Boltzmann Scheme Research Articles

Thermal flux simulations by lattice Boltzmann method; investigation of high Richardson number cross flows over tandem square cylinders

Imposing a constant non-zero heat/mass flux at a boundary is of a rather complicated type of problem with severe convergence issues especially in unsteady conditions. The present report, firstly covers the implementation of thermal lattice Boltzmann scheme and the counter-slip thermal energy density approach, to model the constant flux conditions for a typical flow of fluid over heat generating blocks. Analysis of kinetic relations shows that at certain conditions, the inflating density field could result in inconstancy of fluid heat conductivity leading to unacceptable results. Therefore to minimize such effects, density field normalization at the end of each iteration is suggested. Comparing the test case results of the current study with those of macroscopic finite volume approach in the literature, reveals the promising accuracy of the present method.Secondly, the previously unexplored problem of steady and unsteady mixed convection over inline tandem square cylinders with a constant heat flux at the block boundary is investigated. Due to the employment of Neumann thermal boundary condition on the blocks, modified Richardson number is introduced and its consistency with the conventional definition is checked. Detailed results of the study for 10<Reynolds<100, blockage ratios of 1/3, 1/4, 1/5 and 1/8 and modified Richardson numbers of 0.0, 0.5 and 1.0 are presented. The intense of the superimposed buoyance force (defined by modified Richardson number) and its impact on the onset of vortex shedding and the global quantities such as lift and drag coefficients and Nusselt number is also carefully explored.

A multiple-relaxation-time lattice Boltzmann method for high-speed compressible flows**Project supported by the Innovation Fund for Aerospace Science and Technology of China (Grant No. 2009200066) and the Aeronautical Science Fund of China (Grant No. 20111453012).

This paper presents a coupling compressible model of the lattice Boltzmann method. In this model, the multiple-relaxation-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.

Analysis and improvement of Brinkman lattice Boltzmann schemes: bulk, boundary, interface. Similarity and distinctness with finite elements in heterogeneous porous media.

This work focuses on the numerical solution of the Stokes-Brinkman equation for a voxel-type porous-media grid, resolved by one to eight spacings per permeability contrast of 1 to 10 orders in magnitude. It is first analytically demonstrated that the lattice Boltzmann method (LBM) and the linear-finite-element method (FEM) both suffer from the viscosity correction induced by the linear variation of the resistance with the velocity. This numerical artefact may lead to an apparent negative viscosity in low-permeable blocks, inducing spurious velocity oscillations. The two-relaxation-times (TRT) LBM may control this effect thanks to free-tunable two-rates combination Λ. Moreover, the Brinkman-force-based BF-TRT schemes may maintain the nondimensional Darcy group and produce viscosity-independent permeability provided that the spatial distribution of Λ is fixed independently of the kinematic viscosity. Such a property is lost not only in the BF-BGK scheme but also by "partial bounce-back" TRT gray models, as shown in this work. Further, we propose a consistent and improved IBF-TRT model which vanishes viscosity correction via simple specific adjusting of the viscous-mode relaxation rate to local permeability value. This prevents the model from velocity fluctuations and, in parallel, improves for effective permeability measurements, from porous channel to multidimensions. The framework of our exact analysis employs a symbolic approach developed for both LBM and FEM in single and stratified, unconfined, and bounded channels. It shows that even with similar bulk discretization, BF, IBF, and FEM may manifest quite different velocity profiles on the coarse grids due to their intrinsic contrasts in the setting of interface continuity and no-slip conditions. While FEM enforces them on the grid vertexes, the LBM prescribes them implicitly. We derive effective LBM continuity conditions and show that the heterogeneous viscosity correction impacts them, a property also shared by FEM for shear stress. But, in contrast with FEM, effective velocity conditions in LBM give rise to slip velocity jumps which depend on (i) neighbor permeability values, (ii) resolution, and (iii) control parameter Λ, ranging its reliable values from Poiseuille bounce-back solution in open flow to zero in Darcy's limit. We suggest an "upscaling" algorithm for Λ, from multilayers to multidimensions in random extremely dispersive samples. Finally, on the positive side for LBM besides its overall versatility, the implicit boundary layers allow for smooth accommodation of the flat discontinuous Darcy profiles, quite deficient in FEM.

