Finite Difference Lattice Boltzmann Method Research Articles

From mesoscopic models to continuum mechanics: Newtonian and non-newtonian fluids

A review of the BGK approximation to obtain the equations of motion for an incompressible fluid is presented and its drawbacks are revealed. In order to overcome these inherent problems, new models for the particle distribution functions are needed. Using the Finite Difference Lattice Boltzmann Method (FDLBM) due to Fu and So (2009) [1] and the Thermal Difference Discrete Flux Method (TDDFM) proposed by Fu et al. 2012 [2], it is shown that the newer distribution functions lead to the mass conservation equation, the equations of motion and the energy balance equation for incompressible fluids in two dimensions, employing the D2Q9 lattice as the model. This derivation is extended to compressible fluids as well. Next, using the D3Q15 lattice as an example, the three dimensional equations of continuum mechanics are derived. Since no restrictions are placed on the constitutive equations, the theoretical development applies to all fluids, whether they be Newtonian, or power law fluids, or viscoelastic and viscoplastic fluids. Finally, some comments are offered regarding the numerical scheme to calculate the particle distribution functions to determine the velocity and temperature fields.

差分格子ボルツマン法における高次の非線形スキーム（チャンネル乱流のDNSによる評価）

差分格子ボルツマン法における高次の非線形スキーム（チャンネル乱流のDNSによる評価）

Simulation of double diffusive natural convection and entropy generation of power-law fluids in an inclined porous cavity with Soret and Dufour effects (Part I: Study of fluid flow, heat and mass transfer)

In this paper, double diffusive natural convection of non-Newtonian power-law fluids in an inclined porous cavity in the presence of Soret and Dufour parameters has been analyzed by Finite Difference Lattice Boltzmann Method (FDLBM). This study has been performed for the certain pertinent parameters of thermal Rayleigh number (RaT=104 and 105), Darcy number (Da=10−4, 10−3, and 10−2), power-law index (n=0.6–1.4), Lewis number (Le=2.5 and 5), inclined angles (θ=0°, 40°, 80°, and 120°), Dufour parameter (Df=0, 1, and 5), Soret parameter (Sr=0, 1, and 5) and the buoyancy ratio (N=−1 and 1). Results indicate that the augmentation of the Darcy number causes heat and mass transfer to rise for different power-law indexes. At Da=10−4, the heat and mass transfer increase with the augmentation of the power-law index in the absence of the Soret and Dufour parameters. The rise of the inclined angle from θ=0° to 40° and from θ=80° to 120° provokes heat and mass transfer to augment. As the Soret and Dufour numbers equal zero, the heat transfer enhances with the increment of the power-law index at Da=10−3. The heat transfer increases with the rise of the Dufour parameter and the mass transfer enhances as the Soret parameter increases for different power-law indexes and thermal Rayleigh numbers. In some cases, the augmentation of Soret and Dufour parameters alter the behavior of heat and mass transfer against the alteration of the power-law index.

Simulation of double diffusive natural convection and entropy generation of power-law fluids in an inclined porous cavity with Soret and Dufour effects (Part II: Entropy generation)

In this paper, entropy generation of associated with double diffusive natural convection of non-Newtonian power-law fluids in an inclined porous cavity has been analyzed by Finite Difference Lattice Boltzmann Method (FDLBM). The entropy generations due to fluid friction, heat and mass transfer have been simulated and analyzed for the certain pertinent parameters of thermal Rayleigh number (RaT=104 and 105), Darcy number (Da=10−4, 10−3, and 10−2), power-law index (n=0.6–1.4), Lewis number (Le=2.5 and 5), inclined angles (θ=0°, 40°, 80°, and 120°), Dufour parameter (Df=0, 1, and 5), Soret parameter (Sr=0, 1, and 5) and the buoyancy ratio (N=−1 and 1). Results indicate that the augmentation of the thermal Rayleigh number enhances different entropy generations and declines the average Bejan number. The increase in the Darcy number provokes various irreversibilities to enhance and the average Bejan number decreases significantly. The augmentation of the inclined angle from θ=0° to 40° enhances various total entropy generations and plummets the average Bejan number. The increase in the inclined angle from θ=40° to 80° results in the drop of different total entropy generations and rises the average Bejan number. The rise of Soret and Dufour parameters enhances the entropy generations due to heat transfer and fluid friction. The change of power-law index alters various entropy generations, but the alteration does not follow a specific manner in different studied parameters.

