22
pages

- kolmogorov-like scaling
- cc liquid
- clearly ob- served
- yag laser slaved
- coherent structures
- soap film
- linear polymer
- polymers stretch

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Two dimensional turbulence

with polymer additives

2 2Where : !! (! "" )/8;

!> 0# vortical region, !< 0# strain-dominated region.!

2" ! " " and " =# u +# u .ij ij ij i j j ii,j

! Polymers stretch predominantly in strain-dominated regions.Acknowledgement

1. Dhrubaditya Mitra

2. Dario Vincenzi

3. Roberto Benzi 2d turbulence

2∂ ω+u ·∇ω= ν∇ ω−µω+Ft ω

−5/3E(k)∼k

How polymer additives affect

forward and inverse cascade?

µ

−3+δ(µ)E(k)∼k

Perlekar et al., PRL (2011); Ray et al., PRL (2011);

Wiki: Linear polymer molecule Boffetta et al., ARFM (2012). 2d turbulence: Topological structures

2 2Λ=(ω −σ )/4

How polymer additives affect

the topological properties?

Expts: Daniel and Rutgers, PRL (2002);

Simulations: Perlekar and Pandit, NJP (2009). Soap-film experiment-1/4

week endingPHYSICAL REVIEW LETTERS

PRL 96, 024502 (2006) 20 JANUARY 2006

2/3

100

l

inj

1.810

4

2

<# >

2

3 <" >

1

0 5 10 15 20 25

! (ppm)

0.1 1

*FIG. 1. Experimental setup. A voltage difference V!V ) l (cm)

)V is applied to the ﬁlm generating a uniform current densityJ.

Beneath the ﬁlm is a set of bar magnets with alternating poles. FIG. 2 (color online). The second-order structure function

S #l$. In increasing order of turbulence intensity, the curves2

correspond to "! 24, 12, 9, 3, and 0 ppm. The inset is theKolmogorov forcing generates turbulence in soap-films. goes a sequence of instabilities and becomes turbulent

2 2plot of h' i (circles) and h! i (squares) vs".

whenV>20 V. The soap solution was made of a mixture

of four components (5 cc liquid detergent, 80 g ammonium

Jun et al., PRL, 96, 024502 (2006) chloride, 40 cc glycerol, and 400 cc distilled water). Linear becomes ﬂat forl>l ,indicatingthetruncationofenergyinj

6polymers (polyethylene-oxide, M ! 8"10 , R ’ transfer to these large scales. The blockage of energy tow g

0:4 !m) of varying concentrations (0<"<25 ppm) large scales can also be seen by the decreasing total kinetic

2were used. Within this concentration range, there is no energy v =2, which is the asymptotic value of S #l$ forrms 2

2overlap between polymer coils as evidenced by a small l' l . Figure 2 shows thatv =2 drops sharply for"(inj rms

" dependence of the kinematic viscosity of the soap solu- 10 ppm.

2tion, which we determined to be# ’ 0:02 cm =s. To mea- The abrupt change of turbulent behavior when " in-

sure the velocity ﬁeld v~#x~$, the ﬁlm was seeded with creases is suggestive, indicating that there may be a critical

3hollow glass spheres (diam! 10 !m, $! 1:05 g=cm ). polymer concentration " #’ 10 ppm$ for quenching tur-C

A 12 mJ double-pulsed Nd:YAG laser slaved to a CCD bulence. The observation prompted us to examine other

camera (Redlake,1016"1008 pixels) was used to illumi- signatures that may be used to quantify the effect. One of

2nate the soap ﬁlm. Images (4:5"4:5 cm ) were acquired the prominent features of 2D turbulence is the coherent

at the center of the soap ﬁlm at 30 fps, yielding typically structures, such as vortices and saddle points in the ﬂow. In

410 vectors per velocity ﬁeld. a previous study [10], we investigated the distribution of

1 2 2In the following discussion, ﬁve different polymer con- centers and saddles via the quantity !! #' )! $,2

2 2centrations "! 0;3;9;12;24 ppm were used and the en- which is related to the pressure by)r p!!. Here! !

2 3 1 2 1ergy injection rate " ! 201:4 cm =s was kept ﬁxed byinj #@v )@ v$ and ' ! #@v *@ v$ characterize thei j j i i j j i2 2

maintaining a constantV across the ﬁlm. Figure 2 shows a center and the saddle structures in the ﬂow with the re-

2set of second-order structure functionsS #l$!h%v i mea-2 peated indices implying summations. Statistical distribu-l

2 2sured using different ", where %v is the longitudinall tions of! and' were measured for different" and their

velocity difference on scale l. We found that in all cases probability density functions (pdf) are displayed in Fig. 3.

