一种新型光滑粒子动力学固壁边界施加模型

由于Lagrange粒子法的本质, 固壁边界条件的施加一直是光滑粒子动力学方法的难点之一. 本文从固壁边界的物理原理出发, 应用多层虚粒子表征固壁边界, 提出了一种新型固壁边界施加模型. 将虚粒子看作流体的扩展, 计算中虚粒子密度保持不变, 压力、速度等参数通过对流体粒子的插值获得, 虚粒子有条件的参与控制方程的计算, 对流体的密度/压力产生影响, 通过压力梯度隐式地表征壁面与流体之间的作用强度并对流体粒子施加沿壁面法线方向的斥力作用, 防止流体粒子对壁面的穿透. 数值算例测试结果表明, 与现有固壁边界施加方法相比, 本文方法更加符合流体与固壁边界作用的物理原理, 可以简单、有效地施加固壁边界条件, 方便地应用于具有复杂几何边界的问题, 获得稳定的流场形态、规则的粒子秩序及良好的速度、压力等参量的分布.     

光滑粒子流体动力学方法固壁处理的一种新型排斥力模型

与传统网格法相比, 光滑粒子流体动力学方法不能直接施加壁面边界条件, 这就限制了该方法在工程中的应用.为此, 本文基于Galerkin加权余量法并结合传统排斥力方法, 推导出一种新的排斥力公式来施加壁面边界条件.该方法不含未知参数, 能在不减小边界粒子尺寸的情形下有效地防止流体粒子穿透壁面, 同时可避免邻近边界的流体粒子的速度及压力振荡. 分别通过静止液柱算例、液柱坍塌算例、容器中液体静止算例及溃坝算 例来验证本文方法的有效性, 并与传统边界处理方法进行对比, 结果表明: 本文方法克服了传统方法存在的缺陷, 是一种有效的固壁边界处理方法. 关键词： 光滑粒子流体动力学法 固壁边界 排斥力 加权余量法     

A generalized wall boundary condition for smoothed particle hydrodynamics

In this paper we present a new formulation of the boundary condition at static and moving solid walls in SPH simulations. Our general approach is both applicable to two and three dimensions and is very simple compared to previous wall boundary formulations. Based on a local force balance between wall and fluid particles we apply a pressure boundary condition on the solid particles to prevent wall penetration. This method can handle sharp corners and complex geometries as is demonstrated with several examples. A validation shows that we recover hydrostatic equilibrium conditions in a static tank, and a comparison of the classical dam break simulation with state-of-the-art results in literature shows good agreement. We simulate various problems such as the flow around a cylinder and the backward facing step at Re = 100 to demonstrate the general applicability of this new method.     

Adaptive artificial boundary condition for the two-level Schrödinger equation with conical crossings

In this paper, we present an adaptive approach to design the artificial boundary conditions for the two-level Schrödinger equation with conical crossings on the unbounded domain. We use the windowed Fourier transform to obtain the local wave number information in the vicinity of artificial boundaries, and adopt the operator splitting method to obtain an adaptive local artificial boundary condition. Then reduce the original problem into an initial boundary value problem on the bounded computational domain, which can be solved by the finite difference method. By this numerical method, we observe the surface hopping phenomena of the two-level Schrödinger equation with conical crossings. Several numerical examples are provided to show the accuracy and convergence of the proposed method.     

Accurate Solutions of the Navier-Stokes Equations Despite Unknown Outflow Boundary Data

A very common procedure when constructing boundary conditions for the time-dependent Navier-Stokes equations at artificial boundaries is to extrapolate the solution from grid points near the boundary to the boundary itself. For supersonic outflow, where all the characteristic variables leave the computational domain, this leads to accurate results. In the case of subsonic outflow, where one characteristic variable enters the computational domain, one cannot in general expect accurate solutions by this procedure. The problem with outflow boundary operators of extrapolation type at artificial boundaries with errors in the boundary data of order one will be investigated. Both the problem when the artificial outflow boundary is located in essentially uniform flow and the situation when the artificial outflow boundary is located in a flow field with large gradients are discussed. It will be shown, that in the special case when there are large gradients tangential to the boundary, extrapolation methods can be used even in the subsonic case.     

Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws

We develop a high order finite difference numerical boundary condition for solving hyperbolic conservation laws on a Cartesian mesh. The challenge results from the wide stencil of the interior high order scheme and the fact that the boundary intersects the grids in an arbitrary fashion. Our method is based on an inverse Lax-Wendroff procedure for the inflow boundary conditions. We repeatedly use the partial differential equation to write the normal derivatives to the inflow boundary in terms of the time derivatives and the tangential derivatives. With these normal derivatives, we can then impose accurate values of ghost points near the boundary by a Taylor expansion. At outflow boundaries, we use Lagrange extrapolation or least squares extrapolation if the solution is smooth, or a weighted essentially non-oscillatory (WENO) type extrapolation if a shock is close to the boundary. Extensive numerical examples are provided to illustrate that our method is high order accurate and has good performance when applied to one and two-dimensional scalar or system cases with the physical boundary not aligned with the grids and with various boundary conditions including the solid wall boundary condition. Additional numerical cost due to our boundary treatment is discussed in some of the examples.     

Implementation Aspects of 3D Lattice-BGK: Boundaries, Accuracy, and a New Fast Relaxation Method

In many realistic fluid-dynamical simulations the specification of the boundary conditions, the error sources, and the number of time steps to reach a steady state are important practical considerations. In this paper we study these issues in the case of the lattice-BGK model. The objective is to present a comprehensive overview of some pitfalls and shortcomings of the lattice-BGK method and to introduce some new ideas useful in practical simulations. We begin with an evaluation of the widely used bounce-back boundary condition in staircase geometries by simulating flow in an inclined tube. It is shown that the bounce-back scheme is first-order accurate in space when the location of the non-slip wall is assumed to be at the boundary nodes. Moreover, for a specific inclination angle of 45 degrees, the scheme is found to be second-order accurate when the location of the non-slip velocity is fitted halfway between the last fluid nodes and the first solid nodes. The error as a function of the relaxation parameter is in that case qualitatively similar to that of flat walls. Next, a comparison of simulations of fluid flow by means of pressure boundaries and by means of body force is presented. A good agreement between these two boundary conditions has been found in the creeping-flow regime. For higher Reynolds numbers differences have been found that are probably caused by problems associated with the pressure boundaries. Furthermore, two widely used 3D models, namelyD₃Q₁₅andD₃Q₁₉, are analysed. It is shown that theD₃Q₁₅model may induce artificial checkerboard invariants due to the connectivity of the lattice. Finally, a new iterative method, which significantly reduces the saturation time, is presented and validated on different benchmark problems.     

基于光滑粒子动力学方法的液滴冲击固壁面问题数值模拟

采用改进的光滑粒子动力学(SPH)方法对液滴冲击固壁面问题进行了数值模拟. 为了提高传统SPH方法的计算精度和数值稳定性, 在传统的SPH方法的基础上对粒子方法中的密度和核梯度进行了修正, 采用了考虑黎曼解法的SPH流体控制方程, 构造了一种新型的粒子间相互作用力(IIF)模型来模拟表面张力的影响. 应用改进的SPH方法对液滴冲击固壁面问题进行了数值模拟. 计算结果表明：新型的IIF 模型能够较好地模拟表面张力的影响, 改进的SPH方法能够精细地描述液滴与固壁面相互作用过程中液滴的内部压力场演变和自由面形态变化, 液滴的铺展因子随初始韦伯数的增大而增大, 数值模拟结果与实验得到的结果基本一致. 关键词： 液滴 固壁面 光滑粒子动力学 表面张力     

Exact boundary conditions for the initial value problem of convex conservation laws

The initial value problem of convex conservation laws, which includes the famous Burgers’ (inviscid) equation, plays an important rule not only in theoretical analysis for conservation laws, but also in numerical computations for various numerical methods. For example, the initial value problem of the Burgers’ equation is one of the most popular benchmarks in testing various numerical methods. But in all the numerical tests the initial data have to be assumed that they are either periodic or having a compact support, so that periodic boundary conditions at the periodic boundaries or two constant boundary conditions at two far apart spatial artificial boundaries can be used in practical computations. In this paper for the initial value problem with any initial data we propose exact boundary conditions at two spatial artificial boundaries, which contain a finite computational domain, by using the Lax’s exact formulas for the convex conservation laws. The well-posedness of the initial-boundary problem is discussed and the finite difference schemes applied to the artificial boundary problems are described. Numerical tests with the proposed artificial boundary conditions are carried out by using the Lax–Friedrichs monotone difference schemes.     

A differentially interpolated direct forcing immersed boundary method for predicting incompressible Navier–Stokes equations in time-varying complex geometries

A dispersion-relation-preserving dual-compact scheme developed in Cartesian grids is applied together with the immersed boundary method to solve the flow equations in irregular and time-varying domains. The artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body. We develop in this study a differential-based interpolation scheme which can be easily extended to three-dimensional simulation. The results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems. The accuracy and fidelity of the IB flow solver developed to predict flows with irregular boundaries are therefore demonstrated.     

