Double MRT thermal lattice Boltzmann method for simulating convective flows

A two-dimensional double Multiple Relaxation Time-Thermal Lattice Boltzmann Equation (2-MRT-TLBE) method is developed for predicting convective flows in a square differentially heated cavity filled with air (Pr=0.71). In this Letter, we propose a numerical scheme to solve the flow and the temperature fields using the MRT-D2Q9 model and the MRT-D2Q5 model, respectively. Thus, the main objective of this study is to show the effectiveness of such model to predict thermodynamics for heat transfer. This model is validated by the numerical simulations of the 2-D convective square cavity flow. Excellent agreements are obtained between numerical predictions. These results demonstrate the accuracy and the effectiveness of the proposed methodology.     

Stabilization of thermal lattice Boltzmann models

A three-dimensional thermal lattice-Boltzmann model with two relaxation times to separately control viscosity and thermal diffusion is developed. Numerical stability of the model is significantly improved using Lax-Wendroff advection to provide and adjustable time step. Good agreement with a conventional fiitedifference Navier-Stokes solver is obtained in modeling compressible Rayleigh-Bénard convestion when boundary conditions are treated similarly.     

Advanced lattice Boltzmann scheme for high-Reynolds-number magneto-hydrodynamic flows

Is the lattice Boltzmann method suitable to investigate numerically high-Reynolds-number magneto-hydrodynamic (MHD) flows? It is shown that a standard approach based on the Bhatnagar–Gross–Krook (BGK) collision operator rapidly yields unstable simulations as the Reynolds number increases. In order to circumvent this limitation, it is here suggested to address the collision procedure in the space of central moments for the fluid dynamics. Therefore, an hybrid lattice Boltzmann scheme is introduced, which couples a central-moment scheme for the velocity with a BGK scheme for the space-and-time evolution of the magnetic field. This method outperforms the standard approach in terms of stability, allowing us to simulate high-Reynolds-number MHD flows with non-unitary Prandtl number while maintaining accuracy and physical consistency.     

A hydrodynamically correct thermal lattice Boltzmann model

A three-dimensional lattice-Boltzmann model which yields correct hydrodynamics at the Navier-Stokes level of the Chapman-Enskog expansion requires a minimum of 26 velocities. We present results for a model with one additional velocity, determined by maximizing the equilibrium entropy. For compressible Rayleigh-Bénard convection the model is more accurate but considerably less stable, than a previous, approximate 21-speed model.     

Simulations of Bingham plastic flows with the multiple-relaxation-time lattice Boltzmann model

Fresh cement mortar is a type of workable paste,which can be well approximated as a Bingham plastic and whose flow behavior is of major concern in engineering.In this paper,Papanastasiou’s model for Bingham fluids is solved by using the multiplerelaxation-time lattice Boltzmann model(MRT-LB).Analysis of the stress growth exponent m in Bingham fluid flow simulations shows that Papanastasiou’s model provides a good approximation of realistic Bingham plastics for values of m108.For lower values of m,Papanastasiou’s model is valid for fluids between Bingham and Newtonian fluids.The MRT-LB model is validated by two benchmark problems:2D steady Poiseuille flows and lid-driven cavity flows.Comparing the numerical results of the velocity distributions with corresponding analytical solutions shows that the MRT-LB model is appropriate for studying Bingham fluids while also providing better numerical stability.We further apply the MRT-LB model to simulate flow through a sudden expansion channel and the flow surrounding a round particle.Besides the rich flow structures obtained in this work,the dynamics fluid force on the round particle is calculated.Results show that both the Reynolds number Re and the Bingham number Bn afect the drag coefcients CD,and a drag coefcient with Re and Bn being taken into account is proposed.The relationship of Bn and the ratio of unyielded zone thickness to particle diameter is also analyzed.Finally,the Bingham fluid flowing around a set of randomly dispersed particles is simulated to obtain the apparent viscosity and velocity fields.These results help simulation of fresh concrete flowing in porous media.     

A multiple-relaxation-time lattice Boltzmann method for high-speed compressible flows

This paper presents a coupling compressible model of the lattice Boltzmann method. In this model, the multiplerelaxation-time lattice Boltzmann scheme is used for the evolution of density distribution functions, whereas the modified single-relaxation-time(SRT) lattice Boltzmann scheme is applied for the evolution of potential energy distribution functions. The governing equations are discretized with the third-order Monotone Upwind Schemes for scalar conservation laws finite volume scheme. The choice of relaxation coefficients is discussed simply. Through the numerical simulations,it is found that compressible flows with strong shocks can be well simulated by present model. The numerical results agree well with the reference results and are better than that of the SRT version.     

