Two-dimensional, homogeneous, isotropic fluid turbulence with polymer additives

Anupam Gupta,1, ∗ Prasad Perlekar,2, † and Rahul Pandit2, ‡

1Centre for Condensed Matter Theory, Department of Physics,

Indian Institute of Science, Bangalore 560012, India.2Department of Physics, Department of Mathematics and Computer Science,

Technische Universiteit Eindhoven, Eindhoven, The Netherlands.

We present the most extensive direct numerical simulations. attempted so far, of statisticallysteady, homogeneous, isotropic turbulence in two-dimensional fluid films with air-drag-induced fric-tion and with polymer additives. Our study reveals that the polymers (a) reduce the total fluidenergy, enstrophy, and palinstrophy, (b) modify the fluid energy spectrum both in inverse- andforward-cascade regimes, (c) reduce small-scale intermittency, (d) suppress regions of large vorticityand strain rate and (e) stretch in strain-dominated regions. We compare our results with the earlierexperimental studies; and we propose new experiments.

PACS numbers: 47.27.Gs, 47.27.Ak

Polymer additives have remarkable effects on turbu-lent flows: in wall-bounded flows they lead to the dra-matic phenomenon of drag reduction [1, 2]; in homoge-neous, isotropic turbulence they give rise to dissipationreduction, a modification of the energy spectrum, anda suppression of small-scale structures [3–8]. These ef-fects have been studied principally in three-dimensional(3D) flows; their two-dimensional (2D) analogs have beenstudied only over the past decade in experiments [11, 13]on and direct numerical simulations (DNSs) [14] of fluidfilms with polymer additives. It is important to inves-tigate the differences, both qualitative and quantitative,between 2D and 3D fluid turbulence with polymer ad-ditives because the statistical properties of fluid turbu-lence in 2D and 3D are qualitatively different [16]: theinviscid, unforced 2D Navier-Stokes (NS) equation ad-mits more conserved quantities than its 3D counterpart;one consequence of this is that the energy spectrum for2D fluid turbulence displays an inverse cascade of en-ergy, from the energy-injection length scale linj to largerlength scales, and a forward cascade of enstrophy, fromlinj to smaller length scales. We have, therefore, carriedout the most extensive and high-resolution DNS stud-ies, attempted so far, of homogeneous, isotropic turbu-lence in the incompressible, 2D NS equation with air-drag-induced friction and polymer additives, which wemodel by using the finitely-extensible-nonlinear-elastic-Peterlin (FENE-P) model for the polymer-conformationtensor. Our study yields several interesting results thatwe summarize below. We find that the inverse-cascadepart of the energy spectrum in 2D fluid turbulence issuppressed by the addition of polymers. The effect ofpolymers on the forward-cascade part of the fluid en-ergy spectrum in 2D is similar to their effect on theenergy spectrum in 3D fluid turbulence; in particular,there is slight reduction at intermediate wave-numberand significant enhancement of the fluid energy spectrumin the large-wave-number range. In addition, we findclear manifestations of dissipation-reduction-type phe-

nomena [6, 7]: the addition of polymers to the tur-bulent 2D fluid leads to a reduction of the total en-ergy, and energy- and enstrophy-dissipation rates, a sup-pression of small-scale intermittency, and a decrease inhigh-intensity vortical and strain-dominated regimes. Weshow that the probability distribution functions (PDFs)of σ2 and ω2, the squares of the strain rate and thevorticity, respectively, agree qualitatively with those ob-tained in experiments [13]. We also present PDFs of theOkubo-Weiss parameter Λ = (ω2−σ2)/8, whose sign de-termines whether the flow in a given region is vortical orstrain-dominated [10, 15], and PDFs of the polymer ex-tension; and we show that polymers stretch preferentiallyin strain-dominated regions.The 2D incompressible NS and FENE-P equations can

be written in terms of the stream-function ψ and thevorticity ω = ∇× u(x, t), where u ≡ (−∂yψ, ∂xψ) is thefluid velocity at the point x and time t, as follows:

Dtω = ν∇2ω +µ

τP∇×∇.[f(rP )C]− αω + Fω; (1)

∇2ψ = ω; (2)

DtC = C.(∇u) + (∇u)T .C − f(rP )C − IτP

. (3)