Exact boundary condition for time-dependent wave equation based on boundary integral

An exact non-reflecting boundary conditions based on a boundary integral equation or a modified Kirchhoff-type formula is derived for exterior three-dimensional wave equations. The Kirchhoff-type non-reflecting boundary condition is originally proposed by L. Ting and M.J. Miksis [J. Acoust. Soc. Am. 80 (1986) 1825] and numerically tested by D. Givoli and D. Cohen [J. Comput. Phys. 117 (1995) 102] for a spherically symmetric problem. The computational advantage of Ting–Miksis boundary condition is that its temporal non-locality is limited to a fixed amount of past information. However, a long-time instability is exhibited in testing numerical solutions by using a standard non-dissipative finite-difference scheme. The main purpose of this work is to present a new exact boundary condition and to eliminate the long-time instability. The proposed exact boundary condition can be considered as a limit case of Ting–Miksis boundary condition when the two artificial boundaries used in their method approach each other. Our boundary condition is actually a boundary integral equation on a single artificial boundary for wave equations, which is to be solved in conjunction with the interior wave equation. The new boundary condition needs only one artificial boundary, which can be of any shape, i.e., sphere, cubic surface, etc. It keeps all merits of the original Kirchhoff boundary condition such as restricting the temporal non-locality, free of numerical evaluation of any special functions and so on. Numerical approximation to the artificial boundary condition on cubic surface is derived and three-dimensional numerical tests are carried out on the cubic computational domain.     

声测井中的声压-速度有限差分方法

为了解决薄互储层的声测井问题,提出了声压-速度有限差分方法:用声压和速度矢量做为场变量,分别描述井内流体和井外弹性固体或双相介质。这样选择场变量的优点是:处理脉冲点源(或线源)与套网格技术相比简单得多;在内边界上的差分格式稳定,精度得到了改进;人为边界上的吸收效果较好。用柱坐标分别给出了井壁上流体与弹性固体、流体与双相介质的声压-速度边界条件,并用守恒积分方法处理了井壁上的边界条件。通过用声压-速度有限差分方法模拟弹性固体和双相介质地层的声场,证明了声压-速度有限差分方法的有效性。     

SPH方法气-液界面边界条件研究

针对界面附近粒子光滑函数截断和非物理穿透问题,提出一种气-液界面边界条件的处理方法.当界面附近支持域出现不同材料粒子,每步计算可在支持域设置虚粒子,按照密度分配方法给虚粒子物理量赋值,并对界面附近粒子引入气-液两相阻力.采用SPH方法和Level-Set方法,计算运动激波对气-液界面作用问题,两者计算结果一致,初步验证了气-液界面边界条件处理的适用性.用SPH方法分别计算超声速气流中的圆截面液柱绕流和下落问题,界面两侧粒子压力和法向速度连续,给出弓形激波、回流区和下游回流区等定性合理结果.表明本文方法可适度避免界面附近流体粒子光滑截断和粒子非物理穿透现象、界面附近流场数值振荡.     

Exact partition function and boundary state of 2D massive Ising field theory with boundary magnetic field

We compute the exact partition function, the universal ground-state degeneracy and boundary state of the 2D Ising model with boundary magnetic field at off-critical temperatures. The model has a domain that exhibits states localized near the boundaries. We study this domain of boundary bound state and derive exact expressions for the “g function” and boundary state for all temperatures and boundary magnetic fields. In the massless limit we recover the boundary renormalization group flow between the conformally invariant free and fixed boundary conditions.     

Open and traction boundary conditions for the incompressible Navier–Stokes equations

We present numerical schemes for the incompressible Navier–Stokes equations (NSE) with open and traction boundary conditions. We use pressure Poisson equation (PPE) formulation and propose new boundary conditions for the pressure on the open or traction boundaries. After replacing the divergence free constraint by this pressure Poisson equation, we obtain an unconstrained NSE. For Stokes equation with open boundary condition on a simple domain, we prove unconditional stability of a first order semi-implicit scheme where the pressure is treated explicitly and hence is decoupled from the computation of velocity. Using either boundary condition, the schemes for the full NSE that treat both convection and pressure terms explicitly work well with various spatial discretizations including spectral collocation and C⁰ finite elements. Moreover, when Reynolds number is of O(1) and when the first order semi-implicit time stepping is used, time step size of O(1) is allowed in benchmark computations for the full NSE. Besides standard stability and accuracy check, various numerical results including flow over a backward facing step, flow past a cylinder and flow in a bifurcated tube are reported. Numerically we have observed that using PPE formulation enables us to use the velocity/pressure pairs that do not satisfy the standard inf–sup compatibility condition. Our results extend that of Johnston and Liu [H. Johnston, J.-G. Liu, Accurate, stable and efficient Navier–Stokes solvers based on explicit treatment of the pressure term. J. Comp. Phys. 199 (1) (2004) 221–259] which deals with no-slip boundary conditions only.     

