Power-law decay of the energy spectrum in linearized perturbed … · 2011. 7. 20. · Introduction Initial-value problem formulation Results Conclusions Power-law decay of the energy

IntroductionInitial-value problem formulation

ResultsConclusions

Power-law decay of the energy spectrum inlinearized perturbed systems

Stefania Scarsoglio1

Francesca De Santi2 Daniela Tordella2

1Department of Water Engineering, Politecnico di Torino, Torino, Italy2Department of Aeronautics and Space Engineering, Politecnico di Torino, Torino, Italy

4th International SymposiumBifurcations and Instabilities in Fluid Dynamics

18-21 July 2011, Barcelona, Spain

S. Scarsoglio, BIFD 2011 Power-law decay of the energy spectrum in linearized perturbed systems


ResultsConclusions

Motivation and general aspects

Energy spectrum in fully developed turbulencePhenomenology of turbulence Kolmogorov 1941:−5/3 power-law for the energy spectrum over the inertial range;

It is a common criterium for the production of a fully developedturbulent field to verify such a scaling (e.g. Frisch, 1995; Sreeni-vasan & Antonia, ARFM, 1997; Kraichnan, Phys. Fluids, 1967).

(left) Evangelinos & Karniadakis, JFM 1999. (right) Champagne, JFM 1978.



ResultsConclusions


Energy spectrum in fully developed turbulencePhenomenology of turbulence Kolmogorov 1941:−5/3 power-law for the energy spectrum over the inertial range;It is a common criterium for the production of a fully developedturbulent field to verify such a scaling (e.g. Frisch, 1995; Sreeni-vasan & Antonia, ARFM, 1997; Kraichnan, Phys. Fluids, 1967).




ResultsConclusions


Energy spectrum in fully developed turbulencePhenomenology of turbulence Kolmogorov 1941:−5/3 power-law for the energy spectrum over the inertial range;It is a common criterium for the production of a fully developedturbulent field to verify such a scaling (e.g. Frisch, 1995; Sreeni-vasan & Antonia, ARFM, 1997; Kraichnan, Phys. Fluids, 1967).




ResultsConclusions


Energy spectrum and linear stability analysis

We study the state that precedes the onset of instability and tran-sition to turbulence:

To understand how spectral representation can effectively highlightthe nonlinear interaction among different scales;To quantify the degree of generality on the value of the energy decayexponent of the inertial range;

Different typical perturbed shear systems: plane Poiseuille flowand bluff-body wake.The set of small 3D perturbations:

Constitutes a system of multiple spatial and temporal scales;Includes all the processes of the perturbative Navier-Stokes equa-tions (linearized convective transport, molecular diffusion, linearizedvortical stretching);Leaves aside the nonlinear interaction among the different scales;

The perturbative evolution is ruled by the initial-value problemassociated to the Navier-Stokes linearized formulation.



ResultsConclusions




To understand how spectral representation can effectively highlightthe nonlinear interaction among different scales;

To quantify the degree of generality on the value of the energy decayexponent of the inertial range;






ResultsConclusions










ResultsConclusions





Different typical perturbed shear systems: plane Poiseuille flowand bluff-body wake.

The set of small 3D perturbations:Constitutes a system of multiple spatial and temporal scales;Includes all the processes of the perturbative Navier-Stokes equa-tions (linearized convective transport, molecular diffusion, linearizedvortical stretching);Leaves aside the nonlinear interaction among the different scales;




ResultsConclusions






Constitutes a system of multiple spatial and temporal scales;

Includes all the processes of the perturbative Navier-Stokes equa-tions (linearized convective transport, molecular diffusion, linearizedvortical stretching);Leaves aside the nonlinear interaction among the different scales;




ResultsConclusions






Constitutes a system of multiple spatial and temporal scales;Includes all the processes of the perturbative Navier-Stokes equa-tions (linearized convective transport, molecular diffusion, linearizedvortical stretching);

Leaves aside the nonlinear interaction among the different scales;




ResultsConclusions










ResultsConclusions










ResultsConclusions


Spectral analysis through initial-value problem

The transient linear dynamics offers a great variety of differentbehaviours (Scarsoglio et al., Stud. Appl. Math., 2009; Scarsoglioet al., Phys. Rev. E, 2010):

⇒ Understand how the energy spectrum behaves;Is the linearized perturbative system able to show a power-law scaling for the energy spectrum in an analogous way tothe Kolmogorov argument?We determine the energy decay exponent of arbitrary longitu-dinal and transversal perturbations in their asymptotic statesand we compare it with the -5/3 Kolmogorov decay.



