Ohkitani, K. and Dowker, M. (2012) Numerical study on comparison of Navier-Stokes and Burgers equations. Physics of Fluids, 24 (5). 055113. ISSN 1070-6631
Abstract
We compare freely decaying evolution of the Navier-Stokes equations with that of the 3D Burgers equations with the same kinematicviscosity and the same incompressible initial data by using direct numerical simulations. The Burgers equations are well-known to be regular by a maximum principle [A. A. Kiselev and O. A. Ladyzenskaya, “On existence and uniqueness of the solutions of the nonstationary problem for a viscous incompressible fluid,” Izv. Akad. Nauk SSSR Ser. Mat.21, 655 (1957); A. A. Kiselev and O. A. Ladyzenskaya, Am. Math. Soc. Transl.24, 79 (1957)] unlike the Navier-Stokes equations. It is found in the Burgers equations that the potential part of velocity becomes large in comparison with the solenoidal part which decays more quickly. The probability distribution of the nonlocal term −u⋅∇p−u·∇p, which spoils the maximum principle, in the local energy budget is studied in detail. It is basically symmetric, i.e., it can be either positive or negative with fluctuations. Its joint probability density functions with 12|u|212|u|2 and with 12|ω|212|ω|2 are also found to be symmetric, fluctuating at the same times as the probability density function of −u⋅∇p−u·∇p. A power-law relationship is found in the mathematical bound for the enstrophy growth dQdt+2νP∝(Qa,Pb)α,dQdt+2νP∝QaPbα, where Q and P denote the enstrophy and the palinstrophy, respectively, and the exponents a and b are determined by calculus inequalities. We propose to quantify nonlinearity depletion by the exponent α on this basis.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2012 American Institute of Physics. This is an author produced version of a paper subsequently published in Physics of Fluids. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | calculus; Navier-Stokes equations; numerical analysis; vortices |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 23 Sep 2016 11:47 |
Last Modified: | 15 Apr 2017 00:43 |
Published Version: | http://dx.doi.org/10.1063/1.4719787 |
Status: | Published |
Publisher: | AIP Publishing |
Refereed: | Yes |
Identification Number: | 10.1063/1.4719787 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:78706 |