超弹性柔性结构与流体耦合运动的浸入边界法研究

浸入边界法是模拟大变形柔性弹性结构和粘性流体相互作用的重要数值方法之一。该文有效结合传统的反馈力方法和混合有限元浸入边界方法,对圆柱和方柱绕流后柔性悬臂梁流固耦合振动问题进行数值模拟。其中,固体采用超弹性材料,利用有限单元法求解,流体为不可压缩牛顿流体,使用笛卡尔自适应加密网格,利用有限差分法进行求解。通过数值计算,得到柔性超弹性结构的耦合振动特性和流场动态分布特性,并将计算结果同其他文献计算结果进行比较,验证了该耦合计算方法的可靠性。     

基于投影法求解不可压缩流的高精度紧致格式

本文建立了一种基于投影法的求解不可压缩Navier-Stokes(N-S)方程的高精度紧致差分格式。该方法时间上采用Kim和Moin二阶投影法离散,空间上采用高精度紧致格式离散,并提出了一种新的离散压力边界的紧致格式,同时对计算结果进行分析以验证该投影法的精度和格式稳定性。文中Taylor涡列数值计算结果表明,Kim和Moin投影法能使得压力场和速度场均达到时间二阶精度,且高精度紧致格式投影法也具有空间高阶精度。驱动方腔数值模拟结果显示,本文对N-S方程的离散格式具有很好的可靠性,适用于对复杂流体流动的小尺度问题的数值模拟和研究。     

基于动力特征分区的显式-精细混合积分法

大型动力系统中常因局部的高频振动及非线性等特性限制了系统的积分步长而导致整体计算量激增，针对此问题提出一种分区域异步长显式-精细混合积分方法。在特性复杂的局部区域采取显式积分法，根据精度和稳定性要求取较小的时间步长求解；在其余常规区域则应用精细积分方法，采取可以跨越显式积分区周期的大积分步长求解。对于精细积分区域边界荷载，提出一种基于离散FFT变换的线性项与主频谐波项的组合表示方法，并给出了此种荷载形式下的精细积分计算格式。数值算例结果表明该法能够明显提高计算效率，在显式积分区域和精细积分区域都有很高的精度。     

氧碘化学激光器组分超音速混合反应数值模拟

采用数值方法求解二维可压缩非定常N-S方程及组分连续方程,计算氧碘化学激光器中的组分超音速混合反应流场.空间离散格式为AUSM -up格式,用四步龙格-库塔方法作显式时间推进,湍流粘性系数使用k-ε两方程湍流模型进行求解.假设混合气体为热力学完全而热值非完全气体,化学反应模型采用有限速率反应模型.使用一种松弛迭代的方法来处理化学源项的刚性问题.给出了出口处碘分子的质量分数分布,数值模拟结果与文献结果符合良好.     

复式防波堤断面尺度对波浪爬高的影响研究

数值模拟是求解防波堤爬高问题的重要方法。以Navier-Stokes方程为控制方程,该文建立了基于紧致插值曲线法(CIP法)的二维不可压缩流体的有限差分数值模型。通过分步算法对时间积分求解控制域,采用双曲正切函数自由面捕捉法(THINC法)对自由液面进行捕捉,采用浸入边界法IBM对固体边界进行处理,将波浪爬高问题视为固-液-气多相流问题。通过建立二维数值水槽,对波浪爬高越浪问题进行数值验证;进而在相同波浪条件下,对不同断面尺寸的复式防波堤进行波浪爬高的数值模拟。结果表明:该模型可以较好地模拟波浪过程中的变形、翻卷、破碎等强非线性现象,得出最大爬高与复式防波堤尺寸之间的关系,从而给出了最小爬高时对应的防波堤断面尺度。     

基于Leonard规正变量的修正高分辨率组合格式

通过对规正变量进行重构,本文提出了求解对流扩散方程的修正高分辨率组合格式,它能够求解边界层和大梯度等问题.首先,根据规正变量的定义得出了组合格式的通用表达式,然后对时间项采用二阶中心差分格式,得到了对流扩散方程的离散表达式,对离散化得到的代数方程组采用TDMA算法求解,并推导出了组合格式计算过程迭代收敛时所满足的充分条件.数值实验表明:新格式具有分辨率高,数值耗散较低,总偏差量较小,能很好模拟场变量的大梯度变化,计算结果优于传统格式.     