Numerical stability of explicit off-lattice Boltzmann schemes: A comparative study

The off-lattice Boltzmann (OLB) method consists of numerical schemes which are used to solve the discrete Boltzmann equation. Unlike the commonly used lattice Boltzmann method, the spatial and time steps are uncoupled in the OLB method. In the currently proposed schemes, which can be broadly classified into Runge–Kutta-based and characteristics-based, the size of the time-step is limited due to numerical stability constraints. In this work, we systematically compare the numerical stability of the proposed schemes in terms of the maximum stable time-step. In line with the overall LB method, we investigate the available schemes where the advection approximation is explicit, and the collision approximation is either explicit or implicit. The comparison is done by implementing these schemes on benchmark incompressible flow problems such as Taylor vortex flow, Poiseuille flow, and lid-driven cavity flow. It is found that the characteristics-based OLB schemes are numerically more stable than the Runge–Kutta-based schemes. Additionally, we have observed that, with respect to time-step size, the scheme proposed by Bardow et al. (2006) [1] is the most numerically stable and computationally efficient scheme compared to similar schemes, for the flow problems tested here.

Unexpected coupling between flow and adsorption in porous media.

We study the interplay between transport and adsorption in porous systems under a fluid flow, based on a lattice Boltzmann scheme extended to account for adsorption. We performed simulations on well-controlled geometries with slit and grooved pores, investigating the influence of adsorption and flow on dispersion coefficient and adsorbed density. In particular, we present a counterintuitive effect where fluid flow induces heterogeneity in the adsorbate, displacing the adsorption equilibrium towards downstream adsorption sites in grooves. We also present an improvement of the adsorption-extended lattice Boltzmann scheme by introducing the possibility for saturating Langmuir-like adsorption, while earlier work focused on linear adsorption phenomena. We then highlight the impact of this change in situations of high concentration of adsorbate.

A review of materials science-based models for mixture design and permeability prediction of pervious concretes

Pervious concrete is one of the relatively recent additions to the class of sustainable multifunctional cement-based materials. The material design of pervious concretes relies on trial-and-error-based approaches since the larger porosity and pore size requirements make a minimal porosity-based approach adopted for conventional concretes non-viable. This paper reviews a particle packing-based methodology for pervious concrete material design using a compaction index from compressible packing model of granular particles as the defining parameter. The pore structure features of the thus designed pervious concretes are characterised using well-accepted stereological and morphological methods. A three-dimensional reconstruction procedure, from two-dimensional starting images, used to develop material structures in which performance (permeability) prediction algorithms can be implemented is also reviewed. Permeability of these model structures have been predicted using a Stokes' solver and a Lattice Boltzmann scheme, and compared to the experimentally determined permeability. A stochastic Monte-Carlo simulation is used to quantify the influence of pore structure features on the permeability of pervious concretes.

Predictor-corrector finite-difference lattice Boltzmann schemes

Predictor-corrector finite-difference lattice Boltzmann schemes are proposed. An approach with separate approximation of spatial derivatives in the advective term of kinetic equations and an approach when this term is replaced by a single finite difference are considered. Explicit finite-difference schemes are used at both the stages of the computation process. The lid-driven cavity flow problem and the Taylor — Green problem are solved numerically in a wide range of the Reynolds number. It is shown that the proposed schemes allow a larger time step compared to well-known upwind schemes. Mathematics Subject Classification: 76-04; 76M10; 65N30

The development of thermal lattice Boltzmann models in incompressible limit

In this paper, an incompressible two-dimensional (2-D) and three-dimensional (3-D) thermohydrodynamics for thelattice Boltzmann scheme are developed. The basic idea is to solve the velocity field and the temperature field usingtwo different distribution functions. A derivation of the lattice Boltzmann scheme from the continuous Boltzmannequation for 2-D is discussed in detail. By using the same procedure as in the derivation of the discretised densitydistribution function, we found that new lattice of four-velocity (2-D) and eight-velocity (3-D) models for internal energydensity distribution function can be developed where the viscous and compressive heating effects are negligible.These models are validated by the numerical simulation of the 2-D porous plate Couette flow problem where theanalytical solution exists and the natural convection flows in a cubic cavity.

