Boltzmann Equation Method Research Articles

Analysis of force treatment in lattice Boltzmann equation method

In lattice Boltzmann equation (LBE), density distribution function is apparently relaxed to a given state fi(gs)(ρ,u∗) with an external/internal force, here ρ is fluid density and velocity u∗ can be different with physical velocity u. In the literature, it was shown that fi(0) should be fi(gs)(ρ,u∗), however, present work shows that fi(0) is fi(eq)(ρ,u) with different force scheme in single relaxation time LBE. The corresponding error terms are analyzed at Navier–Stokes level, and numerical results confirm our theoretical analysis.

Heat transfer and flow transitions of a thermal plume generated by a heating element on the enclosure bottom wall

Abstract The two-dimensional buoyancy-induced flow above a heating horizontal element in the center of the bottom of a rectangular enclosure filled with air is numerically studied using the thermal lattice Boltzmann equation method with two distribution functions. The heating element has a higher temperature than that of the remaining walls. The width and height of the enclosure are 5 and 10 times the length of the heating element respectively. The flow is space and time symmetric if after a transient, the flow is symmetric with respect to the vertical axis of the enclosure and is time independent. A suitable quantity, here called the asymmetry, is zero when the flow is time and space symmetric, positive and constant when the first symmetry is lost, and positive and time dependent when both symmetries are broken. Flow transitions are studied as the Rayleigh number, R a , changes in a large interval, 1000 ≤ R a ≤ 350 000 . Eight flow transitions were found, the first ones related to the space and time symmetries. When both symmetries are broken, additional flow transitions were found by studying the Fourier power spectrum of the asymmetry, which allows the identification of the fundamental frequency and the appearance of harmonics and other frequencies. The average Nusselt number as a function of R a is also studied and a transition in their relationship is found at the appearance of harmonics, giving way to two different correlations.

Deformation of bubbles in transformer oil at the action of alternating electric field

Abstract The experimental investigations of the deformation of bubbles floating in transformer oil under the action of uniform electric field were performed. The model based on the lattice Boltzmann equation method for the electrohydrodynamic flows accompanying this process was developed. The degree of deformation of bubbles observed in experiments is in agreement with the one obtained in simulations. Comparison with theoretical calculations show that the experimental data on the deformation of bubbles in transformer oil under the action of strong alternating electric field with a magnitude up to 4 kV/mm are reasonably explained by a model with a changeless surface tension.

LBE simulation of impinging stream inside a T-junction mixer

Lattice Boltzmann Equation (LBE) method is utilized to simulate impinging stream (IS) in a T-junction mixer using a TD2G9 model. It aims to investigate the influence of Reynolds number (Re), aspect ratio of outlet diameter to inlet diameter, ratio of opposite inlet velocities, and the thermal boundary conditions on flow, mixing and heat transfer characteristics. In particular, the vortex evolution, velocity distribution, mixing index and Nusselt number (Nu) distribution in the T-junction mixer are explored in details. Four types of vortices and flow regimes are observed. The instantaneous and time-averaged flow and thermal fields, including vortex structure, transition of flow regimes, streamline and the Nusselt number distribution are discussed. Distinct quantitative transitions, even for dramatic change, are observed near the critical Re. At a low or moderate aspect ratio, the symmetric coherent structure is observed in an unstable flow regime. At a larger aspect ratio, the flow in the T-mixer becomes turbulent and asymmetric. The unequal injections velocities of the nozzles impose significant influence on the flow structure, mixing and heat transfer in vertical tube. Using larger difference between the two inlet velocities can result in more obvious change in flow characteristics. Moreover, mixing index is found to be valid in evaluating the mixing degree under a sinusoidal inlet velocity.