求解Klein-Gordon-Zakharov方程的一个线性化差分格式(英文)

本文针对Klein-Gordon-Zakharov方程运用Crank-Nicolson格式和蛙跳格式的构造方法分别对线性项和非线性项进行离散,得到一个新的半显式有限差分格式.该格式在实际计算中是线性化解耦的,即在具体计算中的每一时间步,只需求解两个独立的三对角线性代数方程组,从而可以大幅提高计算效率.由于难以得到数值解的最大模先验估计,本文引入数学归纳方法并利标准的能量方法和不动点定理得到了数值解的存在唯一性,并证明了格式在时间和空间两个方向的二阶收敛性.数值结果验证了格式的精度和稳定性.     

求解粘弹性问题的时域自适应等几何比例边界有限元法

提出一种基于分段时域自适应算法和等几何分析的求解粘弹性问题的数值方法。利用时域分段展开,建立了递推格式的比例边界元求解方程,环向比例边界采用等几何技术离散,在继承常规比例边界有限元半解析、便于处理应力奇异性/无限域问题等优点的同时,可更准确地描述几何边界,由此进一步提高了计算精度;在时域,通过分段时域自适应计算,保证不同时间步长下的计算精度。通过数值算例,从计算精度、收敛性等方面,对所提方法的有效性进行了验证。     

基于通量非结构网格有限体积法的Level Set方程求解

在非结构网格上,采用基于通量的格心有限体积法求解Level Set方程,并给出了其数值离散格式.对Zalesak圆盘、旋转收缩圆及圆面剪切三个经典界面运动问题进行了数值模拟,其质量盈亏量与收敛阶的计算结果表明,该方法克服了传统数值方法造成的质量不守恒的缺点,能准确有效地求解拓扑结构发生变化的复杂运动界面问题,且计算量较格点格式大为减少.     

缓增分数阶扩散方程的高阶时间离散LDG方法

研究了缓增分数阶扩散方程的高阶时间离散局部间断Galerkin (Local Discontinuous Galerkin, LDG)方法，不是直接求解缓增分数阶扩散方程，而是首先通过变换将其转化成Caputo型时间分数阶扩散方程。接着，采用L1-2差分逼近离散Caputo型分数阶导数，间断有限元离散空间变量，构造求解模型的全离散LDG格式。证明了所建立的全离散格式为无条件稳定的且具有最优误差阶，两个数值算了验证了所建立数值格式的精度和鲁棒性。数值实验结果表明所建立格式在时间和空间方向均具有高精度。     

Solution of three-dimensional viscous flows using integral velocity–vorticity formulation

In this paper, a numerical approach is presented to solve the velocity–vorticity integro-differential formulations for three-dimensional incompressible viscous flow. Both the velocity and pressure are solved in integral formulations and the general numerical method is based on standard finite volume scheme. The velocities needed at the vertexes of each control volume are calculated by a so-called generalized Biot–Savart formula combined with a fast multipole algorithm, which makes the velocity boundary conditions implicitly satisfied by maintaining the kinematic compatibility of the velocity and vorticity fields. The well-known fractional step approaches are used to solve the vorticity transport equation. No-flux boundary conditions on solid objects are satisfied as vorticity Helmholtz equation is solved. The diffusion term in the transport equation is treated implicitly using a conservative finite update. The diffusive fluxes of vorticity into flow domain from solid boundaries are determined by an iterative process in order to satisfy the no tangential-flow boundary condition. As an application example, the impulsively started flow through a sphere with different Reynolds numbers is computed using the method. The calculated results are compared with the experimental data and other numerical results and show good agreement.     

