A two-dimensional lattice Boltzmann model for uniform channel flows

In this paper a novel two-dimensional lattice Boltzmann model (LBM) is developed for uniform channel flows. The axial velocity is solved from a momentum diffusion equation over the cross-sectional plane. An extrapolation boundary condition is also introduced to enhance the no-slip boundary in the momentum equation. This boundary treatment can also be applied to LBM simulations of other diffusion processes. The algorithm and boundary treatment are validated by simulations of steady Poiseuille and pulsatile Womersley flows in circular pipes. The numerical convergence and accuracy are comparable to those of existing models. Moreover, comparison with general three-dimensional lattice Boltzmann simulations demonstrates the advantages of our two-dimensional model, including lower computational resource requirements (memory and time), easier boundary treatment for arbitrary cross-sectional shapes, and no velocity constraint. These features are attractive for practical applications with uniform channel flows.     

Large eddy simulation of turbulent open duct flow using a lattice Boltzmann approach

Large eddy simulations of turbulent open duct flow are performed using the lattice Boltzmann method (LBM) in conjunction with the Smagorinsky sub-grid scale (SGS) model. A smaller value of the Smagorinsky constant than the usually used one in plain channel flow simulations is used. Results for the mean flow and turbulent fluctuations are compared to experimental data obtained in an open duct of similar dimensions. It is found that the LBM simulation results are in good qualitative agreement with the experiments.     

Non-body-fitted Cartesian-mesh simulation of highly turbulent flows using multi-relaxation-time lattice Boltzmann method

This paper presents a lattice Boltzmann method (LBM) based study aimed at numerical simulation of highly turbulent and largely inclined flow around obstacles of curved geometry using non-body-fitted Cartesian meshes. The approach features (1) combining the interpolated bounce-back scheme with the LBM of multi-relaxation-time (MRT) type to enable the use of simple Cartesian mesh for the flow cases even with complex geometries; and (2) incorporating the Spalart–Allmaras (SA) turbulence model into LBM in order to represent the turbulent flow effect. The numerical experiments are performed corresponding to flows around an NACA0012 airfoil at Re=5×10⁵ and around a flat plate at Re=2×10⁴, respectively. The agreement between all simulation results obtained from this study and the data provided by other literature demonstrates the reliability of the enhanced LBM proposed in this paper for simulating, simply on Cartesian meshes, complex flows that may involve bodies of curved boundary, high Reynolds number, and large angle of attack.     

Domain decomposition based coupling between the lattice Boltzmann method and traditional CFD methods—Part I: Formulation and application to the 2-D Burgers’ equation

The lattice Boltzmann method is being increasingly employed in the field of computational fluid dynamics due to its computational efficiency. Floating-point operations in the lattice Boltzmann method involve local data and therefore allow easy cache optimization and parallelization. Due to this, the cache-optimized lattice Boltzmann method has superior computational performance over traditional finite difference methods for solving unsteady flow problems. When solving steady flow problems, the explicit nature of the lattice Boltzmann discretization limits the time step size and therefore the efficiency of the lattice Boltzmann method for steady flows. To quantify the computational performance of the lattice Boltzmann method for steady flows, a comparison study between the lattice Boltzmann method (LBM) and the alternating direction implicit (ADI) method was performed using the 2-D steady Burgers’ equation. The comparison study showed that the LBM performs comparatively poor on high-resolution meshes due to smaller time step sizes, while on coarser meshes where the time step size is similar for both methods, the cache-optimized LBM performance is superior. Because flow domains can be discretized with multiblock grids consisting of coarse and fine grid blocks, the cache-optimized LBM can be applied on the coarse grid block while the traditional implicit methods are applied on the fine grid blocks. This paper finds the coupled cache-optimized lattice Boltzmann-ADI method to be faster by a factor of 4.5 over the traditional methods while maintaining similar accuracy.     

