The existence of drag reduction by polymer additives, well established for
wall-bounded turbulent flows, is controversial in homogeneous, isotropic
turbulence. To settle this controversy we carry out a high-resolution direct
numerical simulation (DNS) of decaying, homogeneous, isotropic turbulence with
polymer additives. Our study reveals clear manifestations of
drag-reduction-type phenomena: On the addition of polymers to the turbulent
fluid we obtain a reduction in the energy dissipation rate, a significant
modification of the fluid energy spectrum especially in the deep-dissipation
range, a suppression of small-scale intermittency, and a decrease in
small-scale vorticity filaments.Comment: 4 pages, 3 figure

C. Canuto

Dhrubaditya Mitra

J. Lumley

M. Tabor

Prasad Perlekar

Rahul Pandit

English

arXiv

The existence of drag reduction by polymer additives, well established for wall-bounded turbulent flows, is controversial in homogeneous, isotropic turbulence. To settle this controversy, we carry out a high-resolution direct numerical simulation of decaying, homogeneous, isotropic turbulence with polymer additives. Our study reveals clear manifestations of drag-reduction-type phenomena: On the addition of polymers to the turbulent fluid, we obtain a reduction in the energy-dissipation rate, a significant modification of the fluid energy spectrum especially in the deep-dissipation range, a suppression of small-scale intermittency, and a decrease in small-scale vorticity filaments

