Multi-level lattice Boltzmann model on square lattice for compressible flows

A compressible lattice Boltzmann model is established on a square lattice. The model allows large variations in the mean velocity by introducing a large particle-velocity set. To maintain tractability, the support set of the equilibrium distribution is chosen to include only four directions and three particle-velocity levels in which the third level is introduced to improve the stability of the model. This simple structure of the equilibrium distribution makes the model efficient for the simulation of flows over a wide range of Mach numbers and gives it the capability of capturing shock jumps. Unlike the standard lattice Boltzmann model, the formulation eliminated the fourth-order velocity tensors, which were the source of concerns over the homogeneity of square lattices. A modified collision invariant eliminates the second-order discretization error of the fluctuation velocity in the macroscopic conservation equation from which the Navier–Stokes equation and energy equation are recovered. The model is suitable for both viscous and inviscid compressible flows with or without shocks. Two-dimensional shock-wave propagations and boundary layer flows were successfully simulated. The model can be easily extended to three-dimensional cubic lattices.     

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.     

Investigation of flow around a pair of side-by-side square cylinders using the lattice Boltzmann method

The low-Reynolds number flow around two square cylinders placed side-by-side is investigated using the lattice Boltzmann method (LBM). The effects of the gap ratio s/d (s is the separation between the cylinders and d is the characteristic dimension) on the flow are studied. These simulations reveal the existence of regimes with either synchronized or non-synchronized vortex-shedding, with transition occurring at s/d ≈ 2, which is larger than for circular cylinders. Detailed results are presented at Re = 73 for s/d = 2.5 and 0.7 corresponding to the synchronized and flip-flop regimes, respectively. Vortex-shedding from the cylinder occurs either in-phase or in-antiphase in the synchronized regime. However, linear stochastic estimate (LSE) calculations show that in-phase locking is the predominant mode. LSE is also employed to educe the underlying modes in the flip-flop regime, where evidence for both in-phase and anti-phase locked vortices is found, indicating that this regime is in a quasi-stable state between these two modes. The merging of the wakes, which is gradual for the synchronized regime, occurs rapidly in the flip-flop regime. The mean pressure on the upstream surface is symmetric and asymmetric for the synchronized and flip-flop regimes, respectively. Differences in results between the two regimes are interpreted in terms of the interaction of the jet formed between the cylinders with the adjoining wakes, the strength of this interaction depending on the spacing.     

Investigation of the LES WALE turbulence model within the lattice Boltzmann framework

Turbulence models which can perform the transition from laminar flow to fully developed turbulent flow are of key importance in industrial applications. A promising approach is the LES WALE model, which can be used without wall functions or global damping functions. The model produces an efficient and fast scheme due to its algebraic character. Additionally, its prediction of the transition from laminar to turbulent regimes has shown promising results. In this work, the LES WALE model is investigated within the lattice Boltzmann framework. For validation purposes, various test cases are presented. First, a channel flow at a Reynolds number of 6876 is investigated. Secondly, the flow around a wall-mounted cube at various Reynolds numbers is determined. The flow regime varies from laminar, to transitional, to fully turbulent conditions at a Reynolds number of 40,000 with respect to the cube height.     

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.     

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.     

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.     

Study of jet precession, recirculation and vortex breakdown in turbulent swirling jets using LES

Large eddy simulations (LES) are used to investigate turbulent isothermal swirling flows with a strong emphasis on vortex breakdown, recirculation and instability behaviour. The Sydney swirl burner configuration is used for all simulated test cases from low to high swirl and Reynolds numbers. The governing equations for continuity and momentum are solved on a structured Cartesian grid, and a Smagorinsky eddy viscosity model with the localised dynamic procedure is used as the sub-grid scale turbulence model. The LES successfully predicts both the upstream first recirculation zone generated by the bluff body and the downstream vortex breakdown bubble. The frequency spectrum indicates the presence of low frequency oscillations and the existence of a central jet precession as observed in experiments. The LES calculations well captured the distinct precession frequencies. The results also highlight the precession mode of instability in the center jet and the oscillations of the central jet precession, which forms a precessing vortex core. The study further highlights the predictive capabilities of LES on unsteady oscillations of turbulent swirling flow fields and provides a good framework for complex instability investigations.     

