Investigation of use of reach-back characteristics method for 2D dispersion equation

The use of the Holly-Preissmann two-point scheme has been very popular for the calculation of the dispersion equation. The key to this scheme is to use the characteristics method incorporating the Hermite cubic interpolation technique to approximate the trajectory foot of the characteristics. This method can avoid the excessive numerical damping and oscillation associated with most finite difference schemes for advection computation. On the basis of the fundamental idea of the Holly-Preissmann two-point scheme, a new technique is introduced herein for the computation of the two-dimensional dispersion equation. This new scheme allows the characteristics projecting back several time steps to fall on the spatial or temporal axis, while the characteristics foot is still solved by the Holly-Preissmann two-point method. The diffusion portion of the dispersion equation is solved by the commonly used Crank-Nicholson method. The calculation for these two processes consisting of advection and diffusion is carried out separately but consecutively in one time step, a method known as the split operator algorithm. A hypothetical model was constructed to demonstrate the applicability of this new technique for the calculation of the pure advection and dispersion equation in two dimensions.     

时间发展平面混合流的三维演化

采用高精度差分方法和群速度控制方法，求解三维可压缩Ｎ Ｓ方程，直接数值模拟了时间发展的平面混合流．研究了平面混合流三维拟序结构的形成及发展．给出了流动失稳后涡的卷起，相邻两涡的对并，激波的形成及发展．指出，涡对并所诱导产生的激波对三维拟序结构的形成及发展过程是重要的．     

自适应无网格法在生物涂层接触问题中的应用

自适应无网格法是针对有限元法无法求解或者不易求解的复杂问题,利用无网格法节点排布灵活、易于增删节点、便于自适应分析等优点发展起来的. 在对自适应无网格法理论基础和发展进行总结基础上,采用基于应变能梯度的自适应无网格—— 有限元耦合方法,对等离子喷涂制备的HA 生物涂层材料的无摩擦接触问题进行分析,对制备的两种不同厚度的生物涂层材料进行求解,分别给出了von Mises 应力分布云图. 结果表明,自适应无网格法能较好地应用于生物涂层接触问题中.     

An operator splitting algorithm for the three-dimensional advection–diffusion equation

Operator splitting algorithms are frequently used for solving the advection–diffusion equation, especially to deal with advection dominated transport problems. In this paper an operator splitting algorithm for the three-dimensional advection–diffusion equation is presented. The algorithm represents a second-order-accurate adaptation of the Holly and Preissmann scheme for three-dimensional problems. The governing equation is split into an advection equation and a diffusion equation, and they are solved by a backward method of characteristics and a finite element method, respectively. The Hermite interpolation function is used for interpolation of concentration in the advection step. The spatial gradients of concentration in the Hermite interpolation are obtained by solving equations for concentration gradients in the advection step. To make the composite algorithm efficient, only three equations for first-order concentration derivatives are solved in the diffusion step of computation. The higher-order spatial concentration gradients, necessary to advance the solution in a computational cycle, are obtained by numerical differentiations based on the available information. The simulation characteristics and accuracy of the proposed algorithm are demonstrated by several advection dominated transport problems. © 1998 John Wiley & Sons, Ltd.     

A Fully Implicit, Compact Finite Difference Method for the Numerical Solution of Unsteady Laminar Flames

Forced laminar diffusion flames form an important class of problems that can help to bridge the significant gap between steady laminar flames in simple burner configurations and the turbulent flames found in many practical combustors. Such flames offer a much wider range of interactions between convection, diffusion, and chemical reaction than can be examined under steady-state conditions, and yet detailed simulations of them should be feasible without having to resort to “modeling” any of the relevant physics, above all without having prematurely to reduce the large kinetic mechanisms typical of hydrocarbon fuels. Nevertheless, the computation of time-dependent laminar diffusion flames with conventional numerical methods is hindered by technical challenges that, while not new, are more troublesome to surmount than in the calculation of otherwise similar, unforced flames. First, the intricate spatiotemporal coupling between fluid dynamics and combustion thermochemistry ensures that spurious numerical diffusion or spatial under-resolution of the mixing process at any stage of the computation can lead to inaccurate prediction of flame characteristics for the remainder thereof. Second, relatively long simulated flow times and extremely short chemical time scales make many standard time integration algorithms impractical on all but the largest parallel computer clusters. This paper introduces a new numerical approach for time-varying laminar flames that addresses these challenges through the use of high order compact finite difference schemes within a robust, fully implicit solver based on a Jacobian-free Newton–Krylov method. The capabilities of this implicit-compact solver are demonstrated on a periodically forced axisymmetric laminar jet diffusion flame with one-step Arrhenius chemistry, and the results are compared to those of a conventional low order finite difference solver.     

