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Binary fluid flow simulations with free energy lattice Boltzmann methods
Abstract
We use free energy lattice Boltzmann methods to simulate shear and extensional flows of a binary fluid in two and three dimensions. To this end, two classical configurations are digitally twinned, namely a parallel-band device for binary shear flow and a four-roller apparatus for binary extensional flow. The free energy lattice Boltzmann method and the test cases are implemented in the open-source parallel C++ framework OpenLB and evaluated for several non-dimensional numbers. Characteristic deformations are captured, where breakup mechanisms occur for critical capillary regimes. Though the known mass leakage for small droplet-domain ratios and large Cahn numbers is observed, suitable mesh sizes show good agreement to analytical predictions and reference results.
... Their work demonstrated the accuracy and versatility of LBM in capturing complex flow patterns and interfacial dynamics. In the context of multi-component systems, Simonis et al. [11] employed LBM to investigate the behavior of binary fluid mixtures. Their study highlighted the accuracy of LBM in predicting interfacial dynamics, phase separation, and the overall behavior of multi-component systems. ...
... To incorporate fluid-solid interactions, an additional term is needed to be included in Eq. (11). This results in the complete Shan-Chen force [23]: ...
... This G c is similar to the term without subscript c i.e. G used in Eq. (11), maintaining consistency with the literature [29]. When G c surpasses the threshold value of 1/(ρ 1 + ρ 2 ), phase separation occurs. ...
The current work presents a novel COllaborative Open-source Lattice Boltzmann Method framework, so-called CooLBM. The computational framework is developed for the simulation of single and multi-component multi-phase problems, along with a reactive interface and conjugate fluid-solid heat transfer problems. CooLBM utilizes a multi-CPU/GPU architecture to achieve high-performance computing (HPC), enabling efficient and parallelized simulations for large scale problems. The code is implemented in C++ and makes extensive use of the Standard Template Library (STL) to improve code modularity, flexibility, and re-usability. The developed framework incorporates advanced numerical methods and algorithms to accurately capture complex fluid dynamics and phase interactions. It offers a wide range of capabilities, including phase separation, interfacial tension, and mass transfer phenomena. The reactive interface simulation module enables the study of chemical reactions occurring at the fluid-fluid interface, expanding its applicability to reactive multi-phase systems. The performance and accuracy of CooLBM are demonstrated through various benchmark simulations, showcasing its ability to capture intricate fluid behaviors and interface dynamics. The modular structure of the code allows for easy customization and extension, facilitating the implementation of additional models and boundary conditions. Finally, CooLBM provides visualization tools for the analysis and interpretation of simulation results. Overall, CooLBM offers an efficient computational framework for studying complex multi-phase systems and reactive interfaces, making it a valuable tool for researchers and engineers in several fields including, but not limited to chemical engineering, materials science, and environmental engineering. CooLBM will be available under open source initiatives for scientific communities.
... To implement and validate these developments, we utilized OpenLB 29 based on version 1.7 30 . This software is a powerful tool for parallel lattice Boltzmann simulations that is employed for example in sub-grid particulate flows 31 , fully resolved particle flows 8,32 , turbulence simulations 33,34 , optimization 35,36 , sub-grid multiphase flows 37 and fully resolved multiphase flows 38 . OpenLB employs unit converters, enabling users to specify physical variables directly in physical units that are then transformed into lattice units by the converter. ...
We derive the partial differential equation (PDE) to which the pseudo-potential lattice Boltzmann method (P-LBM) converges under diffusive scaling, providing a rigorous basis for its consistency analysis. By establishing a direct link between the method's parameters and physical properties-such as phase densities, interface thickness, and surface tension-we develop a framework that enables users to specify fluid properties directly in SI units, eliminating the need for empirical parameter tuning. This allows the simulation of problems with predefined physical properties, ensuring a direct and physically meaningful parametrization. The proposed approach is implemented in OpenLB, featuring a dedicated unit converter for multiphase problems. To validate the method, we perform benchmark tests-including planar interface, static droplet, Galilean invariance, and two-phase flow between parallel plates-using R134a as the working fluid, with all properties specified in physical units. The results demonstrate that the method achieves second-order convergence to the identified PDE, confirming its numerical consistency. These findings highlight the robustness and practicality of the P-LBM, paving the way for accurate and user-friendly simulations of complex multiphase systems with well-defined physical properties.
... 38 Providing distinct advantages HLBM for fluid flow through porous media -part I 5 in terms of parallelizability, the LBM is well-suited for computational fluid dynamics and multiphysics simulations where good scalability on high-performance computing (HPC) facilities is crucial. 34,52,28,58,56,14,53,56,43,16,51,27,55,59,60 Even standard LBM formulations offer an easy to implement and mostly second order accurate, intrinsically matrix-free algorithm in space-time. Those are well-suited for approximating nonstationary and nonlinear problems and, if optimized properly, also capable of saturating modern-day HPC machinery. ...
In this series of studies, we establish homogenized lattice Boltzmann methods (HLBM) for simulating fluid flow through porous media. Our contributions in part I are twofold. First, we assemble the targeted partial differential equation system by formally unifying the governing equations for nonstationary fluid flow in porous media. A matrix of regularly arranged, equally sized obstacles is placed into the domain to model fluid flow through porous structures governed by the incompressible nonstationary Navier--Stokes equations (NSE). Depending on the ratio of geometric parameters in the matrix arrangement, several homogenized equations are obtained. We review existing methods for homogenizing the nonstationary NSE for specific porosities and discuss the applicability of the resulting model equations. Consequently, the homogenized NSE are expressed as targeted partial differential equations that jointly incorporate the derived aspects. Second, we propose a kinetic model, the homogenized Bhatnagar--Gross--Krook Boltzmann equation, which approximates the homogenized nonstationary NSE. We formally prove that the zeroth and first order moments of the kinetic model provide solutions to the mass and momentum balance variables of the macrocopic model up to specific orders in the scaling parameter. Based on the present contributions, in the sequel (part II), the homogenized NSE are consistently approximated by deriving a limit consistent HLBM discretization of the homogenized Bhatnagar--Gross--Krook Boltzmann equation.
... output. Reference simulations are provided in [44]. ...
OpenLB is a generic implementation of lattice Boltzmann methods (LBM) that is shared with the open source community under the terms of the GPLv2 license. Since the first release in 2007, the code continues to be improved and extended, resulting in fifteen releases and counting. The OpenLB framework is written in C++ and covers the full scope of simulations – from pre-processing over parallel and efficient execution to post-processing of results. It offers both the possibility of setting up new simulation cases using the existing rich collection of models and of implementing new custom models. OpenLB supports MPI, OpenMP, AVX(51)2 vectorization and CUDA for parallel execution on systems ranging from low-end smartphones over multi-GPU workstation up to supercomputers.
