Hydrodynamic Stability Without Eigenvalues

Summary

A reconciliation of findings with the traditional analysis is presented based on the "pseudospectra" of the linearized problem, which imply that small perturbations to the smooth flow may be amplified by factors on the order of 105 by a linear mechanism even though all the eigenmodes decay monotonically.

Keywords

Eigenvalues and eigenvectors, Instability, Eigenfunction, Turbulence, Linear stability

References

Perturbation theory for linear operatorsTosio Kato — 1966
A New Mechanism For Linear and Nonlinear Hydrodynamic InstabilityD. J. Benney, Lennart Gustavsson — 1981
Three‐dimensional optimal perturbations in viscous shear flowKathryn M. Butler, Brian F. Farrell — 1992
General Solution for Perturbed Plane Couette FlowGerald Rosen — 1971
Wave Breakdown and TurbulenceMårten T. Landahl — 1975
Three-dimensional instability of a plane free shear layer: an experimental study of the formation and evolution of streamwise vorticesJuan C. Lasheras, Haecheon Choi — 1988
Structure of turbulence at high shear rateMoon Joo Lee, John Kim, Parviz Moin — 1990
Pseudospectra of the Orr-Sommerfeld OperatorSatish C. Reddy, Peter J. Schmid, Dan S. Henningson — 1993
Structure in turbulent mixing layers and wakes using a chemical reactionRobert Breidenthal — 1981
Optimal excitation of three‐dimensional perturbations in viscous constant shear flowBrian F. Farrell, Petros J. Ioannou — 1993
The evolution of a localized disturbance in a laminar boundary layer. Part 1. Weak disturbancesKenneth Breuer, Joseph H. Haritonidis — 1990
Energy growth in viscous channel flowsSatish C. Reddy, Dan S. Henningson — 1993
Stability of linear flowT. Ellingsen, Enok Palm — 1975
Optimal Excitation of Baroclinic WavesBrian F. Farrell — 1989
On the Stability for Three-Dimensional Disturbances of Viscous Fluid Flow between Parallel WallsH. B. Squire — 1933
A flow-visualization study of transition in plane Poiseuille flowDale R. Carlson, Sheila E. Widnall, M. F. Peeters — 1982
Optimal excitation of perturbations in viscous shear flowBrian F. Farrell — 1988
A note on an algebraic instability of inviscid parallel shear flowsM. T. Landahl — 1980
Evolution of wavelike disturbances in shear flows : a class of exact solutions of the Navier-Stokes equationsA. D. D. Craik, W. O. Criminale — 1986
Large-scale complex eigenvalue problemsW. Kerner — 1989

Cited by

Optimal disturbances in boundary layer flowsMartin G. Byström — 2007
Pseudospectra Analysis, Nonlinear Eigenvalue Problems, and Studying Linear Systems with DelaysJane Cullum, Albert E. Ruehli — 2001
Inferring transient dynamics of human populations from matrix non-normalityJosé Tomás Cabot — 2018
Limits of Localized Control in Extended Nonlinear SystemsAndreas Handel — 2004
Transitional and turbulent fibre suspension flowsMathias Kvick — 2014
Analysis and control of transitional shear flows using global modesShervin Bagheri — 2010
On minimizing maximum transient energy growthJames F. Whidborne, John McKernan, Anthony J. Steer — 2005
Étude du couplage aéro-mécanique au sein des moteurs à propergol solideS. Cerqueira — 2012
Flow control of boundary lagers and wakesJens H. M. Fransson — 2003
Laws of Energy Gradient for InstabilitiesHua-Shu Dou — 2005
Approximate Data Assimilation Schemes for Stable and Unstable DynamicsStephen E. Cohn, Ricardo Todling — 1996
Validation of a spatially interconnected model for flow transition controlSaulat S. Chughtai, Gabriela Rusnakova, Herbert Werner — 2009
A Robust Control Approach to Understanding Nonlinear Mechanisms in Shear Flow TurbulenceDennice F. Gayme — 2010
Stability analysis and control design of spatially developing flowsShervin Bagheri — 2008
Blasius 境界層のバイパス遷移の直接シミュレーション(流体工学, 流体機械)Naoya Takahashi, Kiyoshi Yamamoto — 2001
Receptivity of Boundary-Layer Flows over Flat and Curved WallsLars-Uve Schrader — 2010
Nonlinear stability for 2 dimensional plane Couette ﬂowPablo Braz e Silva — 2004
Optimisation and control of boundary layer flowsAntonios Monokrousos — 2009
Receptivity of Boundary Layers under Pressure GradientLars-Uve Schrader — 2008
Hydrodynamic Stability of Newtonian and Non-Newtonian FluidsJulian Mak, Miguel Moyers-González — 2009

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