Rising bubble simulations using lattice Boltzmann method coupled with level set interface capturing

AbstractA multiphase Lattice Boltzmann (LB) scheme coupled with a level set interface capturing module is used for the simulation of multiphase flows, and in particular, rising bubbles under moderate and high density and viscosity ratios. We employ a consistent, robust discretization of the pressure forces which enables us to preserve stability and accuracy. The accuracy of the present scheme is shown to be competitive with that of the benchmark data from the finite element solutions in [3]. (© 2014 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Simulation of strong nonlinear waves with vectorial lattice Boltzmann schemes

We show that a hyperbolic system with a mathematical entropy can be discretized with vectorial lattice Boltzmann schemes using the methodology of kinetic representation of the dual entropy. We test this approach for the shallow water equations in one and two spatial dimensions. We obtain interesting results for a shock tube, reflection of a shock wave and nonstationary two-dimensional propagation. This contribution shows the ability of vectorial lattice Boltzmann schemes to simulate strong nonlinear waves in nonstationary situations.

Diffusion velocity lattice Boltzmann formulation applied to transport in macroscopic porous media

This paper describes the application of a single relaxation time (SRT) lattice Boltzmann scheme to the transport in porous media with large spatial variations of diffusion coefficients. Effective diffusion coefficients can vary substantially within porous media because of their dependence on porosity and tortuosity which can span over several orders of magnitude, depending on pore size and connectivity. Moreover, when mass is transported with pore-water in porous media, the hydrodynamic dispersion, which depends on Darcy's velocity, contributes additionally to the usually anisotropic variation of the dissipative term. In contrast to the traditional treatment of spatially variable diffusion coefficient by the variation of a SRT, here the variability is accommodated through the use of diffusion velocity formulation which allows for larger variabilities of diffusion coefficient. The volume averaged properties of mass transport in macroscopic porous media are resolved through the additional source term which is similar to the existing force adjusting methods. The applicability of both the proposed schemes is demonstrated on two examples. The first demonstrates that the method is accurate for the large variation of diffusion coefficients and porosities. The second example introduces mass diffusion in a real, geometrically complex system with spatially contrasting properties.

Consistent lattice Boltzmann equations for phase transitions.

Unlike conventional computational fluid dynamics methods, the lattice Boltzmann method (LBM) describes the dynamic behavior of fluids in a mesoscopic scale based on discrete forms of kinetic equations. In this scale, complex macroscopic phenomena like the formation and collapse of interfaces can be naturally described as related to source terms incorporated into the kinetic equations. In this context, a novel athermal lattice Boltzmann scheme for the simulation of phase transition is proposed. The continuous kinetic model obtained from the Liouville equation using the mean-field interaction force approach is shown to be consistent with diffuse interface model using the Helmholtz free energy. Density profiles, interface thickness, and surface tension are analytically derived for a plane liquid-vapor interface. A discrete form of the kinetic equation is then obtained by applying the quadrature method based on prescribed abscissas together with a third-order scheme for the discretization of the streaming or advection term in the Boltzmann equation. Spatial derivatives in the source terms are approximated with high-order schemes. The numerical validation of the method is performed by measuring the speed of sound as well as by retrieving the coexistence curve and the interface density profiles. The appearance of spurious currents near the interface is investigated. The simulations are performed with the equations of state of Van der Waals, Redlich-Kwong, Redlich-Kwong-Soave, Peng-Robinson, and Carnahan-Starling.

Benchmark Numerical Simulations of Viscoelastic Fluid Flows with an Efficient Integrated Lattice Boltzmann and Finite Volume Scheme

An efficient IBLF-dts scheme is proposed to integrate the bounce-back LBM and FVM scheme to solve the Navier-Stokes equations and the constitutive equation, respectively, for the simulation of viscoelastic fluid flows. In order to improve the efficiency, the bounce-back boundary treatment for LBM is introduced in to improve the grid mapping of LBM and FVM, and the two processes are also decoupled in different time scales according to the relaxation time of polymer and the time scale of solvent Newtonian effect. Critical numerical simulations have been carried out to validate the integrated scheme in various benchmark flows at vanishingly low Reynolds number with open source CFD toolkits. The results show that the numerical solution with IBLF-dts scheme agrees well with the exact solution and the numerical solution with FVM PISO scheme and the efficiency and scalability could be remarkably improved under equivalent configurations.

