Lattice Boltzmann method applied to the solution of the energy equations of the transient conduction and radiation problems on non-uniform lattices

Application of the lattice Boltzmann method (LBM) to solve the energy equations of conduction–radiation problems is extended on non-uniform lattices. In the LBM on non-uniform lattices, the single relaxation time based on the minimum velocity is used. This minimum velocity corresponds to the smallest size lattice. Because information propagates with the same minimum velocity in the prescribed directions from all the lattice centers, in a given time step, they are not equidistant from the neighboring lattices. Collisions in the LBM take place at the same instant. Therefore, in the LBM on non-uniform lattices, in every time step, interpolation is required to carry the information to the neighboring lattice centers. To validate this very concept in heat transfer problems involving thermal radiation, transient conduction and radiation heat transfer problems in a 1-D planar and a 2-D rectangular geometries containing absorbing, emitting and scattering medium are considered. The finite volume method (FVM) is used to compute the radiative information. In both the geometries, results for the effects of various parameters are compared for LBM–FVM on uniform and non-uniform lattices. To establish the LBM–FVM on non-uniform lattices for the combined conduction and radiation heat transfer problems, numerical experiments were performed with different cluster values. The accurate results were found in all the cases.     

Solving transient heat conduction problems on uniform and non-uniform lattices using the lattice Boltzmann method

The lattice Boltzmann method (LBM) has been used to solve transient heat conduction problems in 1-D, 2-D and 3-D Cartesian geometries with uniform and non-uniform lattices. To study the suitability of the LBM, the problems have also been solved using the finite difference method (FDM). To check the performance of LBM for the non-uniform lattices, the results have been compared with uniform lattices. Cases with volumetric heat generation have also been considered. In 1-D problems, the FDM with implicit scheme was found to take more number of iterations and also the CPU time was more. However, with explicit scheme, with increase in the number of control volumes, the LBM was found faster than the FDM. In 2-D and 3-D problems, with increase in the number of control volumes, the LBM was found faster than the FDM. In 2-D problems, number of iterations in the two methods was comparable, while in 3-D problems, the LBM was found to take less number of iterations. The accurate results were found in all the cases.     

Simulation of Natural Convection in the Presence of Volumetric Radiation Using the Lattice Boltzmann Method

This article deals with the application of the lattice Boltzmann method (LBM) to the analysis of natural convection in the presence of volumetric radiation in a square cavity containing an absorbing, emitting, and scattering medium. Separate particle distribution functions in the LBM are used to calculate the density and velocity fields and the thermal field. The radiative term of the energy equation is computed using the finite-volume method. Streamlines, isotherms, and Nusselt number are analyzed for the effects of different parameters such as Rayleigh number, convection-radiation parameter, extinction coefficient, and scattering albedo.     

A Lattice Boltzmann Formulation for the Analysis of Radiative Heat Transfer Problems in a Participating Medium

Use of the lattice Boltzmann method (LBM) has been extended to analyze radiative transport problems in an absorbing, emitting, and scattering medium. In terms of collision and streaming, the present approach of the LBM for radiative heat transfer is similar to those being used in fluid dynamics and heat transfer for the analyses of conduction and convection problems. However, to mitigate the effect of the isotropy in the polar direction, in the present LBM approach, lattices with more number of directions than those being used for the 2-D system have been employed. The LBM formulation has been validated by solving benchmark radiative equilibrium problems in 1-D and 2-D Cartesian geometry. Temperature and heat flux distributions have been obtained for a wide range of extinction coefficients. The LBM results have been compared against the results obtained from the finite-volume method (FVM). Good comparison has been obtained. The numbers of iterations and CPU times for the LBM and the FVM have also been compared. The number of iterations in the LBM has been found to be much more than the FVM. However, computationally, the LBM has been found to be much faster than the FVM.     