A improved incompressible lattice Boltzmann model for time-independent flows

It is well known that the lattice Boltzmann equation method (LBE) can model the incompressible Navier-Stokes (NS) equations in the limit where density goes to a constant. In a LBE simulation, however, the density cannot be constant because pressure is equal to density times the square of sound speed, hence a compressibility error seems inevitable for the LBE to model incompressible flows. This work uses a modified equilibrium distribution and a modified velocity to construct an LBE which models time-independent (steady) incompressible flows with significantly reduced compressibility error. Computational results in 2D cavity flow and in a 2D flow with an exact solution are reported.     

Convection in multiphase fluid flows using lattice Boltzmann methods

We present high-resolution numerical simulations of convection in multiphase flows (boiling) using a novel algorithm based on a lattice Boltzmann method. We first study the thermodynamical and kinematic properties of the algorithm. Then, we perform a series of 3D numerical simulations changing the mean properties in the phase diagram and compare convection with and without phase coexistence at Rayleigh number Ra～10(7). We show that in the presence of nucleating bubbles non-Oberbeck-Boussinesq effects develop, the mean temperature profile becomes asymmetric, and heat-transfer and heat-transfer fluctuations are enhanced, at all Ra studied. We also show that small-scale properties of velocity and temperature fields are strongly affected by the presence of the buoyant bubble leading to high non-gaussian profiles in the bulk.     

Simulation of micro- and nano-scale flows via the lattice Boltzmann method

We use the lattice Boltzmann method (LBM) for analysis of high and moderate Knudsen number phenomena. Simulation results are presented for microscale Couette and Poiseuille flows. The slip velocity, nonlinear pressure drop, and mass flow rate are compared with previous numerical results and/or experimental data. The Knudsen minimum is successfully predicted for the first time within the LBM framework. These results validate the usage of the LBM based commercial, arbitrary geometry code PowerFLOW for simulating nanoscale problems.     

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.     

Modeling and simulation of thermocapillary flows using lattice Boltzmann method

To understand how thermocapillary forces manipulate droplet motion in microfluidic channels, we develop a lattice Boltzmann (LB) multiphase model to simulate thermocapillary flows. The complex hydrodynamic interactions are described by an improved color-fluid LB model, in which the interfacial tension forces and the Marangoni stresses are modeled in a consistent manner using the concept of a continuum surface force. An additional convection–diffusion equation is solved in the LB framework to obtain the temperature field, which is coupled to the interfacial tension through an equation of state. A stress-free boundary condition is also introduced to treat outflow boundary, which can conserve the total mass of an incompressible system, thus improving the numerical stability for creeping flows.The model is firstly validated against the analytical solutions for the thermocapillary driven convection in two superimposed fluids at negligibly small Reynolds and Marangoni numbers. It is then used to simulate thermocapillary migration of three-dimensional deformable droplet at various Marangoni numbers, and its accuracy is once again verified against the theoretical prediction in the limit of zero Marangoni number. Finally, we numerically investigate how the localized heating from a laser can block the microfluidic droplet motion through the induced thermocapillary forces. The droplet motion can be completely blocked provided that the intensity of laser exceeds a threshold value, below which the droplet motion successively undergoes four stages: constant velocity, deceleration, acceleration, and constant velocity. When the droplet motion is completely blocked, four steady vortices are clearly visible, and the droplet is fully filled by two internal vortices. The external vortices diminish when the intensity of laser increases.     

A lattice Boltzmann method for thermal nonideal fluids

A Lattice Boltzmann Method for van der Waals fluids with variable temperature is described. Thermo-hydrodynamic equations are correctly reproduced at second order of a Chapman-Enskog expansion. The method is applied to study initial stages of phase separation of a fluid quenched by contact with colder walls. Thermal equilibration is favoured by pressure waves which propagate with the sound velocity.     

