Nonlinear Galerkin method for the exterior nonstationary Navier-Stokes equations

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Fully discrete Jacobi-spherical harmonic spectral method for Navier-Stokes equations

A fully discrete Jacobi-spherical harmonic spectral method is provided for the Navier-Stokes equations in a ball. Its stability and convergence are proved. Numerical results show efficiency of this approach. The proposed method is also applicable to other problems in spherical geometry.     

Direct spectral domain decomposition method for 2D incompressible Navier-Stokes equations

An efficient direct spectral domain decomposition method is developed coupled with Chebyshev spectral approximation for the solution of 2D, unsteady and incompressible Navier-Stokes equations in complex geometries. In this numerical approach, the spatial domains of interest are decomposed into several non-overlapping rectangular sub-domains. In each sub-domain, an improved projection scheme with second-order accuracy is used to deal with the coupling of velocity and pressure, and the Chebyshev collocation spectral method (CSM) is adopted to execute the spatial discretization. The influence matrix technique is employed to enforce the continuities of both variables and their normal derivatives between the adjacent sub-domains. The imposing of the Neumann boundary conditions to the Poisson equations of pressure and intermediate variable will result in the indeterminate solution. A new strategy of assuming the Dirichlet boundary conditions on interface and using the first-order normal derivatives as transmission conditions to keep the continuities of variables is proposed to overcome this trouble. Three test cases are used to verify the accuracy and efficiency, and the detailed comparison between the numerical results and the available solutions is done. The results indicate that the present method is efficiency, stability, and accuracy.     

用任意不规则网格求解N-S方程

引入辅助点法开发了新的通量近似计算方法,建立了采用任意不规则畸变网格作为控制体积的单元中心有限体积求解Navier—Stokes的方法。它以同位网格作为变量布置方式,压力一速度耦合采用SIMPLE方法。数值算例表明,该算法对高度不规则的畸变网格适应性强;其改进了传统算法在不规则网格下计算的困难,保证了模型在高扭曲度的网格下的整体计算精度不受网格拉伸畸变和剪切畸变的影响。     

A flux-vector splitting method for steady Navier-Stokes equations

The flux-vector splitting method is applied to the convective part of the steady Navier-Stokes equations for incompressible flow. By the use of partial upwind differences in the split first-order part and central differences in the second-order part, a set of discrete equations is obtained which can be solved by vector variants of classical relaxation schemes. It is shown that accurate results can be obtained on one of the GAMM backward-facing step test problems.     

Finite elements for exterior problems using integral equations

We present some integral methods for exterior problems for the Laplace equation. Then we give finite element approximations for these equations and some errors estimates. Finally, we indicate how these integral equations can be coupled with a usual finite element method on a bounded domain to solve an exterior non-linear problem which is linear far away.     

Local projection stabilized finite element method for Navier-Stokes equations

This paper extends the results of Matthies, Skrzypacz, and Tubiska for the Oseen problem to the Navier-Stokes problem. For the stationary incompressible Navier- Stokes equations, a local projection stabilized finite element scheme is proposed. The scheme overcomes convection domination and improves the restrictive inf-sup condition. It not only is a two-level approach but also is adaptive for pairs of spaces defined on the same mesh. Using the approximation and projection spaces defined on the same mesh, the scheme leads to much more compact stencils than other two-level approaches. On the same mesh, besides the class of local projection stabilization by enriching the approximation spaces, two new classes of local projection stabilization of the approximation spaces are derived, which do not need to be enriched by bubble functions. Based on a special interpolation, the stability and optimal prior error estimates are shown. Numerical results agree with some benchmark solutions and theoretical analysis very well.     

Implementation of the Lagrange-Galerkin method for the incompressible Navier-Stokes equations

This paper is concerned with the implementation of Lagrange-Galerkin finite element methods for the Navier-Stokes equations. A scheme is developed to efficiently handle unstructed meshes with local refinement, using a quad-tree-based algorithm for the geometric search. Several difficulties that arise in the construction of the right-hand side are discussed in detail and some useful tricks are proposed. The resulting method is tested on the lid-driven square cavity and the vortex shedding behind a rectangular cylinder and is found to give satisfactory agreement with previous works. A detailed analysis of the effect of time discretization is included.     