An axisymmetric multiple-relaxation-time lattice Boltzmann scheme

A multiple-relaxation-time (MRT) lattice Boltzmann (LB) scheme developed for axisymmetric flows recovers the complete continuity and Navier–Stokes equations. This scheme follows the strategy of the standard D2Q9 model by using a single particle distribution function and a simple “collision-streaming” updating rule. The extra terms related to axisymmetry in the macroscopic equations are recovered by adding source terms into the LB equation, which are simple and involve no gradients. The compressible effect retained in the Navier–Stokes equations is recovered by introducing a term related to the reversed transformation matrix for MRT collision operator, so as to produce a correct bulk viscosity, making it suitable for compressible flows with high frequency and low Mach number. The validity of the scheme is demonstrated by testing the Hagen–Poiseuille flow and 3D Womersley flow, as well as the standing acoustic waves in a closed cylindrical chamber. The numerical experiments show desirable stability at low viscosities, enabling to simulate a standing ultrasound field in centimeters space.

Lattice Boltzmann method with heat flux boundary condition applied to mixed convection in inclined lid driven cavity

Mixed convection in a inclined cavity has not been investigated by LBM in case of an imposed non zero heat flux. This type of boundary condition, representing very usual situations in physical world, is not simple to model in lattice Boltzmann schemes. In effect, the only boundary condition able to simulate an imposed temperature and an imposed heat flux at a boundary has been presented by D’Orazio et al. in previous works, where the boundary was at rest. In this work, laminar mixed convective heat transfer in two-dimensional rectangular inclined driven cavity is studied numerically by means of a double population thermal Lattice Boltzmann method. The counter-slip internal energy density boundary condition, able to simulate an imposed heat flux at the wall, is applied. Through the top moving lid the heat flux enters the cavity and it leaves the system through the bottom wall; side walls are adiabatic. Results are analyzed over a range of the Richardson numbers and tilting angles of the enclosure, encompassing the dominating forced convection, mixed convection, and dominating natural convection flow regimes. The results show that, as expected, heat transfer rate increases as increases the inclination angle, but this effect is significant for higher Richardson numbers, when buoyancy forces dominate the problem; for horizontal cavity, average Nusselt number decreases with the increase of Richardson number because of the stratified field configuration. This study shows that the counter-slip internal energy density boundary condition can be effectively used to simulate heat transfer phenomena also in case of moving walls and it makes the Lattice Boltzmann Method able to simulate a wide class of cooling process where a given thermal power must be removed.

Pattern formation in liquid-vapor systems under periodic potential and shear.

In this paper the phase behavior and pattern formation in a sheared nonideal fluid under a periodic potential is studied. An isothermal two-dimensional formulation of a lattice Boltzmann scheme for a liquid-vapor system with the van der Waals equation of state is presented and validated. Shear is applied by moving walls and the periodic potential varies along the flow direction. A region of the parameter space, where in the absence of flow a striped phase with oscillating density is stable, will be considered. At low shear rates the periodic patterns are preserved and slightly distorted by the flow. At high shear rates the striped phase loses its stability and traveling waves on the interface between the liquid and vapor regions are observed. These waves spread over the whole system with wavelength only depending on the length of the system. Velocity field patterns, characterized by a single vortex, will also be shown.

Simulation of Flow over a Cavity Using Multi-Relaxation Time Thermal Lattice Boltzmann Method

This article provides numerically study of the multi-relaxation time thermal lattice Boltzmann method (LBM) for compute the flow and isotherm characteristics in the bottom heated cavity located o n a floor of horizontal channel . A double-distribution function (DFF) was coupled with MRT thermal LBM to study the effects of various grashof number (Gr), Reynolds number (Re) and Aspect Ratio (AR) on the flow and isotherm characteristic. The results we re compared with the conventional single-relaxation time lattice Boltzmann scheme and benchmark solution for such flow configuration. The results of the numer ical simulation indicate that multi-relaxation time thermal lattice Boltzmann scheme demonstrated good agreement, which supports its validity in computing fluid flow problem.