格子Boltzmann方法在相变过程中的应用

提出了一种新的描述气液相变过程的单组分格子Boltzmann模型,利用该模型模拟水以及氨分别在R-K,RKS和P-R状态方程控制下的相变过程,发现相对于R-K和RKS状态方程,水以及氨在P-R状态方程控制下模拟结果均与实验值更接近;特别地,P-R状态方程更适合描述氨.为验证该模型处理两相问题的能力,利用该模型模拟不同温度下水以及氨在P-R状态方程控制下的界面密度梯度,所得的结果与经典的界面理论相符.为此,进一步探讨了气泡(液滴)与周围液体(气体)处于力平衡和热平衡时,气泡(液滴)内外压力差在不同温度下与其半径之间的关系,所得的结果满足Laplace定律,并得到了不同温度下水以及氨的表面张力,发现均与实验值符合得很好,且与表面张力临界理论甚为相符.     

A solution-adaptive lattice Boltzmann method for two-dimensional incompressible viscous flows

A stencil adaptive lattice Boltzmann method (LBM) is developed in this paper. It incorporates the stencil adaptive algorithm developed by Ding and Shu [26] for the solution of Navier–Stokes (N–S) equations into the LBM calculation. Based on the uniform mesh, the stencil adaptive algorithm refines the mesh by two types of 5-points symmetric stencils, which are used in an alternating sequence for increased refinement levels. The two types of symmetric stencils can be easily combined to form a 9-points symmetric structure. Using the one-dimensional second-order interpolation recently developed by Wu and Shu [27] along the straight line and the D2Q9 model, the adaptive LBM calculation can be effectively carried out. Note that the interpolation coefficients are only related to the lattice velocity and stencil size. Hence, the simplicity of LBM is not broken down and the accuracy is maintained. Due to the use of adaptive technique, much less mesh points are required in the simulation as compared to the standard LBM. As a consequence, the computational efficiency is greatly enhanced. The numerical simulation of two dimensional lid-driven cavity flows is carried out. Accurate results and improved efficiency are reached. In addition, the steady and unsteady flows over a circular cylinder are simulated to demonstrate the capability of proposed method for handling problems with curved boundaries. The obtained results compare well with data in the literature.     

Modeling electrokinetic flows in microchannels using coupled lattice Boltzmann methods

We present a numerical framework to solve the dynamic model for electrokinetic flows in microchannels using coupled lattice Boltzmann methods. The governing equation for each transport process is solved by a lattice Boltzmann model and the entire process is simulated through an iteration procedure. After validation, the present method is used to study the applicability of the Poisson–Boltzmann model for electrokinetic flows in microchannels. Our results show that for homogeneously charged long channels, the Poisson–Boltzmann model is applicable for a wide range of electric double layer thickness. For the electric potential distribution, the Poisson–Boltzmann model can provide good predictions until the electric double layers fully overlap, meaning that the thickness of the double layer equals the channel width. For the electroosmotic velocity, the Poisson–Boltzmann model is valid even when the thickness of the double layer is 10 times of the channel width. For heterogeneously charged microchannels, a higher zeta potential and an enhanced velocity field may cause the Poisson–Boltzmann model to fail to provide accurate predictions. The ionic diffusion coefficients have little effect on the steady flows for either homogeneously or heterogeneously charged channels. However the ionic valence of solvent has remarkable influences on both the electric potential distribution and the flow velocity even in homogeneously charged microchannels. Both theoretical analyses and numerical results indicate that the valence and the concentration of the counter-ions dominate the Debye length, the electrical potential distribution, and the ions transport. The present results may improve the understanding of the electrokinetic transport characteristics in microchannels.     

Accuracy analysis of high-order lattice Boltzmann models for rarefied gas flows

In this work, we have theoretically analyzed and numerically evaluated the accuracy of high-order lattice Boltzmann (LB) models for capturing non-equilibrium effects in rarefied gas flows. In the incompressible limit, the LB equation is shown to be able to reduce to the linearized Bhatnagar–Gross–Krook (BGK) equation. Therefore, when the same Gauss–Hermite quadrature is used, LB method closely resembles the discrete velocity method (DVM). In addition, the order of Hermite expansion for the equilibrium distribution function is found not to be directly correlated with the approximation order in terms of the Knudsen number to the BGK equation for incompressible flows. Meanwhile, we have numerically evaluated the LB models for a standing-shear-wave problem, which is designed specifically for assessing model accuracy by excluding the influence of gas molecule/surface interactions at wall boundaries. The numerical simulation results confirm that the high-order terms in the discrete equilibrium distribution function play a negligible role in capturing non-equilibrium effect for low-speed flows. By contrast, appropriate Gauss–Hermite quadrature has the most significant effect on whether LB models can describe the essential flow physics of rarefied gas accurately. Our simulation results, where the effect of wall/gas interactions is excluded, can lead to conclusion on the LB modeling capability that the models with higher-order quadratures provide more accurate results. For the same order Gauss–Hermite quadrature, the exact abscissae will also modestly influence numerical accuracy. Using the same Gauss–Hermite quadrature, the numerical results of both LB and DVM methods are in excellent agreement for flows across a broad range of the Knudsen numbers, which confirms that the LB simulation is similar to the DVM process. Therefore, LB method can offer flexible models suitable for simulating continuum flows at the Navier–Stokes level and rarefied gas flows at the linearized Boltzmann model equation level.     

