A comparative study of lattice Boltzmann methods using bounce‐back schemes and immersed boundary ones for flow acoustic problems

In order to find applicable treatments of moving boundary conditions based on the lattice Boltzmann method in flow acoustic problems, three bounce‐back (BB) methods and four kinds of immersed boundary (IB) methods are compared. We focused on fluid–solid boundary conditions for flow acoustic problems especially the simulations of sound waves from moving boundaries. BB methods include link bounce‐back, interpolation bounce‐back and unified interpolation bounce‐back methods. Five IB methods are explicit and implicit direct‐forcing (Explicit‐IB and Implicit‐IB), two kinds of partially saturated computational methods and ghost fluid method. In order to reduce the spurious pressure generated by the fresh grid node changing from solid domain to fluid domain for BB methods and sharp IB methods, we proposed two new kinds of treatments and compared them with two existing ones. Simulations of the benchmark problems prove that the local evolutionary iteration (LI) is the best one in treatments of the fresh nodes. In addition, for standing boundary problems, although BB methods have a little higher accuracy, all the methods have similar accuracy. However, for moving boundary problems, IB methods are more appropriate than BB methods, because IB methods' smooth interpolation of pressure eld produces less disturbing spurious pressure waves. With improved treatments of fresh nodes, BB methods are also acceptable for moving boundary acoustic problems. In comparative tests in respective type, unified interpolation bounce‐back with LI, Implicit‐IB, and ghost fluid with LI are the best choices. Copyright © 2013 John Wiley & Sons, Ltd.     

High-order ghost-point immersed boundary method for viscous compressible flows based on summation-by-parts operators

A high-order immersed boundary method is devised for the compressible Navier-Stokes equations by employing high-order summation-by-parts difference operators. The immersed boundaries are treated as sharp interfaces by enforcing the solid wall boundary conditions via flow variables at ghost points. Two different interpolation schemes are tested to compute values at the ghost points and a hybrid treatment is used. The first method provides the bilinearly interpolated flow variables at the image points of the corresponding ghost points and the second method applies the boundary condition at the immersed boundary by using the weighted least squares method with high-order polynomials. The approach is verified and validated for compressible flow past a circular cylinder at moderate Reynolds numbers. The tonal sound generated by vortex shedding from a circular cylinder is also investigated. In order to demonstrate the capability of the solver to handle complex geometries in practical cases, flow in a cross-section of a human upper airway is simulated.     

A local domain‐free discretization method for simulation of incompressible flows over moving bodies

This paper presents a local domain‐free discretization (DFD) method for the simulation of unsteady flows over moving bodies governed by the incompressible Navier–Stokes equations. The discretization strategy of DFD is that the discrete form of partial differential equations at an interior point may involve some points outside the solution domain. All the mesh points are classified as interior points, exterior dependent points and exterior independent points. The functional values at the exterior dependent points are updated at each time step by the approximate form of solution near the boundary. When the body is moving, only the status of points is changed and the mesh can stay fixed. The issue of ‘freshly cleared nodes/cells’ encountered in usual sharp interface methods does not pose any particular difficulty in the presented method. The Galerkin finite‐element approximation is used for spatial discretization, and the discrete equations are integrated in time via a dual‐time‐stepping scheme based on artificial compressibility. In order to validate the present method for moving‐boundary flow problems, two groups of flow phenomena have been simulated: (1) flows over a fixed circular cylinder, a harmonic in‐line oscillating cylinder in fluid at rest and a transversely oscillating cylinder in uniform flow; (2) flows over a pure pitching airfoil, a heaving–pitching airfoil and a deforming airfoil. The predictions show good agreement with the published numerical results or experimental data. Copyright © 2010 John Wiley & Sons, Ltd.     

