On optimal node spacing for immersed boundary–lattice Boltzmann method in 2D and 3D

A computational study on optimal spacing of Lagrangian nodes discretizing a rigid and immobile immersed body boundary in 2D and 3D is presented in order to show how the density of the Lagrangian points affects the numerical results of the Immersed Boundary–Lattice Boltzmann Method (IB–LBM). The study is based on the implicit velocity correction-based IB–LBM proposed by Wu and Shu (2009, 2010) that allows computing the fluid–body interaction force. However, the (original) method fails for densely spaced Lagrangian points due to ill-conditioned or even singular linear systems that arise from the derivation of the method. We propose a modification that improves the solvability of the linear systems and compare the performance of both methods using several benchmark problems. The results show how the spacing of the Lagrangian points affects the numerical results, mainly the permeability of the discretized body boundary in applications to fluid flows over rigid obstacles and blood flows in arteries in 2D and 3D.     

A lattice Boltzmann based implicit immersed boundary method for fluid–structure interaction

The immersed boundary method is a practical and effective method for fluid–structure interaction problems. It has been applied to a variety of problems. Most of the time-stepping schemes used in the method are explicit, which suffer a drawback in terms of stability and restriction on the time step. We propose a lattice Boltzmann based implicit immersed boundary method where the immersed boundary force is computed at the unknown configuration of the structure at each time step. The fully nonlinear algebraic system resulting from discretizations is solved by an Inexact Newton–Krylov method in a Jacobian-free manner. The test problem of a flexible filament in a flowing viscous fluid is considered. Numerical results show that the proposed implicit immersed boundary method is much more stable with larger time steps and significantly outperforms the explicit version in terms of computational cost.     

Drag of a flexible fiber in a 2D moving viscous fluid

This paper reports the numerical study of the drag of a flexible elastic fiber immersed in a two-dimensional viscous flow using the immersed boundary (IB) method. We found drag reduction of a flexible fiber compared to a stiff one and the drag coefficient decreases with respect to the dimensionless fiber length within a certain range. The results are a starting point for the understanding of the role of flexibility in biological organisms in fluid flows.     

Immersed boundary method and lattice Boltzmann method coupled FSI simulation of mitral leaflet flow

Coupling the immersed boundary (IB) method and the lattice Boltzmann (LB) method might be a promising approach to simulate fluid-structure interaction (FSI) problems with flexible structures and moving boundaries. To investigate the possibility for future IB-LB coupled simulations of the heart flow dynamics, an IB-LB coupling scheme suitable for rapid boundary motion and large pressure gradient FSI is proposed, and the mitral valve jet flow considering the interaction of leaflets and fluid is simulated. After analyzing the respective concepts, formulae and advantages of the IB and LB methods, we first explain the coupling strategy and detailed implementation procedures, and then verify the effectiveness and second-order accuracy of the scheme by simulating a benchmark case, the relaxation of a stretched membrane immersed in fluid. After that, the diastolic filling jet flow between mitral leaflets in a simplified 2D left heart model is simulated. The model consists of the simplified transmitral passage of the heart and two curvilinear leaflets. In the simulation, the atrial and ventricular pressure histories of normal human are specified as boundary conditions, and the leaflets are treated as fibers that interact with the fluid to define their deformations and movements. The resulting opening and closing movements of the leaflets and the flow patterns of the filling jet are qualitatively reasonable and compare well with existing numerical and measured data. It is shown that this IB-LB coupling method is feasible for treating flexible boundary FSI problems with rapid boundary motion and large pressure gradient, the results of the mitral leaflet flow are valuable for understanding the transmitral FSI dynamics, and it is possible to simulate the more realistic 3D heart flow by the scheme in the future.     

Improved coupling of time integration and hydrodynamic interaction in particle suspensions using the lattice Boltzmann and discrete element methods

This paper introduces improvements to the simulation of particle suspensions using the lattice Boltzmann method (LBM) and the discrete element method (DEM). First, the benefit of using a two-relaxation-time (TRT) collision operator, instead of the popular Bhatnagar–Gross–Krook (BGK) collision operator, is demonstrated. Second, a modified solid weighting function for the partially saturated method (PSM) for fluid–solid interaction is defined and tested. Results are presented for a range of flow configurations, including sphere packs, duct flows, and settling spheres, with good accuracy and convergence observed. Past research has shown that the drag, and consequently permeability, predictions of the LBM exhibit viscosity-dependence when used with certain boundary conditions such as bounce-back or interpolated bounce-back, and this is most pronounced when the BGK collision operator is employed. The improvements presented here result in a range of computational viscosities, and therefore relaxation parameters, within which drag and permeability predictions remain invariant. This allows for greater flexibility in using the relaxation parameter to adjust the LBM timestep, which can subsequently improve synchronisation with the time integration of the DEM. This has significant implications for the simulation of large-scale suspension phenomena, where the limits of computational hardware persistently constrain the resolution of the LBM lattice.     

