An efficient swap algorithm for the lattice Boltzmann method

During the last decade, the lattice-Boltzmann method (LBM) as a valuable tool in computational fluid dynamics has been increasingly acknowledged. The widespread application of LBM is partly due to the simplicity of its coding. The most well-known algorithms for the implementation of the standard lattice-Boltzmann equation (LBE) are the two-lattice and two-step algorithms. However, implementations of the two-lattice or the two-step algorithm suffer from high memory consumption or poor computational performance, respectively. Ultimately, the computing resources available decide which of the two disadvantages is more critical. Here we introduce a new algorithm, called the swap algorithm, for the implementation of LBE. Simulation results demonstrate that implementations based on the swap algorithm can achieve high computational performance and have very low memory consumption. Furthermore, we show how the performance of its implementations can be further improved by code optimization.     

Hybrid lattice-Boltzmann finite-difference simulation of convective flows

The principle of lattice-Boltzmann techniques is recalled and some of the difficulties to simulate convective flows are discussed. It is then proposed to use a hybrid scheme with lattice-Boltzmann for fluid velocity variables and finite-difference for the temperature. Convergence studies are presented for particular cases. Some results for two and three-dimensional situations are given. Possible extensions of hybrid schemes are discussed.     

Accelerating geoscience and engineering system simulations on graphics hardware

Many complex natural systems studied in the geosciences are characterized by simple local-scale interactions that result in complex emergent behavior. Simulations of these systems, often implemented in parallel using standard central processing unit (CPU) clusters, may be better suited to parallel processing environments with large numbers of simple processors. Such an environment is found in graphics processing units (GPUs) on graphics cards.This paper discusses GPU implementations of three example applications from computational fluid dynamics, seismic wave propagation, and rock magnetism. These candidate applications involve important numerical modeling techniques, widely employed in physical system simulations, that are themselves examples of distinct computing classes identified as fundamental to scientific and engineering computing. The presented numerical methods (and respective computing classes they belong to) are: (1) a lattice-Boltzmann code for geofluid dynamics (structured grid class); (2) a spectral-finite-element code for seismic wave propagation simulations (sparse linear algebra class); and (3) a least-squares minimization code for interpreting magnetic force microscopy data (dense linear algebra class). Significant performance increases (between 10× and 30× in most cases) are seen in all three applications, demonstrating the power of GPU implementations for these types of simulations and, more generally, their associated computing classes.     

Numerical investigation on drag reduction with superhydrophobic surfaces by lattice-Boltzmann method

The mechanism of drag reduction by using superhydrophobic surfaces whose contact angle is greater than 150° is still an open problem that needs to be investigated. The main purpose of this paper is to reveal how the pressure drop can be decreased. The lattice-Boltzmann method (LBM) is employed to investigate fluid flows through channels with different wettability conditions and topographical surfaces. The drag reduction by superhydrophobic surfaces is determined based on numerical experiments. For the smooth-surface flow, a very thin gas film is observed between the fluid and the superhydrophobic wall; hence, the liquid/solid interface is replaced by the gas/liquid interface. For the rough-surface flow, liquid sweeps over the grooves and the contact area is reduced; therefore, the friction is decreased rapidly. Additionally, the effects of surface wettability and surface roughness are analyzed as well. It is found that introducing roughness elements has a positive effect for reducing the pressure drop for the hydrophobic-surface flow, but has a negative effect for the hydrophilic-surface flow.     

A fast random number generator for stochastic simulations

A discrete random number (DRN) generator for stochastic differential equations is proposed. The generator has exactly 8 states and thus 10 DRN's can be obtained from a single 32-bit random variable. This is advantageous when large numbers of DRN's are needed, as for example in fluctuating lattice-Boltzmann models. The moments of the discrete distribution match those of a Gaussian distribution (zero mean and unit variance) up to 5th order. Numerical tests show that satisfactory statistical properties can be obtained with several 32-bit pseudo random number (PRN) generators.     

Parallel performance of a lattice-Boltzmann/finite element cellular blood flow solver on the IBM Blue Gene/P architecture

We discuss the parallel implementation and scaling results of a hybrid lattice-Boltzmann/finite element code for suspension flow simulations. This code allows the direct numerical simulation of cellular blood flow, fully resolving the two-phase nature of blood and the deformation of the suspended phase. A brief introduction to the numerical methods employed is given followed by an outline of the code structure. Scaling results obtained on Argonne National Laboratories IBM Blue Gene/P (BG/P) are presented. Details include performance characteristics on 512 to 65,536 processor cores.     

