Direct numerical simulations of statistically steady,
homogeneous, isotropic fluid turbulence with polymer
additives
Citation for published version (APA):
Perlekar, P., Mitra, D., & Pandit, R. (2010). Direct numerical simulations of statistically steady, homogeneous, isotropic fluid turbulence with polymer additives. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 82(6), 066313-1/9. [066313]. https://doi.org/10.1103/PhysRevE.82.066313
DOI:
10.1103/PhysRevE.82.066313 Document status and date: Published: 01/01/2010
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Direct numerical simulations of statistically steady, homogeneous, isotropic fluid turbulence
with polymer additives
Prasad Perlekar,1,
*
Dhrubaditya Mitra,2,†and Rahul Pandit3,‡1
Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
2
NORDITA, Roslagstullsbacken 23, 106 91 Stockholm, Sweden 3
Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bangalore 560012, India
共Received 31 July 2010; published 27 December 2010兲
We carry out a direct numerical simulation共DNS兲 study that reveals the effects of polymers on statistically steady, forced, homogeneous, and isotropic fluid turbulence. We find clear manifestations of dissipation-reduction phenomena: on the addition of polymers to the turbulent fluid, we obtain a dissipation-reduction in the energy dissipation rate; a significant modification of the fluid-energy spectrum, especially in the deep-dissipation range; and signatures of the suppression of small-scale structures, including a decrease in small-scale vorticity filaments. We also compare our results with recent experiments and earlier DNS studies of decaying fluid turbulence with polymer additives.
DOI:10.1103/PhysRevE.82.066313 PACS number共s兲: 47.27.Gs, 47.27.Ak
I. INTRODUCTION
The addition of small amounts of polymers to a turbulent fluid leads to dramatic changes that include modifications of the small-scale properties of the flow 关1–4兴 and, in
wall-bounded flows, the phenomenon of drag reduction关5–7兴, in
which polymer additives allow the maintenance of a given flow rate at a lower pressure gradient than is required without these additives. Several experimental, numerical, and ana-lytical studies have investigated drag reduction 关6–14兴 in
wall-bounded flows. These studies have shown that the ad-dition of polymers modifies the turbulence significantly in the region near the wall, and this leads to an increase in the mean velocity in the bulk. By contrast, there have been only some investigations of the effects of polymer additives on homogeneous isotropic turbulence. Examples include recent experiments 关15–17兴, which have been designed to obtain a
high degree of isotropy in the turbulent flow, and shell-model studies and direct numerical simulations 共DNSs兲 关2–4,13,18,19兴. These studies have shown that the addition
of polymers to a turbulent flow leads to a considerable re-duction in small-scale structures; and they have also discov-ered the phenomenon of dissipation reduction, namely, a re-duction in the energy dissipation rate ⑀, in decaying turbulence关2–4兴. In this paper we first elucidate the
phenom-enon of dissipation reduction for the case of statistically steady, homogeneous, and isotropic turbulence with polymer additives; we then study the small-scale properties of such flows.
We do this by conducting a series of high-resolution DNS studies of the three-dimensional Navier-Stokes 共NS兲 equa-tion coupled to an equaequa-tion for the polymer conformaequa-tion tensor, which describes the polymer additives at the level of
the finitely extensible nonlinear elastic-Peterlin 共FENE-P兲 关2,3兴 model. Before we give the details of our study, it is
useful to summarize our principal results; these are in two parts.