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.     

非均匀分布颗粒群中的曳力分布

本文采用格子Boltzmann方法(LBM)在图形处理器(GPU)上计算了由静止圆柱阵列组成的团聚物周期单元内的不可压缩流体流动,流固交界面处采用直接反弹以实现无滑移边界,每个圆柱上的曳力通过统计动量交换直接求得。根据LBM求得的流体速度,对于团聚物中的单圆柱按能量最小多尺度(EMMS)模型计算平均曳力系数,并考察了将聚团近似为均匀悬浮的临界条件。对颗粒雷诺数Re_p在0～10之间的80种固相份额的模拟结果表明,密相空隙率可以表征这种临界条件。当固相份额恒定时,该临界空隙率随着Re_p的增加而降低;当Re_p恒定时,该临界空隙率随着固相份额的增加而降低。     

A unified operator splitting approach for multi-scale fluid–particle coupling in the lattice Boltzmann method

A unified framework to derive discrete time-marching schemes for the coupling of immersed solid and elastic objects to the lattice Boltzmann method is presented. Based on operator splitting for the discrete Boltzmann equation, second-order time-accurate schemes for the immersed boundary method, viscous force coupling and external boundary force are derived. Furthermore, a modified formulation of the external boundary force is introduced that leads to a more accurate no-slip boundary condition. The derivation also reveals that the coupling methods can be cast into a unified form, and that the immersed boundary method can be interpreted as the limit of force coupling for vanishing particle mass. In practice, the ratio between fluid and particle mass determines the strength of the force transfer in the coupling. The integration schemes formally improve the accuracy of first-order algorithms that are commonly employed when coupling immersed objects to a lattice Boltzmann fluid. It is anticipated that they will also lead to superior long-time stability in simulations of complex fluids with multiple scales.     

Lattice-based flow field modeling

We present an approach for simulating the natural dynamics that emerge from the interaction between a flow field and immersed objects. We model the flow field using the lattice Boltzmann model (LBM) with boundary conditions appropriate for moving objects and accelerate the computation on commodity graphics hardware (GPU) to achieve real-time performance. The boundary conditions mediate the exchange of momentum between the flow field and the moving objects resulting in forces exerted by the flow on the objects as well as the back-coupling on the flow. We demonstrate our approach using soap bubbles and a feather. The soap bubbles illustrate Fresnel reflection, reveal the dynamics of the unseen flow field in which they travel, and display spherical harmonics in their undulations. Our simulation allows the user to directly interact with the flow field to influence the dynamics in real time. The free feather flutters and gyrates in response to lift and drag forces created by its motion relative to the flow. Vortices are created as the free feather falls in an otherwise quiescent flow.     

A unified approach for nonslip and slip boundary conditions in the lattice Boltzmann method

This work proposed a unified approach to impose both nonslip and slip boundary conditions for the lattice Boltzmann method (LBM). By introducing the tangential momentum accommodation coefficient (TMAC), the present implementation can determine the change of the tangential momentum on the wall and then impose the correct boundary conditions for LBM. The simulation results demonstrate that this implementation is equivalent to the first-order slip model.     

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.     

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.     

The Lattice-boltzmann method for simulating gaseous phenomena

We present a physically-based, yet fast and simple method to simulate gaseous phenomena. In our approach, the incompressible Navier-Stokes (NS) equations governing fluid motion have been modeled in a novel way to achieve a realistic animation. We introduce the lattice Boltzmann model (LBM), which simulates the microscopic movement of fluid particles by linear and local rules on a grid of cells so that the macroscopic averaged properties obey the desired NS equations. The LBM is defined on a 2D or 3D discrete lattice, which is used to solve fluid animation based on different boundary conditions. The LBM simulation generates, in real-time, an accurate velocity field and can incorporate an optional temperature field to account for the buoyancy force of hot gas. Because of the linear and regular operations in each local cell of the LBM grid, we implement the computation in commodity texture hardware, further improving the simulation speed. Finally, textured splats are used to add small scale turbulent details, achieving high-quality real-time rendering. Our method can also simulate the physically correct action of stationary or mobile obstacles on gaseous phenomena in real-time, while still maintaining highly plausible visual details.     

A Comparison Between the Interpolated Bounce-Back Scheme and the Immersed Boundary Method to Treat Solid Boundary Conditions for Laminar Flows in the Lattice Boltzmann Framework

In this paper, the interpolated bounce-back scheme and the immersed boundary method are compared in order to handle solid boundary conditions in the lattice Boltzmann method. These two approaches are numerically investigated in two test cases: a rigid fixed cylinder invested by an incoming viscous fluid and an oscillating cylinder in a calm viscous fluid. Findings in terms of velocity profiles in several cross sections are shown. Differences and similarities between the two methods are discussed, by emphasizing pros and cons in terms of stability and computational effort of the numerical algorithm.     