Assessment of the 1-fluid method for DNS of particulate flows: Sedimentation of a single sphere at moderate to high Reynolds numbers

This paper introduces an original 1-fluid method for direct simulation of the motion of rigid particles in fluids. The model is based on the implicit treatment of a single fictitious fluid over a fixed grid, and uses an augmented Lagrangian optimization algorithm for the velocity-pressure coupling. The paper focuses on the case of a rigid sphere settling in a viscous medium. For validation purposes, simulations of the transient motion of a sedimenting sphere at Reynolds numbers ranging from 1.5 to 31.9 are compared to the PIV data published by Ten Cate et al. [Ten Cate A, Nieuwstad CH, Derksen JJ, Van den Akker HEA. Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity. Phys Fluids 2002;14(11):4012-25]. Accurate reproduction of the experimental data is obtained. Further simulations are intended to investigate higher Reynolds numbers. Predictions of transient particle sedimentation at Reynolds number 280 are performed and compared with experimental data of the sedimentation trajectory, as well as with simulation results based on the lattice-Boltzmann method.     

Rectangular Lattice-Boltzmann Schemes with BGK-Collision Operator

The usual lattice-Boltzmann schemes for fluid flow simulations operate with square and cubic lattices. Instead of relying on square lattices it is possible to use rectangular and orthorombic lattices as well. Schemes using rectangular lattices can be constructed in several ways. Here we construct a rectangular scheme, with the BGK collision operator, by introducing 2 additional discrete velocities into the standard D2Q9 stencil and show how the same procedure can be applied in three dimensions by extending the D3Q19 stencil. The weights and scaling factors for the new stencils are found as the solutions of the well-known Hermite quadrature problem, assuring isotropy of the lattice tensors up to rank four (Philippi et al., Phys. Rev. E 73(5):056702, 2006) This isotropy is a necessary and sufficient condition for assuring the same second order accuracy of lattice-Boltzmann equation with respect to the Navier–Stokes hydrodynamic equations that is found with the standard D2Q9 and D3Q19 stencils. The numerical validation is done, in the two-dimensional case, by using the new rectangular scheme with D2R11 stencil for simulating the Taylor–Green vortex decay. The D3R23 stencil is numerically validated with three-dimensional simulations of cylindrical sound waves propagating from a point source.     

Benchmark computations based on lattice-Boltzmann,finite element and finite volume methods for laminar flows

The goal of this article is to contribute to the discussion of the efficiency of lattice-Boltzmann (LB) methods as CFD solvers. After a short review of the basic model and extensions, we compare the accuracy and computational efficiency of two research simulation codes based on the LB and the finite-element method (FEM) for two-dimensional incompressible laminar flow problems with complex geometries. We also study the influence of the Mach number on the solution, since LB methods are weakly compressible by nature, by comparing compressible and incompressible results obtained from the LB code and the commercial code CFX. Our results indicate, that for the quantities studied (lift, drag, pressure drop) our LB prototype is competitive for incompressible transient problems, but asymptotically slower for steady-state Stokes flow because the asymptotic algorithmic complexity of the classical LB-method is not optimal compared to the multigrid solvers incorporated in the FEM and CFX code. For the weakly compressible case, the LB approach has a significant wall clock time advantage as compared to CFX. In addition, we demonstrate that the influence of the finite Mach number in LB simulations of incompressible flow is easily underestimated.     

BOUT++: A framework for parallel plasma fluid simulations

A new modular code called BOUT++ is presented, which simulates 3D fluid equations in curvilinear coordinates. Although aimed at simulating Edge Localised Modes (ELMs) in tokamak x-point geometry, the code is able to simulate a wide range of fluid models (magnetised and unmagnetised) involving an arbitrary number of scalar and vector fields, in a wide range of geometries. Time evolution is fully implicit, and 3rd-order WENO schemes are implemented. Benchmarks are presented for linear and non-linear problems (the Orszag-Tang vortex) showing good agreement. Performance of the code is tested by scaling with problem size and processor number, showing efficient scaling to thousands of processors.Linear initial-value simulations of ELMs using reduced ideal MHD are presented, and the results compared to the ELITE linear MHD eigenvalue code. The resulting mode-structures and growth-rate are found to be in good agreement (γBOUT++=0.245ωA, γELITE=0.239ωA, with Alfvénic timescale 1/ωA=R/VA). To our knowledge, this is the first time dissipationless, initial-value simulations of ELMs have been successfully demonstrated.     