Comparative Analysis of Thermal Models in the Lattice Boltzmann Method for the Simulation of Natural Convection in a Square Cavity

In this article, a comparative analysis of thermal models in the lattice Boltzmann method for the simulation of natural convection in a square cavity is presented. A hybrid method, in which the thermal equation is solved by the Navier-Stokes equation method while the mass and momentum equations are solved by the lattice Boltzmann method (LBM), is introduced and its merits are explained. All the governing equations are discretized on a cell-centered, nonuniform grid using the finite-volume method. The convection terms are treated by a second-order central-difference scheme with a deferred-correction method to ensure accuracy of solutions. The resulting algebraic equations are solved by a strongly implicit procedure. The hybrid method is applied to the simulation of natural convection in a square cavity and the predicted results are compared with the benchmark solutions given in the literatures. The predicted results are also compared with those by the double-population LBM and by the Navier-Stokes equation method. In general, the present hybrid method is as accurate as the Navier-Stokes equation method and the double-population LBM. The hybrid method shows better convergence and stability than the double-population LBM. These observations indicate that this hybrid method is an efficient and economic method for the simulation of incompressible fluid flow and heat transfer problems involving complex geometries.     

A Hybrid Lattice Boltzmann and Finite-Volume Method for Melting with Convection

A hybrid lattice Boltzmann and finite-volume model is proposed to solve the natural-convection-controlled melting problem. The lattice Boltzmann method (LBM) is applied to solve the velocity field, while the temperature field is obtained by the finite-volume method (FVM). The D2Q9 model and finite-difference velocity gradient boundary condition are used in the LBM and the SIMPLE algorithm with QUICK scheme is employed in the FVM. An interfacial tracking model based on energy balance at the interface is applied to obtain the location of the solid–liquid interface. The results from the present hybrid method are validated with experimental results, and good agreement is obtained.     

Computation of Turbulent Natural Convection in a Rectangular Cavity with the Lattice Boltzmann Method

A numerical study of a turbulent natural convection in a rectangular cavity with the lattice Boltzmann method (LBM) is presented. The primary emphasis of the present study is placed on investigation of accuracy and numerical stability of the LBM for the turbulent natural-convection flow. A HYBRID method in which the thermal equation is solved by the conventional Reynolds-averaged Navier-Stokes equation (RANS) method while the conservation of mass and momentum equations are resolved by the LBM is employed in the present study. The elliptic-relaxation model is employed for the turbulence model and the turbulent heat fluxes are treated by the algebraic flux model. All the governing equations are discretized on a cell-centered, nonuniform grid using the finite-volume method. The convection terms are treated by a second-order central-difference scheme with the deferred correction method to ensure accuracy and stability of solutions. The present LBM is applied to the prediction of a turbulent natural convection in a rectangular cavity and the computed results are compared with the experimental data commonly used for the validation of turbulence models and those by the conventional finite-volume method. It is shown that the LBM with the present HYBRID thermal model predicts mean velocity components and turbulent quantities which are as good as those by the conventional finite-volume method. It is also found that the accuracy and stability of the solution is significantly affected by the treatment of the convection term, especially near the wall.     

Lattice Boltzmann simulation of natural convection heat transfer in eccentric annulus

In this article, the natural convection flow in eccentric annulus is simulated numerically by Lattice Boltzmann Model (LBM) based on double-population approach. A numerical strategy presents for dealing with curved boundaries of second order accuracy for both velocity and temperature fields. The effect of vertical, horizontal, and diagonal eccentricity at various locations is examined at Ra = 10⁴ and σ = 2. Velocity and temperature distributions as well as Nusselt number are obtained. The results show that the average Nusselt number increases when the inner cylinder moves downward regardless of the radial position. The validation with previous studies shows that double-population approach can evaluate the velocity and temperature fields in curved boundaries with a good accuracy.     

