Nonlinear Galerkin method for the exterior nonstationary Navier-Stokes equations

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Two-stage fourth-order gas-kinetic scheme for three-dimensional Euler and Navier-Stokes solutions

ABSTRACT

For the one-stage third-order gas-kinetic scheme (GKS), successful applications have been achieved for the three-dimensional compressible flows [Pan, L., K. Xu, Q. Li, and J. Li. 2016. “An Efficient and Accurate Two-stage Fourth-order Gas-kinetic Scheme for the Navier-Stokes Equations.” Journal of Computational Physics 326: 197–221]. The high-order accuracy of the scheme is obtained by integrating a multidimensional time-accurate gas distribution function over the cell interface within a time step without using Gaussian quadrature points and Runge-Kutta time-stepping technique. However, to the further increase of the order of the scheme, such as the fourth-order one, the one step formulation becomes very complicated for the multidimensional flow. Recently, a two-stage fourth-order GKS with high efficiency has been constructed for two-dimensional inviscid and viscous flow computations ([Li, J., and Z. Du. 2016. “A Two-stage Fourth Order Time-accurate Discretization for Lax-Wendroff Type Flow Solvers I. Hyperbolic Conservation Laws.” SIAM Journal on Scientific Computing 38: 3046–3069]; Pan et al. 2016), and the scheme uses the time accurate flux function and its time derivatives. In this paper, a fourth-order GKS is developed for the three-dimensional flows under the two-stage framework. Based on the three-dimensional WENO reconstruction and flux evaluation at Gaussian quadrature points on a cell interface, the high-order accuracy in space is achieved first. Then, the two-stage time stepping method provides the high accuracy in time. In comparison with the formal third-order GKS [Pan, L., and K. Xu. 2015. “A Third-order Gas-kinetic Scheme for Three-dimensional Inviscid and Viscous Flow Computations.” Computers & Fluids 119: 250–260], the current fourth-order method not only improves the accuracy of the scheme, but also reduces the complexity of the gas-kinetic flux solver greatly. More importantly, the fourth-order GKS has the same robustness as the second-order shock capturing scheme [Xu, K. 2001. “A Gas-kinetic BGK Scheme for the Navier-Stokes Equations and its Connection with Artificial Dissipation and Godunov Method.” Journal of Computational Physics 171: 289–335]. Numerical results validate the outstanding reliability and applicability of the scheme for three-dimensional flows, such as the cases related to turbulent simulations.     

Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations

Local and parallel finite element algorithms based on two-grid discretization for Navier-Stokes equations in two dimension are presented. Its basis is a coarse finite element space on the global domain and a fine finite element space on the subdomain. The local algorithm consists of finding a solution for a given nonlinear problem in the coarse finite element space and a solution for a linear problem in the fine finite element space, then droping the coarse solution of the region near the boundary. By overlapping domain decomposition, the parallel algorithms are obtained. This paper analyzes the error of these algorithms and gets some error estimates which are better than those of the standard finite element method. The numerical experiments are given too. By analyzing and comparing these results, it is shown that these algorithms are correct and high efficient.     

Optimal Control in Navier-Stokes Equations

This paper presents a formulation for optimal control of a forced convection flow. The state equation that governs the forced convection flow can be expressed as the incompressible Navier-Stokes equations and energy equation. The optimal control can be formulated as finding a control force to minimize a performance function that is defined to evaluate a control object. The stabilized finite element method is used for the spatial discretization, while the Crank-Nicolson scheme is used for the temporal discretization. The Sakawa-Shindo method, which is an iterative procedure, is applied for minimizing the performance function.     

不可压Navier-Stokes方程基于误差估算的网格自适应解法

首先导出了广义Stokes方程Petrov—Galerkin有限元数值解的当地事后误差估算公式;以非连续二阶鼓包（bump）函数空间为速度、压强误差的近似空间,该估算基于求解当地单元上的广义Stokes问题。然后,证明了误差估算值与精确误差之间的等价性。最后,将误差估算方法应用于Navier—Stokes环境,以进行不可压粘流计算中的网格自适应处理。数值实验中成功地捕获了多强度物理现象,验证了本文所发展的方法。     

A total linearization method for solving viscous free boundary flow problems by the finite element method

In this paper a total linearization method is derived for solving steady viscous free boundary flow problems (including capillary effects) by the finite element method. It is shown that the influence of the geometrical unknown in the totally linearized weak formulation can be expressed in terms of boundary integrals. This means that the implementation of the method is simple. Numerical experiments show that the iterative method gives accurate results and converges very fast.     

