Thermal Lattice Boltzmann Equation Research Articles

High-order flux reconstruction thermal lattice Boltzmann flux solver for simulation of incompressible thermal flows.

In this paper, a high-order solver combining the flux reconstruction (FR) method and the thermal lattice Boltzmann flux solver (FRTLBFS) is developed for accurately and efficiently simulating incompressible thermal flows. The conservative differential equations recovered from Chapman-Enskog analysis of the thermal lattice Boltzmann equation are solved by the high-order FR method. The thermal lattice Boltzmann method is only applied to reconstruct the local solution used for evaluating fluxes at the solution and flux points. Unlike the traditional Navier-Stokes-Boussinesq (NSB) solvers where the inviscid and viscous terms are treated separately, the inviscid and viscous fluxes in the current FRTLBFS are coupled and computed uniformly. In comparison with the recently developed high-order flux reconstruction thermal lattice Boltzmann method, the FRTLBFS holds advantages such as high-order accuracy, good stability, and compactness but is more efficient and low storage, since only macroscopic flow variables including density, velocity, and temperature are stored and evolved. In addition, the physical boundary conditions in FRTLBFS can be directly implemented by using the same method as in conventional NSB solvers. Numerical validations of the proposed method are implemented by simulating (a) the porous plate problem, (b) natural convection in a square cavity, (c) unsteady natural convection in a tall cavity, and (d) thermal lid-driven cavity flow. Numerical results demonstrate that the present solver is an attractive tool to simulate incompressible thermal flows due to its high-order accuracy, stability, and low memory cost.

Improved three-dimensional thermal multiphase lattice Boltzmann model for liquid-vapor phase change

Modeling liquid-vapor phase change using the lattice Boltzmann (LB) method has attracted significant attention in recent years. In this paper, we propose an improved three-dimensional thermal multiphase LB model for simulating liquid-vapor phase change. The proposed model has the following features. First, it is still within the framework of the thermal LB method using a temperature distribution function and therefore retains the fundamental advantages of the thermal LB method. Second, in the existing thermal LB models for liquid-vapor phase change, the finite-difference computations of the gradient terms ∇·u and ∇T usually require special treatment at boundary nodes, while in the proposed thermal LB model these two terms are calculated locally. Moreover, in some of the existing thermal LB models, the error term ∂_{t_{0}}(Tu) is eliminated by adding local correction terms to the collision process in the moment space, which causes these thermal LB models to be limited to the D2Q9 lattice in two dimensions and the D3Q15 or D3Q19 lattice in three dimensions. Conversely, the proposed model does not suffer from such an error term and therefore the thermal LB equation can be constructed on the D3Q7 lattice, which simplifies the model and improves the computational efficiency. Numerical simulations are carried out to validate the accuracy and efficiency of the proposed thermal multiphase LB model for simulating liquid-vapor phase change.

A Dirichlet boundary condition for the thermal lattice Boltzmann method

In this work we introduce a boundary condition for thermal lattice Boltzmann simulations that contain a Dirichlet boundary condition by bouncing back the non-equilibrium distribution of the energy distribution function. To this end the thermal lattice Boltzmann equation is modified by introducing an additional collision term that takes into account the thermal diffusivity and local solid volume fraction of a lattice (partially) covered by the solid phase. Asymptotic analysis of the boundary condition confirms that it is of second order accuracy. The method is validated using (i) an analytical solution for the Nusselt number correlation of a single sphere in an unbounded stationary fluid and (ii) direct numerical simulations of the heat transfer between a fluid and individual particles.

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.

A unified thermal lattice Boltzmann equation for conjugate heat transfer problem

In the present paper, an optimal two-relaxation-time (OTRT) thermal lattice Boltzmann equation (TLBE) is proposed to simulate conjugate heat transfer. The derivation process is based on a developed idea of virtual heat capacity correction method. Finally, it is found that a unified TLBE can be achieved. The derivation process shows how the modified equilibrium distribution function works in simulating conjugate heat transfer. It also reveals the relation between the virtual heat capacity correction method and the method of reconstructing the equilibrium distribution function. By using Chapman-Enskog expansion, the proposed OTRT TLBE can recover the standard energy equation up to second-order accuracy. For validation, six test cases including both steady and transient conjugate heat transfer with flat or curved interfaces are calculated. The results show that the OTRT TLBE performs well in simulating conjugate heat transfer problems with both flat and curved interfaces, at both transient and steady states, and the OTRT TLBE could have better numerical accuracy than the single-relaxation-time (SRT) TLBE.

