Does the Kolmogorov scaling bridge hydrodynamic linear stability … · 2011. 11. 20. · Francesca...

Does theKolmogorov

scaling bridgehydrodynamiclinear stability

and turbulence?

S. Scarsoglio,F. De Santi,D. Tordella

Introduction

Motivation andgeneral aspects

Initial-valueproblem

Mathematicalframework

Measure of thegrowth

Linear transientdynamics

Results

Perturbativesystem features

Spectraldistributions

Conclusions

Does the Kolmogorov scaling bridgehydrodynamic linear stability and turbulence?

Stefania ScarsoglioFrancesca De Santi Daniela Tordella

Department of Aeronautics and Space Engineering, Politecnico di Torino, Torino, Italy

American Physical Society

64th Annual DFD Meeting

November 20-22, 2011

Baltimore, Maryland

S. Scarsoglio, F. De Santi, D. Tordella Does the Kolmogorov scaling bridge hydrodynamic linear stability and turbulence?

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Energy spectrum in fully developed turbulence

• Phenomenology of turbulence Kolmogorov 1941:−5/3 power-law for the energy spectrum over the inertial range;

• Common criterium for the production of a fully developed turbu-lent field to verify such a scaling (e.g. Frisch, 1995; Sreenivasan& Antonia, ARFM, 1997; Kraichnan, Phys. Fluids, 1967).

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

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Energy spectrum in fully developed turbulence

• Phenomenology of turbulence Kolmogorov 1941:−5/3 power-law for the energy spectrum over the inertial range;

• Common criterium for the production of a fully developed turbu-lent field to verify such a scaling (e.g. Frisch, 1995; Sreenivasan& Antonia, ARFM, 1997; Kraichnan, Phys. Fluids, 1967).

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

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

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 effectivelyhighlight the nonlinear interaction among different scales;

• To quantify the degree of generality on the value of the energydecay exponent 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;• 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.

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

• 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;

• Leaves aside the nonlinear interaction among the different scales;

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

tions;

• Leaves aside the nonlinear interaction among the different scales;

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Spectral analysis through initial-value problem

• The linear transient dynamics offers a great variety of very differ-ent behaviors (Scarsoglio et al., 2009, 2010, 2011):

⇒ 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 pertur-bations in their asymptotic states and we compare it with the-5/3 Kolmogorov decay.

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

• The linear transient dynamics offers a great variety of very differ-ent behaviors (Scarsoglio et al., 2009, 2010, 2011):⇒ Understand how the energy spectrum behaves;

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Perturbation scheme

• Linear 3D perturbative equations in terms of velocity and vorticity(Criminale & Drazin, Stud. Appl. Math., 1990);

• Laplace-Fourier (wake) and Fourier-Fourier (channel) transformin the x and z directions.

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Perturbation scheme

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Perturbation scheme

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Perturbative equations

• Perturbative linearized system:

∂y2− k2v = Γ

∂t= iα

dy2v − U Γ

(∂2Γ

∂y2− k2Γ

)∂ωy

∂t= −iαUωy − iγ

(∂2ωy

∂y2− k2ωy

)The transversal velocity and vorticity components are v and ωy

respectively, Γ is defined as Γ = ∂xωz − ∂zωx.

• Initial conditions: symmetric and asymmetric inputs;

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

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

∂y2− k2v = Γ

∂t= iα

dy2v − U Γ

(∂2Γ

∂y2− k2Γ

)∂ωy

(∂2ωy

∂y2− k2ωy

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

∂y2− k2v = Γ

∂t= iα

dy2v − U Γ

(∂2Γ

∂y2− k2Γ

)∂ωy

(∂2ωy

∂y2− k2ωy

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

−1 −0.5 0 0.5 1

U(y)asym inputsym input

Channelflow

−5 0 50.4

sym input asym input

Wake flow

U(y)Re=30,x

U(y)Re=100,x

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Perturbation energy

• Kinetic energy density e:

e(t;α, γ) =1

∫ +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) =log[e(t;α, γ)]

• Angular frequency (pulsation) ω:

ω(t; y = y0, α, γ) =dϕ(t; y = y0, α, γ)

dt, ϕ time phase

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Perturbation energy

e(t;α, γ) =1

∫ +yd

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

e(t = 0;α, γ)

