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

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，还对流体计算所必需的数值格式进行了抽象分析。研究包括三个方面：(1)Navier-Stokes方程的数学分析；(2)水波分岔的数值实验；(3)涡旋法欧拉方程的数值计算。(1) Navier-Stokes方程的数学分析。发现了圆柱外Navier-Stokes方程的新精确解；它们是Tamada解和Wang解的推广。研究了Kolmogorov问题，发现当雷诺数趋于无穷时，一些平稳解趋向于C^1而不是C^2向量场。这个解代表了一种内部层，它很可能是理解湍流的关键。研究了用同余超几何函数表示的Navier-Stokes方程的自相似解。考虑了一些有流入和流出的固定解及其稳定性。其中一些对于所有雷诺数都是稳定的。(2)水波分岔的数值实验。考虑具有自由表面的二维无旋流。假定自由表面的轮廓是周期性的，在时间上是永久的。通过改变韦伯数和弗劳德数，我们计算出许多分岔解。我们也计算了负表面张力的水波。它们中的一些，在极限形式下，被发现与欧拉弹性定理相同。(3)用涡旋法对欧拉方程进行数值计算。用涡旋法计算了剪切流中的二维涡旋片。关于涡旋片运动的研究已经有很多，但我们的研究是新的，因为我们研究的是涡旋片与背景剪切流的关系。少

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