A scalable interface-resolved simulation of particle-laden flow using the lattice Boltzmann method

We examine the scalable implementation of the lattice Boltzmann method (LBM) in the context of interface-resolved simulation of wall-bounded particle-laden flows. Three distinct aspects relevant to performance optimization of our lattice Boltzmann simulation are studied. First, we optimize the core sub-steps of LBM, the collision and the propagation (or streaming) sub-steps, by reviewing and implementing five different published algorithms to reduce memory loading and storing requirements to boost performance. For each, two different array storage formats are benchmarked to test effective cache utilization. Second, the vectorization of the multiple-relaxation-time collision model is discussed and our vectorized collision and propagation algorithm is presented. We find that careful use of Intel’s Advance Vector Extensions and appropriate array storage formats can significantly enhance performance. Third, in the presence of many finite-size, moving solid particles within the flow field, three different communication schemes are proposed and compared in order to optimize the treatment of fluid-solid interactions. These efforts together lead to a very efficient LBM simulation code for interface-resolved simulation of particle-laden flows. Overall, the optimized scalable code of particle-laden flow is a factor of 4.0-to-8.5 times faster than our previous implementation.     

Simulation of floating bodies with the lattice Boltzmann method

This paper is devoted to the simulation of floating rigid bodies in free surface flows. For that, a lattice Boltzmann based model for liquid–gas–solid flows is presented. The approach is built upon previous work for the simulation of liquid–solid particle suspensions on the one hand, and on an interface-capturing technique for liquid–gas free surface flows on the other. The incompressible liquid flow is approximated by a lattice Boltzmann scheme, while the dynamics of the compressible gas are neglected. We show how the particle model and the interface capturing technique can be combined by a novel set of dynamic cell conversion rules. We also evaluate the behaviour of the free surface–particle interaction in simulations. One test case is the rotational stability of non-spherical rigid bodies floating on a plane water surface–a classical hydrostatic problem known from naval architecture. We show the consistency of our method in this kind of flows and obtain convergence towards the ideal solution for the heeling stability of a floating box.     

Application of lattice Boltzmann method in free surface flow simulation of micro injection molding

The filling flow in micro injection molding was simulated by using the lattice Boltzmann method (LBM). A tracking algorithm for free surface to handle the complex interaction between gas and liquid phases in LBM was used for the free surface advancement. The temperature field in the filling flow is also analyzed by combining the thermal lattice Boltzmann model and the free surface method. To simulate the fluid flow of polymer melt with a high Prandtl number and high viscosity, a modified lattice Boltzmann scheme was adopted by introducing a free parameter in the thermal diffusion equation to overcome the restriction of the thermal relaxation time. The filling flow simulation of micro injection molding was successfully performed in the study.     

A viscosity counteracting approach in the lattice Boltzmann BGK model for low viscosity flow: Preliminary verification

Due to numerical instability, the lattice Boltzmann model (LBM) with the Bhatnagar–Gross–Krook (BGK) collision operator has some limitations in the simulation of low viscosity flows. In this paper, we propose a viscosity counteracting approach for simulating a moderate viscosity flow. An extra negative viscosity term is introduced to counteract part of the moderate viscosity by using the lattice Boltzmann equation with a source term. The counteracting viscosity term is treated as a non-uniform unsteady source. The stability is enhanced; thus small viscosity flows can be simulated. Model verification consists of benchmark cases such as those of Poiseuille flow, Couette flow, waterhammer waves, Taylor–Green vortex flow, and lid-driven cavity flow. The flow patterns, error characteristics, and representative parameters are carefully analyzed. It is shown that this approach can simulate flows with lower viscosities than may be simulated using the normal LBGK model; the second-order accuracy of the LBGK model is definitely retained, although a little dissipation is added. These preliminary studies prove the effectiveness and accuracy of the model. Sophisticated analysis and further verification of the stability mechanism will be done in the near future.     

A fractional step lattice Boltzmann method for simulating high Reynolds number flows

A fractional step lattice Boltzmann scheme is presented to greatly improve the stability of the lattice Boltzmann method (LBM) in modelling incompressible flows at high Reynolds number. This method combines the good features of the conventional LBM and the fractional step technique. Through the fractional step, the flow at an extreme case of infinite Reynolds number (inviscid flow) can be effectively simulated. In addition, the non-slip boundary condition can be directly implemented.     

