NDLI: A Resolution of the Sommerfeld Paradox

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Lin, Zhiwu Li, Y. Charles
Copyright Year	2011
Abstract	The Sommerfeld paradox roughly says that mathematically Couette linear shear is linearly stable for all Reynolds numbers, but experimentally arbitrarily small perturbations can induce the transition from the linear shear to turbulence when the Reynolds number is large enough. The main idea of our resolution of this paradox is to show that there is a sequence of linearly unstable shears which approaches the linear shear in the kinetic energy norm but not in the enstrophy (vorticity) norm. These oscillatory shears are single Fourier modes in the Fourier series of all the shears. In experiments, such linear instabilities will manifest themselves as transient nonlinear growth leading to the transition from the linear shear to turbulence. Under the Euler dynamics, these oscillatory shears are steady, and cat's eye structures bifurcate from them as traveling waves. The three-dimensional shears $(U(y,z),0,0)$ in a neighborhood of these oscillatory shears are linearly unstable too. Under the NavierStokes dynamics, these oscillatory shears are not steady but rather slowly drifting. When these oscillatory shears are viewed as frozen, the corresponding OrrSommerfeld operator has unstable eigenvalues which approach the corresponding inviscid eigenvalues when the Reynolds number tends to infinity. All the linear instabilities mentioned above offer a resolution to the Sommerfeld paradox and an initiator for the transition from the linear shear to turbulence.
Starting Page	1923
Ending Page	1954
Page Count	32
File Format	PDF
ISSN	00361410
DOI	10.1137/100794912
e-ISSN	10957154
Journal	SIAM Journal on Mathematical Analysis (SJMAAH)
Issue Number	4
Volume Number	43
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2011-08-16
Access Restriction	Subscribed
Subject Keyword	Rayleigh equation shear flow NavierStokes equation OrrSommerfeld equation Sommerfeld paradox Couette flow
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Mathematics

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