Simulating 3D deformable particle suspensions using lattice Boltzmann method with discrete external boundary force

A method for direct numerical analysis of three‐dimensional deformable particles suspended in fluid is presented. The flow is computed on a fixed regular ‘lattice’ using the lattice Boltzmann method (LBM), where each solid particle is mapped onto a Lagrangian frame moving continuously through the domain. Instead of the bounce‐back method, an external boundary force (EBF) is used to impose the no‐slip boundary condition at the fluid–solid interface for stationary or moving boundaries. The EBF is added directly to the lattice Boltzmann equation. The motion and orientation of the particles are obtained from Newtonian dynamics equations. The advantage of this approach is outlined in comparison with the standard and higher‐order interpolated bounce‐back methods as well as the LBM immersed‐boundary and the volume‐of‐fluid methods. Although the EBF method is general, in this application, it is used in conjunction with the lattice–spring model for deformable particles. The methodology is validated by comparing with experimental and theoretical results. Copyright © 2009 John Wiley & Sons, Ltd.     

Using fully implicit conservative statements to close open boundaries passing through recirculations

The numerical solution of the fluid flow governing equations requires the implementation of certain boundary conditions at suitable places to make the problem well‐posed. Most of numerical strategies exhibit weak performance and obtain inaccurate solutions if the solution domain boundaries are not placed at adequate locations. Unfortunately, many practical fluid flow problems pose difficulty at their boundaries because the required information for solving the PDE's is not available there. On the other hand, large solution domains with known boundary conditions normally need a higher number of mesh nodes, which can increase the computational cost. Such difficulties have motivated the CFD workers to confine the solution domain and solve it using artificial boundaries with unknown flow conditions prevailing there. In this work, we develop a general strategy, which enables the control‐volume‐based methods to close the outflow boundary at arbitrary locations where the flow conditions are not known prior to the solution. In this regard, we extend suitable conservative statements at the outflow boundary. The derived statements gradually detect the correct boundary conditions at arbitrary boundaries via an implicit procedure using a finite element volume method. The extended statements are validated by solving the truncated benchmark backward‐facing step problem. The investigation shows that the downstream boundary can pass through a recirculation zone without deteriorating the accuracy of the solution either in the domain or at its boundaries. The results indicate that the extended formulation is robust enough to be employed in solution domains with unknown boundary conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

An improved ghost‐cell immersed boundary method for compressible flow simulations

This study presents an improved ghost‐cell immersed boundary approach to represent a solid body in compressible flow simulations. In contrast to the commonly used approaches, in the present work, ghost cells are mirrored through the boundary described using a level‐set method to farther image points, incorporating a higher‐order extra/interpolation scheme for the ghost‐cell values. A sensor is introduced to deal with image points near the discontinuities in the flow field. Adaptive mesh refinement is used to improve the representation of the geometry efficiently in the Cartesian grid system. The improved ghost‐cell method is validated against four test cases: (a) double Mach reflections on a ramp, (b) smooth Prandtl–Meyer expansion flows, (c) supersonic flows in a wind tunnel with a forward‐facing step, and (d) supersonic flows over a circular cylinder. It is demonstrated that the improved ghost‐cell method can reach the accuracy of second order in L¹ norm and higher than first order in L^∞ norm. Direct comparisons against the cut‐cell method demonstrate that the improved ghost‐cell method is almost equally accurate with better efficiency for boundary representation in high‐fidelity compressible flow simulations. Copyright © 2016 John Wiley & Sons, Ltd.     

Meshfree weak–strong (MWS) form method and its application to incompressible flow problems

A meshfree weak–strong (MWS) form method has been proposed by the authors' group for linear solid mechanics problems based on a combined weak and strong form of governing equations. This paper formulates the MWS method for the incompressible Navier–Stokes equations that is non‐linear in nature. In this method, the meshfree collocation method based on strong form equations is applied to the interior nodes and the nodes on the essential boundaries; the local Petrov–Galerkin weak form is applied only to the nodes on the natural boundaries of the problem domain. The MWS method is then applied to simulate the steady problem of natural convection in an enclosed domain and the unsteady problem of viscous flow around a circular cylinder using both regular and irregular nodal distributions. The simulation results are validated by comparing with those of other numerical methods as well as experimental data. It is demonstrated that the MWS method has very good efficiency and accuracy for fluid flow problems. It works perfectly well for irregular nodes using only local quadrature cells for nodes on the natural boundary, which can be generated without any difficulty. Copyright © 2004 John Wiley & Sons, Ltd.     

