Proceedings of ITC/ISHW2007

Suppression of turbulence by mean flows in two-dimensional fluids
Michael SHATS1) , Hua XIA1) , Horst PUNZMANN1) and Gregory FALKOVICH2)
1)

Australian National University, Canberra ACT 0200, Australia
2)
Weizmann Institute of Science, Rehovot 76100, Israel(Received 15 October 2007)

A review of recent experimental studies of turbulence suppression by mean flows in quasi-two-dimensional
fluids is presented. Large-scale mean flows develop during spectral condensation of 2D turbulence as a resultof the inverse energy cascade in spatially bounded flow. The spectral energy which is accumulated at thelargest scale supports the mean flow which in turn aﬀects turbulence. We show that such a flow can reduce theenergy flux in the inverse energy cascade range via shearing and sweeping of the turbulent eddies. The former
mechanism is more eﬃcient at larger scales, while the latter acts on the smaller scales. Similar suppression of
turbulence has been found in the presence of externally imposed flows. Turbulent (inverse energy) cascade is
reduced in the presence of imposed flow, but still supports Kolmogorov-Kraichnan k −5/3 power law spectrum inthe energy range.
Keywords: two-dimensional fluids, turbulence, mean flow, shear decorrelation, inverse energy cascade
DOI: 10.1585/pfr.1.001

1Introduction

The idea of turbulence suppression in the presence of abackground sheared flow has its origin in the physics ofmagnetically confined plasma. It was proposed in 1990
[1, 2] as a simple hydrodynamic model aimed at explaining
turbulence reduction near transport barriers which form inplasma in the so-called high (H) confinement regime [3].
Since then this concept has received wide recognition in
the plasma literature (e.g. [4, 5, 6, 7]). The model of theshear suppression is supported by substantial indirect experimental evidence, namely, the correlation between theonset of the strong sheared flows in plasma and the reduction in the turbulent transport during transitions to improved confinement.
The main mechanism of the shear suppression is asfollows. When a turbulent eddy is placed in a stable laminar flow whose velocity varies in the direction perpendicular to the flow direction, it becomes stretched and distorted.
The shear suppression can be viewed as the reduction in theeddy’s lifetime. It occurs when the inverse shearing rateτ s ≈ ω−1s becomes shorter than the eddy turnover time, or
its lifetime, τe whatever is shorter, providing that the interaction time between turbulence and flow is longer thanother time scales. A reduction in the spectral power in thepresence of a shear flow is due to the shortened correlation
time of eddies. Dimensional scaling analysis which takes
into account turbulent diﬀusion, shows that this shortened
correlation time τes is related to the shear straining time τ s
2/3
and the eddy turnover time τ e as [8]: τes = τ1/3
e τ s . The
theory of the shear suppression is considered an extensionof the rapid distortion theory into the nonlinear regime [8].
author’s e-mail: Michael.Shats@anu.edu.au

Despite its wide recognition in plasma physics, and attempts to extend its application to other fields (see e.g. [9]),this hydrodynamic phenomenon is not a familiar one in
fluids [10]. This fact has triggered arguments both against[11] and in favor [12] of the shear turbulence suppression
mechanism, casting a shadow on its existence in plasma.
Results reviewed in this paper represent the first experimental evidence of the turbulence suppression by mean
flows in quasi-two-dimensional fluids. First we describe
the experimental setup and methods of the turbulent flow
detection. Suppresion of turbulence is studied (a) during
spectral condensation of the spatially bounded flow, and(b) in the presence of externally imposed mean flow.

2

Experimental setup and results

In the experiments reported here, turbulent flow is generated in stratified thin layers of electrolyte (NaCl water solutions of diﬀerent concentrations, heavier solution at thebottom, total thickness of 6 mm). This setup is similar
to those used in [13] and [14]. Turbulence is generated
by forcing 576 J × B-driven vortices (24 × 24) in a cell.
Spatially varying, vertically directed (normal to the fluidlayers) magnetic field B is produced by a 24 × 24 matrixof permanent magnets placed under the bottom of the fluid
cell (300 × 300 mm 2 ) in a checkerboard fashion (10 mm
between the centers of magnetic dipoles). The schematicof the experimental setup is shown in Fig. 1. Solid Perspex
square boundaries of various sizes are inserted to generate
spectral condensate. Two carbon electrodes on the eitherside of the cell are connected to the power supply. An electric current in the electrolyte can be either modulated, forexample, by driving positive and negative current pulses

