2012 Fiscal Year Final Research Report
Covariant Lyapunov analysis of solutions of the Navier-Stokes equations
| Project/Area Number |
22654014
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Single-year Grants |
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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 |
YAMADA Michio 京都大学, 数理解析研究所, 教授 (90166736)
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| Co-Investigator(Kenkyū-buntansha) |
TAKEHIRO Shin-ichi 京都大学, 数理解析研究所, 准教授 (30274426)
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| Project Period (FY) |
2010 – 2012
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| Keywords | 応用数学 / 流体 / カオス |
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| Research Abstract |
We studied chaotic states of Kolmogorov flows on a 2D flat torus (R/2πZ)2governed by the Navier-Stokes equations of incompressible fluids, by using the covariant Lyapunov analysis. Obtaining the bifurcation diagram of solutions, we performed the traditional Lyapunov analysis, and found that the first Lyapunov number becomes positive at Re/Rc～18, and the second one does at Re/Rc～23. Based on these data, we calculated the covariant Lyapunov vectors by the method of Ginelli et al. (2007), and found that the solution orbit is hyperbolic just after the chaotic transition, but becomes nonhyperbolic at Re/Rc～23, by observing the distribution of the angle between the stable/unstabletangent spaces of the orbit. At the hyperbolic/nonhyperbolic transition point, we found that the fuctional form of the time correlation function of the vorticity changes from oscillatory to non-oscillatory.
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