An Unconditionally Energy Stable Immersed Boundary Method with Application to Vesicle Dynamics

We develop an unconditionally energy stable immersed boundary method, and apply it to simulate 2D vesicle dynamics. We adopt a semi-implicit boundary forcing approach, where the stretching factor used in the forcing term can be computed from the derived evolutional equation. By using the projection method to solve the fluid equations, the pressure is decoupled and we have a symmetric positive definite system that can be solved efficiently. The method can be shown to be unconditionally stable, in the sense that the total energy is decreasing. A resulting modification benefits from this improved numerical stability, as the time step size can be significantly increased (the severe time step restriction in an explicit boundary forcing scheme is avoided). As an application, we use our scheme to simulate vesicle dynamics in Navier-Stokes flow.     

A Direct Forcing Immersed Boundary Method Based Lattice Boltzmann Method to Simulate Flows with Complex Geometry

In the present study, a lattice Boltzmann method based new immersed boundary technique is proposed for simulating two-dimensional viscous incompressible flows interacting with stationary and moving solid boundaries. The lattice Boltzmann method with known force field is used to simulate the flow where the complex geometry is immersed inside the computational domain. This is achieved via direct-momentum forcing on a Cartesian grid by combining "solid-body forcing" at solid nodes and interpolation on neighboring fluid nodes. The proposed method is examined by simulating decaying vortex, 2D flow over an asymmetrically placed cylinder, and in-line oscillating cylinder in a fluid at rest. Numerical simulations indicate that this method is second order accurate, and all the numerical results are compatible with the benchmark solutions.     

求解非定常N-S方程的稳定化分数步长法研究

本文研究了求解非定常Navier-Stokes方程的稳定化分数步长法.首先,通过一阶精度的算子分裂,将非线性项和不可压缩条件分裂到两个不同的子问题中,并对非线性项采用Oseen迭代.格式分为两步:第一步求解一个线性椭圆问题;第二步求解一个广义的Stokes问题.这两个子问题关于速度都满足齐次Dilichlet边界条件.同时,在格式的第二步添加了局部稳定化项,使用等阶序对来加强数值解的稳定性.通过能量估计方法,对速度与压力做了收敛性分析和误差估计.最后,数值实验验证了方法的有效性.     

A decoupled augmented IIM strategy for incompressible two‐phase flows with interfaces on irregular domains

A decoupled augmented immersed interface method for solving incompressible two‐phase flows involving both irregular domains and interfaces is presented. In order to impose the prescribed velocity at the boundary of the irregular domain, singular force as one set of augmented variables is introduced. The velocity components at the two‐fluid interface as another set of augmented variables are introduced to satisfy the continuity condition of the velocity across the interface so that the jump conditions for the velocity and pressure are decoupled across the interface. The augmented variables and/or the forces along the interface/boundary are related to the jumps in both pressure and velocity and the jumps in their derivatives across the interface/boundary and applied to the fluid through jump conditions. The resulting augmented equation is a couple system of these two sets of augmented variables, and the direct application of the GMRES is impractical due to larger iterations. In this work, the novel decoupling of two sets of the augmented variables is proposed, and the decoupled augmented equation is then solved by the LU or the GMRES method. The Stokes equations are discretized via the finite difference method with the incorporation of jump contributions on a staggered Cartesian grid and solved by the conjugate gradient Uzawa‐type method. The numerical results show that second‐order accuracy for the velocity is confirmed. The present method has also been applied to solve for incompressible two‐phase Navier–Stokes flow with interfaces on irregular domains. Copyright © 2011 John Wiley & Sons, Ltd.     

Thin turbomachinery blade design using a finite-volume method

A numerical method is presented for the design of thin turbomachinery blades with specified whirl velocities across the blade span. The numerical scheme involves iteration between the directs solution of a finite-volume method developed earlier on by Soulis¹ and a design solution. The finite-volume method, which is a combination of finite elements and finite differences, solves the three-dimensional, inviscid, steady arid potential flow through turbomachinery blade rows in the incompressible, compressible and transonic flow range. In the design step, the whirl velocity distribution is specified across the blade span (Dirichlet boundary condition). The design procedure yields a new set of co-ordinates for the blade geometry which are used in the next iteration of the direct solution. However, in the present analysis only thin turbomachinery blades are designed, although a fully three-dimensional numerical method is used (the whirl velocity components of the flow field are averaged over the blade suction and pressure surface). The numerical method has been used to design free-vortex thin turbomachinery blades. Results show that the new numerical procedure is a comparatively economic and reliable method for designing thin turbomachinery blades. It may form the baseline for complete three-dimensional turbomachinery blade designs.     