Here Dt ≡ ∂t + u.∇, the uniform solvent density ρ = 1,α is the coefficient of the friction, ν the kinematic vis-cosity of the fluid, µ the viscosity parameter for the so-lute (FENE-P), and τP the polymer relaxation time; tomimic soap-film experiments [13] we use a Kolmogorov-type forcing Fω ≡ kinjF0 cos(kinjy), with amplitudeF0; the energy-injection wave vector is kinj (the lengthscale linj ≡ 2π/kinj); the superscript T denotes a trans-pose, Cβγ ≡ 〈RβRγ〉 are the elements of the polymer-conformation tensor (angular brackets indicate an aver-age over polymer configurations), the identity tensor Ihas the elements δβγ , f(rP ) ≡ (L2 − 2)/(L2 − r2P ) is theFENE-P potential that ensures finite extensibility of thepolymers, and rP ≡

√

Tr(C) and L are, respectively, thelength and the maximal possible extension of the poly-

2

mers; and c ≡ µ/(ν + µ) is a dimensionless measure ofthe polymer concentration [18] .We have developed a parallel MPI code for our DNS,

which uses periodic boundary conditions because westudy homogeneous, isotropic, turbulence; our squaresimulation domain has side L = 2π and N2 collocationpoints. We use a fourth-order Runge-Kutta scheme, withtime step δt, for time marching and an explicit, fourth-order, central-finite-difference scheme in space and theKurganov-Tadmor (KT) shock-capturing scheme [17] forthe advection term in Eq. (3); the KT scheme (see Eq. (7)of Ref. [7]) resolves sharp gradients in Cβγ and thus min-imizes dispersion errors, which increase with L and τP .We solve Eq. (2) in Fourier space by using the FFTWlibrary [19]. We choose δt ≃ 5 × 10−4 to 5 × 10−5 sothat rP does not become larger than L. Table I lists theparameters we use. We preserve the symmetric-positive-definite (SPD) nature of C at all times by adapting totwo dimensions the Cholesky-decomposition scheme ofRefs. [6, 7, 18]. In particular, we define J ≡ f(rP )C, soEq. (3) becomes

DtJ = J .(∇u) + (∇u)T .J − s(J − I) + qJ , (4)

where s = (L2 − 2+ j2)/(τPL2), q = [d/(L2 − 2)− (L2 −

2 + j2)(j2 − 2)/(τPL2(L2 − 2))], j2 ≡ Tr(J ), and d =

Tr[J .(∇u) + (∇u)T .J ]. Given that C and hence J areSPD matrices, we can write J = LLT , where L is alower-triangular matrix with elements ℓij , such that ℓij =0 for j > i; Eq.(4) now yields (1 ≤ i ≤ 2 and Γij ≡ ∂iuj)

Dtℓ11 = Γ11ℓ11 + Γ21ℓ21 +1

2

[

(q − s)ℓ11 +s

ℓ11

]

,

Dtℓ21 = Γ12ℓ11 + Γ21

ℓ222ℓ11

+ Γ22ℓ21

+1

2

[

(q − s)ℓ21 − sℓ21ℓ211

]

,

Dtℓ22 = −Γ21

ℓ21ℓ22ℓ11

+ Γ22ℓ22

+1

2

[

(q − s)ℓ22 −s

ℓ22

(

1 +ℓ221ℓ211

)

]

. (5)

Equation(5) preserves the SPD nature of C if ℓii > 0,which we enforce [6, 7] by considering the evolution ofln(ℓii) instead of ℓii.We use the following initial conditions (super-

script 0): C0βγ(x) = δβγ for all x; and ω0(x) =

ν(

− 4 cos(4y) + 10−4∑2,2

m1=0,m2=0[sin(m1x + m2y) +

cos(m1x+m2y)]m22/√

(m21 +m2

2))

. We maintain a con-stant energy-injection rate Einj ≡< Fu ·u > by a rescal-ing of the amplitude F0 at every time step. Given avalue of Einj , the system attains a nonequilibrium, sta-tistically steady state after ≃ 2− 3τe, where the box-sizeeddy-turnover time τe ≡ L/urms and urms is the root-mean-square velocity.

