外部流动的Oseen耦合方法,I:Oseen耦合逼近

这篇文章考虑了具有非齐次边界条件的二维非定常外部Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程．通过将内部区域的Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程和外部区域的Ｏｓｅｅｎ方程相耦合,得到了Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ问题的逼近问题： Ｏｓｅｅｎ耦合问题,此外,我们证明了 Ｏｓｅｅｎ耦合方程弱解的存在性,唯一性和收敛精度．     

On Nonhomogeneous Slip Boundary Conditions for 2D Incompressible Exterior Fluid Flows

The paper examines the issue of existence of solutions to the steady Navier-Stokes equations in an exterior domain in ℝ². The system is studied with nonhomogeneous slip boundary conditions. The main results proves the existence of weak solutions for arbitrary data.      

The initial boundary value problem for Navier-Stokes equations

By making full use of the estimates of solutions to nonstationary Stokes equations and the method discussing global stability, we establish the global existence theorem of strong solutions for Navier-Stokes equatios in arbitrary three dimensional domain with uniformlyC ³ boundary, under the assumption that |a| _L ²(Θ) + |f| _L ¹(0,∞;L ²(Θ)) or |∇a| _L ²(Θ) + |f| _L ²(0,∞;L ²(Θ)) small or viscosityv large. Herea is a given initial velocity andf is the external force. This improves on the previous results. Moreover, the solvability of the case with nonhomogeneous boundary conditions is also discussed. This work is supported by foundation of Institute of Mathematics, Academia Sinica     

A Non-Conforming Least Squares Spectral Element Formulation for Oseen Equations with Applications to Navier-Stokes Equations

In this article, we propose a non-conforming exponentially accurate least-squares spectral element method for Oseen equations in primitive variable formulation that is applicable to both two- and three-dimensional domains. First-order reformulation is avoided, and the condition number is controlled by a suitable preconditioner for velocity components and pressure variable. A preconditioned conjugate gradient method is used to obtain the solution. Navier-Stokes equations in primitive variable formulation have been solved by solving a sequence of Oseen type iterations. For numerical test cases, similar order convergence has been achieved for all Reynolds number cases at the cost of higher iteration number for higher Reynolds number.     

Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.     

Oseen coupling method for the exterior flow. Part II: Well‐posedness analysis

In this paper, we recall the Oseen coupling method for solving the exterior unsteady Navier–Stokes equations with the non‐homogeneous boundary conditions. Moreover, we derive the coupling variational formulation of the Oseen coupling problem by using of the integral representations of the solution of the Oseen equations at an infinity domain. Finally, we provide some properties of the integral operators over the artificial boundary and the well‐posedness of the coupling variational formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

The local discontinuous Galerkin method for the Oseen equations

We introduce and analyze the local discontinuous Galerkin method for the Oseen equations of incompressible fluid flow. For a class of shape-regular meshes with hanging nodes, we derive optimal a priori estimates for the errors in the velocity and the pressure in - and negative-order norms. Numerical experiments are presented which verify these theoretical results and show that the method performs well for a wide range of Reynolds numbers.

     

First‐order system least squares for the Oseen equations

Following earlier work for Stokes equations, a least squares functional is developed for two‐ and three‐dimensional Oseen equations. By introducing a velocity flux variable and associated curl and trace equations, ellipticity is established in an appropriate product norm. The form of Oseen equations examined here is obtained by linearizing the incompressible Navier–Stokes equations. An algorithm is presented for approximately solving steady‐state, incompressible Navier–Stokes equations with a nested iteration‐Newton‐FOSLS‐AMG iterative scheme, which involves solving a sequence of Oseen equations. Some numerical results for Kovasznay flow are provided. Copyright © 2006 John Wiley & Sons, Ltd.     

Time-dependent coating flows in a strip, Part I: The linearized problem

This work is concerned with time-dependent coating flow in a strip . The Navier-Stokes equations are satisfied in the fluid region, the bottom substrate is moving with fixed velocity , and fluid is entering the strip through the upper boundary . The free boundary has the form for , where is the moving contact point. Our objective is to prove that if the initial data are close to those of a stationary solution (the existence of such a solution was established by the authors in an earlier paper) then the time-dependent problem has a unique solution with smooth free boundary, at least for a small time interval. In this Part I we study the linearized problem, about the stationary solution, and obtain sharp estimates for the solution and its derivatives. These estimates will be used in Part II to establish existence and uniqueness for the full nonlinear problem.

     

The Robin problem for the scalar Oseen equation

We study the Robin problem for the scalar Oseen equation in an open n‐dimensional set with compact Ljapunov boundary. We prescribe two types of Robin boundary conditions, and prove the unique solvability of these problems as well as a representation formula for the solution in form of a scalar Oseen single layer potential. Moreover, we prove the maximum principle for the solution to the Robin problem of the scalar Oseen equation. Copyright © 2013 John Wiley & Sons, Ltd.     

