Effects of small noise on the DMD/Koopman spectrum

Transcription

1 Effects of small noise on the DMD/Koopman spectrum Shervin Bagheri Linné Flow Centre Dept. Mechanics, KTH, Stockholm Sig33, Sandhamn, Stockholm, Sweden May, 29, 2013

2 Questions 1. What are effects of noise on self-sustained oscillations? 2. How sensitive are self-sustained oscillations to noise? 3. Can we quantify the sensitivity? 2

3 Combustion Experiment Thermoacoustic oscillations Feedback loop between acoustic field and unsteady heat release White noise introduced by a loudspeaker Measure pressure and heat release Jegadeesan & Sujith, Proc. Comb. Inst

4 Combustion Experiment Noise modifies limit cycle amplitude and period Increasing noise level Jegadeesan & Sujith, Proc. Comb. Inst

5 Observations: Combustion Observed drift of the phase of pressure oscillations 4 cycles: measuremenst in phase with harmonic signal cycles: phase drifted 45 degrees Lieuwen, J. Sound & Vibration

6 Phase Drift Phase drift increases exponentially with time Lieuwen, J. Sound & Vibration

7 From Experiments White noise induce a new time scale Phase drift increases exponentially with rate 1/τ Auto-correlation functions decreases exponentially with rate 1/τ Noisy limit cycle characterized by two time scales: Limit cycle period: Quality factor: T Q = τ T (Q is number of oscillations which periodicity is maintained) 7

8 Koopman Operator Time evolution of observables governed by Koopman operator (Mezic, 2005, Rowley etal 2009) approximated by Dynamic Mode Decomposition (DMD) (Schmid 2010) can be used to obtain time-correlation functions Thus time scale τ should be present in the spectrum of the Koopman operator 8

9 Stochastic Navier-Stokes Consider u t = f(u)+ ˆξ(t) state: noise amplitude: Gaussian white noise: ˆξ(t)ˆξ(t ) = Qδ(t t ) 9

10 Evolution of Ensemble Average Given an observable g(u) Ensemble average of an observable at time t : probability that a state will found in Write ensemble average with linear evolution operators: Koopman operator 10

11 Eigenvalues of Koopman Operator Koopman operator is linear Eigenvalue decomposition: L t g j (u) =λ j g j (u), j =0, 1, 2,... At leading order the trace formula gives values for limit cycle: (Gaspard, J. Stat. Phys. 2002) λ m = imω S 2T ω2 m 2 + O( 2 ) Koopman eigenvalues for a noisy limit cycle m =0, ±1, ±2,... ω = 2π T S =sensitivity 11

12 Koopman Spectrum Deterministic Koopman eigenvalues: λ m = imω = im 2π T 2nd harmonic modes Global modes Mean flow Global modes 2nd harmonic modes Growth rate σ > 0 2π/T m = 2 m = 1 m =0 m =1 m =2 Frequency Bagheri, JFM

13 Koopman Spectrum Koopman eigenvalues in the presence of noise λ m = imω S 2T ω2 m 2 + O( 2 ) m =0, ±1, ±2,... Non-stationary eigenvalues are damped! New time scale: (rate of phase diffusion) τ S Proportional to (sensitivity) Growth rate 2π/T 1/τ m = 2 m = 1 m =0 m =1 m =2 Frequency

14 DMD of Noisy System: Helium Jet DMD (often) approximates Koopman eigenvalues Input: Sequence of snapshots (from experiments) Schmid, Li, Juniper & Pust, TCFD,

15 DMD of Noisy System: Helium Jet Output: DMD modes (coherent structures) Schmid, Li, Juniper & Pust, TCFD,

16 DMD of Noisy System: Helium Jet Output: DMD values (growth rate and frequency) 2π/T 1/τ (growth rate) (frequency) Parabolas are due to the presence weak noise DMD spectrum reveals two time scales 16

17 Sensitivity How sensitive is a limit cycle to noise? λ m = imω S 2T ω2 m 2 + O( 2 ) Large S high rate of phase diffusion What is the explicit expression for S? 17

18 Stochastic Navier-Stokes The stochastic system u t = f(u)+ ˆξ(t) dim = n can in limit of 0 be written as ṗ =[ f(u)] T p dim = 2n p(t) R n where is an adjoint variable noiseless system system with noise Onsager & Machlup, Phys. Rev

19 Hamiltonian System Hamiltonian: Hamilton equations: Energy conserved along trajectories (constant of motion) dh dt =0 H(u, p) =E Energy is measure of deviation of noisy trajectory from its deterministic prediction Onsager & Machlup, Phys. Rev

20 System with One Limit Cycle Noise-less limit cycle Period Energy The original state-space is now the subspace 20

21 System with Noisy Limit Cycle Noisy limit cycle of period Energy Dimension of original state space is doubled to accommodate for limit cycles of perturbed period 21

22 Sensitivity Consider Energy E = /τ How does a small perturbation of energy affect the time period? define sensitivity as E T δt = T δe = SδE E large a small perturbation of energy, gives large deviation from deterministic time period 22

23 Linearized System Small δt we may linearize system δ u = A(u)+2Qδp δṗ = A T (u)δp Write solution as δu(t )=M T δu(0) + N T δp(0) Floquet expansion δp(t )=(M 1 T )T δp(0) M T = n φ k Λ k ψk T k=1 23

24 Sensitivity One obtains the expression T E = ψt 1 δu(t ) ψ1 T φ 1 if system is non-normal then and ψ T 1 φ 1 1 T S = E 1 24

25 Sensitivity One obtains the expression T E = ψt 1 δu(t ) ψ1 T φ 1 δu(t ) is obtained by solving linearized equations with initial conditions: δu(0) = 0 δp(0) = ψ 1 (Gaspard, J. Stat. Phys. 2002) 25

26 Conclusions Sensitivity Noisy limit cycle may deviate from deterministic cycle if system is highly non-normal Quality factor can be read of the DMD spectrum Look into DMD literature: you observe parabolic branches! Important for control purposes Determine whether randomness is due external noise or some intrinsic dynamics 26