In addition to ω(x, t), ψ(x, t) and C(x, t) weobtain u(x, t), the fluid-energy spectrum E(k) ≡∑

k−1/2<k′≤k+1/2 k′2〈|ψ(k′, t)|2〉t, where 〈〉t denotes a

time average over the stastically steady state, the totalkinetic energy E(t) ≡ 〈1

2|u(x, t)|2〉x, enstrophy Ω(t) ≡

〈12|ω(x, t)|2〉x, palinstrophy P(t) ≡ 〈1

2|∇ × ω(x, t)|2〉x,

where 〈〉x denotes an average over our simulation do-main, the PDF of scaled polymer extensions P (rP /L),the PDFs of ω2, σ2, and Λ2 = (ω2 − σ2)/8, whereσ2 ≡

∑

ij σijσij , and σij ≡ ∂iuj + ∂jui, the PDF ofthe Cartesian components of u, and the joint PDF of Λand r2P . We obtain the order-p, longitudinal, velocitystructure function Sp(r) as follows: We first subtract themean flow from the velocity field, u′ = u − 〈u〉t, thenwe define Sp(r) = 〈[u′(rc + r, t) − u

′(rc, t)] · r/rp〉rc ,where r has magnitude r and rc is an origin, and 〈〉rcdenotes an average over time and the origin (we userc = (i, j), 2 < i, j < 5). We then extract the isotropicpart Sp(r), by using an SO(2) decomposition [9, 10], byintegration over the angle θ that r makes with the x axis,

i.e., Sp(r) ≡∫ 2π

0Sp(r)dθ. We concentrate on S2(r) and

the hyperflatness F6(r) ≡ S6(r)/(S2(r))3; the latter pro-

vides a convenient measure of the intermittency at thescale r.In Fig.(1a) we show how the total kinetic energy

E(t) (top panel), enstrophy Ω(t) (middle panel), andpalinstrophy P(t) (bottom panel) of the fluid fluctuateabout their mean values 〈E(t)〉x, 〈Ω(t)〉x, and 〈P(t)〉xas a function of time t for c = 0 (pure fluid) and 0.4.Clearly 〈E(t)〉x, 〈Ω(t)〉x, and 〈P(t)〉x decrease as c in-creases. This suggests that the addition of polymersincreases the effective viscosity of the solution. How-ever, this naıve conclusion has to be refined becausethe effective viscosity, bacause of the polymers, dependson the length scale [6, 7] as can be seen by comparingthe fluid-energy spectra, with and without polymers, inFig.(1b). Indeed we can define [6, 7] the effective, scale-dependent viscosity νe(k) ≡ ν + ∆ν(k), with ∆ν(k) ≡−µ∑k−1/2<k′≤k+1/2 uk′ ·(∇·J )−k′/[τPk

′2Ep,m(k′)] and

(∇ · J )k the Fourier transform of ∇ · J . The rightinset of Fig.(2a) shows that ∆ν(k) > 0 for k < 30,where Ep(k) < Ef (k), whereas, for large values of k,∆ν(k) < 0, where Ep(k) > Ef (k); the superscripts f andp stand, respectively, for the fluid without and with poly-mers. The left inset of Fig.(2a) shows the suppression,by polymer additives, of Π(k) =

∫∞

k′T (k′)dk′, where

T (k) =∫

ui(k)Pij(k) (u × ω)j(k)dΩ and the projector

Pij(k) = δij − kikj

k2 . The suppression of the spectrumin the small-k regime, which has also been seen in theexperiments [11], and preliminary results in Fig. (4.12)of [12], signifies a reduction of the inverse cascade; theenhancement of the spectrum in the large-k regime leadsto the reduction in Ω and P on the addition of polymers(see the middle and bottom panels of Fig.(1a)).The second-order structure function S2(r) is related

3

simply to the energy spectrum. It is natural to ask, there-fore, how S2(r) is modified by the addition of polymers.In Fig(1c), we plot S2(r) versus r for c = 0 (blue circlesand run R9) and c = 0.2 (green asterisks and run R9);the dashed line, which is shown to guide the eyes has aslope 2; this value is consistent with the S2(r) ∼ r2 formthat we expect, at small r, by Taylor expansion. At largevalues of r, S2(r) deviates from this r2 behavior, moreso for c = 0.2 than for c = 0; this is consistence with theresults of the experiments of Ref. [13]. The inset of Fig.(1c), which contains plots of the hyperflatness F6(r) ver-sus r for c = 0 (blue circles) and c = 0.2 (green asterisks),shows that, on the addition of polymers, small-scale in-termittency decreases as c increases. In Fig. (2a), weshow how Ep(k) changes, as we increase c : at low andintermediate values of k (e.g., k = 1 and 30, respectively),Ep(k) decreases as c increases; but for large values of k(e.g., k = 100) it increases with c. Figure (2b) shows howEp(k) changes, as we increase τP with c held fixed at 0.1.At low values of k (e.g., k = 1), Ep(k) decreases as τPincreases; but for large values of k (e.g., k = 100) it in-creases with τP . In Fig. (2c) we give plots, for c = 0.1, ofthe spectra Ep(k) for L = 100 (red triangles and run R8)and L = 10 (green asterisks and run R9); for comparisonwe also plotc Ef (k) for c = 0 ; as L increases, the differ-ence between Ep(k) and Ef (k) increases at large valuesof k.We turn now to the PDF P (rp/L), which we plot ver-

