Ray, Samriddhi, Frisch, U. (Uriel), 1940-, Nazarenko, Sergei and Matsumoto, Takeshi. (2011) Resonance phenomenon for the Galerkin-truncated Burgers and Euler equations. Physical Review E, Vol.84 (No.1). 016301 . ISSN 1539-3755

Preview

PDF WRAP_Nazarenko_1011.1826v3.pdf - Submitted Version - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (10Mb)

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
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Divisions:	Faculty of 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
Date:	July 2011
Volume:	Vol.84
Number:	No.1
Page Range:	016301
Identification Number:	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)
URI:	http://wrap.warwick.ac.uk/id/eprint/38577

Data sourced from Thomson Reuters' Web of Knowledge

Request changes to a record

Actions (login required)

View Item