Modelling solute transport in shallow water with the lattice Boltzmann method

The lattice Boltzmann method is used to investigate the solute transport in shallow water flows. Shallow water equations are solved using the lattice Boltzmann equation on a D2Q9 lattice with multiple-relaxation-time (MRT-LBM) and Bhatnagar–Gross–Krook (BGK-LBM) terms separately, and the advection–diffusion equation is also solved with a LBM-BGK on a D2Q5 lattice. Three cases: open channel flow with side discharge, shallow recirculation flow and flow in a harbour are simulated to verify the described methods. Agreements between predictions and experiments are satisfactory. In side discharge flow, the reattachment length for different ratios of side discharge velocity to main channel velocity has been studied in detail. Furthermore, the performance of MRT-LBM and BGK-LBM for these three cases has been investigated. It is found that LBM-MRT has better stability and is able to satisfactorily simulate flows with higher Reynolds number. The study shows that the lattice Boltzmann method is simple and accurate for simulating solute transport in shallow water flows, and hence it can be applied to a wide range of environmental flow problems.     

Multiscale modelling strategy using the lattice Boltzmann method for polymer dynamics in a turbulent flow

Polymer dynamics in a turbulent flow is a problem spanning several orders of magnitude in length and time scales. A microscopic simulation covering all those scales from the polymer segment to the inertial scale of turbulence remains improbable within the foreseeable future. We propose a multiscale simulation strategy to enhance the spatio-temporal resolution of the local Lagrangian turbulent flow by matching two different simulation techniques, i.e. direct numerical simulation for the flow as a whole, and the lattice Boltzmann method coupled to polymer dynamics at the Kolmogorov dissipation scale. Local turbulent flows sampled by Lagrangian tracer particles in the direct numerical simulation are reproduced in the lattice Boltzmann model with a finer resolution, by supplying the latter with both the correct initial condition as well as the correct time-dependent boundary condition, sampled from the former. When combined with a Molecular Dynamics simulation of a polymer chain in the lattice Boltzmann model, it provides a strategy to simulate the passive dynamics of a polymer chain in a turbulent flow covering all these scales. Although this approach allows for a fairly realistic model of the macromolecule, the back-coupling to the flow on the large scales is missing.     

Lattice Boltzmann modeling of microchannel flows in the transition flow regime

Owing to its kinetic nature and distinctive computational features, the lattice Boltzmann method for simulating rarefied gas flows has attracted significant research interest in recent years. In this article, a lattice Boltzmann (LB) model is presented to study microchannel flows in the transition flow regime, which have gained much attention because of fundamental scientific issues and technological applications in various micro-electro-mechanical system (MEMS) devices. In the model, a Bosanquet-type effective viscosity is used to account for the rarefaction effect on gas viscosity. To match the introduced effective viscosity and to gain an accurate simulation, a modified second-order slip boundary condition with a new set of slip coefficients is proposed. Numerical investigations demonstrate that the results, including the velocity profile, the non-linear pressure distribution along the channel, and the mass flow rate, are in good agreement with the solution of the linearized Boltzmann equation, the direct simulation Monte Carlo (DSMC) results, and the experimental results over a broad range of Knudsen numbers. It is shown that taking the rarefaction effect on gas viscosity into consideration and employing an appropriate slip boundary condition can lead to a significant improvement in the modeling of rarefied gas flows with moderate Knudsen numbers in the transition flow regime.     

An analysis of natural convection using the thermal finite element discrete Boltzmann equation

The lattice Boltzmann method (LBM) is the simple numerical simulator for fluids because it consists of linear equations. Excluding the higher differential term, the LBM for a temperature field is also achieved as an easy numerical simulation method. However, the LBM is hardly applied to body fitted coordinates for its formulation. It is then difficult to calculate complex lattices using the LBM. In this paper, the finite element discrete Boltzmann equation (FEDBE) is introduced to deal with this weakness of the LBM. The finite element method is applied to the discrete Boltzmann equation (DBE) of the basic equation of the LBM. For FEDBE, the simulation using complex lattices is achieved, and it will be applicable for the development in engineering fields. The natural convection in a square cavity and the Rayleigh–Bernard convection are chosen as the test problem. Each simulation model is accurate enough for the flow patterns, the temperature distribution and the Nusselt number. This method is now considered good for the flow and temperature field, and is expected to be introduced for complex lattices using the DBE.     