结构非平稳随机响应分析的快速虚拟激励法

虚拟激励法能够方便地应用于结构非平稳随机响应分析,但在每个离散频点处都涉及到虚拟激励作用下动力方程的时程积分,对于大型复杂结构,其计算量是难以接受的。将结构动力方程写成状态方程形式,采用精细积分法对状态方程进行数值求解,导出了结构动力响应关于离散时刻处激励的显式线性表达式。利用这一显式表达式,只需要变换离散时刻处的激励数值,就可以方便快捷地求出新的激励作用下的结构动力响应。效率分析和数值算例表明,相对于传统虚拟激励法,本文提出的改进算法在求解非平稳激励下结构随机振动方面具有更高的计算效率。     

Time‐accurate stabilized finite‐element model for weakly nonlinear and weakly dispersive water waves

Introduction of a time‐accurate stabilized finite‐element approximation for the numerical investigation of weakly nonlinear and weakly dispersive water waves is presented in this paper. To make the time approximation match the order of accuracy of the spatial representation of the linear triangular elements by the Galerkin finite‐element method, the fourth‐order time integration of implicit multistage Padé method is used for the development of the numerical scheme. The streamline‐upwind Petrov–Galerkin (SUPG) method with crosswind diffusion is employed to stabilize the scheme and suppress the spurious oscillations, usually common in the numerical computation of convection‐dominated flow problems. The performance of numerical stabilization and accuracy is addressed. Treatments of various boundary conditions, including the open boundary conditions, the perfect reflecting boundary conditions along boundaries with irregular geometry, are also described. Numerical results showing the comparisons with analytical solutions, experimental measurements, and other published numerical results are presented and discussed. Copyright © 2007 John Wiley & Sons, Ltd.     

A hybrid Padé ADI scheme of higher‐order for convection–diffusion problems

A high‐order Padé alternating direction implicit (ADI) scheme is proposed for solving unsteady convection–diffusion problems. The scheme employs standard high‐order Padé approximations for spatial first and second derivatives in the convection‐diffusion equation. Linear multistep (LM) methods combined with the approximate factorization introduced by Beam and Warming (J. Comput. Phys. 1976; 22 : 87–110) are applied for the time integration. The approximate factorization imposes a second‐order temporal accuracy limitation on the ADI scheme independent of the accuracy of the LM method chosen for the time integration. To achieve a higher‐order temporal accuracy, we introduce a correction term that reduces the splitting error. The resulting scheme is carried out by repeatedly solving a series of pentadiagonal linear systems producing a computationally cost effective solver. The effects of the approximate factorization and the correction term on the stability of the scheme are examined. A modified wave number analysis is performed to examine the dispersive and dissipative properties of the scheme. In contrast to the HOC‐based schemes in which the phase and amplitude characteristics of a solution are altered by the variation of cell Reynolds number, the present scheme retains the characteristics of the modified wave numbers for spatial derivatives regardless of the magnitude of cell Reynolds number. The superiority of the proposed scheme compared with other high‐order ADI schemes for solving unsteady convection‐diffusion problems is discussed. A comparison of different time discretizations based on LM methods is given. Copyright © 2009 John Wiley & Sons, Ltd.     

A multigrid method for predicting periodically fully developed flow

The use of multigrid methods in complex fluid flow problems is still under development. In this paper a full multigrid procedure has been incorporated in a finite volume solution for predicting fully developed fluid flow in a streamwise periodic geometry. Steady computations in two-dimensional body fitted co-ordinates have shown considerable savings in computation time by this multigrid method.     

Some further results of the boundary element method for plate bending

Some further results of the boundary element method for the Kirchhoff type plate bending problems are given. The direct boundary integral equation-boundary element scheme with higher conforming properties is used for several computation examples. The results of computation show that the numerical scheme seems to be more economical in computer time and with better accuracy in comparison with some previous results.     

A high convergent precision exact analytic method for differential equation with variable coefficients

The exact analytic method was given by[1].It can be used for arbitrary variable coefficient differential equations and the solution obtained can have the second order convergent precision.In this paper,a new high precision algorithm is given based on[1],through a bending problem of variable cross-section beams.It can have the fourth convergent precision without increasing computation work.The present computation method is not only simple but also fast.The numerical examples are given at the end of this paper which indicate that the high convergent precision can be obtained using only a few elements.The correctness of the theory in this paper is confirmed.     