The first part contains results from our DNS of forced statistically steady fluid turbulence, with polymer additives, at moderate Reynolds numbers 共Re⯝80兲. The forcing is chosen such that the energy injected into the fluid remains fixed关20兴, both with and without polymers; this mimics the
forcing scheme used in the experiments of Refs.关16,17兴. We
find that, on the addition of polymers, the energy in the sta-tistically steady state and the energy-dissipation rate are re-duced. This dissipation reduction increases with an increase in the polymer concentration c at fixed Weissenberg number We, the ratio共see TableI兲 of the polymer time scalePto a
shearing time scale in the turbulent fluid. The dissipation reduction also increases with We if we hold c fixed. The dissipation reduction seen in our simulations should not be confused with the phenomenon of drag reduction seen in wall-bounded flows. In the fluid-energy spectrum we find that the energy content increases marginally at small wave vectors on the addition of polymers, but it decreases for in-termediate wave vectors. In this part of our study we use 2563collocation points and attain a moderate Reynolds
num-ber Re⯝80, but we do not resolve the deep-dissipation range.共We consider the deep-dissipation range in the follow-ing paragraph.兲 We also obtain the structural properties of the fluid with and without polymers and show that polymers suppress large-vorticity and large-strain events; our results here are in qualitative agreement with the experiments of Refs. 关16,17兴. Furthermore, we find, as in our study of
de-caying turbulence 关3兴, that the polymer extension increases
with an increase in the polymer-relaxation timeP. We
com-pare our results, e.g., those for the energy spectrum, with their counterparts in our earlier study of decaying fluid tur-bulence with polymer additives关3兴. In such comparisons, we
use averages over the statistically steady state of our system here; and, for the case of decaying turbulence, we use data obtained at the cascade-completion time, at which a plot of *p.perlekar@tue.nl
†
dhruba.mitra@gmail.com
‡
Also at Jawaharlal Nehru Centre for Advanced Scientific Re-search, Jakkur, Bangalore, India; rahul@physics.iisc.ernet.in
the energy dissipation rate versus time displays a maximum. In the second part of our study we carry out the highest-resolution DNS, attempted so far, of forced statistically steady fluid turbulence with polymer additives; we drive the fluid by an external stochastic force as in Ref.关21兴. This part
of our study has been designed to uncover the effects of polymers on the deep-dissipation range, so the Reynolds numbers is small 共Re⯝16兲. By comparing fluid-energy spectra, with and without polymers, we find that the poly-mers suppress the energy in the dissipation range but in-crease it in the deep-dissipation range. Finally, we calculate the second-order velocity structure function S2共r兲 directly
from the energy spectrum via a Fourier transformation; this shows that S2共r兲 with polymers is smaller than S2共r兲 without
polymers in this range.
The remaining part of this paper is organized as follows. In Sec. II we present the equations we use for the polymer solution and describe the method we use for the numerical integration of these equations. Section III is devoted to a discussion of our results; as we have mentioned above, these are divided into two parts: the first part is contained in Secs.
III A–III C and the second one is contained in Sec. III D.
SectionIVcontains a concluding discussion.
II. EQUATIONS AND NUMERICAL METHODS We model a polymeric fluid solution by using the three-dimensional NS equations for the fluid coupled with the FENE-P equation for the polymer additives关3兴. The polymer
contribution to the fluid is modeled by an extra stress term in the NS equations. The FENE-P equation approximates a polymer molecule by a nonlinear dumbbell, which has a single relaxation time and an upper bound on the maximum extension. The NS and FENE-P共henceforth NSP兲 equations are Dtu =ⵜ2u + P · 关f共rP兲C兴 − p + f, 共1兲 DtC = C · 共u兲 + 共u兲 T ·C −f共rP兲C − I P . 共2兲
Here, u共x,t兲 is the fluid velocity at point x and time t, in-compressibility is enforced by·u=0, Dt=t+ u ·, is the
kinematic viscosity of the fluid,is the viscosity parameter for the solute共FENE-P兲,Pis the polymer-relaxation time,
is the solvent density共set to 1兲, p is the pressure, f共x,t兲 is the external force at point x and time t,共u兲Tis the transpose of 共u兲, C␣⬅具R␣R典 is the elements of the
polymer-conformation tensor C 共angular brackets indicate an average over polymer configurations兲, I is the identity tensor with elements ␦␣, f共rP兲⬅共L2− 3兲/共L2− rP
2兲 is the FENE-P
poten-tial that ensures finite extensibility, rP⬅
冑
Tr共C兲 and L are thelength and the maximum possible extension, respectively, of the polymers, and c⬅/共+兲 a dimensionless measure of the polymer concentration关19兴; c=0.1 corresponds, roughly,
to 100 ppm for polyethylene oxide关7兴. TableIlists the pa-rameters of our simulations.