An explicit level set approach for generalized shape optimization of fluids with the lattice Boltzmann method

This study is concerned with a generalized shape optimization approach for finding the geometry of fluidic devices and obstacles immersed in flows. Our approach is based on a level set representation of the fluid–solid interface and a hydrodynamic lattice Boltzmann method to predict the flow field. We present an explicit level set method that does not involve the solution of the Hamilton–Jacobi equation and allows using standard nonlinear programming methods. In contrast to previous works, the boundary conditions along the fluid–structure interface are enforced by second‐order accurate interpolation schemes, overcoming shortcomings of flow penalization methods and Brinkman formulations frequently used in topology optimization. To ensure smooth boundaries and mesh‐independent results, we introduce a simple, computationally inexpensive filtering method to regularize the level set field. Furthermore, we define box constraints for the design variables that guarantee a continuous evolution of the boundaries. The features of the proposed method are studied by two numeric examples of two‐dimensional steady‐state flow problems. Copyright © 2009 John Wiley & Sons, Ltd.     

A hybrid immersed boundary and material point method for simulating 3D fluid–structure interaction problems

A numerical method is developed for solving the 3D, unsteady, incompressible Navier–Stokes equations in curvilinear coordinates containing immersed boundaries (IBs) of arbitrary geometrical complexity moving and deforming under forces acting on the body. Since simulations of flow in complex geometries with deformable surfaces require special treatment, the present approach combines a hybrid immersed boundary method (HIBM) for handling complex moving boundaries and a material point method (MPM) for resolving structural stresses and movement. This combined HIBM & MPM approach is presented as an effective approach for solving fluid–structure interaction (FSI) problems. In the HIBM, a curvilinear grid is defined and the variable values at grid points adjacent to a boundary are forced or interpolated to satisfy the boundary conditions. The MPM is used for solving the equations of solid structure and communicates with the fluid through appropriate interface‐boundary conditions. The governing flow equations are discretized on a non‐staggered grid layout using second‐order accurate finite‐difference formulas. The discrete equations are integrated in time via a second‐order accurate dual time stepping, artificial compressibility scheme. Unstructured, triangular meshes are employed to discretize the complex surface of the IBs. The nodes of the surface mesh constitute a set of Lagrangian control points used for tracking the motion of the flexible body. The equations of the solid body are integrated in time via the MPM. At every instant in time, the influence of the body on the flow is accounted for by applying boundary conditions at stationary curvilinear grid nodes located in the exterior but in the immediate vicinity of the body by reconstructing the solution along the local normal to the body surface. The influence of the fluid on the body is defined through pressure and shear stresses acting on the surface of the body. The HIBM & MPM approach is validated for FSI problems by solving for a falling rigid and flexible sphere in a fluid‐filled channel. The behavior of a capsule in a shear flow was also examined. Agreement with the published results is excellent. Copyright © 2007 John Wiley & Sons, Ltd.     

自然单元法计算裂纹与材料边界问题

提出一种新的非凸边界上自然单元法形函数计算方法,通过边界结点限制点对间的邻点关系,对包括裂纹和材料边界在内的各种类型的非凸边界具有统一的处理原则,所得到的近似函数在边界结点间具有线性插值性.     