Excess Volume of the Solid/Liquid Interface in Fe-6 at.%Si Bicrystals Wetted by Liquid Zinc

The pressure effect on grain boundary wetting in Fe-6 at.%Si bicrystals of different misorientation angles but constant misorientation axis has been studied. The wetting agent was liquid zinc. It was found that the pressure for the dewetting transition is higher for the near 5 boundary than for the other general boundaries, where is the inverse density of the coincidence sites in the two misoriented crystal lattices. This result was explained assuming a thinner liquid film wetting the near 5 boundary than in the case of nonperiodic grain boundaries. Furthermore, the wetting angle increased with increasing pressure. The wetting angle dependence on pressure could be understood assuming a excess surface volume of the solid/liquid (S/L) interface higher than 0.2 nm. This is considerably higher than the estimated excess volumes of grain boundaries based on computer simulations. To explain this result, it was postulated that in the system studied, where diffusion of Zn, Fe and Si perpendicular to the S/L interface takes place, the S/L interface is relatively thick and the interaction between the two crystals separated by the melt extends over more than 2 nm distance. This long-range interaction was rationalized in terms of clusters of several atoms, detaching from the solid and dissolving in the melt at some distance from the bulk.     

Rapid thermal relaxation in near-critical fluids and critical speeding up: discrepancies caused by boundary effects

We present one-dimensional numerical simulations reporting the temperature evolution of a pure fluid subjected to heating near its liquid-vapor critical point under weightlessness. In this model, thermal boundary conditions are imposed at the outer edges of the solids in contact with the fluid. Our investigations concern the thermal conditions at the edges of the fluid and their consequences on the fluid's global response. The results for piston effect heating are shown to be significantly affected by the simulation of the solid boundaries. Concerning critical speeding up, it is even found that taking conductive solids into account can make the bulk fluid temperature change in a way opposed to that predicted in their absence.     

基于SPH方法的瞬态粘弹性流体的数值模拟

运用SPH(Smoothed Particle Hydrodynamics)方法模拟基于Oldroyd-B模型的平面突然起动Couette流,通过数值解与解析解的比较,验证SPH方法模拟瞬态粘弹性流动的准确性;且对基于Oldroyd-B模型的方腔驱动流进行SPH模拟.采用一种新的固壁边界处理方法,有效地防止了粒子穿透,提高数值计算的准确性.用数值算例验证SPH方法对粘弹性流体模拟的有效性和稳定性.     

Implementation of nonreflecting boundary conditions for the nonlinear Euler equations

Computationally efficient nonreflecting boundary conditions are derived for the Euler equations with acoustic, entropic and vortical inflow disturbances. The formulation linearizes the Euler equations near the inlet/outlet boundaries and expands the solution in terms of Fourier–Bessel modes. This leads to an ‘exact’ nonreflecting boundary condition, local in space but nonlocal in time, for each Fourier–Bessel mode of the perturbation pressure. The perturbation velocity and density are then calculated using acoustic, entropic and vortical mode splitting. Extension of the boundary conditions to nonuniform swirling flows is presented for the narrow annulus limit which is relevant to many aeroacoustic problems. The boundary conditions are implemented for the nonlinear Euler equations which are solved in space using the finite volume approximation and integrated in time using a MacCormack scheme. Two test problems are carried out: propagation of acoustic waves in an annular duct and the scattering of a vortical wave by a cascade. Comparison between the present exact conditions and commonly used approximate local boundary conditions is made. Results show that, unlike the local boundary conditions whose accuracy depends on the group velocity of the scattered waves, the present conditions give accurate solutions for a range of problems that have a wide array of group velocities. Results also show that this approach leads to a significant savings in computational time and memory by obviating the need to store the pressure field and calculate the nonlocal convolution integral at each point in the inlet and exit boundaries.     

Extension of the spectral volume method to high-order boundary representation

In this paper, the spectral volume method is extended to the two-dimensional Euler equations with curved boundaries. It is well-known that high-order methods can achieve higher accuracy on coarser meshes than low-order methods. In order to realize the advantage of the high-order spectral volume method over the low order finite volume method, it is critical that solid wall boundaries be represented with high-order polynomials compatible with the order of the interpolation for the state variables. Otherwise, numerical errors generated by the low-order boundary representation may overwhelm any potential accuracy gains offered by high-order methods. Therefore, more general types of spectral volumes (or elements) with curved edges are used near solid walls to approximate the boundaries with high fidelity. The importance of this high-order boundary representation is demonstrated with several well-know inviscid flow test cases, and through comparisons with a second-order finite volume method.