Numerical methods
We consider homogeneous isotropic turbulence, so we use periodic boundary conditions and solve Eq. 共1兲 by using a
pseudospectral method关22,23兴. We use N3collocation points
in a cubic domain共side L=2兲. We eliminate aliasing errors by the 2/3 rule关22,23兴 to obtain reliable data at small length
scales, and we use a second-order slaved Adams-Bashforth scheme for time marching. In earlier numerical studies of homogeneous isotropic turbulence with polymer additives, it has been shown that sharp gradients are formed during the time evolution of the polymer conformation tensor; this can lead to dispersion errors 关19,24兴. To avoid these dispersion
errors, shock-capturing schemes have been used to evaluate the polymer-advection term 关共u·兲C兴 in Ref. 关24兴. In our
simulations we have modified the Cholesky-decomposition scheme of Ref. 关19兴, which preserves the
symmetric-positive-definite 共SPD兲 nature of the tensor C. We incorpo-rate the large gradients of the polymer conformation tensor by evaluating the polymer-advection term 关共u·兲ᐉ兴 via the Kurganov-Tadmor shock-capturing scheme关25兴. For the
de-rivatives on the right-hand side of Eq.共2兲 we use an explicit
fourth-order central-finite-difference scheme in space, and the temporal evolution is carried out by using an Adams-Bashforth scheme. The numerical error in rP must be con-trolled by choosing a small time step ␦t; otherwise, rP can
become larger than L, which leads to a numerical instability. This time step is much smaller than what is necessary for a
TABLE I. The cube root N of the number of collocation points, the time step␦t, the maximum possible
polymer extension L, the kinematic viscosity, the polymer-relaxation timeP, and the polymer concentra-tion parameter c for our four runs NSP-256A, NSP-256B, and NSP-512. We also carry out DNS studies of the NS equation with the same numerical resolutions as in our NSP runs. The Taylor-microscale Reynolds number Re⬅
冑
20Ef/冑
3⑀f and the Weissenberg number We⬅P冑
⑀f/ are as follows: NSP-256A andNSP-256B: Re⯝80 and NSP-512: Re⯝16; the Kolmogorov dissipation length scale is⬅共3/⑀ f兲1/4. For
our runs NSP-256A-B,⯝1.07␦x; and for run NSP-512,⯝19␦x, where␦x⬅L/N is the grid resolution of
our simulations. The integral length scale lint⬅共3/4兲兺k−1E共k兲/关兺E共k兲兴 and T
eddy⬅urms/lintare as follows:
NSP-256A and NSP-256B: lint⯝1.3 and Teddy⯝1.2; and, for NSP-512, lint⯝2.05 and Teddy⯝4.0.
N ␦t L P c We
NSP-256A 256 5.0⫻10−4 100 5⫻10−3 0.5 0.1 3.5
NSP-256B 256 5.0⫻10−4 100 5⫻10−3 1.0 0.1 7.1
NSP-512 512 10−3 100 5⫻10−2 1.0 0.1 0.9
PERLEKAR, MITRA, AND PANDIT PHYSICAL REVIEW E 82, 066313共2010兲
pseudospectral DNS of the NS equation alone. Table Ilists the parameters we use. We preserve the SPD nature ofC at all times by using 关19兴 the following
Cholesky-decomposition scheme: if we define
J ⬅ f共rP兲C, 共3兲
Eq. 共2兲 becomes
DtJ = J · 共u兲 + 共u兲T·J − s共J − I兲 + qJ, 共4兲
where s =L 2− 3 + j2 PL2 , q =d/共L 2− 3兲 − 共L2− 3 + j2兲共j2− 3兲 关PL2共L2− 3兲兴 , j2⬅ Tr共J兲, d = Tr关J · 共ⵜu兲 + 共ⵜu兲T·J兴.
Here, C and hence J are SPD matrices; we can, therefore, write J=LLT, where L is a lower-triangular matrix with elementsᐉij, such thatᐉij= 0 for j⬎i, and
J ⬅
冢
ᐉ11 2 ᐉ 11ᐉ21 ᐉ11ᐉ31 ᐉ11ᐉ21 ᐉ21 2 +ᐉ222 ᐉ21ᐉ31+ᐉ22ᐉ32 ᐉ11ᐉ31 ᐉ21ᐉ31+ᐉ22ᐉ32 ᐉ312 +ᐉ322 +ᐉ332冣
. 共5兲Equation共4兲 now yields 共1ⱕiⱕ3 and ⌫ij=iuj兲 the follow-ing set of equations:
Dtᐉi1=
兺
k ⌫kiᐉk1+ 1 2冋
共q − s兲ᐉi1+共− 1兲 共i mod 1兲sᐉi1 ᐉ112册
+共␦i3+␦i2兲 ᐉi2 ᐉ11m兺
⬎1 ⌫m1ᐉm2+␦i3⌫i1 ᐉ332 ᐉ11 , for iⱖ 1, Dtᐉi2=兺
mⱖ2 ⌫miᐉm2− ᐉi1 ᐉ11m兺
ⱖ2 ⌫m1ᐉm2+ 1 2冋
共q − s兲ᐉi2 +共− 1兲共i+2兲sᐉi2 ᐉ222冉
1 +ᐉ21 2 ᐉ11 2冊
册
+␦i3冋
ᐉ33 2 ᐉ22冉
⌫32−⌫31 ᐉ21 ᐉ11冊
+ sᐉ21ᐉ31 ᐉ11 2 ᐉ 22册
, for iⱖ 2, Dtᐉ33=⌫33ᐉ33−ᐉ33冋
兺
m⬍3 ⌫3mᐉ3m ᐉmm册
+⌫31ᐉ32ᐉ21ᐉ33 ᐉ11ᐉ22 − sᐉ21ᐉ31ᐉ32 ᐉ112ᐉ22ᐉ33 +1 2冋
共q − s兲ᐉ33+ s ᐉ33冉
1 +兺
m⬍3 ᐉ3m 2 ᐉmm2冊
+ sᐉ21 2ᐉ 32 2 ᐉ112 ᐉ222ᐉ33册
. 共6兲The SPD nature ofC is preserved by Eqs. 共6兲 if ᐉii⬎0, which we enforce explicitly 关19兴 by considering the evolution of
ln共ᐉii兲 instead of ᐉii.