A high-performance lattice Boltzmann implementation to model flow in porous media

We examine the problem of simulating single and multiphase flow in porous medium systems at the pore scale using the lattice Boltzmann (LB) method. The LB method is a powerful approach, but one which is also computationally demanding; the resolution needed to resolve fundamental phenomena at the pore scale leads to very large lattice sizes, and hence substantial computational and memory requirements that necessitate the use of massively parallel computing approaches. Common LB implementations for simulating flow in porous media store the full lattice, making parallelization straightforward but wasteful. We investigate a two-stage implementation consisting of a sparse domain decomposition stage and a simulation stage that avoids the need to store and operate on lattice points located within a solid phase. A set of five domain decomposition approaches are investigated for single and multiphase flow through both homogeneous and heterogeneous porous medium systems on different parallel computing platforms. An orthogonal recursive bisection method yields the best performance of the methods investigated, showing near linear scaling and substantially less storage and computational time than the traditional approach.     

LES of intermittency in a turbulent round jet with different inlet conditions

Large eddy simulation (LES) is a promising technique for accurate prediction of turbulent free shear flows in a wide range of applications. Here the LES technique has been applied to study the intermittency in a high Reynolds number turbulent jet with and without a bluff body. The objective of this work is to study the turbulence intermittency of velocity and scalar fields and its variation with respect to different inlet conditions. Probability density function distributions (pdf) of instantaneous mixture fraction and velocity have been created from which the intermittency has been calculated. The time averaged statistical results for a round jet are first discussed and comparisons of velocity and passive scalar fields between LES calculations and experimental measurements are seen to be good. The calculated probability density distributions show changes from a Gaussian to a delta function with increased radial distance from the jet centreline. The effect of introducing a bluff body into the core flow at the inlet changes the structure of pdfs, but the variation from Gaussian to delta distribution is similar to the jet case. However, the radial variation of the intermittency indicates differences between the results with and without a bluff body at axial locations due the recirculation zone created by the bluff body.     

An adaptive scheme using hierarchical grids for lattice Boltzmann multi-phase flow simulations

The lattice Boltzmann (LB) method is extended and adapted to simulate multi-phase flows on non-uniform tree-type grids. Our model is an extension of the model developed by Gunstensen [Gunstensen AK, Rothman D. Lattice Boltzmann model of immiscible fluids. Phys Rev A 1991;43(8):4320–4327], which is based on the Rothman–Keller model [Rothman DH, Keller JM. Immiscible cellular automaton fluids. J Stat Phys 1988;52:1119–1127]. A first approach we use an a priori grid refinement. We find that the maximum number of possible grid levels for problems with dominant capillary forces is very restricted, if the physical interface is allowed to pass over grid interfaces. Thus a second approach based on adaptive grids was developed, where the physical interface is always discretized on the finest grid level. Efficient and flexible data structures have been developed to manage the remeshing. The application of the scheme for a rising bubble in three dimensions shows very good agreement with the semi-analytical solution and demonstrates the efficiency of our approach.     

Bluff body flow simulation using lattice Boltzmann equation with multiple relaxation time

Numerical simulations using multiple-relaxation-time lattice Boltzmann model (MRT-LBM) are carried out for a long slender rigid circular cylinder in a cross flow to examine three-dimensional wake effect on the flow-induced forces. A mesh refinement technique is applied in the MRT-LBM calculation. The aim is to assess the validity and efficiency of the MRT-LBM model in three-dimensional calculation. In order to simulate the practical situation correctly, wall boundary conditions are specified at both ends of the cylinder. The aspect ratio of the slender cylinder is 16. The calculation is compared with results obtained from a finite volume method (FVM) and a lattice BGK model [Bhatnagar PL, Gross EP, Krook M. A model for collision processes in gases. 1. Small amplitude processes in charged and neutral one-component systems. Phys Rev 1954;94:511–25] with refined grid. Good agreement is obtained. It is found that the MRT-LBM is more efficient and faster in three-dimensional calculations.     