sus the scaled polymer extension rP /L in Fig. (2d) forc = 0.1 and L = 100 (red triangles and run R8), c = 0.4and L = 100 (black squares and run R8), and c = 0.1and L = 10 (green asterisks and run R9). These PDFsfall off sharply (a) near rP /L ≃ 1, because rP 6 L, and(b) rP >

√2, because r2P = Tr(C) > 2; at values of rP /L

that lie between these two extremes, P (rp/L) shows a dis-tinct, power-law regime, with an exponent that dependson c, L, and We; as We increases, this power can go froma negative value to a positive value thus signalling a coil-stretch transition. We will give a detailed study of the Land τP dependence of P (rP /L) elsewhere. Here we showa representative plot, for L = 6 and c = 0.1 (brown dotsand run R1), of P (rP /L) for a case in which the polymersare very nearly stretched.In Figs. (3a), (3b), and (3c) we show PDFs of Λ, σ2,

and ω2, respectively, for c = 0 (blue circles and run R7)and c = 0.2 (red triangles and run R7). These PDFs showthat the addition of polymers suppresses large values ofΛ, σ2, and ω2. The inset of Fig. (3a) shows a filled con-tour plot of the joint PDF of r2P and Λ; this illustratesthat r2P is large where Λ is large and negative, i.e., poly-mer stretching occurs predominantly in strain-dominatedregions; this is evident very strikingly in Fig.(3d), whichcontains a superimposition of contours of r2P on a pseu-docolor plot of Λ; we give a video of a sequence of suchplots at youtube and at rahul .Our extensive DNS study of two-dimensional fluid tur-

bulence with polymer additives yields good qualitativeagreements in low-k regime, with the fluid-energy spec-tra of Ref. [11] and the second-order velocity structurefunction of Ref. [13]. In addition, our study brings outnew results that we hope will stimulate new experiments.In particular, experiments should be able to measure (a)the reduction of 〈E(t)〉x, 〈Ω(t)〉x, and 〈P(t)〉x shown inFig. (1a), (b) the modification of Ep(k) at large k (Fig.(1b)), (c) the c and L dependence of Epk illustrated inFigs. (2a)-(2c), (d) the PDF of (rP /L), Λ, σ

2, and ω2,and (e) the stretching of polymers in strain-dominatedregions as illustrated in Fig. (3d).We thank D. Mitra for discussions, CSIR, UGC, DST

(India), and the COST Action MP0806 for support, andSERC (IISc) for computational resources. PP and RPare members of the International Collaboration for Tur-bulence Research.

∗ anupam@physics.iisc.ernet.in† p.perlekar@tue.nl‡ rahul@physics.iisc.ernet.in;also at Jawaharlal Nehru Centre For Advanced ScientificResearch, Jakkur, Bangalore, India.

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4

FIG. 1. (Color online) (a) Plots versus time t/τe for run R7 of the total kinetic energy of the fluid E (top panel), the enstrophyΩ (middle panel),and the palinstrophy P (bottom panel) for c = 0 (blue circles) and c = 0.4 (black square); (b)Log-log (base10) plots of the energy spectra E(k) versus k for c = 0.2 (red triangles) and c = 0 (blue circles); left inset: energy flux Π(k)versus k for c = 0.2 (red line) and c = 0 (blue line); right inset: polymer contribution to the scale-dependent viscosity ∆ν(k)versus k for c = 0.2 (blue line) for runs R10; bottom (c) Plots of the second order velocity structure function S2(r) versus rversus run R7 for c = 0 (blue circle) and c = 0.2 (red triangles); the dashed line with slope 1.98 is shown for comparision; theinset shows a plot of the hyperflatness F6(r) versus for r for c = 0 (blue circles) and c = 0.2 (red triangles).