2 2there is a well-developed enstrophy range (l<l ) where It is shown that for "<" , P#! $ and P#' $ are unaf-inj C

1:8%0:2S #l$ /l . This scaling relation agrees reasonably fected by ", but they become signiﬁcantly narrower for2

2well with the theoretical prediction S #l$ /l and persists ">" , indicating that strong centers and saddles are2 C

down to the smallest scale (&300 !m) resolvable by the suppressed. The effect is represented by the inset of

2 2PIV. Aside from small changes in the amplitude, the poly- Fig. 2 where h' i and h! i vs " are plotted. We noted

mer appears to have no effect on this scaling behavior. For that in all cases of different",the‘‘topologicalcharges,’’

large scales (l>l ), two classes of behaviors can be averaged over space, are not strictly conserved. The dif-inj

2 2ferences between h' i and h! i result from the ﬁlm beingidentiﬁed: (a) For 0<"<10 ppm, S #l$ increases with2

slightly compressible. Since polymers are mostly de-l and is reminiscent of an inverse energy cascade. Despite a

large Taylor-microscale Reynolds number Re ’ 153, the formed by saddles, it is instructive to compare the&

2=3 distribution of the strain rate ' with the Zimm relaxationKolmogorov-like scaling S #l$ /l was not clearly ob-2

3time(!)R =#k T$ of the polymer. For our system withserved due to the limited inertial range. In spite of this g B

) ’ 0:02 cP and R ’ 0:4 !m, we found ( ’ 16 ms orshortcoming, the magnitude of S #l$ was observed to de- g2

2 3 )2crease as " was increased. (b) For ">10 ppm, S #l$ 1=( ’ 3:91"10 s , which is delineated as the vertical2

024502-2

2 2

S (l) (cm /s )

2

2 2 3 -2

<" >,<# > ($10 s )Soap-film experiment-2/4

week endingPHYSICAL REVIEW LETTERSPRL 96, 024502 (2006) 20 JANUARY 2006

polymer is present, the situations are somewhat different;200

v increases initially, levels off, and then increases again.rms

It forms a plateau for a small range of V between 50 and(a)

150 " 55 volts. This measurement suggests that there exist twop

thresholdsV andV marked by two arrows in the ﬁgure.C1 C2

100 We believed that the lower threshold V corresponds toC1

the onset of the turbulent suppression and the higher"

#50 threshold V corresponds to the saturation of the elasticC2

" ﬁeld. However, for the entire range of V, the measured$

3energy transfer rate (/h$v i) remains positive, indicating0 l0 10 205 15 25 an inverse energy cascade but with a reduced transfer rate

! (ppm) when the polymer is present.

To summarize, the polymer effects on forced 2D turbu-16

lence in freely suspended soap ﬁlms were investigated

quantitatively using two independent control parameters

14 ! and" . The measurement shows that when" is ﬁxed,inj inj

turbulent suppression has a sharp threshold! !’ 10 ppm".C

12 However, when! is ﬁxed, two thresholds can be identiﬁed,

but the transitions in this case are much weaker. We found(b)10 that turbulent suppression occurs concurrently with theV V

C1 C2 elimination of strong saddles. Inspection of Fig. 3(a) re-

veals that those saddles that are eliminated have strength45 50 55 60 65

2 2determined precisely by the relation% * 1=& , indicatingV (volts)

that the time criterion is strictly obeyed in the experiment.

Since polymer-turbulence interactions are primarily viaFIG. 5 (color online). (a) The energy budget vs!. (b) v vsrms

saddles and the weakening of saddles by polymer stretch-V for !$ 0 (circles) and for!$ 15 ppm (squares).

ing has the drastic effect of quenching turbulence, it sug-

gests that this hydrodynamic structure may play a role in

of turbulence and is whereS !l" levels off (see Fig. 2) [12].2 transferring energy from scale to scale. It remains an

An appreciable amount of energy (#30%) is also trans- intriguing possibility that the same mechanism operates

ferred to small scales and consumed by molecular viscos- in 3D as well as in 2D turbulence.

ity. In the absence of polymer, this partition of energy on We acknowledge helpful discussion with Dr. B.

small and large scales is consistent with a previous study Eckhardt. This work is supported by the NSF under

[9]. It is interesting to note that when ! is increased, the Grant No. DMR-0242284.