二维定常不可压缩粘性流动N-S方程的数值流形方法

将流形方法应用于定常不可压缩粘性流动N-S方程的直接数值求解,建立基于Galerkin加权余量法的N-S方程数值流形格式,有限覆盖系统采用混合覆盖形式,即速度分量取1阶和压力取0阶多项式覆盖函数,非线性流形方程组采用直接线性化交替迭代方法和Nowton-Raphson迭代方法进行求解.将混合覆盖的四节点矩形流形单元用于阶梯流和方腔驱动流动的数值算例,以较少单元获得的数值解与经典数值解十分吻合.数值实验证明,流形方法是求解定常不可压缩粘性流动N-S方程有效的高精度数值方法.     

Compact finite difference-Fourier spectral method for three-dimensional incompressible Navier-Stokes equations

A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported. The project supported by the National Natural Science Foundation of China     

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.     

A new method for obtaining the exact solutions of the Navier-Stokes (NS) equations and its applications

In this paper we propose a new method for obtaining the exact solutions of the Mavier-Stokes (NS) equations for incompressible viscous fluid in the light of the theory of simplified Navier-Stokes (SNS) equations developed by the first author^[1,2], Using the present method we can find some new exact solutions as well as the well-known exact solutions of the NS equations. In illustration of its applications, we give a variety of exact solutions of incompressible viscous fluid flows for which NS equations of fluid motion are written in Cartesian coordinates, or in cylindrical polar coordinates, or in spherical coordinates. The project supported by National Natural Science Foundation of China.     

The use of a reformulation of the hydrostatic approximation Navier-Stokes equations for geophysical fluid problems

In order to simulate geophysical general circulation processes, to simplify the governing equations of motion, often the vertical momentum equation of the Navier-Stokes equations is replaced by the hydrostatic approximation equation. The resulting equations are reformulated and a variational formulation of the linearized problem is derived. Iteration schemes are presented to solve this problem. A finite element method is discussed, as well as a finite difference method which is based on a grid that is often used in geophysical general circulation models. The schemes are extended to the non-linear case. Numerical examples are presented to demonstrate the performance of the derived iteration schemes.     

Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations

A global method of generalized differential quadrature is applied to solve the two-dimensional incompressible Navier-Stokes equations in the vorticity-stream-function formulation. Numerical results for the flow past a circular cylinder were obtained using just a few grid points. A good agreement is found with the experimental data.     

A deterministic vortex method for solving the Navier-Stokes equations

In this paper, a new 2-D vortex method is developed, which treats the vorticity diffusion in a deterministical way. The Laplacian operator, which describes vorticity diffusion, is approximated by a contour integral. The numerical results of two model problems show that this method has a good accuracy. A primary error estimation is given, and the self-adaptive vortex blob and the boundary conditions are discussed. The project supported by the National Natural Science Foundation of China     

A spectral method with staggered grid for incompressible Navier-Stokes equations

In order to solve the Navier-Stokes equations by spectral methods, we develop an algorithm using a staggered grid to compute the pressure. On this grid, an iterative process based on an artificial compressibility matrix associates the pressure with the continuity equation. This method is very accurate and avoids naturally most of the effects of parasite modes appearing in classical spectral methods with a velocity—pressure formulation.     

基于局部DQ-RBF不可压缩N-S方程的数值求解

以RBF作为DQ方法的基函数,将迎风机制引入DQ-RBF中,建立了二维不可压缩黏性N-S方程数值求解模型,采用Levenberg-Marquardt算法求解非线性方程组.求解时分析了形状参数对求解精度的影响,改进了边界速度的处理方法.对平板Couette流及有限宽台阶绕流流动问题进行了数值求解.比较了本文方法和FLUE...     

Fractional step method for solution of incompressible Navier-Stokes equations on unstructured triangular meshes

In this paper an implicit fractional step method for the solution of the two-dimensional, time-dependent, incompressible Navier-Stokes equations is presented. The current method was developed for use on an unstructured grid made up of triangles. The basic principles of this method are that the evaluation of the time evolution is split into intermediate steps and that for the spatial discretization of the flow equations a finite volume discretization on an unstructured triangular mesh is used. The present approach has been used to simulate viscous, laminar flows for various Reynolds numbers in test cases such as a backward-facing step, a square cavity and a channel with wavy boundaries.     

A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations

A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming （P1）2 - P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.     

Finite element analysis of the steady Navier-Stokes equations by a multiplier method

In this paper a fully explicit finite element method (FEFEM) is presented for solving steady incompressible viscous flow problems. This full explicitness is achieved by combining the multiplier (or augmented Lagrangian) method with a pseudo-time-iteration method. FEFEM needs no global matrix at all and is of great advantage to large-scale problems because they can be solved within the limit of core memory. The optimum choice of a time increment and a penalty parameter is discussed and the driven cavity flow at a Reynolds number of 1000 is computed with a refined mesh (60 × 60 elements).