FIG. 2. (Color online) (a) Log-log (base 10) plots of the energy spectra E(k) versus k for c = 0 (bluecircles), for c = 0.2 (redtriangles) and c = 0.4 (black squares); left bottom inset: variation of E(k) with polymer concentration for τP = 1 and k = 1;left up inset: variation of E(k) with polymer concentration for τP = 1 and k = 30; right up inset: variation of E(k) withpolymer concentration for τP = 1 and k = 100; run R7; (b) Log-log (base 10) plots of the energy spectra E(k) versus k forτP = 1 (blue line with open circles) , for τP = 0.2 (red line with open triangles) and τP = 4 (black line with open squares);left bottom inset: variation of E(k) with polymer relaxation time for c = 0.4 and k = 1; right up inset: variation of E(k) withpolymer relaxation time fot c = 0.4 and k = 100; run R2 (c) Log-log (base 10) plots, for c = 0.2 and τP = 1, of the energyspectra E(k) versus k for L = 100 (red triangles) and L = 10 (green asterisks); for comparison we put c = 0.0 (blue circles);left bottom inset: ∆ν(k) versus k; right up inset: zoomed picutre of the above said inset, (d) PDFs of polymer extensionsP (r2P/L

2) versus r2P /L2 for c = 0.2 (red triangles), c = 0.4 (black squares) for run R8, c = 0.2 and L = 10 (green asterisks) for

run R9, and c = 0.2 and L = 6 (brown dots) for run R1

−5 0 5 10 15

10−4

10−2

100

Λ

P(Λ

)

(a)

rp2

Λ

500 1000

0

100

200

300

0

2

4

6

8

FIG. 3. (Color online) Probability distribution function (PDFs) of (a) the Okubo-Weiss parameter Λ for run R7; the insetshows a filled contour plot of the joint PDF of Λ and r2P , the square of the polymer extension; (b) σ2, the square of the strainrate, for run R7, and (c) ω2, the square of the vorticity, the inset is PDF of scaled polymer extensions (rP /L); for c = 0 (bluecircles) and c = 0.2 (red triangles) for run R7 (d) A representative pseudocolor plot of Λ superimposed on a contour plot of r2Prun R10.

5

N L τP δt× 104 Einj ν × 10−4 We c Reλ kmaxηd

R1 512 6 2 10.0 0.008 10.0 4.40 0.1 107, 85 3.4, 3.6

R2 1024 100 1, 2, 4 1.0 0.005 5.0 2.11 4.02 6.01 0.1 221, 121, 95 , 68 5.1, 5.3, 5.4, 5.5

R3 2048 100 1 1.0 0.003 5.0 1.60 0.4 147, 60 14.1, 14.8

R4 2048 100 1 1.0 0.0015 5.0 1.26 0.2 86 , 54 13.2, 13.6

R5 4096 100 1 1.0 0.005 5.0 2.08 0.2 233, 90 20.2, 20.9

R6 4096 100 1 1.0 0.002 5.0 2.08 0.2 0.4 108, 62, 45 24.8, 25.8, 26.2

R7 4096 10 1 1.0 0.002 5.0 2.08 0.4 108, 90 24.8, 26.2

R8 4096 100 1 0.5 0.005 1.0 2.81 0.1 1451, 1407 8.0, 8.2

R9 4096 10 1 0.5 0.005 1.0 2.81 0.1 1451, 1407 8.0, 8.2

R10 16384 100 1 0.5 0.002 5.0 1.40 0.2 105, 60 96.4, 102.7

TABLE I. The parameters for our DNS runs R1-R10 for the two-dimensional, incompressible Navier-Stokes equation withair-drag-induced friction and polymer additives modelled via the FENE-P model. In all our runs the polymer relaxation timeτP = 1.0 and the friction coefficient α = 0.01. We also carry out DNS studies of the two-dimensional NS equation with thesame numerical resolutions as used in our runs with polymer additives. N is the number of collocation points, δt the time step,Einj the energy-injection rate, ν the kinematic viscosity, and c the concentration parameter. The Taylor-microscale Reynolds

number is Reλ ≡√2E/

√νǫ and the Weissenberg number is We ≡ τP

√

ǫ/ν, where E is the total kinetic energy of the fluid andǫ the energy dissipation rate per unit mass for the fluid.

6

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[18] T. Vaithianathan and L. Collins, J. Comput. Phys. 187,

1 (2003).[19] http://www.fftw.org