Proceedings of ITC/ISHW2007

electric
current

J

laser sheet
laser

flow modifies the spectral transfer, which is determinedby both the energy and the enstrophy  ω dissipation rates,
and leads to the onset of two inertial ranges. If energy
and enstrophy are injected into the system at k f , then energy in 2D flow cascades towards larger scales, or lower
k < k f (inverse energy cascade range), while the enstrophy
is transferred towards higher k > k f (forward enstrophy
cascade). The former is described by the Kolmogorov law,
E(k) = C k  2/3 k−5/3 ,fluid cellBmagnet arrays

Fig. 1

Schematic of experimental setup.

having random-in-time polarity and the pulse lengths, orit can be DC. Since we are mostly interested in the development of strong coherent condensate flows, a DC modeis chosen in these experiments. Constant forcing is foundto be the most eﬃcient in supporting coherent monopole
condensate. This agrees with findings of numerical simulations of the generation of large coherent structures in
2D turbulence [15]. The J × B forcing is chosen such that
there is no ripple on the free surface, which would violatequasi-two-dimensionality of the flow.
To visualize the flow, imaging particles (polyamid, 50
μm diameter, specific gravity of 1.03) are suspended in
the top layer of the fluid and are illuminated using thin
(1 mm) laser sheet aligned parallel to the free surface ofthe fluid. Laser light scattered by particles is filmed fromabove, using video camera at 25 frames per second. Crosscorrelation based particle image velocimetry technique isused to obtain the velocity fields from the sequence of thevideo frames.
First we discuss the generation of the spectral condensate. It is well known that the evolution of spectra in 3D
flows leads to the transfer of the spectral energy E(k), towards smaller scales (larger k), a forward energy cascade,until it reaches the dissipation scale determined by the viscosity. Spectral regions of the turbulence forcing (k f ) andof the dissipation range (k d ) do not generally coincide. The
spectral range between them is called the inertial range.
Viscous processes determine the energy dissipation rate in the system. Kolmogorov assumed that (a) the statisticalproperties of turbulence in the inertial range (k << k d ) aredetermined only by k and , and (b) that  is the universal
constant of a given flow in time and in space. Dimensional
considerations have led to the famous Kolmogorov law for
the spectral energy: E(k) = C 2/3 k−5/3 . In 2D flows, in addition to the energy conservation, enstrophy, or the volume
integral of the squared vorticity Ω = 1/2 ω2 dV (whereω = ∇ × V is the vorticity and V is velocity) is also conserved [16]. The existence of this second invariant of the(1)(though the energy is transferred in the opposite direction
to that in the 3D turbulence), while the latter is describedbyE(k) = C ω ω2/3 k−3 ,(2)The maximum of the energy spectrum thus lies in the lowk range at k E , and in the absence of the energy dissipation
at large scales kE can not be constant in time. In the presence of damping for large scales, for example via linear
damping μ, the scale corresponding to the maximum of thespectrum, stabilizes at 3 1/2
μ
.
(3)
kE ≈

If the system size is larger than this dissipation scale
λE = π/kE , one should observe the stationary energy spectrum showing two inertial ranges corresponding to the inverse energy cascade (∝ k −5/3 ) and the direct enstrophy
cascade (∝ k−3 ). If however dissipative scale is smaller
than the system size, λ E < L, spectral energy is accumulated at the largest scale. Such a process, in which energy piles up in the largest scale k c , has been predicted
by Kraichnan in 1967 who also noted the similarity between the condensation of the turbulent energy and theBose-Einstein condensation of the 2D quantum gas [17].
A self-generated coherent flow can develop spontaneously during spectral condensation of the bounded 2D
turbulence [17]. In a large domain, inverse cascade proceeds up to the integral scale λ E . When λE is larger than
the size of the boundary L, energy accumulates at the box
scale and self-generation of a large vortex occurs. Thisphenomenon has been confirmed in numerical simulations[18, 19, 20, 21] and has been observed in experiments[22, 13, 14]. For given μ and , the easiest way to achieve
spectral condensation is to reduce the size of the boundary
to satisfy λE ≥ L. In the described experiments, squareboundaries of diﬀerent sizes of L = 90 ÷ 120 mm wereused. The spectral condensation leads to the onset of theself-generated mean flow, which interacts with the background turbulence.
First we consider the eﬀect of the self-generated flowon the bounded (L = 110 mm) turbulence [23]. The time
evolution of the total kinetic energy of the 2D turbulent
flow is shown in Fig. 2(a). The inverse energy cascadeleads to the development of larger eddies and to the growthProceedings of ITC/ISHW2007
10