Numerical study on permeability of gas diffusion layer with porosity gradient using lattice Boltzmann method

In this paper, the lattice Boltzmann method (LBM) has been employed to explore the permeability and internal fluid flow behavior of the gas diffusion layer (GDL). Three different non-uniform porosity distributions are designed as linear type, stepped type, and transitional type and compared with constant porosity samples. Results show that the linear porosity gradient distribution leads to higher permeability values compared with the other two types. For samples with total porosity of 0.65 and 0.75, optimal porosity gradient distributions bring about an enhancement of permeability have been found. The impact of porosity gradient distribution on the velocity field is presented. Dependencies of permeability with porosity and tortuosity are demonstrated through several fitted equations.     

Lattice Boltzmann simulation of heat conduction problems in non‐isothermally heated enclosures

A numerical study following the lattice Boltzmann method (LBM) is performed to solve transient heat conduction problems with and without volumetric heat generation/absorption in 2D and 3D Cartesian geometries. Uniform lattices are considered for both geometries. To validate the correctness of LBM, a finite difference method (FDM) is also used to solve the 2D problem without heat generation/absorption and results are compared with that of LBM. For both 2D and 3D geometries one of the walls is heated and cooled with a sinusoidal function and the rest of the walls are cooled isothermally. Effects of amplitude of the sinusoidal function and volumetric heat generation/absorption on temperature profiles are analyzed. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20406     

Study on forcing schemes in the thermal lattice Boltzmann method for simulation of natural convection flow problems

The present study addresses the effect of various schemes for applying an external force term on the accuracy and performance of the thermal lattice Boltzmann method (LBM) for simulation of free convection problems. Herein, the forcing schemes of Luo, shifted velocity method, Guo, and exact difference method are applied by considering three velocity discrete models of D2Q4, D2Q5, and D2Q9. The accuracy and performance of these schemes are evaluated with the simulation of three natural convection problems, namely, free convection in a closed cavity, in a square enclosure with a hot obstacle inside, and the Rayleigh-Benard problem. The obtained results based on the present thermal LBM with different forcing schemes and velocity discrete models are compared with the existing experimental and numerical data in the literature. This comparison study indicates that imposing all employed forcing schemes leads to similar performance for the simulation of free convection problems studied at the middle range of Rayleigh numbers. It is found that the Luo forcing scheme is simple for implementation in comparison with the other three forcing schemes and provides the results with acceptable accuracy at moderate Rayleigh numbers. At higher Rayleigh numbers, however, the Guo scheme is not only numerically stable but a more precise forcing scheme in comparison with the other three methods. It is illustrated that employing the discrete velocity model of D2Q4 has more appropriate numerical stability along with less computational cost in comparison with two other discrete velocity models for simulation of natural convection heat transfer.     

Lattice Boltzmann Method Applied to Radiative Transport Analysis in a Planar Participating Medium

This article deals with the extension of the usage of the lattice Boltzmann method (LBM) to the analysis of radiative heat transfer with and without conduction in a one-dimensional (1-D) planar participating medium. A novel lattice needed for the calculation of the volumetric radiation spanned over the 4π spherical space has been introduced. The LBM formulation is tested for three benchmark problems, namely, radiative equilibrium, nonradiative equilibrium, and a combined mode conduction–radiation problem in a planar geometry. In the combined mode problem, with radiative information known from the proposed lattice structure, the energy equation is also formulated and solved using the LBM. The D1Q2 lattice is used in the energy equation. For validation, in problems 1 and 2, the LBM results are compared with the finite-volume method (FVM), while in problem 3, the LBM-LBM results are compared with the LBM-FVM in which FVM is used for the computation of radiative information. Comparisons are made for the effects of the governing parameters such as the extinction coefficient, the scattering albedo, and so on, on heat flux and emissive power (temperature) distributions. LBM results are found to be in excellent agreement with the benchmark results.     