We resolve the sharp gradients in the polymer conforma-tion tensor by discretizing the polymer-advecconforma-tion term by using the Kurganov-Tadmor scheme 关25兴. Below we show
the discretization of the advection term uxᐉ, where u ⬅共u,v,w兲 and ᐉ is one of the components of ᐉ␣; the
dis-cretization of the other advection terms in Eq.共6兲 is similar:
uxᐉ =
Hi+1/2,j,k− Hi−1/2,j,k
␦x ,
Hi+1/2,j,k=ui+1/2,j,k关ᐉi+1/2,j,k
+ +ᐉi+1− /2,j,k兴 2 −ai+1/2,j,k关ᐉi+1/2,j,k + −ᐉi+1− /2,j,k兴 2 , ᐉi+1⫾ /2,j,k=ᐉi+1,j,k⫿ ␦x 2共xᐉ兲i+1/2⫾1/2,j,k,
ai+1/2,j,k⬅ 兩ui+1/2,j,k兩, 共7兲
where i , j , k = 0 , . . . ,共N−1兲 denote the grid points and ␦x
=␦y =␦z is the grid spacing along the three directions.
We use the following initial conditions 共superscript 0兲:
Cmn0 共x兲=␦mn for all x and um0共k兲= Pmn共k兲vn0共k兲exp关n共k兲兴, with m , n = x , y , z, Pmn=共␦mn− kmkn/k2兲 as the transverse pro-jection operator, k as the wave vector with components km =共−N/2,−N/2+1, ... ,N/2兲 and magnitude k=兩k兩, n共k兲 as random numbers distributed uniformly between zero and 2, and vn0共k兲 chosen such that the initial kinetic-energy
spec-trum is E0共k兲=k4exp共−2.0k2兲. This initial condition corre-sponds to a state in which the fluid energy is concentrated, to begin with, at small k共large length scales兲, and the polymers are in a coiled state. Our simulations are run for 45Teddyand a statistically steady state is reached in roughly 10Teddy,
where the integral-scale eddy-turnover time Teddy⬅urms/lint,
with urms as the root-mean-square velocity and lint
⬅兺kk−1E共k兲/兺kE共k兲 as the integral length scale. Along with our runs NSP-256A and NSP-256B we also carry out pure-fluid NS simulations until a statistically steady state is reached; this takes about 10Teddy– 15Teddy. Once this
pure-fluid simulation reaches a statistically steady state, we add polymers to the fluid at 27Teddy; i.e., beyond this time we solve the coupled NSP equations 共1兲 and 共2兲 by using the
methods given above. We then allow 5Teddy– 6Teddy to
elapse, so that transients die down, and then we collect data for fluid and polymer statistics for another 25Teddy for our
runs NSP-256A and NSP-256B.
III. RESULTS
We now present the results that we have obtained from our DNS. In addition to u共x,t兲, its Fourier transform uk共t兲,
andC共x,t兲, we monitor the vorticity⬅⫻u, the
kinetic-energy spectrum E共k,t兲⬅兺k−1/2⬍k⬘ⱕk+1/2兩uk⬘
2 共t兲兩, the total
ki-netic energy E共t兲⬅兺kE共k,t兲, the energy-dissipation rate
⑀共t兲⬅兺kk2E共k,t兲, the probability distribution of scaled polymer extensions P共rP2/L2兲, the probability distribution
function共PDF兲 of the strain and the modulus of the vorticity, and the eigenvalues of the strain tensor. For notational con-venience, we do not display the dependence on c explicitly. In Sec.III A we present the time evolution of E and⑀ and provide evidence for dissipation reduction by polymer addi-tives. This is followed by Secs. III B and III C that deal, respectively, with the effects of polymers on fluid-energy spectra and small-scale structures in turbulent flows. In Sec.
III Dwe examine the modification, by polymer additives, of
fluid-energy spectra in the deep-dissipation range.