E (a.u.)

1.2

-5

~k

10

-6

10

-7

10

-8

-3

Etot (k)

10

-6

10

-7

0.8

0.4

(a)

(a)

0
0

t (s)

20

40

60

80

100

(b)

k (m )
-1

kf

1000

100

Efl (k)10

k (m )
-1

-8

100

kf

Fig. 3 Spectrum of the total spectral energy of the flow at t =(9-17)s (a). Spectra of the turbulent velocity fluctuations
before, t = (9-17)s (open squares) and after the formationof a single large vortex, t = (61-79)s (solid circles) (b).

(b)

(c)

Fig. 2 Time evolution of the total kinetic energy (a) and instantaneous velocity fields at t = 13 s (b), and t = 71 s (c).

the inverse energy cascade inertial range [17]. It was suggested in [21] that a k −3 power law is due to the presence oflarge-scale persistent vortices rather than due to the turbulent cascade. To eliminate this eﬀect, we subtract from the
N
V(x, y)instantaneous velocity the mean V = 1/N n=1
obtained by averaging over N instantaneous fields V(x, y).
The resulting spectra,
E f l (k) = 1/Nof the kinetic energy of the system. By about 10 s the kinetic energy reaches 80% of its maximum value. By thistime several large-scale coherent vortices develop in theflow, as seen in Fig. 2(b). These vortices persist for 4-5
turnover times (about 10 s) before they start merging. After this transient stage, large vortices merge to form a single coherent vortex, which then persists in a steady state,
Fig. 2(c). This stable vortex imposes mean flow which affects 2D turbulence.
We compare the turbulence spectra during the transient stage, at t = (9 − 17) s, and after the single vortex
formation, at t = (61 − 79) s. The analysis time in thetransient stage is limited to 8 s during which the flow isquasi-steady. The wave number spectra are averaged overN = 200 realizations (400 in the steady condensate regime)of the “instantaneous” velocity fields (computed every 40
ms using two consecutive video frames):
Etot (k) = 1/N

N

F(V)F ∗ (V),

(4)

n=1

where F denotes Fourier transform and F ∗ is its complex
conjugate. This is a total spectrum which includes bothmean and fluctuating velocity. Before the large vortex formation, this spectrum shows a power-law scaling of E(k) ∝k−3 both above and below the forcing wave number k f =350 m−1 , Fig. 3(a). Such a scaling, which was already observed in the experiments in the spectral condensate regime[14] and in numerical simulations [24, 25, 26, 21], apparently contradicts the E(k) ∝ k −5/3 spectrum expected forN

F(V − V)F ∗ (V − V) ,

(5)

n=1

computed for two time intervals, before and after the generation of the single vortex, are shown in Fig. 3(b). Such
subtraction, proposed in [21], leads to a spectrum less steep
than k−3 , somewhat close to k −5/3 .
After the formation of the single vortex, turbulence
levels are significantly reduced for the wave numbers in
the range of k < 160 m −1 . The explanation of this will begiven below. The level of turbulent fluctuations changes
less between k ≈ 160 m −1 and the injection scale, k f ≈350 m−1 . That interval is too short to distinguish between
E f l (k) ∝  2/3 k−5/3 and E f l (k) ∝ /τ s k that one may expect
assuming that the scale-independent energy transfer is oforder of the shear time τ s . One can see that in the forwardcascade range (k ≥ k f ) fluctuations are also reduced. This
reduction at small k is significant (up to a factor of ten) andreproducible.
Now we turn to the experiment in which a large-scalemean flow was externally imposed on the quasi-2D turbulence [23]. The flow is generated using large permanent
magnet, as illustrated in Fig. 4. In this case the boundarybox exceeds the integral scale (L ≈ 300 mm). We refer tothis configuration as to “unbounded” turbulence.
A large magnet (40 × 40 mm 2) placed 2 mm above
the free surface imposes a large-scale vortex flow, whichslowly decays (for approximately 60 seconds) after themagnet is removed. Instantaneous velocity fields before
and after the generation of this mean flow are shown in
Figs. 5(a,b). Energy spectra shown in Figs. 5(c) are com-

