation of the discretization step to varying fluid dynamic scales is usually achieved by multi-scale lattice Boltzmann schemes, in which the computational domain is decomposed into multiple uniform subdomains with different spatial resolutions. For the sake of connectivity, the resolution factor of adjacent subdomains has to be a multiple of two, introducing an abrupt change of the space-time discretization step at the interface that is prone to trigger instabilites and generate spurious noise sources that contaminate the expected physical pressure signal. In the present PhD thesis, we first elucidate the subject of mesh refinement in the standard lattice Boltzmann method and point out challenges and potential sources of error. Subsequently, we propose a novel hybrid lattice Boltzmann method (HLBM) that combines the stream and collide algorithm with an Eulerian flux-balance algorithm that is obtained from a finite-volume discretization of the discrete velocity Boltzmann equations. The interest of a hybrid lattice Boltzmann method is the pairing of efficiency and low numerical dissipation with an increase in geometrical flexibility. The HLBM allows for non-uniform grids. In the scope of 2D periodic test cases, it is shown that such an approach constitutes a valuable alternative to multi-scale lattice Boltzmann schemes by allowing local mesh refinement of type H. The HLBM properly resolves aerodynamics and aeroacoustics in the interface regions. A further part of the presented work examines the coupling of the stream and collide algorithm with a finite-volume formulation of the isothermal Navier-Stokes equations. Such an attempt bears the advantages that the number of equations of the finite-volume solver is reduced. In addition, the stability is increased due to a more favorable CFL condition. A major difference to the pairing of two kinetic schemes is the coupling in moment space. Here, a novel technique is presented to inject the macroscopic solution of the Navier-Stokes solver into the stream and collide algorithm using a central moment collision. First results on 2D tests cases show that such an algorithm is stable and feasible. Numerical results are compared with those of the previous HLBM.
... c s is the isothermal (or Newtonian) speed of sound, and it corresponds to the lattice constant when the lattice Boltzmann unit system is adopted [122]. Instead of performing collisions in the moment space using the linear matrix form of the collision operator, it is preferred here to work directly with VDFs. ...
Preprint
Full-text available
Over the last decades, several types of collision models have been proposed to extend the validity domain of the lattice Boltzmann method (LBM), each of them being introduced in its own formalism. The present article proposes a formalism that describes all these methods within a common mathematical framework, and in this way allows to draw direct links between them. Here, the focus is put on single and multirelaxation time collision models in either their raw moment, central moment, cumulant or regularized form. In parallel with that, several bases (non orthogonal, orthogonal, Hermite) are considered for the polynomial expansion of populations. General relationships between moments are first derived to understand how moment spaces are related to each other. In addition, a review of collision models further sheds light on collision models that can be rewritten in a linear matrix form. More quantitative mathematical studies are then carried out by comparing explicit expressions for the post collision populations. Thanks to this, it is possible to deduce the impact of both the polynomial basis (raw, Hermite, central, central Hermite, cumulant) and the inclusion of regularization steps on isothermal LBMs. Extensive results are provided for the D1Q3, D2Q9, and D3Q27 lattices, the latter being further extended to the D3Q19 velocity discretization. Links with the most common two and multirelaxation time collision models are also provided for the sake of completeness. The present work ends by emphasizing the importance of an accurate representation of the equilibrium state, independently of the choice of moment space. As an addition to the theoretical purpose of the present article, general instructions are provided to help the reader with the implementation of the most complicated collision models.
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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.
Thesis
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.
Conference Paper
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.