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Studies on mathematical analysis and numerical computation of the Nevier-Stokes equations

内维-斯托克斯方程的数学分析与数值计算研究

基本信息

批准号：
07304019
负责人：
OKAMOTO Hisashi
金额：
$ 1.09万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1996
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07304019/
关键词：
equations of fluid motion singular perturbation vortex motion turbulence internal layr numerical quadrature water wave bifurcation theory 流体力学数値解析モデリング渦法反応拡散系

项目摘要

The present study carried out mathematical and numerical analysis of the Navier-Stokes equations and the Euler equations, which are the master equations of incompressible fluid. In addition, some abstract analysis of numerical schemes which are necessary for the fluid computations. The study consists of three categories : (1) mathematical analysis of the Navier-Stokes equations, (2) numerical experiments on the bifurcation of water waves, and (3) numerical computation of the Euler equations by the vortex method.(1) mathematical analysis of the Navier-Stokes equations. New exact solutions of the Navier-Stokes equation outside a cylinder are discovered ; they are generalizations of Tamada's solution and Wang's solution. Kolmogorov's problem is studied and we find that some stationary solutions tends to C^1 but not C^2 vector field as the Reynolds number tends to infinity. This solution represents a kind of internal layr, which may well serve as a key to the understanding of the turbulent … More power spectra. Some self-similar solutions of the Navier-Stokes equations, which are represented by the congruent hypergeometric functions, are discovered. Some stationary solutions having inflows and outflows and their stability were considered. Some of them are found to be stable for all the Reynolds number.(2) numerical experiments on the bifurcation of water waves. Two-dimensional irrotational flows with free surface are considered. The free surface are assumed to be periodic in its profile and permanent in time. Varying the Weber number and the Froude number, we compute many now bifurcating solutions. We also compute water waves with negative surface tension. Some of them are, in its limiting form, found to be the same as Euler's elastica.(3) numerical computation of the Euler equations by the vortex method. Two-dimensional vortex sheets in shear flows are computed by the vortex method. Many studies on vortex sheet motion are known, but our research is new in that we study the relation between vortex sheet and background shear flow. Less

本文对不可压缩流体的主方程Navier-Stokes方程和Euler方程进行了数学和数值分析。此外，对流体计算所需的数值格式进行了抽象分析。研究内容包括三个方面：（1）Navier-Stokes方程的数学分析，（2）水波分叉的数值实验，（3）Euler方程的涡方法数值计算。(1)纳维-斯托克斯方程的数学分析。发现了柱外Navier-Stokes方程的新的精确解，它们是Tamada解和Wang解的推广。研究了Kolmogorov问题，发现当雷诺数趋于无穷大时，某些定态解趋于C^1而不是C^2向量场。这个解代表了一种内部层，它可以作为理解湍流的一个关键。 ...更多信息功率谱发现了Navier-Stokes方程的一些自相似解，这些解用同余超几何函数表示。考虑了一类具有流入流出的定常解及其稳定性。其中一些被发现是稳定的所有雷诺数。(2)水波分叉的数值实验。考虑了二维自由表面无旋流动。假定自由表面在其轮廓上是周期性的，并且在时间上是永久的。通过改变Weber数和Froude数，我们计算出了许多分支解.我们还计算了负表面张力的水波。其中一些在其极限形式下被发现与欧拉的弹性线相同。(3)用涡方法数值计算欧拉方程。本文用涡方法计算了剪切流中的二维涡面。关于涡面运动的研究已有很多，但我们的研究是新的，因为我们研究了涡面与背景切变流之间的关系。少