Simulation of two-phase liquid-vapor flows using a high-order compact finite-difference lattice Boltzmann method

A high-order compact finite-difference lattice Boltzmann method (CFDLBM) is extended and applied to accurately simulate two-phase liquid-vapor flows with high density ratios. Herein, the He-Shan-Doolen-type lattice Boltzmann multiphase model is used and the spatial derivatives in the resulting equations are discretized by using the fourth-order compact finite-difference scheme and the temporal term is discretized with the fourth-order Runge-Kutta scheme to provide an accurate and efficient two-phase flow solver. A high-order spectral-type low-pass compact nonlinear filter is used to regularize the numerical solution and remove spurious waves generated by flow nonlinearities in smooth regions and at the same time to remove the numerical oscillations in the interfacial region between the two phases. Three discontinuity-detecting sensors for properly switching between a second-order and a higher-order filter are applied and assessed. It is shown that the filtering technique used can be conveniently adopted to reduce the spurious numerical effects and improve the numerical stability of the CFDLBM implemented. A sensitivity study is also conducted to evaluate the effects of grid size and the filtering procedure implemented on the accuracy and performance of the solution. The accuracy and efficiency of the proposed solution procedure based on the compact finite-difference LBM are examined by solving different two-phase systems. Five test cases considered herein for validating the results of the two-phase flows are an equilibrium state of a planar interface in a liquid-vapor system, a droplet suspended in the gaseous phase, a liquid droplet located between two parallel wettable surfaces, the coalescence of two droplets, and a phase separation in a liquid-vapor system at different conditions. Numerical results are also presented for the coexistence curve and the verification of the Laplace law. Results obtained are in good agreement with the analytical solutions and also the numerical results reported in the literature. The study shows that the present solution methodology is robust, efficient, and accurate for solving two-phase liquid-vapor flow problems even at high density ratios.

Simulation of heat transfer and entropy generation of MHD natural convection of non-Newtonian nanofluid in an enclosure

In this paper, heat transfer and entropy generation on laminar natural convection of non-Newtonian nanofluids in the presence of an external horizontal magnetic field in a square cavity has been analyzed by Finite Difference Lattice Boltzmann Method (FDLBM). The cavity is filled with water and nanoparticles of copper (Cu) while the mixture shows shear-thinning behavior. This study has been conducted for the certain pertinent parameters of Rayleigh number (Ra=104–105), power-law index (n=0.6–1), Hartmann number (Ha=0–90) and the volume fraction has been studied from φ=0 to 0.04. Results indicate that the augmentation of the power-law index causes heat transfer to drop in the absence of the magnetic field, by contrast, the heat transfer increases with the rise of power-law index in the presence of the magnetic field. The addition of nanoparticle augments heat transfer for multifarious studied parameters. The heat transfer drops with the increase in Hartmann number generally and also affects the power-law index and nanoparticles influences on heat transfer. Augmentation of the volume fraction and Rayleigh number enhance all kinds of entropy generations of heat transfer, fluid friction, and the magnetic field in different studied parameters. The increase in the Hartmann number causes the total entropy generation to drop and affects the influences of the power-law index and the volume fraction on the entropy generations.

Finite-difference lattice Boltzmann simulation on acoustics-induced particle deposition

Particle manipulation by acoustics has been investigated for many years. By a proper design, particle deposition can be induced by the same principle. The use of acoustics can potentially be developed into an energy-efficient technique for particle removal or filtration system as the pressure drop due to acoustic effects is low and the flow velocity is not necessary to be high. Two nonlinear acoustic effects, acoustic streaming and acoustic radiation pressure, are important. Acoustic streaming introduces vortices and stagnation points on the surface of an air duct and removes the particles by deposition. Acoustic radiation pressure causes particles to form agglomerates and enhances inertial impaction and/or gravitational sedimentation. The objective of this paper is to develop a numerical model to investigate the particle deposition induced by acoustic effects. A three-step approach is adopted and lattice Boltzamnn technique is employed as the numerical method. This is because the lattice Boltzmann equation is hyperbolic and can be solved locally, explicitly, and efficiently on parallel computers. In the first step, the acoustic field and its mean square fluctuation values are calculated. Due to the advantage of the lattice Boltzmann technique, a simple, stable and fast lattice Boltzmann method is proposed and verified. The result of the first step is input into the second step to solve for acoustic streaming. Another finite difference lattice Boltzmann method, which has been validated by a number of flows and benchmark cases in the literature, is used. The third step consists in tracking the particle's motion by a Lagrangian approach where the acoustic radiation pressure is considered. The influence of the acoustics effects on particle deposition is explained. The numerical result matches with an experiment. The model is a useful tool for optimizing the design and helps to further develop the technique.