A. Energy and its dissipation rate
We first consider the effects of polymer additives on the time evolution of the fluid energy E for our runs NSP-256A and NSP-256B; this is shown in Fig. 1. The polymers are added to the fluid at t = 27Teddy. The addition of polymers
leads to a new statistically steady state; specifically, we find for We= 3.5 and We= 7.1 that the average energy of the fluid with polymers is reduced in comparison to the average en-ergy of the fluid without polymers. By using Eq. 共1兲, we
obtain the following energy-balance equation for the fluid with polymer additives:
dE dt =⑀+⑀poly+⑀inj, 共8兲 ⑀= −1 V
冕
u ·ⵜ 2u, ⑀poly=冉
P冊
再
1 V冕
u ·关f共rP兲C兴冎
, ⑀inj= 1 V冕
f · u. 共9兲In the statistically steady state dEdt= 0, and the energy injected is balanced by the fluid dissipation rate ⑀ and the polymer dissipation ⑀poly. Our simulations are designed to keep the
energy injection fixed. Therefore, we can determine how the dissipation gets distributed between the fluid and polymer subsystems in forced statistically steady turbulence.
Before we present our results for the kinetic-energy dissi-pation rate, we first calculate the second-order structure func-tion S2共r兲 via the following exact relation 关26兴:
S2共r兲 =
冕
0⬁
冋
1 −sin共kr兲kr
册
E共k兲dk. 共10兲In Fig. 2 we give a log-log plot of S2共r兲, compensated by
共r/L兲−2, as a function of r/L. We find that, for small r, S2共r兲⬃r2, which implies that our DNS resolves the
analyti-cal range, which follows from a Taylor expansion of S2共r兲 关27兴; this guarantees that energy-dissipation rate has been
calculated accurately. In Fig.3we present plots of⑀共t兲 ver-sus t/Teddyfor We= 3.5 and We= 7.1 with the polymer
con-centration c = 0.1. We find that the average value of ⑀ de-creases as we increase We. This suggests the following natural definition of the percentage dissipation reduction for forced, homogeneous, and isotropic turbulence:
2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 t / T e d d y 2 . 2 2 . 4 2 . 6 2 . 8 3 . 0 3 . 2 3 . 4 E
FIG. 1. 共Color online兲 A plot of the fluid energy E versus the dimensionless time t/Teddy 共runs NSP-256A and NSP-256B兲 for
Weissenberg numbers We= 3.5 共blue circles兲 and We=7.1 共black dashed line兲. The corresponding plot for the pure-fluid case is also shown for comparison 共red solid line兲. The polymers are added to the fluid at t = 27Teddy.
− 2 . 0 − 1 . 5 − 1 . 0 − 0 . 5 l o g 1 0( r / ) 1 . 0 1 . 5 2 . 0 2 . 5 lo g10 [( r/ ) − 2 S 2 (r )]
FIG. 2. 共Color online兲 Log-log 共base 10兲 plots of the second-order structure function S2共r兲, compensated by 共r/L兲−2, versus r/L,
for our run NSP-256B共blue square兲 and for the pure-fluid case 共red circle兲. The regions in which the horizontal black lines overlap with the points indicate the r2scaling ranges.
2 0 2 5 3 0 3 5 4 0 4 5 5 0 5 5 6 0 t / T e d d y 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 εν
FIG. 3. 共Color online兲 Plots of the energy dissipation rate ⑀ versus t/Teddy 共runs NSP-256A and NSP-256B兲 for Weissenberg numbers We= 3.5 共blue circles兲 and We=7.1 共black dashed line兲. The corresponding plot for the pure-fluid case is also shown for comparison共red solid line兲. The polymers are added to the fluid at
t = 27Teddy.
PERLEKAR, MITRA, AND PANDIT PHYSICAL REVIEW E 82, 066313共2010兲
DR⬅
冉
具⑀f典 − 具⑀ P典
具⑀f典
冊
⫻ 100%. 共11兲Here共and henceforth兲, the superscripts f and p stand, respec-tively, for the fluid without and with polymers and the angu-lar brackets denote an average over the statistically steady state. The percentage dissipation reduction DR rises with We; this indicates that ⑀P decreases with We关28兴. Thus, in
contrast to the trend we observed in our decaying-turbulence DNS 关3兴, DR increases with We: for We=3.5, DR⯝30%
and, for We= 7.1, DR⯝50%. Our interpretation is that this increase in DR with We arises because the polymer exten-sions and, therefore, the polymer stresses are much stronger in our forced-turbulence DNS than in our decaying-turbulence DNS 共at least for the Reynolds numbers that we achieved in Ref.关3兴兲. In Fig.4we show the cumulative PDF of the scaled polymer extension; this shows clearly that the extension of the polymers increases with We. In general, the calculation of PDFs from numerical data is plagued by errors originating from the binning of the data to make histograms. Here, instead, we have used the rank-order method to calcu-late the corresponding cumulative PDF, which is free of bin-ning errors 关29兴.