Chapman–Enskog Analyses on the Gray Lattice Boltzmann Equation Method for Fluid Flow in Porous Media

The gray lattice Boltzmann equation (GLBE) method has recently been used to simulate fluid flow in porous media. It employs a partial bounce-back of populations (through a fractional coefficient θ, which represents the fraction of populations being reflected by the solid phase) in the evolution equation to account for the linear drag of the medium. Several particular GLBE schemes have been proposed in the literature and these schemes are very easy to implement; but there exists uncertainty about the need for redefining the macroscopic velocity as there has been no systematic analysis to recover the Brinkman equation from the various GLBE schemes. Rigorous Chapman–Enskog analyses are carried out to show that the momentum equation recovered from these schemes can satisfy Brinkman equation to second order in $$ \epsilon $$ only if $$ \theta = {\rm O}\left( \epsilon \right) $$ in which $$ \epsilon $$ is the ratio of the lattice spacing to the characteristic length of physical dimension. The need for redefining macroscopic velocity is shown to be scheme-dependent. When a body force is encountered such as the gravitational force or that caused by a pressure gradient, different forms of forcing redefinitions are required for each GLBE scheme.

Lattice Boltzmann Simulation of the Hydrodynamic Entrance Region of Rectangular Microchannels in the Slip Regime.

Developing a three-dimensional laminar flow in the entrance region of rectangular microchannels has been investigated in this paper. When the hydrodynamic development length is the same magnitude as the microchannel length, entrance effects have to be taken into account, especially in relatively short ducts. Simultaneously, there are a variety of non-continuum or rarefaction effects, such as velocity slip and temperature jump. The available data in the literature appearing on this issue is quite limited, the available study is the semi-theoretical approximate model to predict pressure drop of developing slip flow in rectangular microchannels with different aspect ratios. In this paper, we apply the lattice Boltzmann equation method (LBE) to investigate the developing slip flow through a rectangular microchannel. The effects of the Reynolds number (1 < Re < 1000), channel aspect ratio (0 < ε < 1), and Knudsen number (0.001 < Kn < 0.1) on the dimensionless hydrodynamic entrance length, and the apparent friction factor, and Reynolds number product, are examined in detail. The numerical solution of LBM can recover excellent agreement with the available data in the literature, which proves its accuracy in capturing fundamental fluid characteristics in the slip-flow regime.

Modeling an Ascending Nitrogen Gas Bubble in a Medium Crude Oil by Lattice Boltzmann Method

The study and modeling of oil biphasic systems, liquid-liquid and liquid-gas, focus mainly on the details of the modifications and application of the numerical methods itself. The correspondence between theoretical and experimental results and the information needed to apply a certain numerical method, usually remain in the background. On the other hand, in the particular case of the prediction of minimum miscibility pressure, extremely important parameter in oil exploration, references that show qualitative and numerical data associated with the characterization of the systems are scarce. The above reasons motivated the realization of this work. We used the Lattice Boltzmann Equation method to model a two-dimensional system of the displacement of a nitrogen gas bubble through a medium crude oil, under different pressure conditions keeping the temperature constant. According to experimental data, the bubble is not miscible by the crude, under a pressure range of 5000 psi to 6500 psi; nevertheless, the bubble is miscible in the range of 7000 psi to 7500 psi. Throughout simulations performed under similar conditions, we showed that it can be inferred the critical pressure range of miscibility of a medium crude oil.

Power-law fluid flow in driven enclosures with undulation using MRT-lattice Boltzmann method

In this article, the flow of non-Newtonian fluid (which is represented by the power-law model) in two-dimensional (2D) driven enclosures is studied. The enclosure consists of regular, rectangular shaped undulations on the bottom wall. Multiple-Relaxation time (MRT) collision model for the lattice Boltzmann equation method (LBM) is employed. First, numerical validation is performed by comparing the MRT-LBM results of power-law fluid flow inside the wall-driven square enclosure (no undulation) and flow inside diagonally flipped L-shaped enclosure with the literature. The strain rate profiles for square enclosure without undulations are compared using various equations available for strain rate calculation in the LBM literature. Further, the effect of different values of the non-hydrodynamic relaxation parameters on the flow is examined. Then, for the undulated enclosures, flow features and eddy dynamics are analyzed and discussed for the variations in the power-law index, n, to represent shear-thinning and shear thickening fluids. The effects of various parameters such as Reynolds numbers, wall undulation heights and wavelength of undulations on the power-law fluid flow are analyzed. Also, the variation of viscosity with spatial location for steady-state flow and total kinetic energy within the computational domain are presented for various values of power-law index.