... The eighteenth ICMMES Conference was held at the University of La Rochelle, La Rochelle, France, from June 28 to July 1, 2022. This special issue of Discrete i ii PREFACE and Continuous Dynamical Systems-Series S (DCDS-S), devoted to this conference, includes 8 selected and peer-reviewed papers on different topics related to the focus areas of ICMMES covering theory and numerical analysis of the LBE and its boundary conditions [1,2,3,4], large-eddy simulations using the LBE [5,6], and numerics and models for multi-phase and multi-component fluids [7,8]. ...
Matter, conceptually classified into fluids and solids, can be completely described by the microscopic physics of its constituent atoms or molecules. However, for most engineering applications, a macroscopic or continuum description has usually been sufficient due to the large disparity between the spatial and temporal scales relevant to these applications and the scales of the underlying molecular dynamics. In this case, the microscopic physics merely determines material properties such as the viscosity of a fluid or the elastic constants of a solid. These material properties cannot be derived within the macroscopic framework, but the qualitative nature of the macroscopic dynamics is usually insensitive to the details of the underlying microscopic interactions.
The traditional picture of the role of microscopic and macroscopic physics is now being challenged as new multi-scale and multi-physics problems begin to emerge. For example, in nano-scale systems, the assumption of scale separation breaks down; thus, macroscopic theory is inadequate, yet microscopic theory may be impractical because it requires computational capabilities far beyond our current reach. This new class of problems poses unprecedented challenges to mathematical modeling as well as numerical simulation and requires new and non-traditional analysis and modeling paradigms. Methods based on mesoscopic theories, which connect the microscopic and macroscopic descriptions of the dynamics, provide a promising approach. They can lead to useful models, possibly requiring empirical inputs to determine some of the model parameters that are sub-macroscopic yet indispensable to the relevant physical phenomena.
The area of complex fluids focuses on materials such as suspensions, emulsions, and gels, where the internal structure is relevant to the macroscopic dynamics. An important challenge will be to construct meaningful mesoscopic models by extracting all the macroscopically relevant information from the microscopic dynamics.
There already exist a few mesoscopic methods such as the Lattice Gas Cellular Automata (LGCA), the Lattice Boltzmann Equation (LBE), Discrete Velocity Models (DVM) of the Boltzmann equation, Gas-Kinetic Schemes (GKS), Smoothed Particle Hydrodynamics (SPH), and Dissipative Particle Dynamics (DPD). Although these methods are sometimes designed for macroscopic hydrodynamics, they are not based upon the Navier-Stokes equations; instead, they are closely related to kinetic theory and the Boltzmann equation. These methods are promising candidates to effectively connect microscopic and macroscopic scales, thereby substantially extending the capabilities of numerical simulations. For this reason, they are the focus of the International Conferences on Mesoscopic Methods in Engineering and Science (ICMMES, http://www.icmmes.org).
The eighteenth ICMMES Conference was held at the University of La Rochelle, La Rochelle, France, from June 28 to July 1, 2022. This special issue of Discrete and Continuous Dynamical Systems–Series S (DCDS-S), devoted to this conference, includes 8 selected and peer-reviewed papers on different topics related to the focus areas of ICMMES covering theory and numerical analysis of the LBE and its boundary conditions [1,2,3,4], large-eddy simulations using the LBE [5,6], and numerics and models for multi-phase and multi-component fluids [7,8].
The editors would like to thank the referees who have helped to review the papers in this special issue. The organizers of ICMMES-2022 and the ICMMES Scientific Committee would like to acknowledge the support from La Rochelle University, the Mathematics Image Applications Laboratory (MIA), Tunis El Manar University, the Old Dominion University Research Foundation, Sugon, and the Beijing Computational Science Research Center (CSRC).
... To realize this potential, we utilize the platform-agnostic LBM open source framework OpenLB [38,39]. Among others, OpenLB has been used for efficient simulations of turbulent fluid flows [29,40], advection-diffusion processes [41][42][43] up to ternary fluid mixture flows [39,44], coupled radiative transfer [45], volume-averaged fluid flow [46], fluid-structure interaction [47,48], and indoor thermal comfort [49]. Thus, to feasibly obtain the consistent benchmark UQ data, we implement MCS within the OpenLB framework as the deterministic kernel (see also [42,50]). ...
... P r e p r i n t n o t p e e r r e v i e w e d open source framework OpenLB [37,38]. Among others, OpenLB has been used for efficient simulations of turbulent fluid flows [28,39], advectiondiffusion processes [40][41][42] up to ternary fluid mixture flows [38,43], coupled radiative transfer [44], volume-averaged fluid flow [45], fluid-structure interaction [46,47], and indoor thermal comfort [48]. Thus, to feasibly obtain the consistent benchmark UQ data, we implement MCS within the OpenLB framework as the deterministic kernel. ...
Efficient modeling and simulation of uncertainties in computational fluid dynamics (CFD) remains a crucial challenge. In this paper, we present the first stochastic Galerkin (SG) lattice Boltzmann method (LBM) built upon the generalized polynomial chaos (gPC). The proposed method offers an efficient and accurate approach that depicts the propagation of uncertainties in stochastic incompressible flows. Formal analysis shows that the SG LBM preserves the correct Chapman-Enskog asymptotics and recovers the corresponding macroscopic fluid equations. Numerical experiments, including the Taylor-Green vortex flow, lid-driven cavity flow, and isentropic vortex con-vection, are presented to validate the solution algorithm. The results demonstrate that the SG LBM achieves the expected spectral convergence and the computational cost is significantly reduced compared to the sampling-based non-intrusive approaches, e.g., the routinely used Monte Carlo method. We obtain a speedup factor of 5.72 compared to Monte Carlo sampling in a randomized two-dimensional Taylor-Green vortex flow test case. By leveraging the accuracy and flexibility of LBM and the efficiency of gPC-based SG, the proposed SG LBM provides a powerful framework for uncertainty quantification in CFD practice.
... To improve on the LGA, in a general LB model, the ensemble-averaged particle distribution functions and the linearized Bhatnagar-Gross-Krook (BGK) approximation is applied to overcome the statistical noise and exponential complexity from which the LGA suffers [1][2][3]. Among the various numerical methods, the LB model has emerged as an impressive candidate because of its efficiency and simplicity in simulating multiphase flows [4][5][6][7][8], compressible flows [9,10], turbulent flows [11,12], combustion [13,14], non-Newtonian flows [15,16] and more. ...