B. Energy spectra
In this section we study fluid-energy spectra Ep共k兲, in the presence of polymer additives, for two different values of the Weissenberg number We and fixed polymer concentration c = 0.1共Fig.5兲. We find that the energy content at intermediate
wave vectors decreases with an increase in We. At small wave-vector magnitudes k, we observe a small increase in the spectrum on the addition of the polymers, but this in-crease is within our numerical two-standard-deviation error bars. Because of the moderate resolution of our simulations we are not able to resolve the dissipation range fully in these simulations. We address this issue by conducting high-resolution low-Reynolds-number simulations in Sec.III D.
C. Small-scale structures
We now investigate how polymers affect small-scale structures in homogeneous isotropic fluid turbulence, and we make specific comparisons with experiments关16,17兴. We
be-gin by plotting the PDFs of the modulus of the vorticity兩兩 and the local energy dissipation rate ⑀loc=兺i,j共iuj+jui兲2/2 in Fig.6. We find that the addition of polymers reduces re-gions of high vorticity and high dissipation共Fig.6兲.
Further-more, we find that, on normalizing兩兩 or⑀locby their
respec-tive standard deviations, the PDFs of these normalized quantities for the fluid with and without polymers collapse
1 0 - 3 1 0 - 2 1 0 - 1 r 2 P / L 2 1 0 - 5 1 0 - 4 1 0 - 3 1 0 - 2 1 0 - 1 1 0 0 P C(r 2 P/L 2)
FIG. 4.共Color online兲 Log-log 共base 10兲 plots of the cumulative PDF PC共r
P
2/L2兲 versus the scaled polymer extension r P
2/L2for We
= 3.5共blue dashed line for run NSP-256A兲 and We=7.1 共full black line for run NSP-256B兲. Note that as We increases, so does the extension of the polymers. These plots are obtained from polymer configurations at t = 60Teddy. 0 0.5 1 1.5 −6 −4 −2 0 log10(k) log 10 [E f (k)] and log 10 [E p (k)]
FIG. 5. 共Color online兲 Log-log plots 共base 10兲 of the energy spectra Ep共k兲 versus k 共runs NSP-256A and NSP-256B兲 for c=0.1
and We= 3.5共blue squares兲 or We=7.1 共black stars兲; we give two-standard-deviation error bars. The corresponding pure-fluid spec-trum Ef共k兲 共red circles兲 is shown for comparison.
0 2 4 6ε 8 1 0 1 2 l o c 1 0 - 6 1 0 - 5 1 0 - 4 1 0 - 3 1 0 - 2 1 0 - 1 1 0 0 1 0 1 P( εloc )
FIG. 6. 共Color online兲 Semilogarithmic plots 共base 10兲 of the PDFs P共兩兩兲 versus 兩兩 共top panel兲 and P共⑀loc兲 versus⑀loc共bottom panel兲, for our run NSP-256B, with 关c=0.1, We=7.1 共blue dashed line兲兴 and without 关c=0 共full red line兲兴 polymer additives.
