Suppression of turbulence by mean flows
in two-dimensional fluids
Michael Shats
Research School of Physical Sciences and Engineering
The Australian National UniversityCanberra
in collaboration withHua Xia, Horst Punzmann, David Byrne (ANU, Canberra, Australia)
Gregory Falkovich (Weizmann Institute of Science, Rehovot, Israel)

OutlineIntroduction to 2D fluid turbulence
Quasi-2D fluid experiment
Suppression of turbulence by mean flow
Self-generated flow
Externally imposed flow

SummaryIntroduction

Energy cascade in 3D turbulence

Self-similar cascade of eddies in a turbulent flow
Kolmogorov: energy cascade is the self-similar‘fractal’ cascade process which transfersenergy from the larger eddies, where it can not
be dissipated, to the smaller ones where it can.
Inertial range

Label an ‘eddy’ by a velocity increment
δul across a distance l: δul = u (x + l ) − u (x )u1ut1Structure functions of the nth order Sn :1u||2u||1u2nn

2

S n (l ) = (δul ) = (u ( x + l ) − u ( x ))ut2E.g., root-mean-square velocityof eddies of scale l = S 2 (l ) =(δul )2Main results of the Kolmogorov 1941 theory
Four-fifths law.
In the limit of infinite Reynolds number, the third order (longitudinal) structure functionof homogeneous isotropic turbulence, evaluated for increments l small compared to theintegral scale, is given in terms of the mean energy dissipation per unit mass ε (assumedto remain finite and nonvanishing) by

S 3 ( l ) = δ v ||3 (r , l ) = −4εl
5Kolmogorov energy spectrum.
The second order structure function follows the l2/3 law :S 2 (l ) = C 2ε 2 / 3l 2 / 3S p (l ) = C p ε p / 3l p / 3−nIf the energy spectrum is a power-law E (k ) = k

then the velocity field has the second-order spatial
structure function which is also a power-law:

E (k ) ~ ε 2 / 3 k −5 / 3(v (r + l ) − v (r ))2 ∝ l n −1
This implies a k -5/3 law for the energy spectrum:

2D turbulence
In a 2D flow both energy and enstrophy are conserved

100

Dual cascade
Energy: inverse cascade, Enstrophy: forward cascade

10-5/3E (k ) = Ck ε 2 / 3 k −5 / 3

k1

E (k ) = Cω ε ω

2/3k −3

R. Kraichnan (1967):

0.01

0.001
10kdis100

Enstrophy
cascadeEnergycascade0.1kinjk

Ω=-321
ωdV2 V∫

enstrophy
Kolmogorov law in 2D:kν

1000

S 3 ( l ) = δ v ||3 (r , l ) =Opposite to 3D, energy flows from smaller to larger structures.
The basis for self-organization

3εl2

Spectral condensation of turbulence in 2D fluids
The maximum of the energy spectrum lies in the low-k range, at kE, and in theabsence of the energy dissipation at large scales can not be constant in time since it
accumulates spectral energy kE = f (ε , t )System size < dissipation scale

At low dissipation in a bound system, at kdis << ksys
spectral energy condenses into large (system size)
coherent structure(s):dipole (periodic boundary),or monopole (no-slip boundary, experiments)

10-5/3

k1

0.1

0.01

Enstrophy
cascade)Energycascade(100

Condensate

Damping for large scales (e.g.lineardamping) μ stabilizes the maximum of the1/ 2spectrum at the scale
kdis ≈ μ 3 εTheory: Kraichnan, 1967- qualitative => Bose condensate]
Experiments: Sommeria (1986), Paret&Tabeling (1998)
Shats et al (2005)

0.001
10ksy skdis

100kinjk-31000

Modelling: Hossain (1983), Smith&Yakhot (1993)…
van Heijst, Clercx, Molenaar (2004-2006), Chertkov et al. (2007)

Turbulence in two dimensions
2D approximation relevant to systems in which a flow velocity in one of the dimensions can beneglected:
2D Navier-Stokes systems: Thin layers of fluids. Rotating fluids. Planetary atmospheres. Oceanic flows
Stratified layers. Conducting fluids in magnetic field. Magnetically confined plasmas.
Antarctic
Circumpolar
Current