Recently, considerable attention has been given to (2+1)-dimensional Kadomtsev-Petviashvili equations due to their extensive applications in solitons that widely exist in nonlinear science. Therefore, developing a reliable numerical algorithm for the Kadomtsev-Petviashvili equations is crucial. The lattice Boltzmann method, which has been an efficient simulation method in the last three decades, is a promising technique for solving Kadomtsev-Petviashvili equations. However, the traditional higher-order moment lattice Boltzmann model for the Kadomtsev-Petviashvili equations suffers from low accuracy because of error accumulation. To overcome this shortcoming, a splitting lattice Boltzmann scheme for (2+1)-dimensional Kadomtsev-Petviashvili-Ⅰ type equations is proposed in this paper. The variable substitution method is applied to transform the Kadomtsev-Petviashvili-Ⅰ type equation into two macroscopic equations. Two sets of distribution functions are employed to construct these two macroscopic equations. Moreover, three types of soliton solutions are numerically simulated by this algorithm. The numerical results imply that the splitting lattice Boltzmann schemes have an advantage over the traditional high-order moment lattice Boltzmann model in simulating the Kadomtsev-Petviashvili-Ⅰ type equations.
... output. Reference simulations are provided in [40]. ...
OpenLB is an object-oriented implementation of LBM. It is the first implementation of a generic platform for LBM programming, which is shared with the open source community (GPLv2). Since the first release in 2007, the code has been continuously improved and extended which is documented by thirteen releases as well as the corresponding release notes which are available on the OpenLB website (https://www.openlb.net). The OpenLB code is written in C++ and is used by application programmers as well as developers, with the ability to implement custom models OpenLB supports complex data structures that allow simulations in complex geometries and parallel execution using MPI, OpenMP and CUDA on high-performance computers. The source code uses the concepts of interfaces and templates, so that efficient, direct and intuitive implementations of the LBM become possible. The efficiency and scalability has been checked and proved by code reviews. This user manual and a source code documentation by DoxyGen are available on the OpenLB project website.
... The most prolific feature of LBM is the suitability for parallelization due to explicitly local calculation of populations. Meanwhile, LBM has been found to provide advanced capabilities for the parallel simulation of turbulent flows [14][15][16][17], advection-diffusion transport [18][19][20], binary fluid flows [21], and in particular, photobioreactors [22], Flettner rotors [23] or Coriolis mass flow meters [24]. As a paragon of effectiveness of the LBM, the comparison between the open-source software packages OpenLB [25][26][27] and OpenFOAM shows 32 times faster computation time of the former by the in-cylinder flow test [25,28]. ...
We derive a novel lattice Boltzmann scheme, which uses a pressure correction forcing term for approximating the volume averaged Navier–Stokes equations (VANSE) in up to three dimensions. With a new definition of the zeroth moment of the Lattice Boltzmann equation, spatially and temporally varying local volume fractions are taken into account. A Chapman–Enskog analysis, respecting the variations in local volume, formally proves the consistency towards the VANSE limit up to higher order terms. The numerical validation of the scheme via steady state and non-stationary examples approves the second order convergence with respect to velocity and pressure. The here proposed lattice Boltzmann method is the first to correctly recover the pressure with second order for space-time varying volume fractions.
The design and optimization of photobioreactor(s) (PBR) benefit from the development of robust and quantitatively accurate computational fluid dynamics (CFD) models, which incorporate the complex interplay of fundamental phenomena. In the present work, we propose a comprehensive computational model for tubular photobioreactors equipped with glass sponges. The simulation model requires a minimum of at least three submodels for hydrodynamics, light supply, and biomass kinetics, respectively. First, by modeling the hydrodynamics, the light–dark cycles can be detected and the mixing characteristics of the flow (besides the mass transport) can be analyzed. Second, the radiative transport model is deployed to predict the local light intensities according to the wavelength of the light and scattering characteristics of the culture. The third submodel implements the biomass growth kinetic by coupling the local light intensities to hydrodynamic information of the CO2 concentration, which allows to predict the algal growth. In combination, the novel mesoscopic simulation model is applied to a tubular PBR with transparent walls and an internal sponge structure. We showcase the coupled simulation results and validate specific submodel outcomes by comparing the experiments. The overall flow velocity, light distribution, and light intensities for individual algae trajectories are extracted and discussed. Conclusively, such insights into complex hydrodynamics and homogeneous illumination are very promising for CFD-based optimization of PBR.
We present the first top-down ansatz for constructing lattice Boltzmann methods (LBM) in d dimensions. In particular, we construct a relaxation system (RS) for a given scalar, linear, d-dimensional advection–diffusion equation. Subsequently, the RS is linked to a d-dimensional discrete velocity Boltzmann model (DVBM) on the zeroth and first energy shell. Algebraic characterizations of the equilibrium, the moment space, and the collision operator are carried out. Further, a closed equation form of the RS expresses the added relaxation terms as prefactored higher order derivatives of the conserved quantity. Here, a generalized (2d+1)×(2d+1) RS is linked to a DdQ(2d+1) DVBM which, upon complete discretization, yields an LBM with second order accuracy in space and time. A rigorous convergence result for arbitrary scaling of the RS, the DVBM and conclusively also for the final LBM is proven. The top-down constructed LBM is numerically tested on multiple GPUs with smooth and non-smooth initial data in d=3 dimensions for several grid-normalized non-dimensional numbers.
The design and optimization of photobioreactor(s) (PBR) benefit from the development of robust and quantitatively accurate computational fluid dynamics (CFD) models, which incorporate the complex interplay of fundamental phenomena. In the present work, we propose a comprehensive computational model for tubular photobioreactors equipped with glass sponges. The simulation model requires a minimum of at least three submodels for hydrodynamics, light supply, and biomass kinetics, respectively. First, by modeling the hydrodynamics, the light–dark cycles can be detected and the mixing characteristics of the flow (besides the mass transport) can be analyzed. Second, the radiative transport model is deployed to predict the local light intensities according to the wavelength of the light and scattering characteristics of the culture. The third submodel implements the biomass growth kinetic by coupling the local light intensities to hydrodynamic information of the CO2 concentration, which allows to predict the algal growth. In combination, the novel mesoscopic simulation model is applied to a tubular PBR with transparent walls and an internal sponge structure. We showcase the coupled simulation results and validate specific submodel outcomes by comparing the experiments. The overall flow velocity, light distribution, and light intensities for individual algae trajectories are extracted and discussed. Conclusively, such insights into complex hydrodynamics and homogeneous illumination are very promising for CFD-based optimization of PBR.