A non‐iterative implicit algorithm for the solution of advection–diffusion equation on a sphere

A numerical algorithm for the solution of advection–diffusion equation on the surface of a sphere is suggested. The velocity field on a sphere is assumed to be known and non‐divergent. The discretization of advection–diffusion equation in space is carried out with the help of the finite volume method, and the Gauss theorem is applied to each grid cell. For the discretization in time, the symmetrized double‐cycle componentwise splitting method and the Crank–Nicolson scheme are used. The numerical scheme is of second order approximation in space and time, correctly describes the balance of mass of substance in the forced and dissipative discrete system and is unconditionally stable. In the absence of external forcing and dissipation, the total mass and L₂‐norm of solution of discrete system is conserved in time. The one‐dimensional periodic problems arising at splitting in the longitudinal direction are solved with Sherman–Morrison's formula and Thomas's algorithm. The one‐dimensional problems arising at splitting in the latitudinal direction are solved by the bordering method that requires a prior determination of the solution at the poles. The resulting linear systems have tridiagonal matrices and are solved by Thomas's algorithm. The suggested method is direct (without iterations) and rapid in realization. It can also be applied to linear and nonlinear diffusion problems, some elliptic problems and adjoint advection–diffusion problems on a sphere. Copyright © 2015 John Wiley & Sons, Ltd.     

Large eddy simulation (2D) using diffusion–velocity method and vortex‐in‐cell

A large eddy Simulation based on the diffusion‐velocity method and the discrete vortex method is presented. The vorticity‐based and eddy viscosity type subgrid scale model simulating the enstrophy transfer between the large and small scale appears as a convective term in the diffusion‐velocity formulation. The methodology has been tested on a spatially growing mixing layer using the two‐dimensional vortex‐in‐cell method and the Smagorinsky subgrid scale model. The effects on the vorticity contours, momemtum thickness, mean streamwise velocity profiles, root‐mean‐square velocity and vorticity fluctuations and negative cross‐stream correlation are discussed. Comparison is made with experiment and numerical work where diffusion is simulated using random walk. Copyright © 2004 John Wiley & Sons, Ltd.     

Simulations of fibre orientation in dilute suspensions with front moving in the filling process of a rectangular channel using level-set method

The simulation of fibre orientation in dilute suspension with front moving is carried out using the projection and level-set methods. The motion of fibres is described using the Jeffery equation, and the contribution of fibres to the flow is accounted for by the configuration-field method. The dilute suspension of short fibres in Newtonian fluids is considered. The governing Navier–Stokes equation for the fluid flow is solved using the projection method with finite difference scheme, while the fibre-related equations are directly solved with the Runge–Kutta method. In the present study for fibres in dilute suspension flow for injection molding, the effects of various flow and material parameters on the fibre orientation, the velocity distributions and the shapes of the leading flow front are found and discussed. Our findings indicate that the presence of fibre motion has little influence on the front shape in the ranges of fibre parameters studied at the fixed Reynolds number. Influence of changing fibre parameters only causes variation of front shape in the region near the wall, and the front shape in the central core area does not vary much with the fibre parameters. On the other hand, the fibre motion has strong influence on the distributions of the streamwise and transverse velocities in the fountain flow. Fibre motion produces strong normal stress near the wall which leads to the reduction of transversal velocity as compared to the Newtonian flow without fibres, which in turn, leads to the increased streamwise velocity near the wall. Thus, the fibre addition to the flow weakens the strength of the fountain flow. The Reynolds number has also displayed significant influence on the distribution of the streamwise velocity behind the flow front for a given fibre concentration. It is also found that the fibre orientation is not always along the direction of the velocity vector in the process of mold filling. In the region of the fountain flow, the fibre near the centreline is more oriented across the streamwise direction compared to that in the region far behind the flow front. This leads to the fact that the fibre near the centreline in the region of fountain flow is more extended along the transverse direction. As the fibre orientation in the suspension flow and the shape of the flow front have great bearing on the quality of the product made from injection molding, this study has much implications for engineering applications. These results can also be useful in other fields dealing with fibre suspensions.     

Transport and deposition of neutral particles in magnetohydrodynamic turbulent channel flows at low magnetic Reynolds numbers

The effect of Lorentz force on particle transport and deposition is studied by using direct numerical simulation of turbulent channel flow of electrically conducting fluids combined with discrete particle simulation of the trajectories of uncharged, spherical particles. The magnetohydrodynamic equations for fluid flows at low magnetic Reynolds numbers are adopted. The particle motion is determined by the drag, added mass, and pressure gradient forces. Results are obtained for flows with particle ensembles of various densities and diameters in the presence of streamwise, wall-normal or spanwise magnetic fields. It is found that the particle dispersion in the wall-normal and spanwise directions is decreased due to the changes of the underlying fluid turbulence by the Lorentz force, while it is increased in the streamwise direction. The particle accumulation in the near-wall region is diminished in the magnetohydrodynamic flows. In addition, the tendency of small inertia particles to concentrate preferentially in the low-speed streaks near the walls is strengthened with increasing Hartmann number. The particle transport by turbophoretic drift and turbulent diffusion is damped by the magnetic field and, consequently, particle deposition is reduced.     