Proceedings of ITC/ISHW2007(a)-6

Efl (k)10
Fig. 4(b)~k

Schematic of the generation of external mean flow

10

~k

-8

10

-1

k (m )-910(c) 100

Analysis and discussion

In the presence of the self-generated large vortex the observed reduction in the spectral power of turbulent eddies
is consistent with the mechanism of the shear turbulence
suppression. We estimate the shear suppression criterions = ωs τe > 1 as follows. The turnover time of an eddy ofthe scale l isll=,|δV(l)| S 1 (l)that is estimated from the mean velocity diﬀerence acrossscale l,δV(l) = V(r0 + l) − V(r0 ).

(7)

The angular brackets denote averaging over all possible positions r0 within the boundary box (or within the computation box in the “unbounded” case), and S 1 = δV isthe first-order structure function averaged over N velocity
fields.
To estimate the shearing rate of the large-scale mean
flows, both self-generated [Fig. 2(c)] and externally forced[Fig. 5(b)], the polar coordinate system with its origin in
the center of the vortex is used. The azimuthal componentof the velocity V θ dominates the flow after the vortex isformed. Its radial distribution is shown in Figs. 6(a,c). In
the case of the self-generated flow radial coordinates r =0 and r = 0.05 m correspond to the vortex center and to
the square boundary respectively. In the case of externallydriven flow r = 0.09 m corresponds to the size of the imaged area of “unbounded” turbulent flow. It is seen thatthe amplitude of velocity of the externally forced flow is afactor of two higher than in the self-organized case. The
shearing rate is determined as follows:ωs = l

dΩ
,
dr

1000

0

0.02 0.04 0.06 0.08
l (m)

(d)

(6)

(8)

externally forced

self-generated

0.02

(c)

(a)

Vq (m/s)

τe ≈

0

kf

with
externalflow

1Fig. 5 Instantaneous velocity fields of “unbounded” turbulence(a) and in the presence of externally generated large scaleazimuthal flow. Spectra of turbulence are computed withmean flow subtracted: before the large flow is imposed(open squares) and in the presence of the mean flow (solid
circles). Third-order structure functions without (opensquares) and with (solid circles) externally imposed mean
flow (d).

0.01

r(m)

r(m)

0
0
dW/dr (1/ms)

3

withexternalflow

3

no flow2-5 / 33

S3 (10 m /s )no flow-7

puted after subtracting the mean flow, using Eq. 5. Bothwith and without the large vortex, spectra are close to thek−5/3 scaling. The mean flow reduces the spectral powerof the turbulent fluctuations everywhere within the inverse
cascade range by a factor of 8.

-73-5 / 3

0

0.02

0.04

0

0.02

0.06

0.1

(d)

(b)

-10

-20
r(m)

r(m)

-30
0

0.02

0.04

0

0.02

0.06

0.1

Fig. 6 Mean azimuthal velocity of the flow after the large-scalevortex generation (a,c) and the derivative of its angular velocity (b,d) during spectral condensation of thebounded turbulence (a,b), and in the case of externallyforced “unbounded” flow (c,d).