Unsteady flow simulation of the Ahmed reference body using a lattice Boltzmann approach

The Ahmed reference body represents a simplified car geometry that can be used to investigate the main flow features in the wake of vehicles. The present work presents unsteady flow simulations at the rear slant angles 25° and 35° using the PowerFLOW 4.0 D3Q19 lattice Boltzmann model. The flow of the wake is discussed and distributions of averaged pressure and velocities are compared with available experimental findings. The resolution requirement is investigated in terms of computational requirement and accuracy achieved. The predictive capability and the feasibility of the very large eddy simulation (VLES) approach within the lattice Boltzmann framework is demonstrated.     

Explicit algebraic Reynolds stress model of incompressible turbulent flow in rotating square duct

An explicit algebraic Reynolds stress model (EARSM) is proposed for the simulation of the incompressible three-dimensional Reynolds averaged Navier-Stokes (RANS) equations. The spatial discretization of the RANS equations is performed by a finite volume method with nonstaggered variable arrangement.The EARSM model which accounts for rotational effects is used to compute the turbulent flows in rotating straight square duct. The Reynolds number of 48,000 is based on the bulk velocity and the hydraulic diameter of the duct and is kept constant in the range of the rotational numbers. The second order closure (EARSM) yields an asymmetric mean velocity profile as well as turbulence properties. Effects of rotation near the cyclonic (suction side) and anticyclonic (pressure side) walls are well observed. Direct numerical simulation and large eddy simulation data are available for this case. The comparison of EARSM results with these accurate simulations shows a very good agreement.     

Multi-relaxation-time lattice Boltzmann model for axisymmetric flows

In this paper, a lattice-Boltzmann equation (LBE) with multi relaxation times (MRT) is presented for axisymmetric flows. The model is an extension of a recent model with single-relaxation-time [Guo et al., Phys. Rev. E 79, 046708 (2009)], which was developed based on the axisymmetric Boltzmann equation. Due to the use of the MRT collision model, the present model can achieve better numerical stability. The model is validated by some numerical tests including the Hagen-Poiseuille flow, the pulsatile Womersley flow, and the external flow over a sphere. Numerical results are in excellent agreement with analytical solutions or other available data, and the improvement in numerical stability is also confirmed.     

Lattice Boltzmann simulation of turbulent flow laden with finite-size particles

The paper describes a particle-resolved simulation method for turbulent flow laden with finite size particles. The method is based on the multiple-relaxation-time lattice Boltzmann equation. The no-slip boundary condition on the moving particle boundaries is handled by a second-order interpolated bounce-back scheme. The populations at a newly converted fluid lattice node are constructed by the equilibrium distribution with non-equilibrium corrections. MPI implementation details are described and the resulting code is found to be computationally efficient with a good scalability. The method is first validated using unsteady sedimentation of a single particle and sedimentation of a random suspension. It is then applied to a decaying isotropic turbulence laden with particles of Kolmogorov to Taylor microscale sizes. At a given particle volume fraction, the dynamics of the particle-laden flow is found to depend mainly on the effective particle surface area and particle Stokes number. The presence of finite-size inertial particles enhances dissipation at small scales while reducing kinetic energy at large scales. This is in accordance with related studies. The normalized pivot wavenumber is found to not only depend on the particle size, but also on the ratio of particle size to flow scales and particle-to-fluid density ratio.     