ResultsConclusions



The transient linear dynamics offers a great variety of differentbehaviours (Scarsoglio et al., Stud. Appl. Math., 2009; Scarsoglioet al., Phys. Rev. E, 2010):⇒ Understand how the energy spectrum behaves;

Is the linearized perturbative system able to show a power-law scaling for the energy spectrum in an analogous way tothe Kolmogorov argument?We determine the energy decay exponent of arbitrary longitu-dinal and transversal perturbations in their asymptotic statesand we compare it with the -5/3 Kolmogorov decay.



ResultsConclusions



The transient linear dynamics offers a great variety of differentbehaviours (Scarsoglio et al., Stud. Appl. Math., 2009; Scarsoglioet al., Phys. Rev. E, 2010):⇒ Understand how the energy spectrum behaves;Is the linearized perturbative system able to show a power-law scaling for the energy spectrum in an analogous way tothe Kolmogorov argument?

We determine the energy decay exponent of arbitrary longitu-dinal and transversal perturbations in their asymptotic statesand we compare it with the -5/3 Kolmogorov decay.



ResultsConclusions



The transient linear dynamics offers a great variety of differentbehaviours (Scarsoglio et al., Stud. Appl. Math., 2009; Scarsoglioet al., Phys. Rev. E, 2010):⇒ Understand how the energy spectrum behaves;Is the linearized perturbative system able to show a power-law scaling for the energy spectrum in an analogous way tothe Kolmogorov argument?We determine the energy decay exponent of arbitrary longitu-dinal and transversal perturbations in their asymptotic statesand we compare it with the -5/3 Kolmogorov decay.



ResultsConclusions

Mathematical frameworkMeasure of the growthVariety of the transient linear dynamics

Perturbation scheme

Linear three-dimensional perturbative equations in terms of veloc-ity and vorticity (Criminale & Drazin, Stud. Appl. Math., 1990);

Laplace-Fourier transform in x and z directions, α complex, γ real.



ResultsConclusions


Perturbation scheme

Linear three-dimensional perturbative equations in terms of veloc-ity and vorticity (Criminale & Drazin, Stud. Appl. Math., 1990);Laplace-Fourier transform in x and z directions, α complex, γ real.



ResultsConclusions


Perturbation scheme

Linear three-dimensional perturbative equations in terms of veloc-ity and vorticity (Criminale & Drazin, Stud. Appl. Math., 1990);Laplace-Fourier transform in x and z directions, α complex, γ real.



ResultsConclusions


Perturbative equations

Perturbative linearized system:

∂2v∂y2

− (k2 − α2i + 2iαrαi )v = Γ

∂Γ

∂t= (iαr − αi )(

d2Udy2

v − UΓ) +1

Re[∂2Γ

∂y2− (k2 − α2

i + 2iαrαi )Γ]

∂ωy

∂t= −(iαr − αi )Uωy − iγ

dUdy

v +1

Re[∂2ωy

∂y2− (k2 − α2

i + 2iαrαi )ωy ]

The transversal velocity and vorticity components are v and ωy

respectively, Γ is defined as Γ = ∂x ωz − ∂z ωx .

Initial conditions: symmetric and asymmetric inputs;Boundary conditions: (u, v , w)→ 0 as y → ±∞ and at walls.



ResultsConclusions




∂2v∂y2

− (k2 − α2i + 2iαrαi )v = Γ

∂Γ

∂t= (iαr − αi )(

d2Udy2

v − UΓ) +1

Re[∂2Γ

∂y2− (k2 − α2

i + 2iαrαi )Γ]

∂ωy


dUdy

v +1

Re[∂2ωy

∂y2− (k2 − α2

i + 2iαrαi )ωy ]



Initial conditions: symmetric and asymmetric inputs;

Boundary conditions: (u, v , w)→ 0 as y → ±∞ and at walls.



ResultsConclusions




∂2v∂y2

− (k2 − α2i + 2iαrαi )v = Γ

∂Γ

∂t= (iαr − αi )(

d2Udy2

v − UΓ) +1

Re[∂2Γ

∂y2− (k2 − α2

i + 2iαrαi )Γ]

∂ωy


dUdy

v +1

Re[∂2ωy

∂y2− (k2 − α2

i + 2iαrαi )ωy ]






ResultsConclusions



−1 −0.5 0 0.5 1

0

0.2

0.4

0.6

0.8

1

y

U(y)asym inputsym input

Channelflow

−5 0 50.4

0.6

0.8

1

y

sym input asym input

Wake flow

U(y)Re=30,x

0=10

U(y)Re=100,x

0=10

U(y)Re=100,x

0=50

U(y)Re=30,x

0=50




ResultsConclusions


Perturbation energy

Kinetic energy density e:

e(t ;α, γ) =12

∫ +yd

−yd

(|u|2 + |v |2 + |w |2)dy

Amplification factor G:

G(t ;α, γ) =e(t ;α, γ)

e(t = 0;α, γ)