Particle Monte Carlo and lattice-Boltzmann methods for simulations of gas-particle flows

A numerical solution concept is presented for simulating the transport and deposition to surfaces of discrete, small (nano-)particles. The motion of single particles is calculated from the Langevin equation by Lagrangian integration under consideration of different forces such as drag force, van der Waals forces, electrical Coulomb forces and not negligible for small particles, under stochastic diffusion (Brownian diffusion). This so-called particle Monte Carlo method enables the computation of macroscopic filter properties as well the detailed resolution of the structure of the deposited particles. The flow force and the external forces depend on solutions of continuum equations, as the Navier-Stokes equations for viscous, incompressible flows or a Laplace equation of the electrical potential. Solutions of the flow and potential fields are computed here using lattice-Boltzmann methods. Essential advantage of these methods are the easy and efficient treatment of three-dimensional complex geometries, given by filter geometries or particle covered surfaces. A number of numerical improvements, as grid refinement or boundary fitting, were developed for lattice-Boltzmann methods in previous studies and applied to the present problem. The interaction between the deposited particle layer and the fluid field or the external forces is included by recomputing of these fields with changed boundaries. A number of simulation results show the influence of different effects on the particle motion and deposition.     

Application of spectral forcing in lattice-Boltzmann simulations of homogeneous turbulence

An efficient numerical method for the direct simulation of homogeneous turbulent flow has been obtained by combining a spectral forcing algorithm for homogeneous turbulence with a lattice-Boltzmann scheme for solution of the continuity and Navier–Stokes equations. The spectral forcing scheme of Alvelius [Alvelius K. Random forcing of three-dimensional homogeneous turbulence. Phys Fluids 1999;11(7):1880–89] is used which allows control of the power input by eliminating the force–velocity correlation in the Fourier domain and enables anisotropic forcing. A priori chosen properties such as the Kolmogorov length scale, the integral length scale and the integral time scale are recovered. This demonstrates that the scheme works accurately with the lattice-Boltzmann method and that all specific features of the forcing scheme are recovered in the lattice-Boltzmann implementation.     

A lattice-Boltzmann simulation study of the drag coefficient of clusters of spheres

Drag coefficients of irregularly shaped particles, constructed from spheres, were measured in lattice-Boltzmann simulations and compared to literature data. The agreement is very well. The distance between the spheres was increased to study the influence of inter-particle distance on the drag force in clusters. The drag coefficient of the clusters was found to increase with inter-particle distance. The drag force on an individual particle in a cluster is lower when that particle is shielded from the flow by other particles.     

A parallel workload balanced and memory efficient lattice-Boltzmann algorithm with single unit BGK relaxation time for laminar Newtonian flows

A parallel workload balanced and memory efficient lattice-Boltzmann algorithm for laminar Newtonian fluid flow through large porous media is investigated. It relies on a simplified LBM scheme using a single unit BGK relaxation time, which is implemented by means of a shift algorithm and comprises an even fluid node partitioning domain decomposition strategy based on a vector data structure. It provides perfect parallel workload balance, and its two-nearest-neighbour communication pattern combined with a simple data transfer layout results in 20-55% lower communication cost, 25-60% higher computational parallel performance and 40-90% lower memory usage than previously reported LBM algorithms. Performance tests carried out using scale-up and speed-up case studies of laminar Newtonian fluid flow through hexagonal packings of cylinders and a random packing of polydisperse spheres on two different computer architectures reveal parallel efficiencies with 128 processors as high as 75% for domain sizes comprising more than 5 billion fluid nodes.     

Development of a parallel semi-implicit two-dimensional plasma fluid modeling code using finite-volume method

In this paper, the development of a two-dimensional plasma fluid modeling code using the cell-centered finite-volume method and its parallel implementation on distributed memory machines is reported. Simulated discharge currents agree very well with the measured data in a planar dielectric barrier discharge (DBD). Parallel performance of simulating helium DBD solved by the different degrees of overlapping of additive Schwarz method (ASM) preconditioned generalized minimal residual method (GMRES) for different modeling equations is investigated for a small and a large test problem, respectively, employing up to 128 processors. For the large test problem, almost linear speedup can be obtained by using up to 128 processors. Finally, a large-scale realistic two-dimensional DBD problem is employed to demonstrate the capability of the developed fluid modeling code for simulating the low-temperature plasma with complex chemical reactions.     