Jupiter’s
zonal flows
GRS

Planetary atmospheresare dominatedby turbulent structures(cyclones, zonalwinds, etc)

Earth: atmospheric zonal flows

Plasma-fluid similarity
The onset of a mean vortex flow (spectral condensation)
seems to coincide with the reduction in the broad-band turbulence
During monopole formation in
2D bounded turbulence

1101010During L-H transition in the transport
Zonal flow
barrier region
1E(k)/EtotalL-1-2

10Inverse-2cascadefP(Vf)-4

10-3

Condensate

f-5-3. 610-4kikc

10fH-610-3. 6-5noise-8

100200400600 k (m-1)

Shats, Xia, Punzman, Phys. Rev. E, 046409 (2005)

0.1

110f (kHz)100Xia, Shats, Punzman, Phys. Rev. Lett. 255003 (2006)

Similarity with the L-H transition physics

Shear flow suppression of turbulence
If the eddy is isolated it stretches into the shapeindicated by the gray shaded curve. In turbulence,the eddy loses coherence in a coherence length,represented as a breakup into two eddies. The lossof coherence reduces the y scale relative to that of
the reference eddy.
P. W. Terry, Rev. Mod. Phys. (2000)

The condition for the turbulence suppression

τ <τseShear suppression model• has its origin in plasma physics;• substantial (indirect) evidence in plasma experiments;• vortex breaking has not been observed in plasma;• not clear if it is universal or uniquely 2D effect;• shear suppression in fluids is never observed;•Biglari, Diamond, Terry, Phys. Fluids B (1990) 1• shear in fluids only generates new instabilities •Shaing, Crume, Houlberg, Phys. Fluids B (1990)(turbulent boundary layers etc.)•P.W. Terry, Phys. Plasmas 7 (2000) 1653

•D.C. Montgomery, Phys. Plasmas 7 (2000) 4785.
•P.W. Terry, Phys. Plasmas 7 (2000) 4787;

Quasi-2D fluid experiment

Quasi 2D fluid: experimental setup
Stratified layer fluid layers

JTurbulence is forced in top layer

Imaging particles (polyamid ρ=1.04,50 um) suspended in the top layerilluminated using laser sheet (1mmthick) Perspex cell 0.3 x 0.3 m2
 Bottom layer: isolator Fluorinert FC-77 (resist. = 2x1015 Ohm cm; SG = 1.78)
 Top layer: electrolyte NaCl solution (SG = 1.04)

2D turbulence forcing
Current in electrolyte interacts withvertical spatially varying B toproduce J x B force acting on fluid
parcels.
900 vortices produce turbulent flow

Video
CameraCylindricallens

BjLaser

Laser sheet

Lighter fluid
Heavier fluid

Magnets

Electrodes

Measuredmagnetic
field

Matrix of 900 magnets (0.35 T)
submerged in the bottom layer

Particle streaks in 2D turbulenceParticle Image Velocimetry
Novel 2-colour particle
image velocimetry :velocity fields derived froma single camera image –use still cameras with highresolution sensors

VideoCamera

BjMagnets

Cylindricallens
Laser sheet

Lighter fluid
Heavier fluid

Laser 2

Laser 1

Electrodes

Suppression of turbulence by mean flow(a) Self-generated flow (condensate)(b) Externally imposed flow

M.G. Shats, H. Xia, H. Punzmann and G. Falkovich
Suppression of turbulence by self-generated and imposed flows,
Physical Review Letters, 99, 164502 (2007)

Evolution turbulence during spectral
condensation

t=9s

t=3st = 60 st = 25 s

Spectra of total velocity
E k (a.u.)

(a)

1.2

Total spectra include mean andfluctuating velocity components

0.8

N

0.4

Etot (k ) = 1 N ∑ F (V )F * (V )
(b)

(c)

0n =1t (s)
-5

020

40

601080E(k) 100-4

Before monopole formation

10E(k)

After monopole formation-510-610

Average of 100 instantaneous
2D Fourier Transform
10spectra,ΔT = 8 s~k-3-610-7-710-810-810-91010100kf

1000-1k (m )10100kf

1000-1k (m )k -3 power law is not due to the turbulent cascade, but rather due to coherent structures[M. Chertkov, C. Connaughton et al. (PRL 2007)]

Spectra of turbulent velocity fields
Average of 100 instantaneous spectraof turbulent velocity fields,
mean flow subtracted