Phase-field lattice Boltzmann modeling of boiling using a sharp-interface energy solver.

The main objective of this paper is to extend an isothermal incompressible two-phase lattice Boltzmann equation method to model liquid-vapor phase change problems using a sharp-interface energy solver. Two discrete particle distribution functions, one for the continuity equation and the other for the pressure evolution and momentum equations, are considered in the current model. The sharp-interface macroscopic internal energy equation is discretized with an isotropic finite difference method to find temperature distribution in the system. The mass flow generated at liquid-vapor phase interface is embedded in the pressure evolution equation. The sharp-interface treatment of internal energy equation helps to find the interfacial mass flow rate accurately where no free parameter is needed in the calculations. The proposed model is verified against available theoretical solutions of the two-phase Stefan problem and the two-phase sucking interface problem, with which our simulation results are in good agreement. The liquid droplet evaporation in a superheated vapor, the vapor bubble growth in a superheated liquid, and the vapor bubble rising in a superheated liquid are analyzed and underlying physical characteristics are discussed in detail. The model is successfully tested for the liquid-vapor phase change with large density ratio up to 1000.

Fluid flow in wall-driven enclosures with corrugated bottom

In this article, incompressible, continuum regime, viscous flow of Newtonian fluid in two-dimensional (2D) wall-driven enclosures consisting of regular, square shaped, corrugations on the bottom wall is studied numerically. Steady and consistent simulation results are obtained using kinetic theory based lattice Boltzmann equation method (LBM) and solution of Navier–Stokes equation based on fictitious domain method (FDM). First, numerical validation is performed by comparing LBM and FDM results for velocity profiles at particular sections inside the enclosures and vertical velocity gradient at the top of the corrugation cavity. Flow features are then compared for variations in Reynolds number, bottom-wall corrugation height and number of these corrugations. Further, complex eddy dynamics with respect to input parameters and geometry is discussed in detail. Flow transition Reynolds numbers showing distinct flow behavior are found. The numerical results obtained are verified and appear to be consistent with the previously published results for 2D flow inside slender and shallow cavity enclosures.

Analysis of force treatment in the pseudopotential lattice Boltzmann equation method.

In this paper, different force treatments are analyzed in detail for a pseudopotential lattice Boltzmann equation (LBE), and the contribution of third-order error terms to pressure tensor with a force scheme is analyzed by a higher-order Chapman-Enskog expansion technique. From the theoretical analysis, the performance of the original force treatment of Shan-Chen (SC), Ladd, Guo etal., and the exact difference method (EDM) are ɛ_{Ladd}<ɛ_{Guo}<ɛ_{EDM}≤ɛ_{SC} with the relaxation time τ≥1, while ɛ_{Ladd}<ɛ_{Guo}<ɛ_{SC}<ɛ_{EDM} with τ<1; here ɛ is a parameter related to the mechanical stability and the subscripts are the corresponding force scheme. To be consistent with the thermodynamic theory, a force term is introduced to modify the coefficients in the pressure tensor. Some numerical simulations are conducted to show that the predictions of modified force treatment of the pseudopotential LBE are all in good agreement with the analytical solution and other predictions.

Pseudopotential lattice Boltzmann equation method for two-phase flow: A higher-order Chapmann-Enskog expansion.

In this paper, a higher order Chapmann-Enskog expansion technique is applied to pseudopotential lattice Boltzmann equation (LBE), and the contribution of third order error terms to pressure tensor is analyzed in detail. To be consistent with the thermodynamic theory, a force term is introduced to modify the coefficients in the pressure tensor. Some numerical simulations are conducted to validate the LBE, and the results show that the predictions of the present LBE agree well with the analytical solution and other predictions.