Solving 2-D nonlinear coupled heat and moisture transfer problems via a self-adaptive precise algorithm in time domain

Introduction Thestudyonnonlineartransienttransferproblemsissignificantpracticallyand theoretically[1,2].Insolvingtheseproblemsdiscretelyinthetimedomain,eitherbyiterative techniques,orbylinearizingapproachesbasedonsomeadditionalassumptions,the adaptabilityofcomputingaccuracytothechangeofthesizeoftimestepneedtobetakeninto account[3].Yang[3]presentedaprecisealgorithminthetimedomaintosolvetransfer problems,themajoradvantagesofthisalgorithmtosolvenonlinearproblemsisthatno additionalassumptionandite…     

NONLINEAR COMPLEX DYNAMIC PHENOMENA OF THE PERTURBED METALLIC BAR CONSIDERING DISSIPATING EFFECT

IntroductionTotheweaklydamped ,periodicallyforcedsine_Gordonequation ,A .R .Bishop[1～ 3]analyzeditssolutionunderperiodicboundaryconditionandconcludedthatitssolutionwouldshowdifferentspatialstructuresandlong_timeasymptoticstatesalongwiththevariationofpara…     

A Precise Computation of Drag Coefficients of a Sphere

We present a computational method for drag coefficients of axisymmetric bodies. It is a kind of consistent flux method but the introduction of a proper test function enables us to establish an error estimate under some assumption. Applying the method, we obtain drag coefficients of a sphere for Reynolds numbers between 10 and 200, which are found between numerical upper and lower bounds.     

DRIVING FORCE AND DEFORMATION ANALYSIS FOR DYNAMIC CRACK PROPAGATION IN GAS PIPELINES UNDER DIFFERENT BOUNDARY CONDITIONS

In the light of a growing need for fracture control of rapid crack propagation(RCP)ingas pipelines,a program PFRAC(Pipeline FRacture Analysis Code)has been developed to analysethc various events.In this paper,by using PFRAC for the simulation of axial crack propagation in gaspipelines,a number of dynamic analysis issues rclating to boundary effects for uncracked and crackedpipes are investigated.This indicates that the boundary conditions along the length of the pipe play animportant role for fracture analysis in the pipe.In contrast.the boundary condutions at thc ends of along pipeline have little effect on the dynamic fracture events.     

四边形等参单元温度场有限元分析中的混合边界条件

在温度场有限元分析中,边界条件的合理确定是一个非常重要的问题.本文以四边形等参单元为基本单元,采用基于微分方程等效积分原理的Galerkin加权余量法,建立了热传导问题的有限元方程,推导出混合边界条件的有限元计算公式.最后,根据求得的温度场可得截面温度应力分布.     

Approximate nonreflecting inflow-outflow boundary conditions for the calculation of navier-stokes equations in internal flows

In this paper, the nonreflecting boundary conditions based upon fundamental ideas of the linear analysis are developed for gas dynamic equations, and the modified boundary conditions for Navier-Stokes equations are proposed as a substitute of the nonreflecting boundary conditions inside boundary layers near rigid walls. These derived boundary conditions are then applied to calculations both for the Euler equations and the Navier-Stokes equations to determine if they can produce acceptable results for the subsonic flows in channels. The numerical results obtained by an implicit second-order upwind difference scheme show the effectiveness and generality of the boundary conditions. Furthermore, the formulae and the analysis performed here may be extended to three dimensional problems. recommended by Prof. Cui Erjie     

A parallel two-level finite element method for the Navier-Stokes equations

Based on domain decomposition, a parallel two-level finite element method for the stationary Navier-Stokes equations is proposed and analyzed. The basic idea of the method is first to solve the Navier-Stokes equations on a coarse grid, then to solve the resulted residual equations in parallel on a fine grid. This method has low communication complexity. It can be implemented easily. By local a priori error estimate for finite element discretizations, error bounds of the approximate solution are derived. Numerical results are also given to illustrate the high efficiency of the method.     