Estimation of a semi-physical GLBE model using dual EnKF learning algorithm coupled with a sensor network design strategy: Application to air field monitoring

In this paper, we present the fusion of two complementary approaches for modeling and monitoring the spatio-temporal behavior of a fluid flow system. We also propose a mobile sensor deployment strategy to produce the most accurate estimate of the true system state. For this purpose, deterministic and statistical information was used. We adopted a filtering method based on a semi-physical model which derives from a fluid flow numerical model known as lattice Boltzmann model (LBM). The a priori physical knowledge was introduced by the Navier–Stokes equations which were discretized by the lattice Boltzmann approach. Moreover, its multiple-relaxation-time (MRT) variant not only improved the stability, but also enabled the introduction of additional degrees of freedom to be estimated like the synaptic weights of a neural network. The statistical knowledge was then introduced into the model by performing a sequential learning of these parameters and an estimation of the speed field of the fluid flow starting from measurements. The low spatial density of measurements, the large amount of data inherent to environmental issues and the nonlinearity of the generalized lattice Boltzmann equations (GLBEs) enjoined us to use the ensemble Kalman filter (EnKF) for the recursive estimation procedure. A dual state-parameter estimation which results in a significantly reduced computation time was used by combining two filters consecutively activated in the same iteration. Finally, we proposed to complete the lack of spatial information of the sparse-observation network by adding a mobile sensor, which was routed to the location where the cell-by-cell output estimation error was the highest. Experimental results in the context of the standard lid-driven cavity problem revealed the presence of few zones of interest, where fixed sensors can be deployed to increase performances in terms of convergence speed and estimation quality. Finally, the study showed the feasibility of introducing some additional parameters which act as degrees of freedom, to perform large-eddy simulation of turbulent flows without numerical instabilities.     

A thermal lattice Boltzmann model with diffuse scattering boundary condition for micro thermal flows

A lattice Boltzmann model for simulating isothermal micro flows has been proposed by us recently [Niu XD, Chew YT, Shu C. A lattice Boltzmann BGK model for simulation of micro flows. Europhys Lett 2004;67(4):600]. In this paper, we extend the model to simulate the micro thermal flows. In particular, the thermal lattice Boltzmann equation (TLBE) [He X, Chen S, Doolen GD. A novel thermal model for the lattice Boltzmann method in incompressible limit. J Comput Phys 1998;146:282] is used with modification of the relaxation times linking to the Knudsen number. The diffuse scattering boundary condition (DSBC) derived in our early model is extended to consider temperature jump at wall boundaries. Simple theoretical analyses of the DSBC are presented and the results are found to be consistent with the conventional velocity slip and temperature jump boundary conditions. Numerical validations are carried out by simulating two-dimensional thermal Couette flows and developing thermal flows in a microchannel, and the obtained results are found to be in good agreement with those given from the direct simulation Monte Carlo (DSMC), the molecular dynamics (MD) approaches and the Maxwell theoretical prediction.     

A Coupled Lattice Boltzmann Method to Solve Nernst–Planck Model for Simulating Electro-osmotic Flows

In this paper, we focus on the nonlinear coupling mechanism of the Nernst–Planck model and propose a coupled lattice Boltzmann method (LBM) to solve it. In this method, a new LBM for the Nernst–Planck equation is developed, a multi-relaxation-time (MRT)-LBM for flow field and an LBM for the Poisson equation are used. And then, we discuss the choice of the model and found that the MRT-LBM is much more stable and accurate than the LBGK model. A reasonable iterative sequence and evolution number for each LBM are proposed by considering the properties of the coupled LBM. The accuracy and stability of the presented coupled LBM are also discussed through simulating electro-osmotic flows (EOF) in micro-channels. Furthermore, to test the applicability of it, the EOF with non-uniform surface potential in micro-channels based on the Nernst–Planck model is simulated. And we investigate the effects of non-uniform surface potential on the pattern of the EOF at different external applied electric fields. Finally, a comparison of the difference between the Nernst–Planck model and the Poisson–Boltzmann model is presented.     