基于算法复杂度理论的拟力法计算效率评价

算法复杂度理论是一种算法效率定量评价方法,该方法通过度量算法的复杂度来客观反映其执行效率,避免了计算机系统性能等因素对效率评价的影响。拟力法是一种高效的结构非线性分析方法,以往研究工作只是在非线性分析过程和运行时间上对计算效率进行了探索,并没有从理论上量化分析。本文采用算法复杂度理论对拟力法和传统变刚度非线性求解方法进行分析,给出了两种方法的时间复杂度函数,并对比了其计算效率进行定量,从数学角度解释了拟力法计算效率高的根本原因。算例对两种方法的时间复杂度和计算时间进行对比分析,直观地说明了拟力法在计算效率方面的优越性。     

谱元方法求解环形空间内自然对流换热

提出了将谱元方法应用到极坐标系下，利用极坐标系下的谱元方法求解环形空间内自然对流问题。具体求解了原始变量速度和压力的不可压缩Navier-Stokes方程和能量方程，通过在时间方向采用时间分裂方法和空间采用谱元方法对方程进行离散求解，取得了与基准解较一致的计算结果。     

弹塑性力学问题的神经网络计算模型

为了提高弹塑性力学问题的计算速度,给出了其新的势能变分原理形式,利用神经网络的并行、分布处理的特点进行寻优,并给出了相应网络的结构形式和网络参数。     

Using digital holographic microscopy for simultaneous measurements of 3D near wall velocity and wall shear stress in a turbulent boundary layer

A digital holographic microscope is used to simultaneously measure the instantaneous 3D flow structure in the inner part of a turbulent boundary layer over a smooth wall, and the spatial distribution of wall shear stresses. The measurements are performed in a fully developed turbulent channel flow within square duct, at a moderately high Reynolds number. The sample volume size is 90 × 145 × 90 wall units, and the spatial resolution of the measurements is 3–8 wall units in streamwise and spanwise directions and one wall unit in the wall-normal direction. The paper describes the data acquisition and analysis procedures, including the particle tracking method and associated method for matching of particle pairs. The uncertainty in velocity is estimated to be better than 1 mm/s, less than 0.05% of the free stream velocity, by comparing the statistics of the normalized velocity divergence to divergence obtained by randomly adding an error of 1 mm/s to the data. Spatial distributions of wall shear stresses are approximated with the least square fit of velocity measurements in the viscous sublayer. Mean flow profiles and statistics of velocity fluctuations agree very well with expectations. Joint probability density distributions of instantaneous spanwise and streamwise wall shear stresses demonstrate the significance of near-wall coherent structures. The near wall 3D flow structures are classified into three groups, the first containing a pair of counter-rotating, quasi streamwise vortices and high streak-like shear stresses; the second group is characterized by multiple streamwise vortices and little variations in wall stress; and the third group has no buffer layer structures.     

Numerical simulations of high-speed chemically reacting flow

The essentially nonoscillatory (ENO) shock-capturing scheme for the solution of hyperbolic equations is extended to solve a system of coupled conservation equations governing two-dimensional, time-dependent, compressible chemically reacing flow with full chemistry. The thermodynamic properties of the mixture are modeled accurately, and stiff kinetic terms are separated from the fluid motion by a fractional step algorithm. The methodology is used to study the concept of shock-induced mixing and combustion, a process by which the interaction of a shock wave with a jet of low-density hydrogen fuel enhances mixing through streamwise vorticity generation. Test cases with and without chemical reaction are explored here. Our results indicate that, in the temperature range examined, vorticity generation as well as the distribution of atomic species do not change significantly with the introduction of a chemical reaction and subsequent heat release. The actual diffusion of hydrogen is also relatively unaffected by the reaction process. This suggests that the fluid mechanics of this problem may be successfully decoupled from the combustion processes, and that computation of the mixing problem (without combustion chemistry) can elucidate much of the important physical features of the flow.This work has been supported by the Aerospace Corporation through a Corporate Fellowship, by NASA Dryden Flight Research Center under Grant NCC 2-374, by ONR Grant N00014-86-K-0691, by NSF Grant DMS 88-11863, and by a DARPA Grant in the ACMP Program.