On the implementation of symmetric and antisymmetric periodic boundary conditions for incompressible flow

In this paper we consider symmetric and antisymmetric periodic boundary conditions for flows governed by the incompressible Navier-Stokes equations. Classical periodic boundary conditions are studied as well as symmetric and antisymmetric periodic boundary conditions in which there is a pressure difference between inlet and outlet. The implementation of this type of boundary conditions in a finite element code using the penalty function formulation is treated and also the implementation in a finite volume code based on pressure correction. The methods are demonstrated by computation of a flow through a staggered tube bundle.     

A finite element method for the one‐dimensional extended Boussinesq equations

A new finite element method for Nwogu's (O. Nwogu, ASCE J. Waterw., Port, Coast., Ocean Eng., 119 , 618–638 (1993)) one‐dimensional extended Boussinesq equations is presented using a linear element spatial discretisation method coupled with a sophisticated adaptive time integration package. The accuracy of the scheme is compared to that of an existing finite difference method (G. Wei and J.T. Kirby, ASCE J. Waterw., Port, Coast., Ocean Eng., 121 , 251–261 (1995)) by considering the truncation error at a node. Numerical tests with solitary and regular waves propagating in variable depth environments are compared with theoretical and experimental data. The accuracy of the results confirms the analytical prediction and shows that the new approach competes well with existing finite difference methods. The finite element formulation is shown to enable the method to be extended to irregular meshes in one dimension and has the potential to allow for extension to the important practical case of unstructured triangular meshes in two dimensions. This latter case is discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

High‐order compact scheme for Boussinesq equations: implementation and numerical boundary condition issue

Application of the three‐point fourth‐order compact scheme to spatial differencing of the vorticity‐stream function‐density formulation of the two‐dimensional incompressible Boussinesq equations is presented. The details for the derivation of difference relations at boundaries to generate accurate and stable solutions are also given. To assess the numerical accuracy, two linear prototype test problems with known exact solution are used. The two‐dimensional planar and cylindrical lock‐exchange flow configurations are used to conduct the numerical experiments for the Boussinesq equations. Quantitative measures for the two linear prototype test problems and comparison of the results of this work with the published results for the planar lock‐exchange flow indicates the validity and accuracy of the three‐point fourth‐order compact scheme for numerical solution of two‐dimensional incompressible Boussinesq equations. In addition, the study of using different high‐order numerical boundary conditions for the implementation of the no‐penetration boundary condition for the density at no‐slip walls is considered. It is shown that the numerical solution is sensitive to the choice of difference relation for the density at boundaries and using an inappropriate difference relation leads to spurious numerical solution. Copyright © 2011 John Wiley & Sons, Ltd.     

An h4 accurate vorticity-velocity formulation for calculating flow past a cylinder

A method of solution for the two-dimensional Navier-Stokes equations for incompressible flow past a cylinder is given in which the euquation of continuity is solved by a step-by-step integration procedure at each stage of an interative process. Thus the formulation involves the solution of one first-order and one second-order equation for the velocity components, together with the vorticity transport equation. the equations are solved numerically by h⁴-accurate methods in the case of steady flow past a circular cylinder in the Reynolds number range 10–100. Results are in satisfactory agreement with recent h⁴-accurate calculations. An improved approximation to the boundary conditions at large distance is also considered.     

A kind of distortion of mean velocity profile in pipe Poiseuille flow and its stability behaviour

This paper presents a kind of distortion of Hagen-Poiseuille velocity profile in pipe Poiseuille flow. This distortion can be regarded as a general expression of the influence on the mean flow by nonlinear interaction of various components of axisymmetric perturbations. Through the investigation of the stability behaviour of this velocity profile, this paper obtains unstable result induced by axisymmetric perturbations for the first time, and thus presents a new possible approach which leads to instability of Hagen-Poiseuille flow.