Temporal growth rate r :

r(t ;α, γ) =|dG/dt |

G

Angular frequency (pulsation) ω (Whitham, 1974):

ω(t ;α, γ) =dϕ(t)

dt, ϕ time phase



ResultsConclusions


Perturbation energy


e(t ;α, γ) =12

∫ +yd

−yd

(|u|2 + |v |2 + |w |2)dy


G(t ;α, γ) =e(t ;α, γ)

e(t = 0;α, γ)



G



dt, ϕ time phase



ResultsConclusions


Perturbation energy


e(t ;α, γ) =12

∫ +yd

−yd

(|u|2 + |v |2 + |w |2)dy


G(t ;α, γ) =e(t ;α, γ)

e(t = 0;α, γ)



G



dt, ϕ time phase



ResultsConclusions


Perturbation energy


e(t ;α, γ) =12

∫ +yd

−yd

(|u|2 + |v |2 + |w |2)dy


G(t ;α, γ) =e(t ;α, γ)

e(t = 0;α, γ)



G



dt, ϕ time phase



ResultsConclusions


Relevant transient behaviours

0 150 300 450

10−2

100

101

t

G10 40 70

10−10

100

|G−

<G

>|

Channel flow Re=500 k=1

φ=0

φ=π/4

φ=π/2

1000 3000 40000

1

2x 10

4

t

G

k=2

k=1

2.5<k<1000

k=1.5 Channel flow Re=10000

sym φ= π/2

0 150 450

10−5

100

104

t

G 50 100 20010

−8

10−1

|G−

<G

>|

asym

sym

φ=π/2

Wake flow Re=100 x

0=10 k=0.7

100 200 3000

2

4

t

G

0.7 ≤ k ≤ 0.9

3 ≤ k ≤ 70

0.9 ≤ k ≤ 2.5

WakeflowRe=100x

0=50

symφ=π/4



ResultsConclusions

Perturbative system featuresSpectral distributions

Spectral representation

The energy spectrum is evaluated as the wavenumber distributionof the perturbation kinetic energy density, G(k);

The spectral representation is determined by comparing theenergy of the waves when they are exiting their transientstate;

Every perturbation has its characteristic transient exiting time, Te;

The asymptotic condition is reached when the perturbative waveexceeds the transient exiting time, Te, that is when r ∼ const issatisfied for stable and unstable waves.

Scarsoglio, De Santi & Tordella, ETC XIII, 2011.



ResultsConclusions










ResultsConclusions










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ResultsConclusions


Energy G(k) at the asymptotic state (r ∼const)

100

101

102

103

10−8

10−6

10−4

10−2

k

G

A φ=π/2A φ=π/4A φ=0S φ=π/2S φ=π/4S φ=0

Channel flow Re=500

k−2

k−5/3

100

101

102

103

10−5

100

105

k

G

A φ=π/2A φ=π/4A φ=0S φ=π/2S φ=π/4S φ=0E

uu

Evv

Eww

k−2Channel flowRe=10000

Moser et al., 1999 Re=12390

k−5/3

100

101

102

103

104

10−10

10−5

100

105

k

G

A φ=π/2

A φ=π/4

A φ=0

S φ=π/2

S φ=π/4

S φ=0

E(k) unfE(k) fil

Wake flow Re=30 x

0=50

k−5/3

k−5/3

k−2

Cerutti &Meneveau, 2000Re=572080

10−1

100

101

102

103

10−4

10−8

100

104

k

G

A φ=π/2A φ=π/4A φ=0S φ=π/2S φ=π/4S φ=0E

u

Ev

Wake flowRe=100 x

0=10 k−2

k−4Ong &Wallace,1996Re=3900

k−5/3

k−5/3



ResultsConclusions


Pulsation ω(k) at the asymptotic state (r ∼const)

100

101

102

103

10−1

100

101

102

103

k

ω

A φ=π/4A φ=0

S φ=π/4S φ=0

Channel flowRe=500

lab Nishioka et al. (1975)Re=3000, 4000, 5000theor Ito (1974)Re=3000, 4000, 5000lab/theor Asai & Floryan (2006)Re=4000, 5000

k

100

101

102

103

10−1

100

101

102

103

k

ω

A φ=π/4A φ=0

S φ=π/4S φ=0

Channel flowRe=10000

lab Nishioka et al. (1975)Re=6000, 7000theor Ito (1974)Re=6000, 8000lab/theor Asai & Floryan (2006)Re=6000

k

10−1

100

101

102

103

10−1

100

101

102

103

k

ω

A φ=π/4A φ=0S φ=π/4S φ=0

Wake flowRe=30 x

0=50

loc Pier (2003)loc Giannetti & Luchini (2007)