fraction of energy consumed by the ﬂuid viscosity remains

almost constant [see the heights of the dark-hatched area in

Fig. 5(a)] until! crosses! , where " suffers a jump ofC "

#28:7%. The effect is more dramatic for the energy trans-

[1] B. A. Toms, in Proceedings of the International Congressfer to large scales as indicated by the heights of the light-

on Rheology (North-Holland, Amsterdam, 1949).hatched area in the same ﬁgure. Here one observes that"#

[2] J. L. Lumley, J. Polym. Sci. 7, 263 (1973).keeps decreasing with ! and drops precipitously at ! .C

[3] T. Tabor and P.-G. de Gennes, Europhys. Lett. 2, 519Such a strong ! dependence is due to the signiﬁcant

(1986).change in v when the polymer was introduced into therms [4] R. Benzi, E. De Angelis, R. Govindarajan, and I.

ﬂow as seen in Fig. 2. Since" is constant, it follows thatinj Procaccia, Phys. Rev. E 68, 016308 (2003).

more energy is sequestered by polymer’s elastic deforma- [5] J. M. J. de Toonder, M. A. Hulsen, G. D. C. Kuiken, and

tion, and " increases markedly around ! as delineated F. T. M. Nieuwstadt, J. Fluid Mech. 337, 193 (1997).p C

by the heights of white area in Fig. 5(a). [6] J. R. Cressman, Q. Bailey, and W. Goldburg, Phys. Fluids

13, 867 (2001).The above measurement shows that for a given" , thereinj

[7] Y. Amarouchene and H. Kelley, Phys. Rev. Lett. 89,exists a sharp change in the turbulence behavior when !

104502 (2002).crosses! . Is the converse true? To ﬁnd out, we conductedC [8] G. Boffetta, A. Celani, and S. Musachio, Phys. Rev. Lett.an experiment in which!$ 15 ppm was ﬁxed but" wasinj 91, 034501 (2003).

varied by changing the applied voltage (46<V<65 V). [9] M. Rivera and X. L. Wu, Phys. Rev. Lett. 85, 976 (2000).

For comparison, an independent run was also carried out [10] M. Rivera, X. L. Wu, and C. Yeung, Phys. Rev. Lett.

with !$ 0. Figure 5(b) shows that, in the absence of 87, 044501 (2001).

polymers, the turbulent intensity characterized by the [11] E. Lindborg, J. Fluid Mech. 326, 343 (1996).

v is a smooth increasing function of V. When the [12] M. Rivera and X. L. Wu, Phys. Fluids 14, 3098 (2002).rms

024502-4

2 3

" (cm /s )

v (cm/s)

rms Soap-film experiment-3/4

week endingPHYSICAL REVIEW LETTERSPRL 96, 024502 (2006) 20 JANUARY 2006

Suppression of both large and small scales

2/3

100

l

inj

1.810

4

2

<# >

2

3 <" >

1

0 5 10 15 20 25

! (ppm)

0.1 1

*FIG. 1. Experimental setup. A voltage difference V!V ) l (cm)

)V is applied to the ﬁlm generating a uniform current densityJ.

Beneath the ﬁlm is a set of bar magnets with alternating poles. FIG. 2 (color online). The second-order structure function

S #l$. In increasing order of turbulence intensity, the curves2

correspond to "! 24, 12, 9, 3, and 0 ppm. The inset is thegoes a sequence of instabilities and becomes turbulent 2 2plot of h' i (circles) and h! i (squares) vs".

whenV>20 V. The soap solution was made of a mixture

of four components (5 cc liquid detergent, 80 g ammonium

chloride, 40 cc glycerol, and 400 cc distilled water). Linear becomes ﬂat forl>l ,indicatingthetruncationofenergyinj

6polymers (polyethylene-oxide, M ! 8"10 , R ’ transfer to these large scales. The blockage of energy tow g

0:4 !m) of varying concentrations (0<"<25 ppm) large scales can also be seen by the decreasing total kinetic

2were used. Within this concentration range, there is no energy v =2, which is the asymptotic value of S #l$ forrms 2

2overlap between polymer coils as evidenced by a small l' l . Figure 2 shows thatv =2 drops sharply for"(inj rms

" dependence of the kinematic viscosity of the soap solu- 10 ppm.