Double emulsion droplets (DEs) are water/oil/water droplets that can be sorted via fluorescence-activated cell sorting (FACS), allowing for new opportunities in high-throughput cellular analysis, enzymatic screening, and synthetic biology. These applications require stable, uniform droplets with predictable microreactor volumes. However, predicting DE droplet size, shell thickness, and stability as a function of flow rate has remained challenging for monodisperse single core droplets and those containing biologically-relevant buffers, which influence bulk and interfacial properties. As a result, developing novel DE-based bioassays has typically required extensive initial optimization of flow rates to find conditions that produce stable droplets of the desired size and shell thickness. To address this challenge, we conducted systematic size parameterization quantifying how differences in flow rates and buffer properties (viscosity and interfacial tension at water/oil interfaces) alter droplet size and stability, across 6 inner aqueous buffers used across applications such as cellular lysis, microbial growth, and drug delivery, quantifying the size and shell thickness of >22 000 droplets overall. We restricted our study to stable single core droplets generated in a 2-step dripping-dripping formation regime in a straightforward PDMS device. Using data from 138 unique conditions (flow rates and buffer composition), we also demonstrated that a recent physically-derived size law of Wang et al. can accurately predict double emulsion shell thickness for >95% of observations. Finally, we validated the utility of this size law by using it to accurately predict droplet sizes for a novel bioassay that requires encapsulating growth media for bacteria in droplets. This work has the potential to enable new screening-based biological applications by simplifying novel DE bioassay development.
We provide a first investigation of using lattice Boltzmann methods (LBM) for temporal large eddy simulation (TLES). The temporal direct deconvolution model (TDDM) is injected as a closure for the filtered discrete velocity Bhatnagar–Gross–Krook (BGK) Boltzmann equation with orthogonal multiple-relaxation-time (MRT) collision. The novel combination of methods is calibrated for decaying homogeneous isotropic turbulence. The Taylor–Green vortex flow is used as a benchmark for the Reynolds numbers 800 and 3000. Various turbulence quantities are numerically evaluated. The numerical results obtained with the MRT LBM are validated against a well-established spectral element method both, with and without the proposed turbulence model. A qualitatively good agreement to reference results of direct numerical simulation is observed in terms of dissipation rate, energy spectrum and dissipation spectrum. Whereas the latter two show marginal differences compared to underresolved simulations without TLES, the dissipation rate exhibits substantial improvement through the model in capturing the peak region for low and intermediate resolutions. The consistency of the TDDM with first and second order discretization is numerically demonstrated for single-relaxation-time and MRT collision via computing the total dissipation rate error. Measuring the interaction between subgrid activity and energy spectrum error serves as a proof of concept for the proposed MRT LBM TLES. Conclusively, the model recovers and enhances the expected numerical features of LBM with respect to the target equation.
Multiple-relaxation-time (MRT) lattice Boltzmann methods (LBM) based on orthogonal moments exhibit lattice Mach number dependent instabilities in diffusive scaling.
The present work renders an explicit formulation of stability sets for orthogonal moment MRT LBM.
The stability sets are defined via the spectral radius of linearized amplification matrices of the MRT collision operator with variable relaxation frequencies.
Numerical investigations are carried out for the three-dimensional Taylor–Green vortex benchmark at Reynolds number 1600.
Extensive brute force computations of specific relaxation frequency ranges for the full test case are opposed to the von Neumann stability set prediction.
Based on that, we prove numerically that a scan over the full wave space, including scaled mean flow variations, is required to draw conclusions on the overall stability of LBM in turbulent flow simulations.
Further, the von Neumann results show that a grid dependence is hardly possible to include in the notion of linear stability for LBM.
Lastly, via brute force stability investigations based on empirical data from a total number of 22696 simulations, the existence of a deterministic influence of the grid resolution is deduced.
In this paper, we use a fluid–structure interaction (FSI) approach to simulate a Coriolis mass flowmeter (CMF). The fluid dynamics is calculated by the open-source framework OpenLB, based on the lattice Boltzmann method (LBM). For the structural dynamics we employ the open-source software Elmer, an implementation of the finite element method (FEM). A staggered coupling approach between the two software packages is presented. The finite element mesh is created by the mesh generator Gmsh to ensure a complete open source workflow. The Eigenmodes of the CMF, which are calculated by modal analysis, are compared with measurement data. Using the estimated excitation frequency, a fully coupled, partitioned, FSI simulation is applied to simulate the phase shift of the investigated CMF design. The calculated phase shift values are in good agreement to the measurement data and verify the suitability of the model to numerically describe the working principle of a CMF.
We present the OpenLB package, a C++ library providing a flexible framework for lattice Boltzmann simulations. The code is publicly available and published under GNU GPLv2, which allows for adaption and implementation of additional models. The extensibility benefits from a modular code structure achieved e.g. by utilizing template meta-programming. The package covers various methodical approaches and is applicable to a wide range of transport problems (e.g. fluid, particulate and thermal flows). The built-in processing of the STL file format furthermore allows for the simple setup of simulations in complex geometries. The utilization of MPI as well as OpenMP parallelism enables the user to perform those simulations on large-scale computing clusters. It requires a minimal amount of dependencies and includes several benchmark cases and examples. The package presented here aims at providing an open access platform for both, applicants and developers, from academia as well as industry, which facilitates the extension of previous implementations and results to novel fields of application for lattice Boltzmann methods. OpenLB was tested and validated over several code reviews and publications. This paper summarizes the findings and gives a brief introduction to the underlying concepts as well as the design of the parallel data structure.
Modelling double emulsion formation in planar flow-focusing microchannels - Volume 895 - Ningning Wang, Ciro Semprebon, Haihu Liu, Chuhua Zhang, Halim Kusumaatmaja
The connection of relaxation systems and discrete velocity models is essential to the progress of stability as well as convergence results for lattice Boltzmann methods. In the present study we propose a formal perturbation ansatz starting from a scalar one-dimensional target equation, which yields a relaxation system specifically constructed for its equivalence to a discrete velocity Boltzmann model as commonly found in lattice Boltzmann methods. Further, the investigation of stability structures for the discrete velocity Boltzmann equation allows for algebraic characterizations of the equilibrium and collision operator. The methods introduced and summarized here are tailored for scalar, linear advection–diffusion equations, which can be used as a foundation for the constructive design of discrete velocity Boltzmann models and lattice Boltzmann methods
to approximate different types of partial differential equations.
