Internal Report
Bounds for threshold amplitudes in subcritical shear flows.
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Dan Henningson, Anders Lundbladh, G. Kreiss 
Technical report 
1993 
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Id 
ISSN 0348467X 
ISRN KTH/MEK/TR93/07SE 
Abstract
A general theory which can be used to derive bounds on solutions to the NavierStokes equations is presented. The behaviour of the resolvent of the linear operator in the unstable halfplane is used to bound the energy growth of the full nonlinear problem. Plane Couette flow is used as an example. The norm of the resolvent in plane Couette flow in the unstable halfplane is proportional to the square of the Reynolds number (R). This is now used to predict the asymptotic behaviour of the threshold amplitude below which all disturbances eventually decay. A lower bound is found to be R/sup 21/4/. Examples, obtained through direct numerical simulation, give an upper bound on the threshold curve, and predict a threshold of R/sup 1/. The discrepancy is discussed in the light of a model problem.