2tion, which we determined to be# ’ 0:02 cm =s. To mea- The abrupt change of turbulent behavior when " in-

sure the velocity ﬁeld v~#x~$, the ﬁlm was seeded with creases is suggestive, indicating that there may be a critical

3hollow glass spheres (diam! 10 !m, $! 1:05 g=cm ). polymer concentration " #’ 10 ppm$ for quenching tur-C

A 12 mJ double-pulsed Nd:YAG laser slaved to a CCD bulence. The observation prompted us to examine other

camera (Redlake,1016"1008 pixels) was used to illumi- signatures that may be used to quantify the effect. One of

2nate the soap ﬁlm. Images (4:5"4:5 cm ) were acquired the prominent features of 2D turbulence is the coherent

at the center of the soap ﬁlm at 30 fps, yielding typically structures, such as vortices and saddle points in the ﬂow. In

410 vectors per velocity ﬁeld. a previous study [10], we investigated the distribution of

1 2 2In the following discussion, ﬁve different polymer con- centers and saddles via the quantity !! #' )! $,

2

2 2centrations "! 0;3;9;12;24 ppm were used and the en- which is related to the pressure by)r p!!. Here! !

2 3 1 2 1ergy injection rate " ! 201:4 cm =s was kept ﬁxed byinj #@v )@ v$ and ' ! #@v *@ v$ characterize thei j j i i j j i2 2

maintaining a constantV across the ﬁlm. Figure 2 shows a center and the saddle structures in the ﬂow with the re-

2set of second-order structure functionsS #l$!h%v i mea-2 peated indices implying summations. Statistical distribu-l

2 2sured using different ", where %v is the longitudinall tions of! and' were measured for different" and their

velocity difference on scale l. We found that in all cases probability density functions (pdf) are displayed in Fig. 3.

2 2there is a well-developed enstrophy range (l<l ) where It is shown that for "<" , P#! $ and P#' $ are unaf-inj C

1:8%0:2S #l$ /l . This scaling relation agrees reasonably fected by ", but they become signiﬁcantly narrower for2

2well with the theoretical prediction S #l$ /l and persists ">" , indicating that strong centers and saddles are2 C

down to the smallest scale (&300 !m) resolvable by the suppressed. The effect is represented by the inset of

2 2PIV. Aside from small changes in the amplitude, the poly- Fig. 2 where h' i and h! i vs " are plotted. We noted

mer appears to have no effect on this scaling behavior. For that in all cases of different",the‘‘topologicalcharges,’’

large scales (l>l ), two classes of behaviors can be averaged over space, are not strictly conserved. The dif-inj

2 2ferences between h' i and h! i result from the ﬁlm beingidentiﬁed: (a) For 0<"<10 ppm, S #l$ increases with2

slightly compressible. Since polymers are mostly de-l and is reminiscent of an inverse energy cascade. Despite a

formed by saddles, it is instructive to compare thelarge Taylor-microscale Reynolds number Re ’ 153, the&

2=3 distribution of the strain rate ' with the Zimm relaxationKolmogorov-like scaling S #l$ /l was not clearly ob-2

3time(!)R =#k T$ of the polymer. For our system withserved due to the limited inertial range. In spite of this g B

) ’ 0:02 cP and R ’ 0:4 !m, we found ( ’ 16 ms orshortcoming, the magnitude of S #l$ was observed to de- g2

2 3 )2crease as " was increased. (b) For ">10 ppm, S #l$ 1=( ’ 3:91"10 s , which is delineated as the vertical2

024502-2

2 2

S (l) (cm /s )

2

2 2 3 -2

<" >,<# > ($10 s )Soap-film experiment-4/4

week endingPHYSICAL REVIEW LETTERSPRL 96, 024502 (2006) 20 JANUARY 2006

All the terms in this equation can be evaluated from the-1 -1

10 10 measured velocity ﬁeld. In particular, the pressure ﬁeld can

2be solved based on the equation r p%x~&"#!%x~& using a

-2 -2 Fourier method. Figure 4(a) shows the lhs (L , circles) and10 10 yy

$<$ the right-hand side (rhs) (R , lines) of Eq. (2) for!" 0. It$<$ yyCC

-3 -3 is found that in the absence of polymer, the two sides are

10 10 #1matched if&" 0:7s . In contrast, if the same calculation

$>$ $>$

C C is carried out for !" 12 ppm as in Fig. 4(b), there is a

-4 -4

10 10 signiﬁcant discrepancy between the lhs and the rhs of

(b) Eq. (2). Such discrepancy is expected because the(a)