求解Oseen流的交替线松弛多重网格方法

利用Riemann解的通量差分分裂法——Godunov方法对Oseen流控制方程进行离散,得到了基于一阶上迎风格式的离散方程,并给出了使用多重网格方法求解该离散方程的V-循环算法和W-循环算法的收敛性分析.通过局部Fourier分析方法,对获得的离散方程的聚对称交替线GaussSeidel松弛的光滑性质进行了研究.结果表明:使用多重网格的两层网格及三层网格算法求解具有不同Reynolds数的Oseen流,即便是在高Reynolds数情况下,聚对称交替线Gauss-Seidel松弛具有很好的光滑性质,多重网格W-循环算法收敛性比V-循环算法好.     

Thorough analysis of the Oseen system in 2D exterior domains

We construct L_p‐estimates for the inhomogeneous Oseen system studied in a two‐dimensional exterior domain Ω with inhomogeneous slip boundary conditions. The kernel of the paper is a result for the half space ?. Analysis of this model system shows us a parabolic character of the studied problem, resulting as an appearance of the wake region behind the obstacle. Main tools are given by the Fourier analysis to obtain the maximal regularity estimates. The results imply the solvability for the Navier–Stokes system for small velocity at infinity. Copyright © 2009 John Wiley & Sons, Ltd.     

Shape optimization of a body in the Oseen flow

This article is concerned with a numerical simulation of shape optimization of the Oseen flow around a solid body. The shape gradient for shape optimization problem in a viscous incompressible flow is computed by the velocity method. The flow is governed by the Oseen equations with mixed boundary conditions containing the pressure. The structure of continuous shape gradient of the cost functional is derived by using the differentiability of a minimax formulation involving a Lagrange functional with a function space parametrization technique. A gradient type algorithm is applied to the shape optimization problem. Numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A numerical method for the viscous incompressible Oseen flow in shape reconstruction

This paper is concerned with the shape reconstruction of a bounded domain with a viscous incompressible fluid driven by the Oseen equations. For the approximate solution of the ill-posed and nonlinear problem we propose a regularized Gauss-Newton method. A theoretical foundation for the method is given by establishing the differentiability of the boundary value problem with respect to the boundary in the sense of the domain derivative. The results of several numerical experiments show that our theory is useful for practical purpose, and the proposed algorithm is feasible.     

Global L~2 Stability of the Nonhomogeneous Incompressible Navier–Stokes Equations

In this paper, the problem of the global L^2 stability for large solutions to the nonhomogeneous incompressible Navier-Stokes equations in 3D bounded or unbounded domains is studied. By delicate energy estimates and under the suitable condition of the large solutions, it shows that if the initial data are small perturbation on those of the known strong solutions, the large solutions are stable.     

Far field boundary conditions for incompressible flows computation

Many far field boundary conditions are proposed in the literature to solve Navier-Stokes equations. It is necessary to distinguish the streamwise or outlet boundary conditions and the spanwise boundary conditions. In the first case the flow crosses the artificial frontier and it is required to avoid reflections that can change significantly the flow. In the second case the Navier-slip boundary condition is often used but if the frontier is not far enough the boundary is both inlet and outlet. Thus the Navier-slip boundary condition is not well suited as it imposes no flux through the frontier. The aim of this work is to compare some well-known boundary conditions, to quantify to which extend the artificial frontier can be close to the bodies in two- and three-dimensions and to take into account the flow rate through the spanwise directions.     

Lq‐estimates for the stationary Oseen equations on the exterior of a rotating obstacle

We study the Dirichlet problem for the stationary Oseen equations around a rotating body in an exterior domain. Our main results are the existence and uniqueness of weak and very weak solutions satisfying appropriate L^q‐estimates. The uniqueness of very weak solutions is shown by the method of cut‐off functions with an anisotropic decay. Then our existence result for very weak solutions is deduced by a duality argument from the existence and estimates of strong solutions. From this and interior regularity of very weak solutions, we finally establish the complete D^1,r‐result for weak solutions of the Oseen equations around a rotating body in an exterior domain, where 4/3<r <4. Here, D^1,r is the homogeneous Sobolev space.     

A stabilized Oseen iterative finite element method for stationary conduction–convection equations

A stabilized Oseen iterative finite element method for the stationary conduction–convection equations is investigated in this paper. The stability and iterative error estimates are analyzed, which show that the presented method is stable and has good precision. Numerical results are shown to support the developed theory analysis and demonstrate the efficiency of the given method. Copyright © 2011 John Wiley & Sons, Ltd.     

二维N－S方程的Fourier非线性Galerkin方法

本文对周期边界条件Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程，证明了其Ｆｏｕｒｉｅｒ非线性Ｇａｌｅｒｋｉｎ逼近解的存在唯一性，同时给出了逼近解的误差估计．