1996 Fiscal Year Final Research Report Summary
Studies on mathematical analysis and numerical computation of the Nevier-Stokes equations
| Project/Area Number |
07304019
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 総合 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
OKAMOTO Hisashi Research Institute for Mathematical Sciences, Kyoto Univ., Prof., 数理解析研究所, 教授 (40143359)
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| Co-Investigator(Kenkyū-buntansha) |
SHOJI Mayumi College of Science and Technology, Nihon Univ., Lecturer, 理工学部, 講師 (10216161)
NAKAKI Tatsuyuki Faculty of Science, Hiroshima Univ., associate Professor, 理学部, 助教授 (50172284)
IKEDA Hideo Faculty of Science, Toyama Univ., Associate Professor, 理学部, 助教授 (60115128)
SUGIHARA Masaki Faculty of Engineering, The Univ.of Tokyo, Associate Prof., 工学部, 助教授 (80154483)
NISHIMASA Yasumasa Research Institute for Electronic Science, Hokkaido Univ., Prof., 電子科学研究所, 教授 (00131277)
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| Project Period (FY) |
1995 – 1996
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| Keywords | equations of fluid motion / singular perturbation / vortex motion / turbulence / internal layr / numerical quadrature / water wave / bifurcation theory |
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| Research Abstract |
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
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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
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