Analytical solution of axi-symmetrical lattice Boltzmann model for cylindrical Couette flows

Analytical solution for the axi-symmetrical lattice Boltzmann model is obtained for the low-Mach number cylindrical Couette flows. In the hydrodynamic limit, the present solution is in excellent agreement with the result of the Navier–Stokes equation. Since the kinetic boundary condition is used, the present analytical solution using nine discrete velocities can describe flows with the Knudsen number up to 0.1. Meanwhile, the comparison with the simulation data obtained by the direct simulation Monte Carlo method shows that higher-order lattice Boltzmann models with more discrete velocities are needed for highly rarefied flows.     

Stochastic finite difference lattice Boltzmann method for steady incompressible viscous flows

With the advent of state-of-the-art computers and their rapid availability, the time is ripe for the development of efficient uncertainty quantification (UQ) methods to reduce the complexity of numerical models used to simulate complicated systems with incomplete knowledge and data. The spectral stochastic finite element method (SSFEM) which is one of the widely used UQ methods, regards uncertainty as generating a new dimension and the solution as dependent on this dimension. A convergent expansion along the new dimension is then sought in terms of the polynomial chaos system, and the coefficients in this representation are determined through a Galerkin approach. This approach provides an accurate representation even when only a small number of terms are used in the spectral expansion; consequently, saving in computational resource can be realized compared to the Monte Carlo (MC) scheme. Recent development of a finite difference lattice Boltzmann method (FDLBM) that provides a convenient algorithm for setting the boundary condition allows the flow of Newtonian and non-Newtonian fluids, with and without external body forces to be simulated with ease. Also, the inherent compressibility effect in the conventional lattice Boltzmann method, which might produce significant errors in some incompressible flow simulations, is eliminated. As such, the FDLBM together with an efficient UQ method can be used to treat incompressible flows with built in uncertainty, such as blood flow in stenosed arteries. The objective of this paper is to develop a stochastic numerical solver for steady incompressible viscous flows by combining the FDLBM with a SSFEM. Validation against MC solutions of channel/Couette, driven cavity, and sudden expansion flows are carried out.     

A spectral-element discontinuous Galerkin lattice Boltzmann method for nearly incompressible flows

We present a spectral-element discontinuous Galerkin lattice Boltzmann method for solving nearly incompressible flows. Decoupling the collision step from the streaming step offers numerical stability at high Reynolds numbers. In the streaming step, we employ high-order spectral-element discontinuous Galerkin discretizations using a tensor product basis of one-dimensional Lagrange interpolation polynomials based on Gauss–Lobatto–Legendre grids. Our scheme is cost-effective with a fully diagonal mass matrix, advancing time integration with the fourth-order Runge–Kutta method. We present a consistent treatment for imposing boundary conditions with a numerical flux in the discontinuous Galerkin approach. We show convergence studies for Couette flows and demonstrate two benchmark cases with lid-driven cavity flows for Re = 400–5000 and flows around an impulsively started cylinder for Re = 550–9500. Computational results are compared with those of other theoretical and computational work that used a multigrid method, a vortex method, and a spectral element model.     

Direct assessment of lattice Boltzmann hydrodynamics and boundary conditions for recirculating flows

A hydrodynamic boundary condition is developed for lattice Boltzmann hydrodynamics using a square, orthogonal grid. A constraint based on energy considerations is developed to provide closure for the equations which govern the particle distribution at the boundaries. This boundary condition is applied to the two-dimensional, steady flow of an incompressible fluid behind a grid, known as Kovasznay flow. The results are compared to those using alternate boundary conditions using the known exact solution. The hydrodynamic boundary condition produces quadratic spatial convergence, while alternate techniques fail to maintain this second-order accuracy.