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Pseudospectra of the Linear Navier-Stokes Evolution Operator and Instability of Plane Poiseuille and Couette Flows: (preliminary report)

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This is a rough, interim report on some new results concerning the stability of plane Poiseuille and Couette fluid flows, following upon recent work by Henningson and Reddy, Butler and Farrell, Gustavsson and others. We emphasize that the conclusions proposed here have not yet been checked carefully and are subject to change. Our principal results are as follows: 1. Plots of the spectra of the "full" Navier-Stokes operator for Poiseuille and Couette flows, i.e., without restriction to a wave number pair (α,β) or to even or odd modes ($\S\S4,5).2.Analogousplotsforthepseudospectraofthisoperator.Comparisonofthepseudospectrawiththespectragivesanewinterpretationofwhythephysicsoftheselinearflowproblemsisnotcontrolledbythelocationofthemostunstableeigenvalue(\S\S4,5).3.DemonstrationthatthesepseudospectrapredicttheButlerFarrell"optimal"transientenergygrowthratiostowithinafactorofabout2(\S6).4.Demonstrationthatabout90\times3linearmodelobtainedbyprojectingtheNavierStokesproblemontothespacespannedbythreedominanteigenmodes,forCouetteflow,orfourinthecaseofPoiseuilleflow(\S8).5.Demonstrationthatalthough1OrrSommerfeldmodeand3Squiremodessufficeforthe4\times4modelinthePoiseuillecase,inkeepingwitharecentresultofGustavsson,onecandoequallywellwith2modesofeachkindorwith3OrrSommerfeldmodesand1Squiremode(\S$8). 6. Demonstration that the minimal operator perturbation required to destabilize a stable flow has norm of order R2, where R is the Reynolds number, though the distance of the least stable eigenvalue from the real axis is O(R1) ($\S7).7.Presentationofa2\times$2 model illustrating that if the linear problems described above are capable of transient energy growth of order M (e.g., $M\approx$1000 according to Butler and Farrell), a weak and intrinsically energy-conserving nonlinear term can "bootstrap" that growth to a higher order such as M2. This supports the view that although nonlinear terms are of course essential to the subcritical instability of fluid flows, the detailed nature of the nonlinear interactions may sometimes be relatively unimportant ($\S2).8.Adaptationofthis"bootstrapping"ideatothefluidflowsconsideredearlier,particularlythe3\times$3 approximation for Couette flow with R=1000.

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1992-06

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Cornell University

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computer science; technical report

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http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.cs/TR92-1291

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technical report

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