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Spectral Representation of Oscillators
Shervin Bagheri
Linné Flow Centre
Mechanics, KTH, Stockholm
APS/DFD 2012,
San Diego, Nov 18-20
Supercritical Flow Dynamics
2
Spectral Expansion
• Expansion of flow field into – Operator approach: Koopman Operator
– Computational approach - Dynamic Mode Decomposition
3
Mezic, 2005, Nonlin. Dyn. Rowley etal 2009, JFM Schmid, 2010, JFM
Koopman modes
Koopman eigenvalues
Ritz values
Ritz vectors
Observables
An observable with governed by
Define Koopman operator
Spectral decomposition
4 Mezic, Nonlin. Dyn. 2005 Mezic, Ann. Rev. Fluid Mech. 2013
Oscillator – Cylinder flow (Re=50)
• Navier-Stokes near the critical threshold for oscillation
– Unstable equilibrium
– Attracting limit cycle
• Frequency • Growth rate
5
Koopman Eigenvalues
– Formulas based on trace
or
provides the Koopman eigenvalues
6 Cvitanovic & Eckhardt 1991 Gaspard 1998
DMD of Attractor and Near Attractor
7
Ritz values (DMD)
Koopman eigenvalues
Koopman Modes
• Two steps 1. Scale separation: fast limit cycle period but slow saturation
2. Spectral expansion of S-L equation
Provides Koopman Modes
8
Leading Koopman Modes
9
Mean flow (asymptotic)
Global mode (asymptotic)
Subharmonic mode (asymptotic)
Shift mode (transient)
Global mode (asymptotic)
Subharmonic mode (transient)
Ritz Vectors
10
Mean flow
Global mode
Subharmonic mode
Shift mode
Transient Global mode
Transient subharmonic
Asymptotic modes Transient modes
DMD of Attractor and Near Attractor
Ritz cluster and form branches due to large algebraic growth
11
Ritz values (DMD)
Koopman eigenvalues
Conclusions
12
• Analytical results – Koopman eigenvalues form a lattice – Koopman modes correspond to mean flow, shift mode, global
modes,..
• Computational results – Ritz vectors/values good approximation near and on attractor – Algebraic dynamics generates clusters/branches in spectrum