A unified thermal lattice Boltzmann equation based on MRT model for conjugate heat transfer in anisotropic media

In the present paper, a unified thermal lattice Boltzmann equation (TLBE) based on multiple-relaxation-time (MRT) model, which can simulate conjugate heat transfer in anisotropic media directly without any additional modification, is derived. A series of test cases containing both steady and transient conjugate heat transfer with flat or curved interfaces are calculated to validate the unified MRT-TLBE. The results show that the equation performs well in simulating conjugate heat transfer in anisotropic media. It is feasible for both steady and transient conjugate hear transfer problems with flat or curved interfaces. In addition, the comparisons between the calculated results of the present equation and those of the finite volume method (FVM) indicate that the unified MRT-TLBE has better continuity than FVM in simulating conjugate heat transfer in anisotropic media.

Thermohydrodynamics of an evaporating droplet studied using a multiphase lattice Boltzmann method.

In this paper, the thermohydrodynamics of an evaporating droplet is investigated by using a single-component pseudopotential lattice Boltzmann model. The phase change is applied to the model by adding source terms to the thermal lattice Boltzmann equation in such a way that the macroscopic energy equation of multiphase flows is recovered. In order to gain an exhaustive understanding of the complex hydrodynamics during evaporation, a single droplet is selected as a case study. At first, some tests for a stationary (non-)evaporating droplet are carried out to validate the method. Then the model is used to study the thermohydrodynamics of a falling evaporating droplet. The results show that the model is capable of reproducing the flow dynamics and transport phenomena of a stationary evaporating droplet quite well. Of course, a moving droplet evaporates faster than a stationary one due to the convective transport. Our study shows that our single-component model for simulating a moving evaporating droplet is limited to low Reynolds numbers.

A kinetic theory based thermal lattice Boltzmann equation model

A thermal lattice Boltzmann equation (LBE) model within the framework of double distribution function (DDF) method is proposed from the continuous DDF Boltzmann equation, which has a clear physical significance. Since the discrete velocity set in present LBE model is not space filled, a Lax–Wendroff scheme is applied to solve the evolution equations by which the spatial interpolation of two distribution functions is overcome. To validate the model, some classical numerical tests include thermal Couette flow and natural convection flow are simulated, and the results agree well with the analytic solutions and other numerical results, which showed that the present model had the ability to describe the thermal fluid flow phenomena.

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.

Lattice-Boltzmann simulations of the thermally driven 2D square cavity at high Rayleigh numbers

The thermal lattice Boltzmann equation (TLBE) with multiple-relaxation-times (MRT) collision model is used to simulate the steady thermal convective flows in the two-dimensional square cavity with differentially heated vertical walls at high Rayleigh numbers. The MRT-TLBE consists of two sets of distribution functions, i.e., a D2Q9 model for the mass-momentum equations and a D2Q5 model for the temperature equation. The dimensionless flow parameters are the following: the Prandtl number Pr=0.71 and the Rayleigh number Ra=106,107, and 108. The D2Q9+D2Q5 MRT-TLBE is shown to be second-order accurate and to be capable of yielding results of benchmark quality, including various Nusselt numbers and local hydrodynamic intensities. Our results also agree well with existing benchmark data obtained by other methods.

Thermal lattice Boltzmann flux solver and its application for simulation of incompressible thermal flows

A thermal lattice Boltzmann flux solver (TLBFS) is developed in this work for simulation of incompressible thermal flows. In TLBFS, the thermal lattice Boltzmann method (TLBM) is only applied to reconstruct the local solution of TLBM for evaluation of fluxes at the cell interface. Meanwhile, the macroscopic flow variables at cell centers are obtained by using the finite volume method to solve conservative differential equations recovered from Chapman–Enskog analysis of thermal lattice Boltzmann equation. The physical boundary conditions in TLBFS can be directly implemented using the same way as in conventional Navier–Stokes (N–S) solvers. The present solver eliminates the constraints associated with conventional lattice Boltzmann method such as limitation to uniform Cartesian mesh, tie-up between the time step and the mesh spacing, as well as implementation of boundary conditions for distribution functions. TLBFS is validated through numerical examples of natural convection in enclosures, including a square cavity and a cylindrical annulus, and mixed convection from a heated cylinder. Through numerical validation, it is shown that TLBFS can be effectively and flexibly applied to solve thermal flow problems with curved boundaries.