改进虚拟边界算法在超声速流动问题求解中的应用

提出了一种改进的虚拟单元浸没边界法, 并与一种高阶格式的有限差分算法相结合, 运用于求解超声速复杂几何绕流问题.该算法的核心思想在于在固体边界的内部和外部分别施加满足边界关系的作用点, 使得几何边界离散更加细化, 起到了壁面附近网格局部加密的作用.采用源空间内流体点作为反距离插值算法的重构点, 有效避免了插值点数目过少而与作用点相重合情况.通过对二维激波反射现象 (马赫数为 2.81) 和三维超声速球体绕流问题 (马赫数为 1.2) 的数值模拟, 与实验结果对比表明, 本文改进算法相对一般的虚拟边界法来说能显著提高数值精度, 减小计算误差.计算结果揭示了球体绕流中剪切层、压缩波系和尾迹的相互作用导致自由剪切层失稳的机理.剪切层厚度和湍流雷诺脉动经历了线性增长、大幅度震荡和小幅度波动三个阶段, 导致剪切层表面褶皱因子变化呈指数规律增长.其湍流结构表现出明显的各向异性, 具体在流向雷诺正应力在湍流脉动中占主导地位, 激波的压缩作用对不同方向雷诺正应力的影响存在空间迟滞效应.      

A method for direct simulation of flexible fiber suspensions using lattice Boltzmann equation with external boundary force

The computational method presented here can be used to study the effect of volume fraction and particle deformation on the rheology and microstructure of deformable fibers suspended in Newtonian fluid. In this method, the flow is computed on a fixed regular ‘lattice’ using the lattice Boltzmann method, where each solid particle is mapped onto a Lagrangian frame moving continuously through the domain. Instead of the standard bounce-back method, an external boundary force is used to impose the no-slip boundary condition at the fluid–solid interface for stationary or moving boundaries. The motion and orientation of the fiber are obtained from Newtonian dynamics equations. Although the external boundary force method is general, in this application it is used in conjunction with a flexible fiber model, which calculates the flexible fiber deformation by the real material properties. The methodology is validated by comparing with experimental and theoretical results.     

格子Boltzmann方法处理复杂边界新进展

格子玻尔兹曼方法是相对较新的计算流体力学方法,具有其独特的优点,如自然并行计算以及复杂边界的处理．在日益重要的高性能计算上,将发挥越来越大的作用．在已经较为成熟的计算模型基础上,为计算复杂几何边界对流场的作用,大量学者提出了各类针对格子玻尔兹曼方法的边界处理格式．边界处理方式对计算的影响主要有三个方面：计算精度,算法稳定性以及并行性．本文简单介绍格子玻尔兹曼方法在边界处理上的一些最新进展．并通过对实际算例的计算比较各类边界处理方式的优劣．     

A lattice Boltzmann model for the compressible Euler equations with second‐order accuracy

In this paper, we propose a new lattice Boltzmann model for the compressible Euler equations. The model is based on a three‐energy‐level and three‐speed lattice Boltzmann equation by using a method of higher moments of the equilibrium distribution functions. In order to obtain second‐order accuracy, we employ the ghost field distribution functions to remove the non‐physical viscous parts. We also use the conditions of the higher moment of the ghost field equilibrium distribution functions to obtain the equilibrium distribution functions. In the numerical examples, we compare the numerical results of this scheme with those obtained by other lattice Boltzmann models for the compressible Euler equations. Copyright © 2008 John Wiley & Sons, Ltd.     

A hybrid immersed‐boundary and multi‐block lattice Boltzmann method for simulating fluid and moving‐boundaries interactions

A numerical method is developed for modelling the interactions between incompressible viscous fluid and moving boundaries. The principle of this method is introducing the immersed‐boundary concept in the framework of the lattice Boltzmann method, and improving the accuracy and efficiency of the simulation by refining the mesh near moving boundaries. Besides elastic boundary with a constitutive law, the method can also efficiently simulate solid moving‐boundary interacting with fluid by employing the direct forcing technique. The method is validated by the simulations of flow past a circular cylinder, two cylinders moving with respect to each other and flow around a hovering wing. The versatility of the method is demonstrated by the numerical studies including elastic filament flapping in the wake of a cylinder and fish‐like bodies swimming in quiescent fluid. Copyright © 2006 John Wiley & Sons, Ltd.     