In this paper, we present a high-order polynomial free energy for the phase-field model of two-phase incompressible fluids. The model consists of the Navier–Stokes (NS) equation and the Cahn–Hilliard (CH) equation with a high-order polynomial free energy potential. In practice, a quartic polynomial has been used for the bulk free energy in the CH equation. It is well known that the CH equation does not satisfy the maximum principle and the phase-field variable takes shifted values in the bulk phases instead of taking the minimum values of the double-well potential. This phenomenon substantially changes the original volume enclosed by the isosurface of the phase-field function. Furthermore, it requires fine resolution to keep small shapes. To overcome these drawbacks, we propose high-order (higher than fourth order) polynomial free energy potentials. The proposed model is tested for an equilibrium droplet shape in a spherical symmetric configuration and a droplet deformation under a simple shear flow in a fully three-dimensional fluid flow. The computational results demonstrate the superiority of the proposed model with a high-order polynomial potential to the quartic polynomial function in volume conservation property.
Stability, consistency and accuracy of various lattice Boltzmann schemes are investigated by means of numerical experiments on decaying homogeneous isotropic turbulence (DHIT). Therefore, the Bhatnagar–Gross–Krook (BGK), the entropic lattice Boltzmann (ELB), the two-relaxation-time (TRT), the regularized lattice Boltzann (RLB) and the multiple-relaxation-time (MRT) collision schemes are applied to the three-dimensional Taylor–Green vortex, which represents a benchmark case for DHIT. The obtained turbulent kinetic energy, the energy dissipation rate and the energy spectrum are compared to reference data. Acoustic and diffusive scaling is taken into account to determine the impact of the lattice Mach number. Furthermore, three different Reynolds numbers [Formula: see text], [Formula: see text] and [Formula: see text] are considered. BGK shows instabilities, when the mesh is highly underresolved. The diverging simulations for MRT are ascribed to a strong lattice Mach number dependency. Despite the fact that the ELB modifies the bulk viscosity, it does not mimic a turbulence model. Therefore, no significant increase of stability in comparison to BGK is observed. The TRT “magic parameter” for DHIT at moderate Reynolds numbers is estimated with respect to the energy contribution. Stability and accuracy of the TRT scheme is found to be similar to BGK. For small lattice Mach numbers, the RLB scheme exhibits lowered energy contribution in the dissipation range compared to an analytical model spectrum. Overall, to enhance stability and accuracy, the lattice Mach number should be chosen with respect to the applied collision scheme.
In this work we study the deformation of clean and surfactant-laden droplets in laminar shear-flow. The simulations are based on Direct Numerical Simulation of the Navier–Stokes equations coupled with a Phase Field Method to describe interface topology and surfactant concentration. Simulations are performed considering both 2D (circular droplet) and 3D (spherical droplet) domains. First, we focus on clean droplets and we characterize the droplet shape and deformation. This enables us to define the range of parameters in which theoretical models well predict the results obtained from 2D and 3D simulations. Then, surfactant-laden droplets are considered; the main factors leading to larger droplet deformation are carefully described and quantified. Results obtained indicate that the average surface tension reduction and the accumulation of surfactant at the tips of the deformed droplet have a dominant role, while tangential stresses at the interface (Marangoni stresses) have a limited effect on the overall droplet deformation. Finally, the distribution of surfactant over the droplet surface is examined in relation to surface deformation and shear stress distribution.
Binary fluid mixtures are examples of complex fluids whose microstructure and flow are strongly coupled. For pairs of simple fluids, the microstructure consists of droplets or bicontinuous demixed domains and the physics is controlled by the interfaces between these domains. At continuum level, the structure is defined by a composition field whose gradients which are steep near interfaces drive its diffusive current. These gradients also cause thermodynamic stresses which can drive fluid flow. Fluid flow in turn advects the composition field, while thermal noise creates additional random fluxes that allow the system to explore its configuration space and move towards the Boltzmann distribution. This article introduces continuum models of binary fluids, first covering some well-studied areas such as the thermodynamics and kinetics of phase separation, and emulsion stability. We then address cases where one of the fluid components has anisotropic structure at mesoscopic scales creating nematic (or polar) liquid-crystalline order; this can be described through an additional tensor (or vector) order parameter field. We conclude by outlining a thriving area of current research, namely active emulsions, in which one of the binary components consists of living or synthetic material that is continuously converting chemical energy into mechanical work.
We present a new ternary free energy lattice Boltzmann model. The
distinguishing feature of our model is that we are able to analytically derive
and independently vary all fluid-fluid surface tensions and the solid surface
contact angles. We carry out a number of benchmark tests: (i) double emulsions
and liquid lenses to validate the surface tensions, (ii) ternary fluids in
contact with a square well to compare the contact angles against analytical
predictions, and (iii) ternary phase separation to verify that the
multicomponent fluid dynamics is accurately captured. Additionally we also
describe how the model here presented here can be extended to include an
arbitrary number of fluid components.
One characteristic of multiphase lattice Boltzmann equation (LBE) methods is that the interfacial region has a finite (i.e., noninfinitesimal) thickness known as a diffuse interface. In simulations of, e.g., bubble or drop dynamics, for problems involving nonideal gases, one frequently observes that the diffuse interface method produces a spontaneous, nonphysical shrinkage of the bubble or drop radius. In this paper, we analyze in detail a single-fluid two-phase model and use a LBE model for nonideal gases in order to explain this fundamental problem. For simplicity, we only investigate the static bubble or droplet problem. We find that the method indeed produces a density shift, bubble or droplet shrinkage, as well as a critical radius below which the bubble or droplet eventually vanishes. Assuming that the ratio between the interface thickness D and the initial bubble or droplet radius r0 is small, we analytically show the existence of this density shift, bubble or droplet radius shrinkage, and critical bubble or droplet survival radius. Numerical results confirm our analysis. We also consider droplets on a solid surface with different curvatures, contact angles, and initial droplet volumes. Numerical results show that the curvature, contact angle, and the initial droplet volume have an effect on this spontaneous shrinkage process, consistent with the survival criterion.
We consider the deformation and burst of small fluid droplets in steady linear, two-dimensional motions of a second immiscible fluid. Experiments using a computer-controlled, four-roll mill to investigate the effect of flow type are described, and the results compared with predictions of several available asymptotic deformation and burst theories, as well as numerical calculations. The comparison clarifies the range of validity of the theories, and demonstrates that they provide quite adequate predictions over a wide range of viscosity ratio, capillary number, and flow type.
