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

• 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

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Navier-Stokes方程和Euler方程进行了数学和数值分析，这些方程是不可压缩流体的主方程。此外，对流体计算所需的数值方案的一些抽象分析。该研究包括三类：（1）Navier-Stokes方程的数学分析，（2）关于水波分叉的数值实验，以及（3）通过Vortex方法对Euler方程进行数值计算。（1）Navier-Stokes方程的数学分析。发现了圆柱体外的Navier-Stokes方程的新的精确解决方案；它们是塔玛达解决方案和王的解决方案的概括。研究了Kolmogorov的问题，我们发现某些固定溶液倾向于C^1，但由于Reynolds的数量趋于无穷大。该解决方案代表了一种内部外行，很可能是理解湍流的关键……更多的功率谱。发现了由一致的超测量功能表示的Navier-Stokes方程的一些自相似解。考虑了一些具有流入和出口及其稳定性的固定解决方案。发现其中一些在所有雷诺数中都是稳定的。（2）关于水波分叉的数值实验。考虑了与自由表面的二维无关流。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。改变Weber号码和Froude号码，我们计算了许多现在分叉的解决方案。我们还计算出负表面张力的水波。其中一些以限制形式被发现与Euler的Elastica相同。（3）通过Vortex方法对Euler方程进行数值计算。剪切流中的二维涡流表由Vortex方法计算。许多关于涡旋板运动的研究是已知的，但是我们的研究是新的，因为我们研究了涡流纸和背景剪切流之间的关系。较少的

项目成果

M. taniguchi and Y. Nishiura: "Stability and Characteristic wavelength of Planar interfaces in the large diffusion limit of the inhibitor" to appear in Proc. Roy. Soc. Edingburgh.

M. taniguchi 和 Y. Nishiura：“抑制剂大扩散极限中平面界面的稳定性和特征波长”出现在 Proc 中。

今井仁司,石村直之,中村正彰: "磁気ベナ-ル問題のカオス" 日本物理学会学会誌. 50. 697-703 (1995)

Hitoshi Imai、Naoyuki Ishimura、Masaaki Nakamura：“磁贝纳德问题的混沌”日本物理学会杂志 50. 697-703 (1995)。

H. Okamoto and M. Shoji: "Two Dimensional, Periodic, Capillary-Gravity Waves With Negative Surfase Tension" Proc. IUTAM Conf. Structure and Dynamics of Nonlinear Waves in Fluids. 363-369 (1995)

H. Okamoto 和 M. Shoji：“具有负表面张力的二维、周期性毛细管重力波”Proc。

Y. Nishiura, et al.: "Dynamics of inhibitory pulse-coupled oscillators" Applicable Analysis“Dynamical Systems and applications". 4. 549-562 (1995)

Y. Nishiura 等人：“抑制性脉冲耦合振荡器的动力学”适用分析“动态系统和应用”4. 549-562 (1995)。

H. Imai, and et al.: "On Spectral Collocation Methods in Space and Time for Free Boundary Problems" Computational Mechanics '95 (Proc. of ICES '95). 1. 798-803 (1995)

H. Imai 等人：“自由边界问题的空间和时间频谱搭配方法”计算力学 95（Proc. of ICES 95）。