Proceedings of ITC/ISHW2007
where l is the radial extent of the eddy. The derivative of
the radially localized angular velocity Ω = V θ /r is determined asdΩ 1 dVθ Vθ
=
− 2,
dr
r dr
r

N

S

N
S

(9)

which is zero for the solid-body mean flow rotation andnonzero for the sheared flow. Figs. 6(b,d) show dΩ/dr forthe self-generated and externally driven shear flows. Sinceboth ω s and τe grow with l, the shear aﬀects larger scales
first.
For the case of the self-generated flow dΩ/dr ≈ 15(ms)−1 , S 1 ≈ 8 × 10−3 m/s, and the shearing parameters ≈ 2 × 103 l2 . The criterion for the shear suppression,s > 1, is satisfied for the scales l > 0.022 m. This gives
an estimate of the aﬀected wave number range k = π/l ≤145 m−1 , which is in agreement with the observation of
the turbulence suppression in the wave number range ofk ≤ 160 m−1 seen in Fig. 3(b).
For the externally forced mean flow dΩ/dr ≈ 22(ms)−1 , S 1 ≈ 2 × 10−3 m/s, and the shearing parameters ≈ 1.1 × 104 l2 . The suppression criterion is satisfied forthe scales l > 0.0095 m, which extends very close to theforcing scale l f ≈ 9 mm (k f = 350m−1). Again, this isin agreement with our observation that the spectral energy
is reduced everywhere within the inverse energy cascadeinertial range, Fig. 5(c).
The externally driven flow must be strong enough toaﬀect the energy flux through the k < k f inertial range.
To test this we computed the third-order structure function
S 3 (l) =< δV(l)3 > to estimate the energy flux  from theKolmogorov law,3S 3 (l) = − l.
2

V

(10)

Similarly to S 1 , the third-order structure function S 3 iscomputed by averaging over the boundary box and then by
averaging S 3 in time over 100 subsequent velocity fields.
It should be noted that δV(l) represents here the longitudinal velocity increment, δV || (l), defined in Kolmogorov’s
theory [16]. Also, S 3 is computed allowing positive andnegative values of the velocity increments δV || (l). Suchcomputations require very large statistical averaging to obtain converged results. We obtained satisfactory convergence for the steady-state “unbounded” turbulence and forthe turbulence in the presence of the slowly decaying externally driven flow. The result is illustrated in Fig. 5(d).
Both before and after the mean flow is imposed, S 3 is alinear function of the scale l. As a result, the energy flux
 = −(2/3)S 3/l is constant to within 15% for all scales
in the energy inertial range. This flux  is reduced in thepresence of the flow by one order of magnitude comparedto the case without the flow.
The reduction in the energy flux can be attributed totwo phenomena in this case. First, it is the shearing of
the forcing scale vortices discussed above. However, the

N
S

N

S

Fig. 7 Schematic illustration of the eﬀect of sweeping of theforcing scale vortices by mean flow relative to the magnets

force-connected vortices (with k ≈ k f ) must be more resistant to shearing than the inertial scale eddies at k < k f .
Second, the force-fed vortices are swept by the mean flow
relative to the magnets, which must also reduce the energy
input. This eﬀect is schematically illustrated in Fig. 7.
One can define a dimensionless sweeping parametersw = ωsw τe , where the sweeping rate is given by ω sw =Vθ /l. Since
sw =

Vθ l
∼ Vθ (l)−1/3 ,
l S1

(11)

sweeping acts more eﬃciently on the smaller scales (while
shearing is more eﬀective on larger scales). At the forcing
scale l f , this parameter is sw ≈ 0.75 for the self-generated
flow and it is sw ≈ 7 with externally forced mean flow.
Thus the sweeping can be responsible for the reductionin the energy flux through the inverse cascade range inthe presence of an externally-induced flow. The dominantrole of sweeping in this case is also supported by the factthat the spectrum of the inverse cascade remains k −5/3 , justshifted down as shown in Fig. 5(c). Such modifications tothe spectrum are also consistent with a ten-fold decrease in
the energy flux , since E(k) = C k  2/3 k−5/3 .
Let us stress the qualitative diﬀerence between
Fig. 3(b) (strong decrease at small k) and Fig. 5(c) (uniform
decrease for all k) which shows that there are two mechanisms of suppression. Sweeping may also be responsible
for the reduction in the enstrophy flux through the forwardcascade (k > k f ) in the presence of the self-generated flow[see Fig. 3(b)]. In this case, we could not obtain statistically converged computations of S 3 during spectral condensation to compare  before and after the formation of
the large vortex.