Comment on “An improved gray Lattice Boltzmann model for simulating fluid flow in multi-scale porous media”: Intrinsic links between LBE Brinkman schemes

In this Comment on the recent work (Zhu and Ma, 2013) [11] by Zhu and Ma (ZM) we first show that all three local gray Lattice Boltzmann (GLB) schemes in the form (Zhu and Ma, 2013) [11]: GS (Chen and Zhu, 2008; Gao and Sharma, 1994) [1,4], WBS (Walsh et al., 2009) [12] and ZM, fail to get constant Darcy’s velocity in series of porous blocks. This inconsistency is because of their incorrect definition of the macroscopic velocity in the presence of the heterogeneous momentum exchange, while the original WBS model (Walsh et al., 2009) [12] does this properly. We improve the GS and ZM schemes for this and other related deficiencies. Second, we show that the “discontinuous velocity” they recover on the stratified interfaces with their WBS scheme is inherent, in different degrees, to all LBE Brinkman schemes, including ZM scheme. None of them guarantees the stress and the velocity continuity by their implicit interface conditions, even in the frame of the two-relaxation-times (TRT) collision operator where these two properties are assured in stratified Stokes flow, Ginzburg (2007) [5]. Third, the GLB schemes are presented in work (Zhu and Ma, 2013) [11] as the alternative ones to direct, Brinkman-force based (BF) schemes (Freed, 1998; Nie and Martys, 2007) [3,8]. Yet, we show that the BF-TRT scheme (Ginzburg, 2008) [6] gets the solutions of any of the improved GLB schemes for specific, viscosity-dependent choice of its one or two local relaxation rates. This provides the principal difference between the GLB and BF: while the BF may respect the linearity of the Stokes–Brinkman equation rigorously, the GLB-TRT cannot, unless it reduces to the BF via the inverse transform of the relaxation rates. Furthermore, we show that, in limited parameter space, “gray” schemes may run one another. From the practical point of view, permeability values obtained with the GLB are viscosity-dependent, unlike with the BF. Finally, the GLB shares with the BF a so-called anisotropy (Ginzburg, 2008; Nie and Martys, 2007) [6,8], that is, flow-direction-dependency in their effective viscosity corrections, related to the discretized spatial variation of the resistance forcing.

Lattice Boltzmann scheme for electrolytes by an extended Maxwell-Stefan approach.

This paper presents an extended multicomponent lattice Boltzmann model for the simulation of electrolytes. It is derived by means of a finite discrete velocity model and its discretization. The model recovers momentum and mass transport according to the incompressible Navier-Stokes equation and Maxwell-Stefan formulation, respectively. It includes external driving forces (e.g., electric field) on diffusive and viscous scales, concentration-dependent Maxwell-Stefan diffusivities, and thermodynamic factors. The latter take into account nonideal diffusion behavior, which is essential as electrolytes involve charged species and therefore nonideal long and short-range interactions among the molecules of the species. Furthermore, we couple our scheme to a finite element method to include electrostatic interactions on the macroscopic level. Numerical experiments show the validity of the presented model.

Single-relaxation-time lattice Boltzmann scheme for advection-diffusion problems with large diffusion-coefficient heterogeneities and high-advection transport.

The paper presents an approach that extends the flexibility of the standard lattice Boltzmann single relaxation time scheme in terms of spatial variation of dissipative terms (e.g., diffusion coefficient) and stability for high Péclet mass transfer problems. Spatial variability of diffusion coefficient in SRT is typically accommodated through the variation of relaxation time during the collision step. This method is effective but cannot deal with large diffusion coefficient variations, which can span over several orders of magnitude in some natural systems. The approach explores an alternative way of dealing with large diffusion coefficient variations in advection-diffusion transport systems by introducing so-called diffusion velocity. The diffusion velocity is essentially an additional convective term that replaces variations in diffusion coefficients vis-à-vis a chosen reference diffusion coefficient which defines the simulation time step. Special attention is paid to the main idea behind the diffusion velocity formulation and its implementation into the lattice Boltzmann framework. Finally, the performance, stability, and accuracy of the diffusion velocity formulation are discussed via several advection-diffusion transport benchmark examples. These examples demonstrate improved stability and flexibility of the proposed scheme with marginal consequences on the numerical performance.