Mesoscopic approach for steady‐state free convection in a diamond array

The purpose of the present paper is testing an in‐house efficiency algorithm based on lattice Boltzmann method (LBM) and using it to resolve the obtained coupled nondimensional governing equations to analyze two‐dimensional free convection inside a cold outer cavity subjected to a heated cylindrical diamond array. Steady state or oscillatory results are obtained using the Bhatnagar‐Gross‐Krook collision model associated to the thermal LBM. Both the velocity and temperature fields are solved using the D2Q9 models. With different Rayleigh numbers (Ra), the tested free convection can either achieve to steady state or oscillatory. We extended our in house Fortran 90 code using curved boundary conditions and implemented them into a cavity with a diamond array. The numerical simulations were done using distinct Ra (10⁶ and 10 ⁷) and distances between the four neighboring circular cylinders aligned in a diamond array. The effects of several physical parameters, including Ra and position of the hot body array on flow and heat transfer characteristics are investigated. The obtained results are highlighted in the form of streamlines, isotherms, and velocities plots. We show in this paper the stability and the efficiency of the LBM to deal with a complex geometry and its ability to reach suitable convergence criteria for high Ra (10 ⁶ and 10 ⁷). The numerical results indicate that LBM can simulate numerical problems with a high Ra reaching a steady state where we can depict the change of the flow pattern and enhancement of the heat transfer in the presence of heated diamond array.     

Application of the lattice Boltzmann method for solving the energy equation of a 2-D transient conduction-radiation problem

A lattice Boltzmann method (LBM) is used to solve the energy equation in a test problem involving thermal radiation and to thus investigate the suitability of scalar diffusion LBM for a new class of problems. The problem chosen is transient conductive and radiative heat transfer in a 2-D rectangular enclosure filled with an optically absorbing, emitting and scattering medium. The energy equation of the problem is solved alternatively with a previously used finite volume method (FVM) and with the LBM, while the radiative transfer equation is solved in both cases using the collapsed dimension method. In a parametric study on the effects of the conduction-radiation parameter, extinction coefficient, scattering albedo, and enclosure aspect ratio, FVM and LBM are compared in each case. It is found that, for given level of accuracy, LBM converges in fewer iterations to the steady-state solution, independent of the influence of radiation. On the other hand, the computational cost per iteration is higher for LBM than for the FVM for a simple grid. For coupled radiation-diffusion, the LBM is faster than the FVM because the radiative transfer computation is more time-consuming than that of diffusion.     

Probing the Rayleigh–Benard convection phase change mechanism of low-melting-point metal via lattice Boltzmann method

A double distribution function lattice Boltzmann method (LBM) with multirelaxation time is implemented to simulate the Rayleigh–Benard convection melting of a typical low-melting-point metal in a rectangular cavity. Typical cases frequently encountered in practice with constant heat flux/constant temperature boundary conditions are parametrically investigated, with corresponding dimensionless results outlined; the influence of inclination angle of the cavity is also clarified. The computational speed of the current LBM would reach about 40 times faster than that of conventional finite volume method as performed by commercial software Fluent. The obtained results would be valuable for guiding practical thermal design.     

Analysis of combined mode heat transfer in a porous medium using the lattice Boltzmann method

ABSTRACT

Application of the lattice Boltzmann method (LBM) in solving a combined mode conduction, convection, and radiation heat transfer problem in a porous medium is extended. Consideration is given to a 1-D planar porous medium with a localized volumetric heat generation zone. Three particle distribution functions, one each for the solid temperature, the gas temperature, and the intensity of radiation, are simultaneously used to solve the gas- and the solid-phase energy equations. The volumetric radiation source term appears in the solid-phase energy equation, and it is also computed using the LBM. To check the accuracy of the LBM results, the same problem is also solved using the finite volume method (FVM). Effects of convective coupling, flow enthalpy, solid-phase conductivity, scattering albedo porosity, and emissivity on axial temperature distribution are studied and compared with the FVM results. Effects of flow enthalpy, solid-phase conductivity, and emissivity are also studied on radiative output. LBM results are in excellent agreement with those of the FVM.     