FDLBM simulation of entropy generation in double diffusive natural convection of power-law fluids in an enclosure with Soret and Dufour effects

In this paper, double-diffusive natural convection with Soret and Dufour effects in a square cavity filled with non-Newtonian power-law fluid has been simulated by Finite Difference Lattice Boltzmann Method (FDLBM) while entropy generations through fluid friction, heat transfer, and mass transfer are analyzed. This study has been conducted for certain pertinent parameters of Rayleigh number (Ra=104 and 105), power-law index (n=0.6–1.4), Lewis number (Le=2.5 and 5), Dufour parameter (Df=0–1), Soret parameter (Sr=0–1) and the buoyancy ratio (N=−1,0.1,1). Results indicate that the augmentation of the power-law index causes heat and mass transfer to drop. Entropy generation due to fluid friction, heat and mass transfer rises with increase in Rayleigh number. Augmentation of Lewis number enhances entropy generation due to mass transfer. At N=0.1 and 1, growth of power-law index causes different types of entropy generation to drop. The rise of buoyancy ratio from N=−1 to 1 augments the total irreversibility. The Dufour parameter provokes heat and mass transfer to rise while results in the enhancement of the total irreversibility. The Soret parameter augments mass transfer, but it does not affect heat transfer and total entropy generation considerably.

Numerical simulation of natural convection between hot and cold microtubes in a cylinder enclosure

Natural convection around a pair of hot and cold microtubes parallelized horizontally in an adiabatic cylindrical enclosure was numerically simulated, using a hybrid Lattice-Boltzmann Finite-Difference Method (LBFDM), coupled with Direct Forcing (DF) technique. The sharp interface DF scheme was used for the curve boundaries of both the microtubes and the cylindrical enclosure. The computation code was validated by using a case of natural convection in an eccentric annulus, and a similar previous study using large Rayleigh numbers. In addition to flow patterns and temperature profiles around those microtubes, a correlation between the Nusselt number (Nu) and the Rayleigh number (Ra) and their relative position was proposed in the form of Nu = f (Ra, Pr, θ). The results indicate that the effective heat transfer coefficient between the micro-tubes is the smallest when hot tube is placed directly over the cold one, and the maximum can be achieved with the cold tube above the hot tube, but with the inclination angle 60° from the upward direction when Ra < 1400.

FDLBM simulation of magnetic field effect on mixed convection in a two sided lid-driven cavity filled with non-Newtonian nanofluid

In this paper, laminar mixed convection of non-Newtonian nanofluids in a two sided lid-driven enclosure in the presence of a horizontal magnetic field has been analyzed by finite difference Lattice Boltzmann method (FDLBM). The cavity is filled with water and nanoparticles of Alumina (Al2O3) while the mixture shows shear-thinning behavior. This study has been conducted for the certain pertinent parameters of Richardson number (Ri=0.001–1), power-law index (n=0.2–1), and Hartmann numbers (Ha=0–60) and volume fraction has been studied from φ=0 to 0.09. Results indicate that the augmentation of Richardson number decreases heat transfer. The fall of the power law index declines heat transfer for different studied Richardson numbers. The addition of nanoparticle augments heat transfer for multifarious studied parameters. The increase in Hartmann number drops heat transfer generally and also affects the power-law index and nanoparticle influences on heat transfer.

FDLBM simulation of mixed convection in a lid-driven cavity filled with non-Newtonian nanofluid in the presence of magnetic field

In this paper, laminar mixed convection of non-Newtonian nanofluids in a square lid-driven cavity in the presence of a vertical magnetic field has been analyzed by Finite Difference Lattice Boltzmann Method (FDLBM). The cavity is filled with water and nanoparticles of copper (Cu) while the mixture shows shear-thinning behavior. This study has been conducted for the certain pertinent parameters of Richardson number (Ri = 0.001–1), power-law index (n = 0.2–1), and the volume fraction from φ = 0 to 0.09. Results indicate that the augmentation of Richardson number decreases heat transfer. The fall of the power law index declines heat transfer for different studied Richardson numbers. The addition of nanoparticle augments heat transfer for multifarious studied parameters although the effect of nanoparticle on the enhancement of heat transfer varies in different power-law indexes. The magnetic field declines heat transfer generally and also changes the effect degree of nanoparticles on the increase in heat transfer.