A flow topology optimization method for steady state flow using transient information of flow field solved by lattice Boltzmann method

A topology optimization method for fluid flow using transient information is proposed. In many conventional methods, the design domain is updated using steady state information which is obtained after solving the flow field equations completely. Hence we must solve the flow field at each iterative which leads to high computational cost. In contrast, the proposed method updates the design domain using transient information of flow field. Hence the flow field is solved only once. The flow field is solved by lattice Boltzmann method (LBM). It is found that, by using LBM, the flow field is stably computed even though the design domain drastically changes during the computation. The design domain is updated according to sensitivity analysis. In many conventional methods, the sensitivity of objective functionals under lattice Boltzmann equations is obtained using additional adjoint equations. However, in the proposed method, the sensitivity is explicitly formulated and computed without using adjoint variables. In this paper, we show some numerical examples for low Reynolds number flows. The results demonstrate good convergence property in small computation time.     

Isotropic color gradient for simulating very high-density ratios with a two-phase flow lattice Boltzmann model

This study presents the integration of isotropic color gradient discretization into a lattice Boltzmann Rothman–Keller (RK) model designed for two-phase flow simulation. The proposed model removes one limitation of the RK model, which concerns the handling of O(1000) large density ratios between the fluids for a wide range of parameters. Taylor’s series expansions are used to characterize the difference between an isotropic gradient discretization and the commonly used anisotropic gradient. The proposed color gradient discretization can reduce, by one order of magnitude, the spurious current problem that affects the interface between the phases. A set of numerical tests is conducted to show that a rotationally invariant discretization enables widening of the parameter range for the surface tension. Surface tensions from O(10⁻²) to O(10⁻⁸), depending on the density ratio, are accurately simulated. An extreme density ratio of O(10,000) is successfully tested for a steady bubble with an error of 0.5% for Laplace’s law across a sharp interface, with a thickness of about 5–6 lattice units.     

LES validation of turbulent rotating buoyancy- and surface tension-driven flow against DNS

The paper is concerned with the validation and error analysis of predictions for the flow and heat transfer in a silicon melt (Pr=0.013) found in a Czochralski (Cz) apparatus for crystal growth. This system resembles turbulent Rayleigh-Bénard-Marangoni convection. Since for practical applications predictions based on direct numerical simulations (DNS) require too many resources to conduct parametric studies or optimizations, nowadays in practice the method of choice is the large-eddy simulation (LES). The case considered consists of an idealized cylindrical crucible of 170 mm radius with a rotating crystal of 50 mm radius. Boundary conditions from experimental data were applied, which lead to the dimensionless numbers of , and Ra=2.8×10⁷. The filtered Navier-Stokes equations were solved based on a finite-volume scheme for curvilinear block-structured grids and an explicit time discretization. For a comprehensive error analysis, different grid sizes, subgrid-scale models, and discretization schemes were employed. The results were compared to reference DNS data of the same case recently generated by the authors (Int J Heat Mass Transfer, 51 (2008) 6219-6234) for validation. For the finest LES grid (10⁶ control volumes) using a standard Smagorinsky model with van Driest damping or a dynamic model, both with central discretization, the results agree well with the DNS reference while the computational effort could be reduced by a factor of 20. When using an upwind scheme even of formally second-order accuracy, significant deviations occur. Further stepwise reductions of the grid size decrease the CPU time drastically, but also lead to larger aberrations. When the grid is coarsened by a factor of 32 (resulting in ca. 130,000 CVs), even qualitative differences between the LES and the DNS solution appear.It could be shown in the present work that the LES method is an efficient tool to model the turbulent flow and heat transfer in Rayleigh-Bénard-Marangoni configurations. However, care should be taken in the choice of the grid resolution and discretization scheme for the nonlinear convective terms, as too coarse meshes in combination with upwind schemes lead to significant numerical errors. Finally, a quantified relation between the achievable accuracy and the necessary computational effort is presented.