Recently there has been a huge effort in the scientific community to miniaturise fluidic operations to micron and nanoscales
[1]. This has changed the way scientists think about fluids, and it potentially has far-reaching technological implications,
analogous to the miniaturization of electronics. The goal is to engineer “lab on a chip” devices, where numerous biological
and chemical experiments can be performed rapidly, and in parallel, while consuming little reagent.
We present the details of a lattice Boltzmann approach to phase separation in nonideal one- and two-component fluids. The collision rules are chosen such that the equilibrium state corresponds to an input free energy and the bulk flow is governed by the continuity, Navier-Stokes, and, for the binary fluid, a convection-diffusion equation. Numerical results are compared to simple analytic predictions to confirm that the equilibrium state is indeed thermodynamically consistent and that the kinetics of the approach to equilibrium lie within the expected universality classes. The approach is compared to other lattice Boltzmann simulations of nonideal systems. © 1996 The American Physical Society.
We show that discrete lattice effects must be considered in the introduction of a force into the lattice Boltzmann equation. A representation of the forcing term is then proposed. With the representation, the Navier-Stokes equation is derived from the lattice Boltzmann equation through the Chapman-Enskog expansion. Several other existing force treatments are also examined.
The late-stage demixing following spinodal decomposition of a three-dimensional symmetric binary fluid mixture is studied numerically, using a thermodynamicaly consistent lattice Boltzmann method. We combine results from simulations with different numerical parameters to obtain an unprecendented range of length and time scales when expressed in reduced physical units. Using eight large (256^3) runs, the resulting composite graph of reduced domain size l against reduced time t covers 1 < l < 10^5, 10 < t < 10^8. Our data is consistent with the dynamical scaling hypothesis, that l(t) is a universal scaling curve. We give the first detailed statistical analysis of fluid motion, rather than just domain evolution, in simulations of this kind, and introduce scaling plots for several quantities derived from the fluid velocity and velocity gradient fields.
We derive a novel lattice Boltzmann scheme, which uses a pressure correction forcing term for approximating the volume averaged Navier–Stokes equations (VANSE) in up to three dimensions. With a new definition of the zeroth moment of the Lattice Boltzmann equation, spatially and temporally varying local volume fractions are taken into account. A Chapman–Enskog analysis, respecting the variations in local volume, formally proves the consistency towards the VANSE limit up to higher order terms. The numerical validation of the scheme via steady state and non-stationary examples approves the second order convergence with respect to velocity and pressure. The here proposed lattice Boltzmann method is the first to correctly recover the pressure with second order for space-time varying volume fractions.
Lattice Boltzmann methods (LBM) are well suited to highly parallel computational fluid dynamics simulations due to their separability into a perfectly parallel collision step and a propagation step that only communicates within a local neighborhood. The implementation of the propagation step provides constraints for the maximum possible bandwidth-limited performance, memory layout and usage of vector instructions. This article revisits and extends the work on implicit propagation on directly addressed grids started by A-A and its shift-swap-streaming (SSS) formulation by reconsidering them as transformations of the underlying space filling curve. In this work, a new periodic shift (PS) pattern is proposed that imposes minimal restrictions on the implementation of collision operators and utilizes virtual memory mapping to provide consistent performance across a range of targets. Various implementation approaches as well as time dependency and performance anisotropy are discussed. Benchmark results for SSS and PS on SIMD CPUs including Intel Xeon Phi as well as Nvidia GPUs are provided. Finally, the application of PS as the propagation pattern of the open source LBM framework OpenLB is summarized.
Am Karlsruher Institut für Technologie (KIT) findet seit 2010 jedes Sommersemester das projektorientierte Softwarepraktikum statt. Wir bieten es gemeinschaftlich vom Institut für Angewandte und Numerische Mathematik (IANM) und dem Institut für Mechanische Verfahrenstechnik und Mechanik (MVM) an. Seit dem Frühjahr 2020 werden praktische Lehrveranstaltungen durch die Einschränkungen zur Bekämpfung der Corona-Pandemie extrem beeinträchtigt. Unser Artikel beschreibt, wie wir es unter Pandemiebedingungen trotzdem geschafft haben, Studierende mehrerer Studiengänge zu begeistern, sodass wir schließlich sogar vom KIT Präsidium ausgerechnet für ein Praktikum ausgezeichnet wurden.
Traditional Lattice–Boltzmann modelling of advection-diffusion flow is affected by numerical instability if the advective term becomes dominant over the diffusive (i.e., high-Péclet flow). To overcome the problem, two 3D one-way coupled models are proposed. In a traditional model, a Lattice–Boltzmann Navier–Stokes solver is coupled to a Lattice–Boltzmann advection-diffusion model. In a novel model, the Lattice–Boltzmann Navier–Stokes solver is coupled to an explicit finite-difference algorithm for advection-diffusion. The finite-difference algorithm also includes a novel approach to mitigate the numerical diffusivity connected with the upwind differentiation scheme.
The models are validated using two non-trivial benchmarks, which includes discontinuous initial conditions and the case Peg→∞ for the first time, where Peg is the grid Péclet number. The evaluation of Peg alongside Pe is discussed. Accuracy, stability and the order of convergence are assessed for a wide range of Péclet numbers. Recommendations are then given as to which model to select depending on the value Peg—in particular, it is shown that the coupled finite-difference/Lattice–Boltzmann provide stable solutions in the case Pe→∞, Peg→∞.
Detailed numerical analyses of temperature and air velocity distributions are relevant to assess thermal comfort in a wide range of applications. Until now mainly simulations based on Reynolds-averaged Navier–Stokes equations (RANS) are used, whereby fluctuations as well as anisotropy of the turbulence are represented with insufficient precision. This paper applies a thermal large eddy lattice Boltzmann method (LES-LBM) as an efficient and accurate transient modeling of turbulence. The benchmark case Manikin Heat Loss for Thermal Comfort Evaluation is studied and the model of Predicted Mean Vote (PMV) is applied for estimating thermal sensation. The results for the air velocity, the temperature field and the PMV show a satisfactory agreement with both, the experiment and the results from RANS simulations. The accuracy and the model quality of the simulation are increased further by considering the buoyancy and an inlet seeding. This suggests a successful evaluation of the present model, whereby additional transient flow field data are provided. The obtained transient flow field data, however, motivates future work to study thermal comfort in the present manner. The investigation of the influence of fluctuations on thermal comfort as well as the application to more complex problems seem promising.
This review summarizes the rigorous mathematical theory behind the lattice Boltzmann equation (LBE). Relevant properties of the Boltzmann equation and a derivation of the LBE from the Boltzmann equation are presented. A summary of some important LBE models is provided. Focus is given to results from the numerical analysis of the LBE as a solver for the nearly incompressible Navier-Stokes equations with appropriate boundary conditions. A number of numerical results are provided to demonstrate the efficacy of the lattice Boltzmann method.