Numerical prediction of effective thermal conductivity of ceramic fiber board using lattice Boltzmann method

Abstract

Application of the lattice Boltzmann method has been extended for the analysis of combined transient conduction and radiation heat transfer through highly porous fibrous insulation media. Firstly, LBM has been employed for the analysis of combined mode of transient conduction radiation heat transfer in a 2?D rectangular enclosure containing an absorbing, emitting and scattering medium and results are compared with already published ones. The results have been found in good accord for different values of radiation-conduction parameter, scattering albedo and south (hot) wall emissivity. Furthermore, the proposed LBM for the calculation of effective thermal conductivity of ceramic fiber board has been employed. A random-generation growth method for generating micro morphology of natural ceramic fiber board has been selected. The conductive, radiative and effective thermal conductivity has been numerically estimated using the present LBM. It is found that the predicted effective thermal conductivity for different values of fibrous bulk density is in good agreement with the experimental data.     

A new algorithm for solving transient heat and mass transfer in metal–hydrogen reactor

A new algorithm based on the lattice Boltzmann method (LBM) is proposed as a potential solver for two-dimensional and dynamic heat and mass transfer in metal–hydrogen reactor. To check the validity of this algorithm, computational results have been compared with the experimental data and a good agreement is obtained. The advantages of the proposed numerical approach include, among others, simple implementation on a computer, accurate CPU time, and capability of stable simulation.     

Performance evaluation of four radiative transfer methods in solving multi-dimensional radiation and/or conduction heat transfer problems

This article reports results of the four popular and widely used numerical methods, viz., the Monte Carlo method (MCM), the discrete transfer method (DTM), the discrete ordinates method (DOM) and the finite volume method (FVM) used to calculate radiative information in any thermal problem. Different classes of problems dealing with radiation and/or conduction heat transfer problems in a 2-D rectangular absorbing, emitting and scattering participating medium have been considered. In radiative equilibrium and non-radiative equilibrium cases, the MCM results have been used as the benchmark data for comparing the performances of the DTM, the DOM and the FVM. In the combined radiation and conduction mode problem, the energy equation has been formulated using the lattice Boltzmann method (LBM). To compare the performance of the DTM, the DOM and the FVM, the required radiative field data computed using these methods have been provided to the LBM formulation. Temperature distributions obtained using the four methods and those obtained from the LBM in conjunction with the DTM, the DOM and the FVM have been compared for different parameters such as the extinction coefficient, the scattering albedo, the conduction-radiation parameter, the wall emissivity, the aspect ratio and heat generation rate. In all the cases, results of these methods have been found in good agreements. Computationally, the DTM was found the most time consuming, and the DOM was computationally the most efficient.     

A fractional-step lattice Boltzmann flux solver for axisymmetric thermal flows

ABSTRACT

A fractional-step lattice Boltzmann flux solver is proposed in this work for effective simulation of axisymmetric thermal flows with rotating walls. The predictor and corrector steps are introduced in the solver. In the predictor step, excluding axisymmetric effects, the intermediate flow variables are predicted by the lattice Boltzmann flux solver (LBFS), which applies the finite-volume method to discretize the conservative equations recovered by the standard lattice Boltzmann method (LBM). The fluxes of the LBFS at the cell interfaces are reconstructed by local application of the lattice Boltzmann (LB) model with three distribution functions. These three distribution functions are used respectively for calculating axial and radial velocities, azimuthal velocity, and internal energy. In the corrector step, the intermediate flow variables are corrected by considering the axisymmetric effects. The present method not only retains the simplicity of the LBM but also eliminates the complicated derivation process in the axisymmetric LB model. The reliability of the proposed solver is examined by its application to simulate natural convection in an annulus, the Rayleigh-Benard convection, mixed convection in a vertical tall annulus, and Wheeler’s benchmark problem in crystal growth. The numerical results obtained agree well with the published data.