-5 -5 polymer-ﬂuid interaction is not included in the equation.10 100 1 2 0 1 23 In the single-point limit (r~ ! 0), Eq. (2) gives the energy2 4 -2 2 4 -2

# ("10 s ) ! ("10 s ) balance [11]:

2FIG. 3 (color online). The pdfs of saddles P%" & (a) and " " " $" $" ; (3)inj % & p2centers P%! & (b). For !<! ’ 10 ppm, the pdfs are nearlyC

identical but for!>! ,thepdfsbecomenarrower.TheverticalC

where the energy injection rate " " hu F i (see Fig. 1)inj x xline in (a) corresponds to the square of the Zimm relaxation rate

#2 and the energy dissipation rates due to ﬂuid viscosity" "# . %

2 2%h" i and due to air friction " "&hv i are all standard&

deﬁnitions and can be evaluated. The rate of energy uptake

line in Fig. 3(a). A simple calculation shows that for!< by the polymer " is included and will be found byp! , !38% of saddle points satisfy the time criterionC measurements. In the experiment, " was kept constant,inj("#>1), and this fraction drops to !29% for !>! .C whereas " and " were measured by varying!. Figure 5% &We also noted that the vertical line coincides approxi- 2 3summarizes the result: for !" 0, " ’ 57:3 cm =s and%mately with the point where the two groups of pdfs cross 2 3 2 3" ’ 144:1 cm =s , yielding" ’ 201:4 cm =s . The fact& injeach other. The signiﬁcance of this observation is dis-

that" >" is consistent with the physical picture that the& %cussed in the summary. A weakened saddle distribution

injected energy is predominantly transferred to large scalesmust be accompanied by a weakened vorticity distribution

3 1=32 2 l>l . Here l ’%" =& & =8 ’ 3 cm is the outer scale0 0 injsince h" i!h! i. This is clearly delineated by Fig. 3(b).

To quantitatively assess the fraction of injected energy

" that is ultimately transferred to the polymer’s degreesinj

of freedom and how this fraction changes with !, we 100 (a)

measured the overall energy budget of the system. The

EM cell is well suited for this task since the full velocity

ﬁeld can be measured using the PIV, allowing various 50

energy rates to be calculated. To start with, we used the

´ ´Karman-Howarth relation:

! 0@ 1 @@ 0 0 0 0 0hu u i" hu u u #u u u i# # hpu ii i s i sj j j j@r $ @r@t s i

" 2@ @0 0 0$ hpui $2% hu u i$hu Fi 100i i ij j (b)2@r @rj s

0 0$hu F i#2&hu u i; (1)i ij j

50

where & is the air drag coefﬁcient and the prime and

unprimed quantities correspond to locations x~$r~ and x~,

respectively. Because of the steady-state condition, the 0

left-hand side (lhs) vanishes. For the inertial range, the

viscosity term may also be ignored. Since the Lorentz force

0 2 31is in the x direction, the above equation can be further

simpliﬁed if only they component is evaluated. This yields l (cm)

! "

@ 1 @ @ FIG. 4 (color online). Comparison between the lhs (L ,0 0 0 0 0 yyhu u u #u u u i# # hpu i$ hpu iy s y y s y y y circles) and the rhs (R , lines) of Eq. (2) for !" 0 (a) and@r $ @r @r yys y y

#1for !" 12 ppm (b). The air friction coefﬁcient & ’ 0:7s is

0"#2&hu u i: (2)y y used in both cases.

024502-3

2

P(# )

2

P(! )

2 3

L , R (cm /s )

yy yyModeling polymer solutions

FENE-P Model

∂uα

+(u ∂ )u =−∂ p+ν∂ u +∂ Tγ γ α α γγ α γ αγ

∂t

∂C 1αβ

+(u ∂ )C =(∂ u )C +C (∂ u )− Tγ γ αβ γ α γβ αγ γ β αβ

∂t µ

2L −2f(r)C − δαβ αβ f(r)=T = µαβ 2 2L −rτP

Oldroyd-B Model

C2L →∞

f(r)→ 1 Assumption: Smooth flow around

polymer. week endingPHYSICAL REVIEW LETTERS

18 JULY 2003VOLUME 91, NUMBER 3

asto compensatefor thefactor" in Eq. (5), resulting ina

ﬁnite effect also in the inﬁnite Re limit. Since energy is

essentially dissipated by linear friction, the depletion of

2hjuj i entails immediately the reduction of energy dissi-

pation. The main difference between two-dimensional

‘‘friction reduction’’ and three-dimensional drag reduc-

tion resides in the length scales involved in the energy

drain—large scales in 2D vs small scales in 3D.