ω(t; y = y0, α, γ) =dϕ(t; y = y0, α, γ)

dt, ϕ time phase

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Perturbation energy

e(t;α, γ) =1

∫ +yd

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

e(t = 0;α, γ)

ω(t; y = y0, α, γ) =dϕ(t; y = y0, α, γ)

dt, ϕ time phase

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Perturbation energy

e(t;α, γ) =1

∫ +yd

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

e(t = 0;α, γ)

ω(t; y = y0, α, γ) =dϕ(t; y = y0, α, γ)

dt, ϕ time phase

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Relevant transient behaviors

0 1000 3000 40000

k=2k=1.5

k=1Channel flow

Re=10000Sym ic, φ=π/2

2.5<k<1000

0 500 1000 1500

10−10

10−5

Wake flow Asym ic φ=0 Re=100 x

0.45≤ k≤ 500

0 10 20 300

Sym φ=0Asym φ=0Sym φ=π/4Asym φ=π/4

Channel flow k=15Re=10000 y

0 500 1000 1500

Sym φ=0Asym φ=0Sym φ=π/4Asym φ=π/4

Wake flowRe= 100x

0=50 k=0.7

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Spectral representation

• The energy spectrum is evaluated as the wavenumber distributionof the amplification factor, G(k);

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

• Every perturbation has a characteristic transient exit time, Te;

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

Scarsoglio, De Santi & Tordella, submitted to Phys. Rev. Lett., 2011.

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

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

10−4

10−8

k−5/3 k−5/3

Channel flowRe=500

10−5

Channel flowRe=10000

k−5/3k−2

k−5/3

Present results (Re values: see panels)Asym ic, φ=π/2 Asym ic, φ=π/4 Asym, φ=0 Sym ic, φ=π/2 Sym ic, φ=π/4 Sym, φ=0

(Ex, E

z) Moser et al. (MKM), 1999, Re=12390, DNS

Saddoughi & Veeravalli (SV), 1993, Reδ*=4994, lab data

(Eu, E

v) Ong & Wallace (OW), 1996, Re=3900, lab data

Cerutti & Meneveau (CM), 2000, Re=572080, DNS & LES

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

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

10−8

10−4

k−5/3

k−2Wake flowRe=30 x

10−5

k−5/3

Wake flowRe=100 x

k−5/3

Present results (Re values: see panels)Asym ic, φ=π/2 Asym ic, φ=π/4 Asym, φ=0 Sym ic, φ=π/2 Sym ic, φ=π/4 Sym, φ=0

(Ex, E

z) Moser et al. (MKM), 1999, Re=12390, DNS

Saddoughi & Veeravalli (SV), 1993, Reδ*=4994, lab data

(Eu, E

v) Ong & Wallace (OW), 1996, Re=3900, lab data

Cerutti & Meneveau (CM), 2000, Re=572080, DNS & LES

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Transient exit time Te(k)

10−2

k−5/3

Channel flowRe=500

k−5/3

k−2Channel flowRe=10000

10−2

k−5/3

Wake flow Re=30 x

Wake flowRe=100 x

k−5/3

Does theKolmogorov

and turbulence?

Introduction

Results

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 −5/3 spectral power-law scaling of inertial waves seemsto be a general and intrinsic dynamical property of the NSsolutions encompassing the nonlinear interaction.

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Concluding remarks

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Concluding remarks

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Concluding remarks

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Coming next...

• Analysis of the perturbation transient dynamics in the 2Dand 3D boundary layer (W. O. Criminale, University of Wash-ington);

• Analytical integration of the kinetic energy equation basedon the perturbed velocity and vorticity field (G. Staffilani,MIT)⇒ Study of the intermediate and long term asymptotics.

Does theKolmogorov

and turbulence?

Introduction

Results

Conclusions

Coming next...

• Analysis of the perturbation transient dynamics in the 2Dand 3D boundary layer (W. O. Criminale, University of Wash-ington);

• Analytical integration of the kinetic energy equation basedon the perturbed velocity and vorticity field (G. Staffilani,MIT)⇒ Study of the intermediate and long term asymptotics.

Does the Kolmogorov scaling bridge hydrodynamic linear stability … · 2011. 11. 20. · Francesca...

Documents

Transcript of Does the Kolmogorov scaling bridge hydrodynamic linear stability … · 2011. 11. 20. · Francesca...