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arXiv 1306.3950math.AP
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有界区域上Navier-Stokes- $α$方程对湍流的近似

On the approximation of turbulent fluid flows by the Navier-Stokes-$α$ equations on bounded domains

Juan Vicente Gutiérrez-Santacreu, Marko Antonio Rojas-Medar

摘要:

Navier-Stokes- $α$方程属于LES(大涡模拟)模型族,其基本思想是捕捉小尺度对大尺度的影响,而不计算流中存在的所有整个范围。常数$α$是具有模型可分辨的最小尺度的维度的状态流参数。因此,当$α=0$时,人们恢复了粘性不可压缩牛顿流体流的经典Navier-斯托克斯方程。此外,Navier-Stokes- $α$方程也可以被解释为斯托克斯方程的正则化,其中$α$代表正则化参数。本文首先利用Helmholtz算子的Leray正则化方法,给出了有界区域上的Navier-Stokes- $α$方程的无滑移边界条件。然后研究了在有界区域上无滑移边界条件下Navier-Stokes- $α$方程的Galerkin近似与Navier-Stokes-斯托克斯方程解的性质之间的关系问题。Galerkin方法是通过使用与Stokes算子相关联的本征函数来进行的。我们将得到局部和全球的时间误差估计测量的政权参数$α$和特征值。特别是,为了获得全球的时间误差估计,我们将工作的概念的稳定性的解决方案的斯托克斯方程的$L^2$规范。

英文摘要:

The Navier-Stokes-$α$ equations belong to the family of LES (Large Eddy Simulation) models whose fundamental idea is to capture the influence of the small scales on the large ones without computing all the whole range present in the flow. The constant $α$ is a regime flow parameter that has the dimension of the smallest scale being resolvable by the model. Hence, when $α=0$, one recovers the classical Navier-Stokes equations for a flow of viscous, incompressible, Newtonian fluids. Furthermore, the Navier-Stokes-$α$ equations can also be interpreted as a regularization of the Navier-Stokes equations, where $α$ stands for the regularization parameter. In this paper we first present the Navier-Stokes-$α$ equations on bounded domains with no-slip boundary conditions by means of the Leray regularization using the Helmholtz operator. Then we study the problem of relating the behavior of the Galerkin approximations for the Navier-Stokes-$α$ equations to that of the solutions of the Navier-Stokes equations on bounded domains with no-slip boundary conditions. The Galerkin method is undertaken by using the eigenfunctions associated with the Stokes operator. We will derive local- and global-in-time error estimates measured in terms of the regime parameter $α$ and the eigenvalues. In particular, in order to obtain global-in-time error estimates, we will work with the concept of stability for solutions of the Navier-Stokes equations in terms of the $L^2$ norm.

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