Lattice Boltzmann simulations of impedance tube flows

An original time-domain surface acoustic impedance condition for Lattice Boltzmann methods has been developed. The basis for this method is the extension proposed by Delattre et al. [Delattre G, Manoha E, Redonnet S, Sagaut P. Time-domain simulation of sound absorption on curved wall. 13th AIAA/CEAS Aeroacoustics conference, Rome, Italy, AIAA-2007-3493; 2007] of the z-transform approach suggested by Özyörük et al. [Özyörük Y, Long LN, Jones M. Time-domain numerical simulation of a flow impedance tube. J Comput Phys 1998;146:29-57]. Using this boundary condition that links the normal velocity and the pressure, the basic idea consists in calculating the Lattice Boltzmann populations at a boundary node thanks to the gradients of the fluid velocity. This paper describes the proposed LBM boundary conditions and its assessment on the NASA Langley flow-impedance tube with a constant depth ceramic tubular liner. We performed both single and broadband-frequency simulations, without mean flow and with sheared mean flows. Excellent agreement is shown with both experimental data and other simulation results at various frequencies up to a Mach number equal to 0.5.     

Viscous flow and colloid transport near air–water interface in a microchannel

In order to understand the transport behavior of colloids near an air–water interface (AWI), two computational methods are applied to simulate the local water flow field near a moving AWI in a 2D microfluidic channel. The first method is a mesoscopic multicomponent and multiphase lattice Boltzmann (LBM) model and the second is the macroscopic, Navier–Stokes based, volume-of-fluid interface tracking method. In the LBM, it is possible to predict the dynamic contact angles after the static contact angle is correctly set, and the predicted dynamic contact angles are in good agreement with previous observations. It is demonstrated that the two methods can yield a similar flow velocity field if they are applied properly. The flow field relative to AWI depends on the direction of the flow, and exhibits curved streamlines that transport fluid between the center of the channel and the wall region. Using the obtained flow, the motion of sub-micron colloids in a de-ionized water solution is then studied by a Lagrangian approach. The observed colloid trajectories are in qualitative agreement with our visualizations using a confocal microscope.     

Topology optimization of flow domains using the lattice Boltzmann method

We consider the optimal design of two- (2D) and three-dimensional (3D) flow domains using the lattice Boltzmann method (LBM) as an approximation of Navier-Stokes (NS) flows. The problem is solved by a topology optimization approach varying the effective porosity of a fictitious material. The boundaries of the flow domain are represented by potentially discontinuous material distributions. NS flows are traditionally approximated by finite element and finite volume methods. These schemes, while well established as high-fidelity simulation tools using body-fitted meshes, are effected in their accuracy and robustness when regular meshes with zero-velocity constraints along the surface and in the interior of obstacles are used, as is common in topology optimization. Therefore, we study the potential of the LBM for approximating low Mach number incompressible viscous flows for topology optimization. In the LBM the geometry of flow domains is defined in a discontinuous manner, similar to the approach used in material-based topology optimization. In addition, this non-traditional discretization method features parallel scalability and allows for high-resolution, regular fluid meshes. In this paper, we show how the variation of the porosity can be used in conjunction with the LBM for the optimal design of fluid domains, making the LBM an interesting alternative to NS solvers for topology optimization problems. The potential of our topology optimization approach will be illustrated by 2D and 3D numerical examples.     

Lattice Boltzmann simulation of the dispersion of aggregated particles under shear flows

The lattice Boltzmann method (LBM) for multicomponent immiscible fluids is applied to simulations of the deformation and breakup of a particle-cluster aggregate in shear flows. In the simulations, the solid particle is modeled by a droplet with strong interfacial tension and large viscosity. The van der Waals attraction force is taken into account for the interaction between the particles. The ratio of the hydrodynamic drag force to cohesive force, I, is introduced, and the effect of I on the aggregate deformation and breakup in shear flows is investigated. It is found that the aggregate is easier to deform and to be dispersed when I is over 100.