onto each other共within our numerical error bars兲 as shown in Fig. 7. Our results for these PDFs are in qualitative agree-ment with the results of Refs.关16,17兴 共see Fig. 2 of Ref. 关16兴
and Fig. 3 of Ref. 关17兴兲. Earlier high-resolution large-Re
DNS studies of homogeneous isotropic fluid turbulence
with-out polymer additives共see, e.g., Refs. 关30,31兴 and references
therein兲 have established that iso-兩兩 surfaces are filamentary for large values of兩兩. In Fig.8 we show how such iso-兩兩
surfaces change on the addition of polymers 共c=0.1, We = 3.5 or 7.1兲. In particular, the addition of polymers sup-presses a significant fraction of these filaments共compare the top and middle panels of Fig. 8兲, and this suppression
be-comes stronger as We increases 共middle and bottom panels of Fig.8兲. In addition to suppressing events which contribute
to large fluctuations in the vorticity, the addition of polymers also affects the statistics of the eigenvalues of the rate-of-strain matrix 关Sij=共iuj+jui兲/
冑
2兴, namely, ⌳n, with n = 1 , 2 , 3. They provide a measure of the local stretching and compression of the fluid. In our study, these eigenvalues are arranged in decreasing order, i.e.,⌳1⬎⌳2⬎⌳3.Incompress-ibility implies that 兺i⌳i= 0; therefore, for an incompressible fluid, one of the eigenvalues 共⌳1兲 must be positive and one 共⌳3兲 must be negative. The intermediate eigenvalue ⌳2 can
either be positive or negative. In Figs.9 and10we plot the PDFs of these eigenvalues. The tails of these PDFs shrink on the addition of polymers. This indicates that the addition of the polymers leads to a substantial decrease in the regions where there is large strain, a result that is in qualitative
agreement with the experiments of Ref.关16兴 共see Fig. 3共b兲 of
Ref.关16兴兲. Evidence for the suppression of small-scale
struc-tures on the addition of polymers can also be obtained by examining the attendant change in the topological properties of a three-dimensional turbulent flow. For incompressible ideal fluids in three dimensions there are two topological invariants: Q⬅−Tr共A2兲/2 and R⬅−Tr共A3兲/3, where A is the velocity-gradient tensor u 关32兴. Topological properties of
such a flow can be classified关33,34兴 by a Q-R plot, which is
a contour plot of the joint PDF of Q and R. In Fig. 11 we
0 2 4 6 8 1 0 1 2 1 4 1 6 | ω | / σ ω 1 0 - 6 1 0 - 5 1 0 - 4 1 0 - 3 1 0 - 2 1 0 - 1 1 0 0 P( |ω |/ σω ) 0 5 1 0 1 5 2 0 2 5 3 0 εl o c/ σ ε 1 0 - 6 1 0 - 5 1 0 - 4 1 0 - 3 1 0 - 2 1 0 - 1 1 0 0 P( εloc /σε )
FIG. 7. 共Color online兲 Semilogarithmic plots 共base 10兲 of the scaled PDFs P共兩兩/兲 versus 兩兩/ 共top panel兲 and P共⑀loc/⑀兲 versus ⑀loc/⑀ 共bottom panel兲, where and ⑀ are the standard
deviations for兩兩 and⑀loc, respectively, for our run NSP-256B, with
关c=0.1, We=7.1 共dashed line兲兴 and without 关c=0 共solid line兲兴 poly-mer additives. These plots are normalized such that the area under each curve is unity.
FIG. 8. 共Color online兲 Constant-兩兩 isosurfaces for 兩兩=兩兩 + 2 at t⬇60Teddy without 共top panel兲 and with 关middle panel
We= 3.5 共run 256A兲 and bottom panel We=7.1 共run NSP-256B兲兴 polymers; 兩兩 is the mean and is the standard deviation of兩兩.
PERLEKAR, MITRA, AND PANDIT PHYSICAL REVIEW E 82, 066313共2010兲
give Q-R plots from our DNS studies with and without poly-mers; although the qualitative shape of these joint PDFs re-mains the same, the regions of large R and Q are dramati-cally reduced on the additions of polymers; this is yet another indicator of the suppression of small-scale structures.
D. Effects of polymer additives on deep-dissipation-range spectra
In the previous sections we have studied the effects of polymer additives on the structural properties of a turbulent fluid at moderate Reynolds numbers. We now investigate the effects of polymer additives on the deep-dissipation range. To uncover such deep-dissipation-range effects, we conduct a very-high-resolution, but low-Re共=16兲, DNS study 共NSP-512兲. The parameters used in our run NSP-512 are given in Table I. The fluid is driven by using the stochastic-forcing scheme of Ref.关21兴. In Fig.12we plot fluid-energy spectra with and without polymer additives. The general behavior of these energy spectra is similar to that in our decaying-turbulence study 关3兴. We find that, on the addition of
poly-mers, the energy content at intermediate wave vectors de-creases, whereas the energy content at large wave vectors
increases significantly. We have checked explicitly that this increase in the energy spectrum in the deep-dissipation range is not an artifact of aliasing errors: note first that this increase starts at wave vectors whose magnitude is considerably lower than the dealiasing cutoff kmax in our DNS;
further-more, the enstrophy spectrum k2Ep共k兲, which we plot versus
k in Fig.13, decays at large k; this indicates that the
