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Resonance phenomenon for the Galerkin-truncated Burgers and Euler equations
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Ray, Samriddhi, Frisch, U., Nazarenko, Sergey and Matsumoto, Takeshi (2011) Resonance phenomenon for the Galerkin-truncated Burgers and Euler equations. Physical Review E, Vol.84 (No.1). 016301 . doi:10.1103/PhysRevE.84.016301 ISSN 1539-3755.
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Official URL: http://dx.doi.org/10.1103/PhysRevE.84.016301
Abstract
It is shown that the solutions of inviscid hydrodynamical equations with suppression of all spatial Fourier modes having wave numbers in excess of a threshold K(G) exhibit unexpected features. The study is carried out for both the one-dimensional Burgers equation and the two-dimensional incompressible Euler equation. For large K(G) and smooth initial conditions, the first symptom of truncation, a localized short-wavelength oscillation which we call a "tyger," is caused by a resonant interaction between fluid particle motion and truncation waves generated by small-scale features (shocks, layerswith strong vorticity gradients, etc.). These tygers appear when complex-space singularities come within one Galerkin wavelength lambda(G) = 2 pi/K(G) from the real domain and typically arise far away from preexisting small-scale structures at locations whose velocities match that of such structures. Tygers are weak and strongly localized at first-in the Burgers case at the time of appearance of the first shock their amplitudes and widths are proportional to K(G)(-2/3) and K(G)(-1/3), respectively-but grow and eventually invade the whole flow. They are thus the first manifestations of the thermalization predicted by T. D. Lee [Q. J. Appl. Math. 10, 69 (1952)]. The sudden dissipative anomaly-the presence of a finite dissipation in the limit of vanishing viscosity after a finite time t(star)-which is well known for the Burgers equation and sometimes conjectured for the three-dimensional Euler equation, has as counterpart, in the truncated case, the ability of tygers to store a finite amount of energy in the limit K(G)->infinity. This leads to Reynolds stresses acting on scales larger than the Galerkin wavelength and thus prevents the flow from converging to the inviscid-limit solution. There are indications that it may eventually be possible to purge the tygers and thereby to recover the correct inviscid-limit behavior.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Hydrodynamics, Burgers equation, Resonance | ||||
Journal or Publication Title: | Physical Review E | ||||
Publisher: | American Physical Society | ||||
ISSN: | 1539-3755 | ||||
Official Date: | July 2011 | ||||
Dates: |
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Volume: | Vol.84 | ||||
Number: | No.1 | ||||
Page Range: | 016301 | ||||
DOI: | 10.1103/PhysRevE.84.016301 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Funder: | European Cooperation in the Field of Scientific and Technical Research (Organization) (COST), France. Agence nationale de la recherche (ANR), Japan. Monbu Kagakushō [Japan. Ministry of Education, Culture, Sports, Science and Technology] (MK), India. Dept. of Science and Technology, India. University Grants Commission, France. Ministère de l'éducation nationale | ||||
Grant number: | MP0806 (COST), BLAN07-2_183172 (ANR) |
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