The dynamics of a droplet suspended in a medium fluid is determined by the hydrodynamic forces and surface tension force, and the Taylor analogy had been successfully employed to predict droplet deformation and breakup in a spray. This paper aims to extend the Taylor analogy for prediction of droplet dynamics in planar extensional flow which has great significance in droplet-based microfluidic systems. Performance of the proposed model is compared with the three-dimensional numerical simulation results over a wide range of capillary number and viscosity ratio. Experimental data available in the literature is also compared with the prediction results for verification purpose. The proposed model could describe both the time scale and the magnitude of droplet deformation accurately.
Theory and Application of Multiphase Lattice Boltzmann Methods presents a comprehensive review of all popular multiphase Lattice Boltzmann Methods developed thus far and is aimed at researchers and practitioners within relevant Earth Science disciplines as well as Petroleum, Chemical, Mechanical and Geological Engineering. Clearly structured throughout, this book will be an invaluable reference on the current state of all popular multiphase Lattice Boltzmann Methods (LBMs). The advantages and disadvantages of each model are presented in an accessible manner to enable the reader to choose the model most suitable for the problems they are interested in. The book is targeted at graduate students and researchers who plan to investigate multiphase flows using LBMs. Throughout the text most of the popular multiphase LBMs are analyzed both theoretically and through numerical simulation. The authors present many of the mathematical derivations of the models in greater detail than is currently found in the existing literature. The approach to understanding and classifying the various models is principally based on simulation compared against analytical and observational results and discovery of undesirable terms in the derived macroscopic equations and sometimes their correction. A repository of FORTRAN codes for multiphase LBM models is also provided.
This book is an introduction to the theory, practice, and implementation of the Lattice Boltzmann (LB) method, a powerful computational fluid dynamics method that is steadily gaining attention due to its simplicity, scalability, extensibility, and simple handling of complex geometries. The book contains chapters on the method's background, fundamental theory, advanced extensions, and implementation.
To aid beginners, the most essential paragraphs in each chapter are highlighted, and the introductory chapters on various LB topics are front-loaded with special "in a nutshell" sections that condense the chapter's most important practical results. Together, these sections can be used to quickly get up and running with the method. Exercises are integrated throughout the text, and frequently asked questions about the method are dealt with in a special section at the beginning. In the book itself and through its web page, readers can find example codes showing how the LB method can be implemented efficiently on a variety of hardware platforms, including multi-core processors, clusters, and graphics processing units. Students and scientists learning and using the LB method will appreciate the wealth of clearly presented and structured information in this volume.
New experimental results describing the breakup of drops suddenly subjected to shearing forces sufficient to cause drop breakup are presented. The experiments examine fluids which are immiscible and Newtonian under the conditions studied. These results and other breakup mechanism results are then discussed in the context of combining them with kinematic information on more complex flows in order to predict drop size distributions in processing flows. By analyzing the time scales of various breakage mechanisms studied in simple shear flow and comparing them to the appropriate time scales in more complex flows, the relevant importance of a given mechanism in determining the daughter drop size distribution is predicted. This type of time scale analysis is performed for flow in a cavity and flow in an extruder.
The behavior of a single liquid drop suspended in another liquid and subjected to simple shear flow is studied numerically using a diffuse interface free energy lattice Boltzmann method. The system is fully defined by three physical, and two numerical dimensionless numbers: a Reynolds number ReRe, a capillary number CaCa, the viscosity ratio λλ, an interface-related Peclet number Pe, and the ratio of interface thickness and drop size (the Cahn number Ch). The influence of Pe,ChPe,Ch and mesh resolution on accuracy and stability of the simulations is investigated. Drops of moderate resolution (radius less than 30 lattice units) require smaller interface thickness, while a thicker interface should be used for highly resolved drops. The Peclet number is controlled by the mobility coefficient ΓΓ. Based on the results, the simulations are stable when ΓΓ is in the range 1–15. In addition, the numerical tool is verified and validated in a wide range of physical conditions: Re=0.0625-50,λ=1,2,3Re=0.0625-50,λ=1,2,3 and a capillary number range over which drops deform and break. Good agreement with literature data is observed.
A spherical drop, placed in a second liquid of the same density, is subjected to shearing between parallel plates. The subsequent flow is investigated numerically with a volume-of-fluid (VOF) method. The scheme incorporates a semi-implicit Stokes solver to enable computations at low Reynolds number. Our simulations compare well with previous theoretical, numerical, and experimental results. For capillary numbers greater than the critical value, the drop deforms to a dumbbell shape and daughter drops detach via an end-pinching mechanism. The number of daughter drops increases with the capillary number. The breakup can also be initiated by increasing the Reynolds number.
Numerical computations are employed to study the flow field produced by a four‐roll mill. The radius of the cylinders a, the cylinder spacing 2b, and the size 2l of the square container are varied to assess the effects on the kinematics of the flow field. It is found that a ratio of a/b=0.625 with l/b≥3.0 produces the best approximation to a pure extensional flow. With these parameter values, the extension rate remains constant with an error of less than 1% over an axial region x/b≤0.5. By contrast, the commonly accepted design a/b=0.772 suggested by Fuller and Leal [J. Polym. Sci. Polym. Phys. 19, 557 (1981)] produces a variation in extension rate of 50% over the same region. Streamline patterns and velocity gradient error contours are presented for these two designs.
We investigate drop breakup in dilute Newtonian emulsions in simple shear flow using high-speed video microscopy over a wide range of viscosity ratio (0.0017<λ<3.5), focusing on high capillary number (Ca up to 12 Cac, Cac is the critical capillary number). The final drop size distribution of emulsions is found to be intimately linked to the drop breakup mechanism, which depends on Ca and λ. Drop breakup is caused by end pinching at Ca<2 Cac. For Ca<2 Cac, breakup dynamics are strongly controlled by λ. For 0.1<λ<1, capillary instability is the dominant drop breakup mechanism, and thread radius and wavelength at breakup are independent of the initial drop sizes. Fairly monodisperse emulsions are obtained, and the average drop size is inversely proportional to the shear rate. For 1.0<λ<3. 5, long wavelength capillary instability generates large satellite drops, resulting in emulsions with bimodal distribution. For λ<0.1, a new drop re-breaking mechanism is observed, producing polydisperse emulsions. The polydispersity increases with decreasing λ. The capillary number based on the thread radius at breakup CaT is about 2.5 Cac and shows a minimum at λ=1.0. The measured CaT agrees with slender body theory for λ<0.1. Drops deform pseudo affinely for 0.1<λ<1.0, but deformation deviates from being pseudo affine otherwise.