Nn =1) ()1.0-610(E fl (k ) = 1 N ∑ F V − V N F * V − V N
0.8Efl ( k)before

0.6
0.4
0.2

-7

0.0
0.0
1.0

10

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.8

after

0.6

k (m-1)

-8

100.4

kf

100

0.2
0.0
0.0

Large scales with k < 150 m-1 suppressed

Mean flow and its shear
1.00.01θVθ ( m/s)

0.8

0.008

0.6
0.006

r
0.4

0.004

0.2

0.002

(a)
0

0.0
0

10

0.2

0.4

0.6

0.8 ρ

10.0s (1/ ms)ωs = l-10

0.6

0.8

1.0dΩ ⎡ 1 dVθ Vθ ⎤= l⎢− 2 ⎥ = sldr⎣ r dr r ⎦Average shear of s = 15 m-1s-1
suppresses eddies with k = π / l < 160 m-1-20(b)-30
0.20.4

Ω = Vθ r

000.2

0.4

0.6

0.8

ρ

1Suppression means reduction in the eddy lifetime

Scale reduction by shear in turbulence
10-6

Efl ( k)beforeShear suppression condition-7

10is satisfied in the middle ofthe energy inertial range
afterk (m-1)-810100kf• Development of spectral condensate in the form of large vortex produces shear flow
• Shear is sufficient to affect eddies with l >20 mm and lifetimes ~ (1-2) s• Broadband spectrum of the turbulent velocity fluctuations is reduced at k < 160m-1

The first experimental evidence of the shear turbulence
suppression in fluids

Suppression of turbulence by mean flow(a) Self-generated flow (condensate)(b) Externally imposed flow

Flow externally imposed on turbulence
Nlargemagnet

Large magnet generates large vortex
superimposed onto turbulence.
The monopole vortex has slow decay time t>40sec.

STurbulence with externallyimposed mean flow

Particle streak photoof turbulent flow

Imposed mean flow
externally forced

self-generated

0.02(c)Vθ (m/s)

(a)

0.01

r(m)
r(m)

00

0.02

0.04

Self-organized

00.02

0.06

Externally imposed

1.1

Imposed flow can affectturbulence in two ways:
Shearing
Sweeping

7

If the flow velocity is large,
sweeping seems to play adominant role

Shearing rate at theforcing scalel 2 dΩ
S1 dr

0.2

Sweeping rate at
the forcing scale
V l Vθsw = ω swτ e = θ=l S1 S10.75

s = ω sτ e =0.1

Imposed flow affects scales
down to the forcing scale

Sweeping due to mean flow
Mean flow sweeps forcing scale vortices relative to magnets
Sweepingbecomes important if sw (l) >1
SNNS

Eddy life timeτ e = l / δu (l ) = l / S1

SNN

SShearing acts more efficiently on large scales:s = ω sτ e ~ l 5 / 3

Sweeping is more efficient on small scales:sw = ω swτ e =Eddy life time:Vθ∝ l −1/ 3S1

τ e = l / δu (l ) = l / S1

Imposed flow reduces turbulence
Externally imposed flow leads to turbulence reduction but retains k-5/3 scaling

Energy spectra

3rd order structure function-73 33S3 ( 10 m / s )-5

10Efl (k)no flow-610~k-7

10~k-5 / 3-5 / 3

10withexternal
flow

1

with
externalflow-8

no flow

2-1

k (m )

0-9

10

10100

kf

01000

0.02 0.04 0.06 0.08l (m)The energy flux through inertial range ε = −2 / 3 S3 (l ) / l

ε is constant for all scales lε is reduced by a factor of ~10 in the presence of the strong flow
Mean flow reduces energy injected into turbulent cascade ε, leads to E (k ) drop:E ( k ) = C k ε 2 / 3 k −5 / 3

Summary• Mean flows (both self-generated and imposed) suppress turbulence in 2D flows• Weaker flow: shearing of larger turbulent eddies in the inertial range distorts theE (k) spectrum• Stronger flow affects energy injection at the forcing scale via shearing and sweeping• The inverse energy cascade persists in the presence of mean flow
• Energy spectra of turbulence do not change in the presence of strong flows• No evidence in support of “eddy-breaking” in the presence of shear