Heat Transfer Evaluation on Curved Boundaries in Thermal Lattice Boltzmann Equation Method

An efficient and accurate approach for heat transfer evaluation on curved boundaries is proposed in the thermal lattice Boltzmann equation (TLBE) method. The boundary heat fluxes in the discrete velocity directions of the TLBE model are obtained using the given thermal boundary condition and the temperature distribution functions at the lattice nodes close to the boundary. Integration of the discrete boundary heat fluxes with effective surface areas gives the heat flow rate across the boundary. For lattice models with square or cubic structures and uniform lattice spacing the effective surface area is constant for each discrete heat flux, thus the heat flux integration becomes a summation of all the discrete heat fluxes with constant effective surface area. The proposed heat transfer evaluation scheme does not require a determination of the normal heat flux component or a surface area approximation on the boundary; thus, it is very efficient in curved-boundary simulations. Several numerical tests are conducted to validate the applicability and accuracy of the proposed heat transfer evaluation scheme, including: (i) two-dimensional (2D) steady-state thermal flow in a channel, (ii) one-dimensional (1D) transient heat conduction in an inclined semi-infinite solid, (iii) 2D transient heat conduction inside a circle, (iv) three-dimensional (3D) steady-state thermal flow in a circular pipe, and (v) 2D steady-state natural convection in a square enclosure with a circular cylinder at the center. Comparison between numerical results and analytical solutions in tests (i)–(iv) shows that the heat transfer is second-order accurate for straight boundaries perpendicular to one of the discrete lattice velocity vectors, and first-order accurate for curved boundaries due to the irregularly distributed lattice fractions intersected by the curved boundary. For test (v), the computed surface-averaged Nusselt numbers agree well with published results.

Univariate polynomial equation providing on-lattice higher-order models of thermal lattice Boltzmann theory

A univariate polynomial equation is presented. It provides on-lattice higher-order models of the thermal lattice Boltzmann equation. The models can be accurate up to any required level and can be applied to regular lattices, which allow efficient and accurate approximate solutions of the Boltzmann equation. We derive models approaching the complete Galilean invariant and providing accuracy of the fourth-order moment and beyond. We simulate one-dimensional thermal shock tube problems to illustrate the accuracy of our models. Moreover, we show the remarkably enhanced stability obtained by our models and our discretized equilibrium distributions.

Boundary conditions for thermal lattice Boltzmann equation method

We propose a thermal boundary condition treatment based on the “bounce-back” idea and interpolation of the distribution functions for both the Dirichlet and Neumann (normal derivative) conditions in the thermal lattice Boltzmann equation (TLBE) method. The coefficients for the distribution functions involved are determined to satisfy the Dirichlet or Neumann condition with second-order accuracy. For the Dirichlet condition there is an adjustable parameter in the treatment and three particular schemes are selected for demonstration, while for the Neumann condition the second-order accurate scheme is unique. When applied to inclined or curved boundaries, the Dirichlet condition treatment can be directly used, while the Neumann condition given in the normal direction of the boundary should be converted into derivative conditions in the discrete velocity directions of the TLBE model. A spatially coupled formula relating the boundary temperature, boundary normal heat flux, and the distribution functions near the boundary is thus derived for the Neumann problems on curved boundaries. The applicability and accuracy of the present boundary treatments are examined with several numerical tests for which analytical solutions are available, including the 2-dimensional (2-D) steady-state channel flow, the 1-D transient heat conduction in an inclined semi-infinite solid, the 2-D steady-state and transient heat conduction inside a circle and the 3-D steady-state circular pipe flow. While the Dirichlet condition treatment leads to second-order accuracy for the temperature field, it only gives a first-order accurate boundary heat flux because of the irregularity of the cuts by the curved boundary with the lattices. With the Neumann condition on the curved boundary, the accuracy for the temperature field obtained is first-order. When the tangential temperature gradient on the boundary is decoupled, second-order convergence of the temperature field can be obtained with Neumann conditions.

Lattice Boltzmann simulations of thermal convective flows in two dimensions

In this paper we study the lattice Boltzmann equation (LBE) with multiple-relaxation-time (MRT) collision model for incompressible thermo-hydrodynamics with the Boussinesq approximation. We use the MRT thermal LBE (TLBE) to simulate the following two flows in two dimensions: the square cavity with differentially heated vertical walls and the Rayleigh–Bénard convection in a rectangle heated from below. For the square cavity, the flow parameters in this study are the Rayleigh number Ra=103–106, and the Prandtl number Pr=0.71; and for the Rayleigh–Bénard convection in a rectangle, Ra=2⋅103, 104 and 5⋅104, and Pr=0.71 and 7.0.