希薄気体流れを対象としたトポロジー最適化

We construct a topology optimization method for two dimensional rarefied gas flow problems, based on level-set boundary expressions. The degree of rarefaction is expressed by the Knudsen number, which is the ratio of the mean free path and the characteristic length of the system. As the Knudsen number approaches 0 in the limit, flow behaviors can be described by Navier-Stokes equations and topology optimization methods for such flows have already been proposed. On the other hand, the governing equation for flows which have a large rarefaction is the Boltzmann equation and topology optimization methods for such flows have not been seen. This paper presents the topology optimization method for rarefied gas flows whose Knudsen number is approximately 1, aiming at an application for the design of flow channels in micromachines. We use the Bhatnagar-Gross-Krook (BGK) model of the Boltzmann equation and extend it to the entire design domain that includes both rarefied gas and solid domains. First, we briefly discuss the Boltzmann equation and the level set-based topology optimization method. Second, an optimization problem is formulated to address the design of flow channels that aim to maximize the flow velocity induced along a temperature gradient. Finally, several numerical examples demonstrate the validity and usefulness of the proposed method.

Breakdown Electric Field of Hot 30% CF3I/CO2 Mixtures at Temperature of 300–3500 K During Arc Extinction Process**supported by National Natural Science Foundation of China (No. 10875093)

We calculated the uniform dielectric breakdown field strength of residual 30% CF3I/CO2 gas mixtures during the arc extinction process over the temperature range 300–3500 K at 0.1 MPa. The limiting reduced field strengths are decided by a balance of electron generation and loss based on chemical reactions estimated by the electron energy distribution function (EEDF), which employs the Boltzmann equation method with two-term expanding approximation in the steady-state Townsend (SST) condition. During the insulation recovery phase, the hot CF3I/CO2 gas mixtures have maximum dielectric strength at a temperature of about 1500 K. At room temperature 300 K, the electric strength after arc extinction (90.3 Td, 1 Td=10−21 V·m2) is only 38% of the original value before arc (234.9 Td). The adverse insulation recovery ability of CF3I/CO2 gas mixtures in arc extinction hinders its application in electric circuit breakers and other switchgears as an arc quenching and insulating medium.

Comparison of the lattice Boltzmann equation and discrete unified gas-kinetic scheme methods for direct numerical simulation of decaying turbulent flows.

The main objective of this work is to perform a detailed comparison of the lattice Boltzmann equation (LBE) and the recently developed discrete unified gas-kinetic scheme (DUGKS) methods for direct numerical simulation (DNS) of the decaying homogeneous isotropic turbulence and the Kida vortex flow in a periodic box. The flow fields and key statistical quantities computed by both methods are compared with those from the pseudospectral method at both low and moderate Reynolds numbers. The results show that the LBE is more accurate and efficient than the DUGKS, but the latter has a superior numerical stability, particularly for high Reynolds number flows. In addition, we conclude that the DUGKS can adequately resolve the flow when the minimum spatial resolution parameter k_{max}η>3, where k_{max} is the maximum resolved wave number and η is the flow Kolmogorov length. This resolution requirement can be contrasted with the requirements of k_{max}η>1 for the pseudospectral method and k_{max}η>2 for the LBE. It should be emphasized that although more validations should be conducted before the DUGKS can be called a viable tool for DNS of turbulent flows, the present work contributes to the overall assessment of the DUGKS, and it provides a basis for further applications of DUGKS in studying the physics of turbulent flows.

Effects of tangential-type boundary condition discontinuities on the accuracy of the lattice Boltzmann method for heat and mass transfer.

We present a systematic study on the effects of tangential-type boundary condition discontinuities on the accuracy of the lattice Boltzmann equation (LBE) method for Dirichlet and Neumann problems in heat and mass transfer modeling. The second-order accurate boundary condition treatments for continuous Dirichlet and Neumann problems are directly implemented for the corresponding discontinuous boundary conditions. Results from three numerical tests, including both straight and curved boundaries, are presented to show the accuracy and order of convergence of the LBE computations. Detailed error assessments are conducted for the interior temperature or concentration (denoted as a scalar ϕ) and the interior derivatives of ϕ for both types of boundary conditions, for the boundary flux in the Dirichlet problem and for the boundary ϕ values in the Neumann problem. When the discontinuity point on the straight boundary is placed at the center of the unit lattice in the Dirichlet problem, it yields only first-order accuracy for the interior distribution of ϕ, first-order accuracy for the boundary flux, and zeroth-order accuracy for the interior derivatives compared with the second-order accuracy of all quantities of interest for continuous boundary conditions. On the lattice scale, the LBE solution for the interior derivatives near the singularity is largely independent of the resolution and correspondingly the local distribution of the absolute errors is almost invariant with the changing resolution. For Neumann problems, when the discontinuity is placed at the lattice center, second-order accuracy is preserved for the interior distribution of ϕ; and a "superlinear" convergence order of 1.5 for the boundary ϕ values and first-order accuracy for the interior derivatives are obtained. For straight boundaries with the discontinuity point arbitrarily placed within the lattice and curved boundaries, the boundary flux becomes zeroth-order accurate for Dirichlet problems; and all three quantities, including the interior and boundary ϕ values and the interior derivatives, are only first-order accurate for Neumann problems.