The effect of polymer additives cannot be merely rep-

resentedbya rescalingof velocityﬂuctuationsbyagiven

factor. In Fig. 4,weshow theprobabilitydistributionofaFIG. 2. Snapshots of the vorticity ﬁeld r'u in the

velocity component, u . The choice of the x direction isxNewtonian (left) and in the viscoelastic case with strong feed-

immaterial by virtue of statistical isotropy. In theback (right). Notice the suppressionof large-scale structures in

Newtonian case, the distribution is remarkably close tothe latter case. The ﬁelds are obtained by a fully dealiased

2 3pseudospectralsimulationofEqs. (1)and(2)at resolution 256 . the sub-Gaussian density Nexp#"cju j $ stemming fromx

"3Theviscosityis"! 1:5'10 ,!! 0:2,therelaxationtimeis the balance between forcing and nonlinear terms in the

#! 4, and the energy input is F! 3:5. As customary, an Navier-Stokesequation, inagreement withtheprediction

artiﬁcial stress-diffusivity term is added to Eq. (2) to prevent by Falkovich and Lebedev [23]. On the contrary, the

numerical instabilities [21]. The corresponding Schmidt num- distribution in the viscoelastic case is markedly super-

ber is Sc! 0:25.

Gaussian, with approximately exponential tails. This

strong intermittency in the velocity dynamics is due to

the alternation of quiescent low-velocity phases ruled byEq. (4), we multiply Eq. (1) by u, add to it the trace of

polymer feedbackandburstingeventswhereinertialnon-Eq. (2) times !"=#, and average over space and time.

linearities take over.Since in two dimensions kinetic energy ﬂows towards

large scales, it is mainly drained by friction, and viscous Dilutepolymersalsoaltersigniﬁcantly thedistribution

dissipation isvanishingly small inthelimit of verylarge of ﬁnite-time Lyapunov exponents P#&;t$. In Fig. 5, the

"1´Reynolds numbers [16]. Neglecting $ and observing that Cramer rate function S#&$ / t lnP#&;t$ is shown for

the Newtonian and for the viscoelastic case. Since inintheNewtoniancase(!! 0)thebalance(4)yieldsF!

2 the former situation the Lyapunov exponent' is greater%hjuj i , we obtain NN

than 1=#, were the polymers passive all moments of

2!"

2 2 elongation would grow exponentially fast. However, thehjuj i! hjuj i " #htr!i"tr1$: (5)N 2 Earlier studies: Simulations %# feedback can damp stretching so effectively that after

polymer addition' lies below 1=#. This implies a strongAs a consequence of incompressibility and chaoticity of

reduction of Lagrangian chaos and a decreased mixingtheﬂow,it canbeshown from Eq.(2)thattr!% tr1,and

2 2 3we ﬁnally have hjuj i& hjuj i , in agreement with nu-N Homogeneous isotropic turbulence, 256 DNS

merical results.Thissimpleenergybalanceargument can 0

10be generalized to nonlinear elastic models. As viscosity

Oldroyd-B model tendstozero,theaveragepolymerelongation increasesso

-1

10 Passive polymers

50 Wi= λτ >1300 -2

10

40 200 Unbounded growth in polymer

-3 extension. No steady state. 10030 10

0

0 5 10 15 Active polymers 20 t /! -4

10

-15 -10 -5 0 5 10 15 Wi= λτ <1

10

ux

1. Presence of back-reaction 0 3P(u )∼ exp(−c|u| )-5 0 5 10 15 dramatically alters the steady FIG. 4. Intermittency Puofre fveloluidcit y ﬂuctuations induced byx

polymer additives. The probability density function P#u $ oft / ! x state.

the velocity component u for the Newtonian (solid line) andx 2. Steady state for polymer

FIG. 3. Dilute polymers reduce the level of velocity ﬂuctua- for the viscoelastic case with strong feedback (dashed line).R extension. 2tions ju#x;t$j dx. Polymers are introduced in the ﬂow at Same parameters as in Fig. 2. Also shown is the distributionR Fluid + polymers

3=2 3 "3 3. No coil-stretch transition! t! 0. In the inset, the mean square elongation tr!#x;t$dx !#2=3$3 exp#"cju j $=#4(c$ with c! 2:1'10 (dottedx

as a function of time. line).Boffetta, Celani, and Mussachio, PRL, 034501 (2003)

034501-3 034501-3

2

< u >

< tr" >

P(u )

x