4

Conclusions

We have shown in [23] that turbulence is quasi-2D flow
is significantly reduced in the presence of a large coherent
vortex. In the case of a self-generated vortex, larger scales
are aﬀected more than the smaller ones. This qualitatively

Proceedings of ITC/ISHW2007
agrees with the description of the shear turbulence suppression mechanism as a reduction in the eddy life-time [1]. In
the presence of externally imposed flow two eﬀects may be
responsible for the observed strong reduction in the turbulence level. The vortex sweeping by the mean flow seems
to play an important role here. In this case the shape of thespectrum is not modified but the (inverse) spectral energy
flux is substantially reduced.
It should be noted that three conditions needed for the
shear turbulence suppression in fluids discussed in [10] aresatisfied in our experiment. The shear flow must be stable
in a sense that the time during which turbulence remains in
the region of flow shear should exceed both the eddy lifetime τe and the shear straining time τ s . The shear flow is
stable in our experiment. Due to the dominant flow in theazimuthal direction after the monopole formation, turbulence stays in the region of shear, ρ = (0.2 − 0.9), Fig. 6.
Finally, the 2D dynamics of the flow is imposed on thesystem by the stratification of thin fluid layers.
The mechanism of the shear suppression in plasma isoften described as the loss of coherence by a turbulent eddy
and a breakup into two eddies of the smaller scale (e.g.
[9]). One would expect in this case an increase in the spectral power of eddies of intemediate scales. We have notfound any evidence in support of the eddy breakup. Observations in 2D fluids are more consistent with the ideaof reduction in the turnover time of the larger scale eddies[8]. In this case the inverse energy cascade is arrested at thescales aﬀected by the shear flow, which presumably leads
to the reduction in the spectral energy, similar to that seenin Fig. 3(b).
[1] H. Biglari, P.H. Diamond and P.W. Terry, Phys. Fluids B 2(1990) 1.
[2] K.C. Shaing, E.J. Crume and W.A. Houlberg, Phys. Fluids B
2 (1990) 1492.
[3] F. Wagner et al. Phys. Rev. Lett. 49 (1982) 1408.
[4] T.S. Hahm and K.H. Burrell, Phys. Plasmas 2 (1995) 1648.
[5] T.S. Hahm, Phys. Plasmas 4 (1997) 4074.
[6] K.H. Burrell, Phys. Plasmas 6 (1999) 4418.
[7] K.H. Burrell, Phys. Plasmas 4 (1997) 1499.
[8] P.W. Terry, Rev. Mod. Phys. 72 (2000) 109.
[9] E.-J. Kim, Astron. and Astrophysics 441 (2005) 763.
[10] P.W. Terry, Phys. Plasmas 7 (2000) 1653.
[11] D.C. Montgomery, Phys. Plasmas 7 (2000) 4785.
[12] P.W. Terry, Phys. Plasmas 7 (2000) 4787.
[13] J. Paret and P. Tabeling, Phys.Fluids 10 (1998) 3126.
[14] M.G. Shats, H. Xia, H. Punzmann, Phys. Rev. E 71 (2005)
046409.
[15] N.N. Kukharkin, J. Sci. Comp. 10 (1995) 409.
[16] U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov(Cambridge University Press, Cambridge,1995).
[17] R.H. Kraichnan, Phys. Fluids 10 (1967) 1417.
[18] M. Hossain, W. H. Matthaeu, D. Montgomery, J. Plasma
Phys. 30 (1983) 479.
[19] L. M. Smith, V. Yakhot, Phys. Rev. Lett. 71 (1993) 352.
[20] D. Molenaar, H.J.H. Clercx and G.J.F. van Heijst, Physica
D 196 (2004) 329.
[21] M.Chertkov, C. Connaughton, I. Kolokolov and V. Lebedev,
Phys. Rev. Lett. 99 084501 (2007).

[22] J. Sommeria, J. Fluid Mech. 170 (1986) 139.
[23] M.G. Shats, H. Xia, H. Punzmann and G. Falkovich Phys.
Rev. Lett. 99 164502 (2007).
[24] V. Borue, Phys. Rev. Lett. 72 (1994) 1475.
[25] L.M. Smith and F. Waleﬀe, Phys. Fluids 11 (1999) 1608.
[26] C. V. Tran and J. C. Bowman, Phys. Rev. E 69 (2004)
036303.