A direct-forcing immersed boundary method for the thermal lattice Boltzmann method

In this study, a direct-forcing immersed boundary method (IBM) for thermal lattice Boltzmann method (TLBM) is proposed to simulate the non-isothermal flows. The direct-forcing IBM formulas for thermal equations are derived based on two TLBM models: a double-population model with a simplified thermal lattice Boltzmann equation (Model 1) and a hybrid model with an advection–diffusion equation of temperature (Model 2). As an interface scheme, which is required due to a mismatch between boundary and computational grids in the IBM, the sharp interface scheme based on second-order bilinear and linear interpolations (instead of the diffuse interface scheme, which uses discrete delta functions) is adopted to obtain the more accurate results. The proposed methods are validated through convective heat transfer problems with not only stationary but also moving boundaries – the natural convection in a square cavity with an eccentrically located cylinder and a cold particle sedimentation in an infinite channel. In terms of accuracy, the results from the IBM based on both models are comparable and show a good agreement with those from other numerical methods. In contrast, the IBM based on Model 2 is more numerically efficient than the IBM based on Model 1.

On the finite-volume Lattice Boltzmann modeling of thermo-hydrodynamics

In this paper, Thermal Finite-Volume Lattice Boltzmann Method is developed. To demonstrate the temperature field, the Double Distribution Function (DDF) of thermal lattice Boltzmann equation is used. The upwind biasing factors based on pressure and temperature are defined and applied as flux corrector in the thermo-hydrodynamic lattice Boltzmann equations. A consistent open and solid boundary treatment of flow is also addressed. The unknown energy distribution at the boundary cells are decomposed into its equilibrium and non-equilibrium parts. Then the non-equilibrium part is approximated with extrapolation of the non-equilibrium part of the populations at the neighboring nodes. This treatment enlarges the domain stability and led up to faster convergence. Two test cases namely, thermo-hydrodynamic in a backward-facing step and around a circular cylinder inserted within a backward-facing step are carried out. The results are compared with the available solutions in the technical literature.

Kinetic theory based lattice Boltzmann equation with viscous dissipation and pressure work for axisymmetric thermal flows

A lattice Boltzmann equation (LBE) for axisymmetric thermal flows is proposed. The model is derived from the kinetic theory which exhibits several features that distinguish it from other previous LBE models. First, the present thermal LBE model is derived from the continuous Boltzmann equation, which has a solid foundation and clear physical significance; Second, the model can recover the energy equation with the viscous dissipation term and work of pressure which are usually ignored by traditional methods and the existing thermal LBE models; Finally, unlike the existing thermal LBE models, no velocity and temperature gradients appear in the force terms which are easy to realize in the present model. The model is validated by thermal flow in a pipe, thermal buoyancy-driven flow, and swirling flow in vertical cylinder by rotating the top and bottom walls. It is found that the numerical results agreed excellently with analytical solution or other numerical results.

Thermal levitation

A particle with a density slightly larger than that of the fluid in which it is immersed will sediment. However, if the particle's temperature is higher than that of the fluid, the terminal velocity of sedimentation will be smaller and can even change sign. When the terminal velocity is zero we say there is thermal levitation. Thermal levitation can also occur when the density and temperature of the particle are smaller than those of the fluid. Using a two-component thermal lattice Boltzmann equation method, we study this phenomenon and show it can be stable or unstable.

THERMAL CURVED BOUNDARY TREATMENT FOR THE THERMAL LATTICE BOLTZMANN EQUATION

In this paper, a recent curved non-slip wall boundary treatment for isothermal Lattice Boltzmann equation (LBE) [Z. Guo, C. Zheng and B. Shi, Phys. Fluids14(6) (2002)] is extended to handle the thermal curved wall boundary for a double-population thermal lattice Boltzmann equation (TLBE). The unknown distribution population at a wall node which is necessary to fulfill streaming step is decomposed into its equilibrium and non-equilibrium parts. The equilibrium part is evaluated according to Dirichlet and Neumann boundary constraints, and the non-equilibrium part is obtained using a first-order extrapolation from fluid lattices. To validate the thermal boundary condition treatment, we carry out numerical simulations of Couette flow between two circular cylinders, the natural convection in a square cavity, and the natural convection in a concentric annulus between an outer square cylinder and an inner circular cylinder. The results agree very well with analytical solution or available data in the literature. Our numerical results also demonstrate that the TLBE together with the present boundary scheme is of second-order accuracy.