A non-reflecting boundary in fluid-structure interaction

A very efficient non-reflecting boundary condition is derived for the seismic response analysis of a submerged structure, such as a dam or an offshore structure, interacting with a compressible fluid domain of unbounded extent. The fluid-structure system is assumed to be two-dimensional and the analysis is conducted in the frequency domain. In the finite element discretization, pressure and displacements are considered to be the basic nodal unknowns for the fluid domain and the structure, respectively. The implementation of the proposed boundary condition in any existing finite element code, based on such a formulation, is extremely simple. Some fluid-structure systems are analysed to demonstrate the effectiveness and efficiency of the proposed method.     

Two-dimensional simulation of fluid–structure interaction using lattice-Boltzmann methods

The development of flow instabilities due to high Reynolds number flow in artificial heart-value geometries inducing high strain rates and stresses often leads to hemolysis and related highly undesired effects. Geometric and functional optimization of artificial heart valves is therefore mandatory. In addition to experimental work in this field it is meanwhile possible to obtain increasing insight into flow dynamics by computer simulation of refined model problems. Here we present two-dimensional simulation results of the coupled fluid–structure problem defined by a model geometry of an artificial heart value with moving leaflets exposed to a channel flow driven by transient boundary conditions representing a physiologically relevant regime. A modified lattice-Boltzmann approach is used to solve the coupled problem.     

Robust treatment of no-slip boundary condition and velocity updating for the lattice-Boltzmann simulation of particulate flows

In the past decade, the lattice-Boltzmann method (LBM) has emerged as a very useful tool in studies for the direct-numerical simulation of particulate flows. The accuracy and robustness of the LBM have been demonstrated by many researchers; however, there are several numerical problems that have not been completely resolved. One of these is the treatment of the no-slip boundary condition on the particle-fluid interface and another is the updating scheme for the particle velocity. The most common used treatment for the solid boundaries largely employs the so-called “bounce-back” method (BBM). [Ladd AJC. Numerical simulations of particulate suspensions via a discretized Boltzmann equation Part I. Theoretical foundation. J Fluid Mech (1994);271:285; Ladd AJC. Numerical simulations of particulate suspensions via a discretized Boltzmann equation Part II. Numerical results. J Fluid Mech (1994);271:311.] This often causes distortions and fluctuations of the particle shape from one time step to another. The immersed boundary method (IBM), which assigns and follows a series of points in the solid region, may be used to ensure the uniformity of particle shapes throughout the computations. To ensure that the IBM points move with the solid particles, a force density function is applied to these points. The simplest way to calculate the force density function is to use a direct-forcing scheme. In this paper, we conduct a complete study on issues related to this scheme and examine the following parameters: the generation of the forcing points; the choice of the number of forcing points and sensitivity of this choice to simulation results; and, the advantages and disadvantages associated with the IBM over the BBM. It was also observed that the commonly used velocity updating schemes cause instabilities when the densities of the fluid and the particles are close. In this paper, we present a simple and very effective velocity updating scheme that does not only facilitate the numerical solutions when the particle to fluid density ratios are close to one, but also works well for particle that are lighter than the fluid.     

A pressure-evolution-based multi-relaxation-time high-density-ratio two-phase lattice-Boltzmann model

Two of the major challenges in extending the application of multiphase lattice-Boltzmann (LB) models to realistic fluid flow simulations are their numerical instability at high liquid-to-gas density ratios, and at low viscosities. In this paper, a model, recently presented in the literature, for simulating high-density ratios is extended to lower viscosities by employing multiple-relaxation-times for particle collision. In applying the multiple-relaxation-time (MRT) model, the collision term is treated explicitly and the three-step solution procedure suggested by the prior author is employed. The model is evaluated by verifying the Laplace-Young relation for a static infinitely long liquid cylinder, comparing frequency of oscillations of an initially elliptic cross-section liquid cylinder with analytic values and comparing the behavior of a two-dimensional (2D) planar drop impinging on a wet wall with prior results. The results show satisfactory agreement with available data, and a significant gain in numerical stability.     

Numerical modeling of electrowetting processes in digital microfluidic devices

We use a three-dimensional multiphase lattice-Boltzmann model to study basic operations such as transport, merging and splitting of nanoliter water droplets actuated by electrowetting in digital microfluidic devices. In a first step, numerical and analytical predictions for the droplet transport velocity are compared and very good agreement is obtained for a wide range of contact angles. The same algorithm is employed then to study the dynamics of the splitting processes at different contact angles and different geometries of the cell. The configuration of the liquid droplet involved in a splitting process and the dependence of the splitting time on the transport velocity are also investigated and phenomenological laws describing these processes are also proposed.