Extension of domain‐free discretization method to simulate compressible flows over fixed and moving bodies

This paper is the first endeavour to present the local domain‐free discretization (DFD) method for the solution of compressible Navier–Stokes/Euler equations in conservative form. The discretization strategy of DFD is that for any complex geometry, there is no need to introduce coordinate transformation and the discrete form of governing equations at an interior point may involve some points outside the solution domain. The functional values at the exterior dependent points are updated at each time step to impose the wall boundary condition by the approximate form of solution near the boundary. Some points inside the solution domain are constructed for the approximate form of solution, and the flow variables at constructed points are evaluated by the linear interpolation on triangles. The numerical schemes used in DFD are the finite element Galerkin method for spatial discretization and the dual‐time scheme for temporal discretization. Some numerical results of compressible flows over fixed and moving bodies are presented to validate the local DFD method. Copyright © 2006 John Wiley & Sons, Ltd.     

幂律流体圆柱绕流的格子波尔兹曼模拟

基于插值补充格子波尔兹曼方法和幂律流体的本构方程,建立了贴体坐标系下适用于幂律流体的格子波尔兹曼模型,模拟了幂律流体的圆柱绕流问题,采用非平衡外推格式处理圆柱表面的速度无滑移边界,利用应力积分法确定曳力系数和升力系数,并与基于标准的格子波尔兹曼方法和有限容积法获得的数值数据进行对比,吻合良好. 进行了网格无关性验证之后,分析了稳态流动时,不同雷诺数下幂律指数对于尾迹长度、分离角、圆柱表面黏度分布、表面压力系数及曳力系数的影响,以及非定常流动中,幂律指数对于流场、曳力系数、升力系数和斯特劳哈尔数的影响. 获得的变化规律与基于其他数值模拟方法得到的结果相一致,充分验证了模型的有效性和正确性. 结果表明:插值补充格子波尔兹曼方法可以用来模拟幂律流体在具有复杂边界流场内的流动问题,通过引入不同的非牛顿流体本构方程,该方法还可以进一步应用于其他类型的非牛顿流体研究中.      

A new approach in modeling of constant temperature boundary condition in thermal lattice‐Boltzmann method

The lattice‐Boltzmann method is being applied to a diversity of fluid flow and heat transfer problems nowadays. Because of its microscale nature, strict attention should be paid when introducing macroscopic inputs to the model. One of the challenging issues dealing with macroscale and microscale treatment is the implementation of boundary conditions. In this regard constant‐temperature boundaries are frequently used in energy transfer problems. Such boundaries are simply modeled in Navier–Stokes based solvers, but they are not so harnessed in lattice‐Boltzmann models. One of the problems is that the calculated tangential heat flux is not zero along such boundaries in most of the previous models. In the present paper, a model has been developed, which has the capability of controlling tangential heat flux along the constant‐temperature boundaries. It aims to set the heat flux nearly zero along the boundary in midplane grid schemes. Copyright © 2012 John Wiley & Sons, Ltd.     

An improved immersed boundary‐lattice Boltzmann method based on force correction technique

In this paper, an improved immersed boundary‐lattice Boltzmann method based on the force correction technique is presented for fluid‐structure interaction problems including the moving boundary interfaces. By introducing a force correction coefficient, the non‐slip boundary conditions are much better enforced compared with the conventional immersed boundary‐lattice Boltzmann methods. In addition, the implicit and iterative calculations are avoided; thus, the computational cost is reduced dramatically. Several numerical experiments are carried out to test the efficiency of the method. It is found that the method has the second‐order accuracy, and the non‐slip boundary conditions are enforced indeed. The numerical results also show that the present method is a suitable tool for fluid‐structure interaction problems involving complex moving boundaries.