On enhanced non-linear free surface flow simulations with a hybrid LBM–VOF model

In this paper, we present extensions, extensive validations and applications of our previously published hybrid volume-of-fluid-based (VOF) model for the simulation of free-surface flow problems. For the solution of the flow field, the lattice Boltzmann method is used, where the free surface is represented by a VOF approach. The advection equation for the VOF fill level is discretized with a finite volume method, on the basis of a 3D Piecewise Linear Interface Reconstruction (PLIC) algorithm. The model is validated for several standard free surface benchmarks, such as breaking dam scenarios and a free falling jet. Finally, the hybrid algorithm is applied to the simulation of a wave breaking by overturning during shoaling, which is considered to be a demanding test case, especially for VOF solvers. In this case, the flow field is initialized early in the shoaling process with a solitary wave solution from inviscid, irrotational potential flow. The wave breaking process is then simulated with the 3D transient and turbulent LBM–VOF solver. All validation and benchmark tests confirm the accuracy of the proposed hybrid model.     

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.     

Three-dimensional discrete-velocity BGK model for the incompressible Navier–Stokes equations

The lattice Boltzmann method (LBM) has been widely used for the simulations of the incompressible Navier–Stokes (NS) equations. The finite difference Boltzmann method (FDBM) in which the discrete-velocity Boltzmann equation is solved instead of the lattice Boltzmann equation has also been applied as an alternative method for simulating the incompressible flows. The particle velocities of the FDBM can be selected independently from the lattice configuration. In this paper, taking account of this advantage, we present the discrete velocity Boltzmann equation that has a minimum set of the particle velocities with the lattice Bharnagar–Gross–Krook (BGK) model for the three-dimensional incompressible NS equations. To recover incompressible NS equations, tensors of the particle velocities have to be isotropic up to the fifth rank. Thus, we propose to apply the icosahedral vectors that have 13 degrees of freedom to the particle velocity distributions. Validity of the proposed model (D3Q13BGK) is confirmed by numerical simulations of the shear-wave decay problem and the Taylor–Green vortex problem. With respect to numerical accuracy, computational efficiency and numerical stability, we compare the proposed model with the conventional lattice BGK models (D3Q15, D3Q19 and D3Q27) and the multiple-relaxation-time (MRT) model (D3Q13MRT) that has the same degrees of freedom as our proposal. The comparisons show that the compressibility error of the proposed model is approximately double that of the conventional lattice BGK models, but the computational efficiency of the proposed model is superior to that of the others. The linear stability of the proposed model is also superior to that of the lattice BGK models. However, in non-linear simulations, the proposed model tends to be less stable than the others.     

Numerical simulation of viscous flow over non-smooth surfaces

The incompressible viscous flow over several non-smooth surfaces is simulated numerically by using the lattice Boltzmann method. A numerical strategy for dealing with a curved boundary with second-order accuracy for velocity field is presented. The drag evaluation is performed by the momentum-exchange method. The streamline contours are obtained over surfaces with different shapes, including circular concave, circular convex, triangular concave, triangular convex, regular sinusoidal wavy and irregular sinusoidal wavy, are obtained. The flow patterns are discussed in detail. The velocity profiles over different kinds of non-smooth surfaces are investigated. The numerical results show that the lattice Boltzmann method is reliable, accurate, easy to implement, and can provide valuable help for some engineering practices.     

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.     

Adjoint-based fluid flow control and optimisation with lattice Boltzmann methods

A lattice Boltzmann (LB) framework to solve fluid flow control and optimisation problems numerically is presented. Problems are formulated on a mesoscopic basis. In a side condition, the dynamics of a Newtonian fluid is described by a family of simplified Boltzmann-like equations, namely BGK–Boltzmann equations, which are linked to an incompressible Navier–Stokes equation. It is proposed to solve the non-linear optimisation problem by a line search algorithm. The needed derivatives are obtained by deriving the adjoint equations, referred to as adjoint BGK–Boltzmann equations. The primal equations are discretised by standard lattice Boltzmann methods (LBM) while for the adjoint equations a novel discretisation strategy is introduced. The approach follows the main ideas behind LBM and is therefore referred to as adjoint lattice Boltzmann methods (ALBM). The corresponding algorithm retains most of the basic features of LB algorithms. In particular, it enables a highly-efficient parallel implementation and thus solving large-scale fluid flow control and optimisation problems. The overall solution strategy, the derivation of a prototype adjoint BGK–Boltzmann equation, the novel ALBM and its parallel realisation as well as its validation are discussed in detail in this article. Numerical and performance results are presented for a series of steady-state distributed control problems with up to approximately 1.6 million unknown control parameters obtained on a high performance computer with up to 256 processing units.