NMA Zebib (1987)glob Giannetti & Luchini (2007)

k

10−1

100

101

102

103

10−1

100

101

102

103

k

ω

A φ=π/4A φ=0S φ=π/4S φ=0

Wake flowRe=100 x

0=10

loc Pier (2003)loc and glob Giannetti & Luchini (2007)

lab Williamson (1989)

DNS Pier (2003) LSA Barkley (2006) LSA Noack et al. (2003)lab Norberg (1994)lab Nishioka & Sato (1978)

k



ResultsConclusions


Transient exiting time Te(k)

100

101

102

103

10−4

10−2

100

102

104

k

Te

A φ=π/2A φ=π/4A φ=0S φ=π/2S φ=π/4S, φ=0

Channel flowRe=500

k−5/3

100

101

102

103

10−2

100

102

104

106

k

Te


Channel flowRe=10000

k−5/3

10−1

100

101

102

103

10−4

10−2

100

102

104

k

Te


Wake flow Re=30 x

0=50

k−5/3

10−1

100

101

102

103

10−4

10−2

100

102

104

k

Te

A φ=π/2A φ=π/4A φ=0

S φ=π/2S φ=π/4S φ=0

Wake flowRe=100 x

0=10

k−5/3



ResultsConclusions

Conclusions

Concluding remarks

Spectrum determined by evaluating the energy of the waves whenthey are exiting their transient state;

Regardless the symmetry and obliquity of perturbations, there ex-ists an intermediate range of wavenumbers in the spectrumwhere the energy decays with the same exponent observedfor fully developed turbulent flows (−5/3), where the nonlinearinteraction is considered dominant;Scale-invariance of G and Te at different (stable and unstable)Reynolds numbers and for different shear flows;The spectral power-law scaling of inertial waves is a generaldynamical property which encompasses the nonlinear inter-action;The −5/3 power-law scaling in the intermediate range seemsto be an intrinsic property of the Navier-Stokes solutions.



ResultsConclusions

Conclusions

Concluding remarks

Spectrum determined by evaluating the energy of the waves whenthey are exiting their transient state;Regardless the symmetry and obliquity of perturbations, there ex-ists an intermediate range of wavenumbers in the spectrumwhere the energy decays with the same exponent observedfor fully developed turbulent flows (−5/3), where the nonlinearinteraction is considered dominant;

Scale-invariance of G and Te at different (stable and unstable)Reynolds numbers and for different shear flows;The spectral power-law scaling of inertial waves is a generaldynamical property which encompasses the nonlinear inter-action;The −5/3 power-law scaling in the intermediate range seemsto be an intrinsic property of the Navier-Stokes solutions.



ResultsConclusions

Conclusions

Concluding remarks

Spectrum determined by evaluating the energy of the waves whenthey are exiting their transient state;Regardless the symmetry and obliquity of perturbations, there ex-ists an intermediate range of wavenumbers in the spectrumwhere the energy decays with the same exponent observedfor fully developed turbulent flows (−5/3), where the nonlinearinteraction is considered dominant;Scale-invariance of G and Te at different (stable and unstable)Reynolds numbers and for different shear flows;

The spectral power-law scaling of inertial waves is a generaldynamical property which encompasses the nonlinear inter-action;The −5/3 power-law scaling in the intermediate range seemsto be an intrinsic property of the Navier-Stokes solutions.



ResultsConclusions

Conclusions

Concluding remarks

Spectrum determined by evaluating the energy of the waves whenthey are exiting their transient state;Regardless the symmetry and obliquity of perturbations, there ex-ists an intermediate range of wavenumbers in the spectrumwhere the energy decays with the same exponent observedfor fully developed turbulent flows (−5/3), where the nonlinearinteraction is considered dominant;Scale-invariance of G and Te at different (stable and unstable)Reynolds numbers and for different shear flows;The spectral power-law scaling of inertial waves is a generaldynamical property which encompasses the nonlinear inter-action;

The −5/3 power-law scaling in the intermediate range seemsto be an intrinsic property of the Navier-Stokes solutions.



ResultsConclusions

Conclusions

Concluding remarks

Spectrum determined by evaluating the energy of the waves whenthey are exiting their transient state;Regardless the symmetry and obliquity of perturbations, there ex-ists an intermediate range of wavenumbers in the spectrumwhere the energy decays with the same exponent observedfor fully developed turbulent flows (−5/3), where the nonlinearinteraction is considered dominant;Scale-invariance of G and Te at different (stable and unstable)Reynolds numbers and for different shear flows;The spectral power-law scaling of inertial waves is a generaldynamical property which encompasses the nonlinear inter-action;The −5/3 power-law scaling in the intermediate range seemsto be an intrinsic property of the Navier-Stokes solutions.


Documents

Power-law decay of the energy spectrum in linearized perturbed … · 2011. 7. 20. · Introduction Initial-value problem formulation Results Conclusions Power-law decay of the energy