Mesoscopic simulation of magnetic field effect on Bingham fluid in an internal flow

In this paper, effect of magnetic field on Bingham fluid in a lid-driven cavity has been analyzed by finite difference lattice Boltzmann method (FDLBM). This study has been performed for the certain pertinent parameters of Reynolds number (Re = 400, 1000, and 5000), Stuart number (N = 0, 1, and 10) and Bingham number (Bn = 1 and 10) as the magnetic field is applied vertically. Results show that the increment of Bingham number augments the unyielded section in the cavity. Furthermore, the increase in Stuart number enhances the unyielded zone while the shapes of the streamlines alter entirely. The rise of Reynolds number provokes the influence of the magnetic field on the extension of the unyielded part to grow.

Simulation of vertical and horizontal magnetic fields effects on non-Newtonian power-law fluids in an internal flow using FDLBM

In this paper, the effect of a magnetic field on non-Newtonian fluid flow in a lid-driven cavity has been analyzed by Finite Difference Lattice Boltzmann Method (FDLBM). It was assumed that the viscosity of the fluid flow is shear-dependent as the power-law model has been selected for the viscosity of the flow. This study has been performed for the certain pertinent parameters of Reynolds number (Re=100–1000), Stuart number (N=0–50) and power-law index (n=0.6–1.4) as the magnetic field is applied at different inclinations of γ=0° and 90°. Results show that the increment of Reynolds number augments the effect of magnetic field on the fluid flow. Furthermore, the drop in the power-law indexes increases the influence of the magnetic field at the inclined angle of γ=90°. The greatest effect of the horizontal magnetic field (γ=0°) is observed for the power-law index of n=1.

Magnetic field effect on heat and mass transfer of mixed convection of shear-thinning fluids in a lid-driven enclosure with non-uniform boundary conditions

In this paper, the effect of a magnetic field on mixed convection of shear-thinning fluids in a square lid-driven cavity with sinusoidal boundary conditions under the combined buoyancy effects of thermal and mass diffusion has been analyzed by finite difference lattice Boltzmann method (FDLBM). This study has been conducted for the certain pertinent parameters of Richardson number (Ri = 0.00062–1), Hartmann number (Ha = 0–100), Lewis number (Le = 5 and 50), power-law index (n = 0.2–1) as the buoyancy ratio is studied from N = −10 to 10. Results indicate that the augmentation of Richardson number causes heat and mass transfer to drop. The enhancement of Hartmann number declines heat and mass transfer for multifarious buoyancy ratios and power-law indexes steadily at Ri = 0.00062 and 0.01. The fall of the power law index declines heat and mass transfer at Ri = 0.00062 and 0.01 for various Hartmann numbers and buoyancy ratios. The increase in power-law index and Hartmann number influence heat and mass transfer differently at Ri = 1 for various buoyancy ratios and Lewis numbers. The growth of Lewis number enhances mass transfer for the studied Hartmann numbers and buoyancy ratios.

Mesoscopic simulation of mixed convection on non-Newtonian nanofluids in a two sided lid-driven enclosure

In this paper, laminar mixed convection of non-Newtonian nanofluids in a square two sided lid-driven cavity has been analyzed by Finite Difference Lattice Boltzmann Method (FDLBM). The cavity is filled with water and nanoparticles of alumina (Al2O3) while the mixture shows shear-thinning behavior. This study has been conducted for the certain pertinent parameters of Richardson number (Ri=0.001–1), power-law index (n=0.2–1), and the volume fraction has been studied from φ=0–0.09. Results indicate that the augmentation of Richardson number causes the heat transfer to drop. The heat transfer drops for different Richardson numbers as the power-law index falls. The addition of nanoparticle augments heat transfer for multifarious studied parameters. The greatest effect of nanoparticle on the enhancement of heat transfer was observed at power-law index of n=0.2 for various Richardson numbers.

Chebyshev collocation spectral lattice Boltzmann method for simulation of low-speed flows.