Local membrane length conservation in two-dimensional vesicle simulation using a multicomponent lattice Boltzmann equation method.

We present a method for applying a class of velocity-dependent forces within a multicomponent lattice Boltzmann equation simulation that is designed to recover continuum regime incompressible hydrodynamics. This method is applied to the problem, in two dimensions, of constraining to uniformity the tangential velocity of a vesicle membrane implemented within a recent multicomponent lattice Boltzmann simulation method, which avoids the use of Lagrangian boundary tracers. The constraint of uniform tangential velocity is carried by an additional contribution to an immersed boundary force, which we derive here from physical arguments. The result of this enhanced immersed boundary force is to apply a physically appropriate boundary condition at the interface between separated lattice fluids, defined as that region over which the phase-field varies most rapidly. Data from this enhanced vesicle boundary method are in agreement with other data obtained using related methods [e.g., T. Krüger, S. Frijters, F. Günther, B. Kaoui, and J. Harting, Eur. Phys. J. 222, 177 (2013)10.1140/epjst/e2013-01834-y] and underscore the importance of a correct vesicle membrane condition.

Cahn-Hilliard方程式に対する格子ボルツマン法の精度評価

Cahn-Hilliard方程式に対する格子ボルツマン法の精度評価

Analysis of the insulation characteristics of C3F8 gas mixtures with N2 and CO2 using boltzmann equation method

This paper studied the insulation characteristics of C3F8 gas mixtures with N2 and CO2 from the point view of electron swarm parameters. The density-normalized effective ionization coefficient (α-η)/N and electron drift velocity Ve in binary C3F8-N2 and C3F8- CO2 gas mixtures are calculated by solving the Boltzmann equation in the condition of steady-state Townsend discharge experiment. Values of the limiting electric field strength for which the electron ionization exactly balances the attachment are derived from the effective ionization coefficient data. It is found that the limiting field strength of C3F8-N2 gas mixture is greater than that of C3F8-CO2 for C3F8 gas content k varied from 0% to 100%, and the C3F8-N2 gas mixture containing 20% proportion C3F8 can achieve 60% dielectric strength of pure C3F8. Additionally, the synergistic effects of the insulation strength for C3F8 gas mixtures are also evaluated and compared with that of SF6-N2 mixture. The calculation results, together with the discussions on liquefaction temperature and global warming potential indicate that gas mixtures containing C3F8 are feasible for applying in electric power apparatus as insulation medium.

Continuous surface force based lattice Boltzmann equation method for simulating thermocapillary flow

In this paper, we extend a lattice Boltzmann equation (LBE) with continuous surface force (CSF) to simulate thermocapillary flows. The model is designed on our previous CSF LBE for athermal two phase flow, in which the interfacial tension forces and the Marangoni stresses as the results of the interface interactions between different phases are described by a conception of CSF. In this model, the sharp interfaces between different phases are separated by a narrow transition layers, and the kinetics and morphology evolution of phase separation would be characterized by an order parameter via Cahn–Hilliard equation which is solved in the frame work of LBE. The scalar convection–diffusion equation for temperature field is resolved by thermal LBE. The models are validated by thermal two layered Poiseuille flow, and two superimposed planar fluids at negligibly small Reynolds and Marangoni numbers for the thermocapillary driven convection, which have analytical solutions for the velocity and temperature. Then thermocapillary migration of two/three dimensional deformable droplet are simulated. Numerical results show that the predictions of present LBE agreed with the analytical solution/other numerical results.