Embed Size (px)
Citation preview
Experimental study of wave turbulence in a plate set into chaotic vibration
Olivier Cadot, Cyril TouzéUnité de Mécanique de l'ENSTA-Paristech, Palaiseau
Arezki BoudaoudLaboratoire de Physique Statistique de l'ENS, Paris
Outline
• Introduction
• Experiment
• Description of the wave turbulence state of the plate• Energy spectrum scaling
• Dissipation and cut - off frequency
• Statistical intermittency of the velocity
• Characterization of the injected power statistics• PDF of the injected power
• Statistics of the time averaged injected power (Gallavotti-Cohen theorem)
• Conclusion
Geometric non-linearity
Relation strain-displacement
Imperfect plate equation
Non-linearities of the dynamics
+ Linear Elasticity
Camier et al. EJMA/S 2009
Cubic Quadratic
Ideal plate : and only four-wave interaction
Imperfect plate allows three-wave interaction
Experiment
I
F
AmpliUC
I
R LAmpliUC
I
R L
F(t) = K I(t) Thomas et al. JSV 2003
EMT 140 reverberating plate from Radio France
Description of the wave turbulence state
Wave turbulence state
fc~<p>1/3
⇒
Boudaoud et al. PRL 2008, Mordant PRL 2008
From WT theory, the spectrum should scale as :
ε1/3 for four-wave interaction
ε1/2 for three-wave interaction
Kinetic energy spectrum of the displacement velocity: v = dw/dt
Experiment suggests : ε2/3
Wave turbulence stateDissipation vs Injection
Damping factor measured in linear
regime
γ(s-1)
Using the damping, the dissipated power can be estimated as :
and compared to the injected power :
0
500
1000
1500
2000
2500
3000
3500
4000
4500
0 5000 10000 15000
diss
ipat
ed (
mm
/s)3
injected (mm/s)3
Wave turbulence stateCut-off frequency ? - role of the damping.
Damping factor
γ(f)~f1/2
γ(s-1)
fc~<p>1/3
consistent with measurements
Balance : influx in the cascade and total dissipated energy
using h and epsilon : Mordant 2008
Wave turbulence stateStatistical intermittency of the temporal velocity differences
fc
forcing : 50 ms
No evolution of the PDF with timescale
No intermittency
Characterization of the injected power statistics
Response at the forcing point for different forcing
: Gaussian white noise [5Hz-f0], f0 =75Hz
sinusoidal
randomForce spectrum
-3
-2
-1
0
1
2
-3-2-10123
-2
0
2
4
6
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
t (s)
Force - F
Displacement velocity- v
Injected power - P=vF
•Sinus•Random
Evolution with mean injected power
Sinus Random
Kinetic energy spectrum of the displacement velocity: v = dw/dt
Mean dissipation <P> = <F V> vs the forcing amplitude
<P> = <FV> = r σF σV r = <FV>/ σF σV
σF
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.5 1 1.5
Sig FσF
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0 0.5 1 1.5
Sig FσF
Mean power Correlation F-V
•random•sinus
σF2
PDF( F/σF)
PDF of force and velocity
ForcePDF( V/σV)Velocity
Sinus Random
Power statistics PDF( P/<P>)
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
-5 0 5 10
p/<p>
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
-10 -5 0 5 10 15 20
p/<p>
sinusoidal forcing
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
-5 0 5 10
p/<p>
u : random feedback
Injected power model
only depends upon r
Model for PDF( P/<P>)
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
-10 -5 0 5 10 15 20
p/<p>
Random forcing
v : v.a. Gaussienne
F: v.a. Gaussienne
F and v correlated r : binormal law
Pumir POF 1996Falcon et al. PRL 2008Bandi & Connaughton PRE 2008
only depends upon r
Injected power model
Model for PDF( P/<P>)
Intermediate forcing
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
-5 0 5 10 15 20
p/<p>
1.E-02
1.E-01
1.E+00
-2 0 2 4 6
p/<p>
α =1
α=0.9
α=0.8
α=0.7
α=0.6
α=0
u : random feedback
Injected power model
only depends upon r and α
Model for PDF( P/<P>)
Time-averaged power statistics
SinusRandom
PDF of averaged power
cadot , Boudaoud , Touzé EPJB 2008
Farago 2002Falcon et al. 2008 Phase space contraction 700 Hz
Time-averaged power
SinusRandom
(Gallavotti-Cohen)
Estimating phase space contraction from the sum of the damping factor of the excited modes :
Time-averaged power
All the excited modes of frequency < 225 Hz gives a total phase space contraction of 700 Hz. It is the order of magnitude of the cut-off frequency of the WT spectrum.
Conclusions
• Energy spectrum scales as ε2/3 f-1/2, disagrements with 4-wave theory : plate imperfections ? and ?
• Frequency cut off of the energy spectrum consistent with a viscous dissipating scale given by fc ~ ε1/3/h.
• No statistical intermittency of the velocity differences
• A unique model for the PDF of the injected power for periodic,random or intermediate forcing based on a decomposition in a linear response to the excitation and a random feedback due to WT.
• The nature of the forcing, periodic or random condition the conclusions of the Gallavotti-Cohen theorem.
• For the sinus forcing, the phase space contraction rate corresponds to the sum of the damping factors of all of the excited modes of the wave turbulence.