In this study, the Chebyshev collocation spectral lattice Boltzmann method (CCSLBM) is developed and assessed for the computation of low-speed flows. Both steady and unsteady flows are considered here. The discrete Boltzmann equation with the Bhatnagar-Gross-Krook approximation based on the pressure distribution function is considered and the space discretization is performed by the Chebyshev collocation spectral method to achieve a highly accurate flow solver. To provide accurate unsteady solutions, the time integration of the temporal term in the lattice Boltzmann equation is made by the fourth-order Runge-Kutta scheme. To achieve numerical stability and accuracy, physical boundary conditions based on the spectral solution of the governing equations implemented on the boundaries are used. An iterative procedure is applied to provide consistent initial conditions for the distribution function and the pressure field for the simulation of unsteady flows. The main advantage of using the CCSLBM over other high-order accurate lattice Boltzmann method (LBM)-based flow solvers is the decay of the error at exponential rather than at polynomial rates. Note also that the CCSLBM applied does not need any numerical dissipation or filtering for the solution to be stable, leading to highly accurate solutions. Three two-dimensional (2D) test cases are simulated herein that are a regularized cavity, the Taylor vortex problem, and doubly periodic shear layers. The results obtained for these test cases are thoroughly compared with the analytical and available numerical results and show excellent agreement. The computational efficiency of the proposed solution methodology based on the CCSLBM is also examined by comparison with those of the standard streaming-collision (classical) LBM and two finite-difference LBM solvers. The study indicates that the CCSLBM provides more accurate and efficient solutions than these LBM solvers in terms of CPU and memory usage and an exponential convergence is achieved rather than polynomial rates. The solution methodology proposed, the CCSLBM, is also extended to three dimensions and a 3D regularized cavity is simulated; the corresponding results are presented and validated. Indications are that the CCSLBM developed and applied herein is robust, efficient, and accurate for computing 2D and 3D low-speed flows. Note also that high-accuracy solutions obtained by applying the CCSLBM can be used as benchmark solutions for the assessment of other LBM-based flow solvers.

FDLBM simulation of entropy generation due to natural convection in an enclosure filled with non-Newtonian nanofluid

In this paper, heat transfer and entropy generation due to laminar natural convection in a square cavity filled with non-Newtonian nanofluid have been analyzed by Finite Difference Lattice Boltzmann Method (FDLBM). The cavity is filled with water and nanoparticles of copper (Cu) while the mixture shows shear-thinning behavior. This study has been conducted for the certain pertinent parameters of Rayleigh number (Ra=104–105), power-law index (n=0.6–1), and the volume fraction has been studied from φ=0 to 0.06. Results indicate that the augmentation of the power-law index causes heat transfer to drop while increase in volume fraction of nanoparticles augments it. Entropy generation due to fluid friction and heat transfer rises as the Rayleigh number enhances. Augmentation of volume fraction enhances entropy generations due to heat transfer and fluid friction in different power-law indexes. The total entropy generation declines slightly as power-law index increases.

差分格子ボルツマン法による流れ方向に斜めに配置された2次元フィン列を過ぎる流れから生じる空力音の直接計算

差分格子ボルツマン法による流れ方向に斜めに配置された2次元フィン列を過ぎる流れから生じる空力音の直接計算

Three-dimensional simulation of interfacial convection in CO2–ethanol system by hybrid lattice Boltzmann method with experimental validation

By using a hybrid lattice-Boltzmann–finite-difference method (hybrid LBM–FDM method), three-dimensional simulations of solutal interfacial convection were conducted for the process of CO2 absorption into ethanol. A self-renewal interface model is adopted as an interfacial perturbation model. The simulation results revealed some three-dimensional features of the induced interfacial convection, such as the development of diverging cellular flow and Rayleigh plume-like convection in liquid phase. The concentration distribution of the simulation result is validated and found to be in well agreement with the Schlieren visualization results qualitatively. Additionally, the mass transfer enhancements by interfacial convection were investigated via both simulation and experiment for the absorption process, and the mass transfer is shown to be enhanced by the interfacial convection by about two-fold comparing with that by diffusion.

Spatial Scale Effects on Rayleigh Convection and Interfacial Mass Transfer Characteristics in CO2 Absorption

AbstractConcentration gradient‐induced Rayleigh convections in the CO2 absorption process were investigated by the hybrid Lattice‐Boltzmann/finite‐difference method (LBM‐FDM). The spatial scale effects on Rayleigh convection were studied by simulating Rayleigh convections with different liquid layers. The scale of convection increased with the liquid layer height but the average mass transfer coefficient showed the adverse tendency. The Rayleigh convection had a pronounced effect on the surface renewal. For better assessment of the renewal intensity, two statistical quantities were proposed. The transient variations of these quantities provided a good following feature with the mass transfer coefficient which confirms their accuracy and precision in characterizing the mass transfer process.