This communication presents experimental data for steady and transient deformations of a purely elastic drop (PIB/PB Boger fluid) suspended in a Newtonian fluid (PDMS) undergoing a planar extensional flow produced by a four-roll mill. The viscoelastic effects alter the steady drop shape from being ellipsoidal to drop shapes with more blunt ends. The results reflect a balance between the direct tensile stress contribution of the viscoelastic fluid to the normal stress balance, and modifications of the viscous (i.e. Newtonian) stress and pressure due to viscoelastic changes in the flow. This is qualitatively consistent with the conclusion from recent simulation results [S. Ramaswamy, L.G. Leal, The deformation of a viscoelastic drop subjected to steady uniaxial extensional flow of a Newtonian fluid, Journal of Non-Newtonian Fluid Mechanics 85 (2–3) (1999) 127–163]. There is no overshoot of the drop deformation upon start-up of the flow, and a relative insensitivity to the Deborah number. However, when the external flow is turned off, there is a broad spectrum of relaxation times for the drop to return to a spherical shape, between the characteristic capillary timescale for a Newtonian fluid, aμ/σ, to the longer polymer relaxation time, τR.
In this note we examine the implications of Cahn-Hilliard diffusion on mass conservation when using a phase-field model for simulating two-phase flows. Even though the phase-field variable φ is conserved globally, a drop shrinks spontaneously while φ shifts from its expected values in the bulk phases. Those changes are found to be proportional to the interfacial thickness, and we suggest guidelines for minimizing the loss of mass. Moreover, there exists a critical radius below which drops will eventually disappear. With a properly chosen mobility parameter, however, this process will be much slower than the physics of interest and thus has little ill effect on the simulation.
The hydrodynamic interaction between a droplet immersed in Couette flow and the containing walls is studied. The analysis is based on the assumptions that the disturbance flow induced by the droplet is without inertia, that the droplet maintains its nearly spherical shape and that the radius of the droplet is small compared with the distance between the walls. Based on Lorentz's reflection method, a first-order simple analytical solution is derived for the case of a droplet in close vicinity to one wall. An integral solution is given for the general configuration of a droplet interacting with two walls. First-order corrections for wall effects are obtained for the drag force and the droplet's deviation from sphericity.
This paper is an experimental investigation of the deformation and relaxation of a Newtonian drop suspended in a PIB/PB Boger fluid. The suspending fluid is undergoing a planar extensional flow produced in a four-roll mill. We show that increasing elasticity of the suspending fluid has a pronounced effect on both the deformation and relaxation of a drop. For steady flows, as the strength of viscoelastic effects in the suspending fluid is increased, the drops become more deformed, with ends that are generally more pointed. This leads to a decrease in the maximum (“critical”) capillary number for the existence of a steady, deformed drop shape. In transient startup and step flows, the elasticity of the suspending fluid produces a large deformation shape that is more pointed at its ends and more tubular in its midsection than is observed for a drop in a Newtonian fluid (bulbous ends with necking at the waist). This enables a drop in the PIB/PB suspending fluid to be extended to a longer length without breaking upon flow cessation. However, at smaller deformations, the elasticity of the suspending fluid retards the relaxation of the drop. The observed viscoelastic effects on the steady and transient deformation, as well as the relaxation of drops in the PIB/PB suspending fluid, cannot be explained by viscoelastic modifications of the global, undisturbed flow field. Instead, our results suggest the existence of a non-linear coupling between the drop shape, the local disturbance flow, and the polymer configuration in the vicinity of the drop. This coupling enhances elastic effects, such that a drop can display significant non-Newtonian behavior prior to any changes in the global, undisturbed flow field.
We present a new phase field model for three-component immiscible liquid flows with surface tension. In the phase field approach, the classical sharp-interface between the two immiscible fluids is replaced by a transition region across which the properties of fluids change continuously. The proposed method incorporates a chemical potential which can eliminate the unphysical phase field profile and a continuous surface tension force formulation from which we can calculate the pressure field directly from the governing equations. The capabilities of the method are demonstrated with several examples. We compute the ternary phase separation via spinodal decomposition, equilibrium phase field profiles, pressure field distribution, and a three-interface contact angle resulting from a spreading liquid lens on an interface. The numerical results show excellent agreement with analytical solutions.
We propose a simple and effective iterative procedure to generate consistent initial conditions for the lattice Boltzmann equation (LBE) for incompressible flows with a given initial velocity field u0. Using the Chapman-Enskog analysis we show that not only the proposed procedure effectively solves the Poisson equation for the pressure field p0 corresponding to u0, it also generates at the same time the initial values for the nonequilibrium distribution functions {fα} in a consistent manner. This procedure is validated for the decaying Taylor–Green vortex flow in two dimensions and is shown to be particularly effective when using the generalized LBE with multiple relaxation times.
A kinetic theory approach to collision processes in ionized and neutral gases is presented. This approach is adequate for the unified treatment of the dynamic properties of gases over a continuous range of pressures from the Knudsen limit to the high-pressure limit where the aerodynamic equations are valid. It is also possible to satisfy the correct microscopic boundary conditions. The method consists in altering the collision terms in the Boltzmann equation. The modified collision terms are constructed so that each collision conserves particle number, momentum, and energy; other characteristics such as persistence of velocities and angular dependence may be included. The present article illustrates the technique for a simple model involving the assumption of a collision time independent of velocity; this model is applied to the study of small amplitude oscillations of one-component ionized and neutral gases. The initial value problem for unbounded space is solved by performing a Fourier transformation on the space variables and a Laplace transformation on the time variable. For uncharged gases there results the correct adiabatic limiting law for sound-wave propagation at high pressures and, in addition, one obtains a theory of absorption and dispersion of sound for arbitrary pressures. For ionized gases the difference in the nature of the organization in the low-pressure plasma oscillations and in high-pressure sound-type oscillations is studied. Two important cases are distinguished. If the wavelengths of the oscillations are long compared to either the Debye length or the mean free path, a small change in frequency is obtained as the collision frequency varies from zero to infinity. The accompanying absorption is small; it reaches its maximum value when the collision frequency equals the plasma frequency. The second case refers to waves shorter than both the Debye length and the mean free path; these waves are characterized by a very heavy absorption.
- S Simonis
- M J Krause
S. Simonis and M. J. Krause, Limit consistency for lattice Boltzmann equations, preprint,
arXiv:2208.06867, 2022.