How close are shell models to the 3D Navier-Stokes equations?
File(s)
Accepted version
Author(s)
Vincenzi, Dario
Gibbon, John D
Type
Journal Article
Abstract
Shell models have found wide application in the study of hydrodynamic turbulence because they are easily solved numerically even at very large Reynolds numbers. Although bereft of spatial variation, they accurately reproduce the main statistical properties of fully-developed homogeneous and isotropic turbulence. Moreover, they enjoy regularity properties which still remain open for the three-dimensional (3D) Navier–Stokes equations (NSEs). The goal of this study is to make a rigorous comparison between shell models and the NSEs. It turns out that only the estimate of the mean energy dissipation rate is the same in both systems. The estimates of the velocity and its higher-order derivatives display a weaker Reynolds number dependence for shell models than for the 3D NSEs. Indeed, the velocity-derivative estimates for shell models are found to be equivalent to those corresponding to a velocity gradient averaged version of the 3D Navier–Stokes equations (VGA-NSEs), while the velocity estimates are even milder. Numerical simulations over a wide range of Reynolds numbers confirm the estimates for shell models.
Date Issued
2021-08-01
Date Acceptance
2021-01-27
Citation
Nonlinearity, 2021, 34 (8), pp.5821-5843
ISSN
0951-7715
Publisher
IOP Publishing
Start Page
5821
End Page
5843
Journal / Book Title
Nonlinearity
Volume
34
Issue
8
Copyright Statement
© 2021 IOP Publishing Ltd & London Mathematical Society Printed in the UK. This is an author-created, un-copyedited version of an article accepted for publication in Nonlinearity. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher authenticated version is available online at https://doi.org/10.1088/1361-6544/abe096.
Identifier
http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000672960700001&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=1ba7043ffcc86c417c072aa74d649202
Subjects
Science & Technology
Physical Sciences
Mathematics, Applied
Physics, Mathematical
Mathematics
Physics
turbulence
shell models
high-order moments of the velocity derivatives
ENERGY-DISSIPATION
WEAK SOLUTIONS
LENGTH SCALES
TURBULENCE
MULTIFRACTALITY
STATISTICS
THEOREMS
SPECTRUM
MOMENTS
BOUNDS
Publication Status
Published
Date Publish Online
2021-07-13