dissipa-tion range has been resolved adequately in our DNS. For homogeneous isotropic turbulence, the relationship between the second-order structure function and the energy spectrum is given in Eq. 共10兲 关26兴. Using this relationship
and the data for the energy spectrum shown in Fig. 12, we have obtained the second-order structure function S2共r兲 for our run NSP-512. We find that the addition of polymers leads to a decrease in the magnitude of S2共r兲. Our plots for S2共r兲
are similar to those found in the experiments of Ref.关15兴. In
our simulations we are able to reach much smaller values of
r/than has been possible in experimental studies on these systems 关15兴; however, we have not resolved the inertial
range very well in these runs. Note that the spectra Ep共k兲, with polymers, and Ef共k兲, without polymers, cross each other
as shown in Fig. 12. But such a crossing is not observed in the corresponding plots of second-order structure functions
R Q −400 −200 0 200 400 −200 0 200 2 3 4 5 R Q −400 −200 0 200 400 −100 0 100 200 2 3 4 5 6
FIG. 11. 共Color online兲 Contour plots of the joint PDF P共R,Q兲 from our DNS studies with共left兲 and without 共right兲 polymer addi-tives. In this Q-R plot, Q = −Tr共A2兲/2 and R=−Tr共A3兲/3 are the
invariants of the velocity-gradient tensor u. Note that P共R,Q兲 shrinks on the addition of polymers; this indicates a depletion of small-scale structures. The contour levels are logarithmically spaced and are drawn at the following values: 1.3, 2.02, 2.69, 3.36, 4.04, 4.70, 5.38, and 6.05. 1 0 0 1 0 1 1 0 2 k 1 0 - 1 2 1 0 - 9 1 0 - 6 1 0 - 3 1 0 0 E f(k ) an d E p(k )
FIG. 12. 共Color online兲 Log-log 共base 10兲 plots of the fluid-energy spectrum Ep共k兲 versus the magnitude of the wave vector k
for our run NSP= 512共full black line with squares兲 for c=0.1 and P= 1. The corresponding plot for the pure fluid共full red line with
circles兲 is also shown for comparison. FIG. 9. 共Color online兲 Semilogarithmic 共base 10兲 plots of the
PDF P共⌳1兲 versus the first eigenvalue ⌳1of the strain-rate tensor S
for the run NSP-256B, with关We=7.1 共blue dashed line兲兴 and with-out关c=0 共full red line兲兴 polymer additives. These plots are normal-ized such that the area under each curve is unity.
FIG. 10. 共Color online兲 Semilogarithmic 共base 10兲 plots of the PDF P共⌳2兲 versus the second eigenvalue ⌳2of the strain-rate
ten-sor S for the run NSP-256B, with关We=7.1 共blue dashed line兲兴 and without 关c=0 共full red line兲兴 polymer additives. These plots are normalized such that the area under each curve is unity.
共Fig.14兲. This can be understood by noting that S2共r兲
com-bines large- and small-k parts关35兴 of the energy spectrum.
IV. CONCLUSIONS
We have presented an extensive numerical study of the effects of polymer additives on statistically steady, homoge-neous, and isotropic fluid turbulence. Our study comple-ments, and extends considerably, our earlier work 关3兴.
Fur-thermore, our results compare favorably with several recent experiments.
Our first set of results show that the average viscous en-ergy dissipation rate decreases on the addition of polymers. This allows us to extend the definition of dissipation reduc-tion, introduced in Ref.关3兴, to the case of statistically steady,
homogeneous, and isotropic fluid turbulence with polymers. We find that this dissipation reduction increases with an in-crease in the Weissenberg number We at fixed polymer con-centration c. We obtain PDFs of the modulus of the vorticity, of the eigenvalues⌳n, of the rate-of-strain tensor S, and Q-R plots; we find that these are in qualitative agreement with the experiments of Refs.关16,17兴.
Our second set of results deal with a high-resolution DNS that we have carried out to elucidate the deep-dissipation-range forms of共a兲 energy spectra and 共b兲 the related second-order velocity structure functions. We find that this deep-dissipation-range behavior is akin to that in our earlier DNS of decaying, homogeneous, and isotropic fluid turbulence with polymers关3兴. Furthermore, the results we obtain for the
scaled second-order velocity structure S2共r兲 yield trends that are in qualitative agreement with the experiments of Ref. 关15兴. We hope that the comprehensive study that we have
presented here will stimulate further detailed experimental studies of the statistical properties of homogeneous isotropic fluid turbulence with polymer additives.
ACKNOWLEDGMENTS
We thank J. Bec, E. Bodenschatz, F. Toschi, and H. Xu for discussions, Leverhume Trust, European Research Council under the AstroDyn Research Project No. 227952, CSIR, DST, and UGC 共India兲 for support, and SERC 共IISc兲 for computational resources. Two of us共R.P. and P.P.兲 acknowl-edge the International Collaboration for Turbulence Research and support from the COST Action Grant No. MP0806. P.P. acknowledges IISc.
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PERLEKAR, MITRA, AND PANDIT PHYSICAL REVIEW E 82, 066313共2010兲
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