A combined BEM–FEM model for the velocity–vorticity formulation of the Navier–Stokes equations in three dimensions

This paper describes a combined boundary element and finite element model for the solution of velocity–vorticity formulation of the Navier–Stokes equations in three dimensions. In the velocity–vorticity formulation of the Navier–Stokes equations, the Poisson type velocity equations are solved using the boundary element method (BEM) and the vorticity transport equations are solved using the finite element method (FEM) and both are combined to form an iterative scheme. The vorticity boundary conditions for the solution of vorticity transport equations are exactly obtained directly from the BEM solution of the velocity Poisson equations. Here the results of medium Reynolds number of up to 1000, in a typical cubic cavity flow are presented and compared with other numerical models. The combined BEM–FEM model are generally in fairly close agreement with the results of other numerical models, even for a coarse mesh.     

A numerical method for the analysis of flexible bodies in unsteady viscous flows

A two‐dimensional numerical model for unsteady viscous flow around flexible bodies is developed. Bodies are represented by distributed body forces. The body force density is found at every time‐step so as to adjust the velocity within the computational cells occupied by the body to a prescribed value. The method combines certain ideas from the immersed boundary method and the volume of fluid method. The main advantage of this method is that the computations can be effected on a Cartesian grid, without having to fit the grid to the body surface. This is particularly useful in the case of flexible bodies, in which case the surface of the object changes dynamically, and in the case of multiple bodies moving relatively to each other. The capabilities of the model are demonstrated through the study of the flow around a flapping flexible airfoil. The novelty of this method is that the surface of the airfoil is modelled as an active flexible skin that actually drives the flow. The accuracy and fidelity of the model are validated by reproducing well‐established results for vortex shedding from a stationary as well as oscillating rigid cylinder. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical study for the effects of thermophoresis and variable thermal conductivity on heat and mass transfer over an accelerating surface with heat source

A modified numerical solution scheme, for local similarity boundary layer analysis, is used to study the effects of thermophoresis and variable thermal conductivity on heat and mass transfer over an accelerating surface with heat source in the presence of suction and blowing. This numerical scheme is efficient and accurate and it can be programmed and applied easily and its application is illustrated, step by step, by studying the above mentioned problem. The resulting boundary layer equations are solved numerically by Chebyshev finite difference method. Numerical results for the velocity, temperature and concentration as well as for the skin friction, Nusselt and Sherwood numbers are obtained and reported graphically for various parametric conditions to show interesting aspects of the solution.     

A three‐dimensional hybrid meshfree‐Cartesian scheme for fluid–body interaction

A numerical method based on a hybrid meshfree‐Cartesian grid is developed for solving three‐dimensional fluid–solid interaction (FSI) problems involving solid bodies undergoing large motion. The body is discretized and enveloped by a cloud of meshfree nodes. The motion of the body is tracked by convecting the meshfree nodes against a background of Cartesian grid points. Spatial discretization of second‐order accuracy is accomplished by the combination of a generalized finite difference (GFD) method and conventional finite difference (FD) method, which are applied to the meshfree and Cartesian nodes, respectively. Error minimization in GFD is carried out by singular value decomposition. The discretized equations are integrated in time via a second‐order fractional step projection method. A time‐implicit iterative procedure is employed to compute the new/evolving position of the immersed bodies together with the dynamically coupled solution of the flow field. The present method is applied on problems of free falling spheres and tori in quiescent flow and freely rotating spheres in simple shear flow. Good agreement with published results shows the ability of the present hybrid meshfree‐Cartesian grid scheme to achieve good accuracy. An application of the method to the self‐induced propulsion of a deforming fish‐like swimmer further demonstrates the capability and potential of the present approach for solving complex FSI problems in 3D. Copyright © 2011 John Wiley & Sons, Ltd.