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

本研究对不可压缩流体的主方程纳维-斯托克斯方程和欧拉方程进行了数学和数值分析，此外，还对流体计算所必需的数值格式进行了一些抽象分析。三个类别：（1）纳维-斯托克斯方程的数学分析，（2）水波分岔的数值实验，（3）涡旋对欧拉方程的数值计算(1)对Navier-Stokes方程进行数学分析。发现了圆柱外Navier-Stokes方程的新精确解；它们是Tamada解和Kolmogorov问题的推广，并发现了一些平稳解。趋向于 C^1 而不是 C^2 向量场，因为雷诺数趋于无穷大。这个解代表了一种内部层，这很可能是理解湍流的关键。 …更多功率谱。发现了一些由全等超几何函数表示的纳维-斯托克斯方程的自相似解，并且发现了一些具有流入和流出的平稳解，其中一些解被认为是稳定的。对于所有的雷诺数。(2) 考虑了具有自由表面的二维无旋流的数值实验。假设自由表面的轮廓是周期性的并且随时间变化。韦伯数和弗劳德数，我们计算了许多现在的分岔解。我们还计算了具有负表面张力的水波，其中一些在其极限形式上与欧拉弹性数相同。(3) 的数值计算。涡流法的欧拉方程剪切流中的二维涡流片是通过涡流法计算的，许多关于涡流片运动的研究是已知的，但我们的研究是新的，因为我们研究了它们之间的关系。涡片和背景剪切流较少。