Perlekar, Prasad

Dhrubaditya, Mitra

Pandit, Rahul

Publications of the IAS Fellows

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Manifestations of Drag Reduction by Polymer Additives
in Decaying, Homogeneous, Isotropic Turbulence
Prasad Perlekar,1, ∗ Dhrubaditya Mitra,2, † and Rahul Pandit1, ‡
1Centre for Condensed Matter Theory, Department of Physics,
Indian Institute of Science, Bangalore 560012, India.
2Observatoire de la Coˆte d‘Azur, BP 4229, 06304 Nice Cedex 4, France.
The existence of drag reduction by polymer additives, well established for wall-bounded turbulent
flows, is controversial in homogeneous, isotropic turbulence. To settle this controversy we carry out
a high-resolution direct numerical simulation (DNS) of decaying, homogeneous, isotropic turbulence
with polymer additives. Our study reveals clear manifestations of drag-reduction-type phenomena:
On the addition of polymers to the turbulent fluid we obtain a reduction in the energy dissipation
rate, a significant modification of the fluid energy spectrum especially in the deep-dissipation range,
a suppression of small-scale intermittency, and a decrease in small-scale vorticity filaments.
PACS numbers: 47.27.Gs, 47.27.Ak
The dramatic reduction of drag by the addition of
small concentrations of polymers to a turbulent fluid con-
tinues to engage the attention of engineers and physicists.
Significant advances have been made in understanding
drag reduction both experimentally [1, 2, 3] and theo-
retically [4, 5, 6, 7] in channel flows or the Kolmogorov
flow [8]. However, the existence of drag-reduction-type
phenomena in turbulent flows that are homogeneous and
isotropic [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] remains
controversial. Some experimental [15, 16, 17, 18], numer-
ical [11, 12, 13, 14], and theoretical [9, 10] studies have
suggested that drag reduction should occur even in ho-
mogeneous, isotropic turbulence; but other studies have
refuted this claim [19].
To settle this controversy we have initiated an exten-
sive direct numerical simulation (DNS) of decaying, ho-
mogeneous, isotropic turbulence in the presence of poly-
mer additives. We monitor the decay of turbulence from
initial states in which the kinetic energy of the fluid is
concentrated at small wave vectors; this energy then cas-
cades down to large wave vectors where it is dissipated
by viscous effects; the energy-dissipation rate ǫ attains
a maximum at tm, roughly the time at which the cas-
cade is completed. A recent shell-model study [13] has
suggested that this peak in ǫ can be used to quantify
drag reduction by polymer additives. Since shell mod-
els are far too simple to capture the complexities of real
flows, we have studied decaying turbulence in the Navier-
Stokes (NS) equation coupled to the Finitely Extensible
Nonlinear Elastic Peterlin (FENE-P) model [20] for poly-
mers. Our study, designed specifically to uncover drag-
reduction-type phenomena, shows that the position of
the maximum in ǫ depends only mildly on the polymer
concentration c; however, the value of ǫ at this maxi-
mum falls as c increases. We use this decrease of ǫ to de-
fine the percentage drag (or dissipation) reduction DR in
decaying homogeneous, isotropic turbulence; we also ex-
plore other accompanying physical effects and show that
they are in qualitative accord with drag-reduction exper-
iments [18, 21]: In particular, DR increases with c (upto
25% in one of our simulations). For small values of c
the energy spectrum of the fluid is modified appreciably
only in the dissipation range; however, this suffices to
yield significant drag reduction. We show that vorticity
filaments and intermittency are reduced at small spatial
scales and that the extension of the polymers decreases
as c increases.
The NS and FENE-P (henceforth NSP) equations are
Dtu = ν∇
2
u+
µ
τP
∇.[f(rP )C]−∇p; (1)
DtC = C.(∇u) + (∇u)
T .C −
f(rP )C − I
τP
. (2)
Here u(x, t) is the fluid velocity at point x and time t,
incompressibility is enforced by ∇.u = 0, Dt = ∂t+u.∇,
ν is the kinematic viscosity of the fluid, µ the viscos-
ity parameter for the solute (FENE-P), τP the polymer
relaxation time, ρ the solvent density (set to 1), p the
pressure, (∇u)T the transpose of (∇u), Cαβ ≡ 〈RαRβ〉
the elements of the polymer-conformation tensor C (an-
gular brackets indicate an average over polymer con-
figurations), I the identity tensor with elements δαβ ,
f(rP ) ≡ (L
2 − 3)/(L2 − r2P ) the FENE-P potential that
ensures finite extensibility, rP ≡
√
Tr(C) and L the
length and the maximum possible extension, respectively,
of the polymers, and c ≡ µ/(ν+µ) a dimensionless mea-
sure of the polymer concentration [22]. c = 0.1 corre-
sponds, roughly, to 100ppm for polyethylene oxide [1].
We consider homogeneous, isotropic, turbulence, so we
use periodic boundary conditions and solve Eq. (1) by
using a massively parallel pseudospectral code [23] with
N3 collocation points in a cubic domain (side L = 2π).
We eliminate aliasing errors [23] by the 2/3 rule, to
obtain reliable data at small length scales, and use a
second-order, slaved Adams-Bashforth scheme for time
marching. For Eq. (2) we use an explicit sixth-order
central-finite-difference scheme in space and a second-
order Adams-Bashforth method for temporal evolution.
2The numerical error in rP must be controlled by choosing
a small time step δt, 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 pseudospec-
tral DNS of the NS equation alone. Table I lists the
parameters we use. We preserve the symmetric-positive-
definite (SPD) nature of C at all times by using[22] the
following Cholesky-decomposition scheme: If we define
J ≡ f(rP )C, Eq. (2) becomes
DtJ = J .(∇u) + (∇u)
T .J − s(J − I) + qJ , (3)
where s = (L2− 3+ j2)/(τPL
2), q = [d/(L2 − 3)− (L2 −
3 + j2)(j2 − 3)/(τPL
2(L2 − 3))], j2 ≡ Tr(J ), and d =
Tr[J .(∇u) + (∇u)T .J ]. Since C and hence J are SPD
matrices, we can write J = LLT , where L is a lower-
triangular matrix with elements ℓij , such that ℓij = 0 for
j > i. Thus Eq.(3) now yields (1 ≤ i ≤ 3 and Γij = ∂iuj)
Dtℓi1 =
∑
k
Γkiℓk1 +
1
2
[
(q − s)ℓi1 + (−1)
(i mod 1) sℓi1
ℓ211
]
+ (δi3 + δi2)
ℓi2
ℓ11
∑
m>1
Γm1ℓm2
+ δi3Γi1
ℓ233
ℓ11
, for i ≥ 1;
Dtℓi2 =
∑
m>2
Γmiℓm2 −
ℓi1
ℓ11
∑
m>2
Γm1ℓm2
+
1
2
[
(q − s)ℓi2 + (−1)
(i+2)s
ℓi2
ℓ222
(
1 +
ℓ221
ℓ211
)]
+ δi3
[ℓ233
ℓ22
(
Γ32 − Γ31
ℓ21
ℓ11
)
+ s
ℓ21ℓ31
ℓ211ℓ22
]
, for i ≥ 2;
Dtℓ33 = Γ33ℓ33 − ℓ33
[ ∑
m<3
Γ3mℓ3m
ℓmm
]
+
Γ31ℓ32ℓ21ℓ33
ℓ11ℓ22
− s
ℓ21ℓ31ℓ32
ℓ211ℓ22ℓ33
+
1
2
[
(q − s)ℓ33
+
s
ℓ33
(
1 +
∑
m<3
ℓ23m
ℓ2mm
)
+
sℓ221ℓ
2
32
ℓ211ℓ
2
22ℓ33
]
. (4)
The SPD nature of C is preserved by Eq.(4) if ℓii > 0,
which we enforce explicitly [22] by considering the evolu-
tion of ln(ℓii) instead of ℓii.
We use the following initial conditions (superscript
0): C0mn(x) = δmn for all x; and u
0
m(k) =
Pmn(k)v
0
n(k) exp(iθn(k)), with m,n = x, y, z, Pmn =
(δmn − kmkn/k
2) the transverse projection operator, k
the wave-vector with components km = (−N/2,−N/2+
1, . . . , N/2), k = |k|, θn(k) random numbers distributed
uniformly between 0 and 2π, and v0n(k) chosen such
that the initial kinetic-energy spectra are either of type
I, with EI(k) = k2 exp(−2k4), or of type II, with
EII(k) = k4 exp(−2k2).
In addition to u(x, t), its Fourier transform uk(t), and
C(x, t) we monitor the vorticity ω ≡ ∇ × u, the kinetic-
energy spectrum E(k, t) ≡
∑
k−1/2<k′≤k+1/2 |u
2
k′
(t)|, the
total kinetic energy E(t) ≡
∑
k E(k, t), the energy-
dissipation-rate ǫ(t) ≡ ν
∑
k k
2E(k, t), the cumulative
probability distribution of scaled polymer extensions
PC(r2P /L
2), and the hyperflatness F6(r) ≡ S6(r)/S
3
2 (r),
where Sp(r) ≡ 〈{[u(x+ r)− u(x)] · r/r}
p〉 is the order-p
longitudinal velocity structure function and the angular
brackets denote an average over our simulation domain
at tm. For notational convenience, we do not display the
dependence on c explicitly.
Figure (1a) shows that ǫ first increases with time,
reaches a peak, and then decreases; for c = 0 this peak
occurs at t = tm. The position of this peak changes
mildly with c but its height goes down significantly as
c increases. This suggests the following natural defini-
tion [13] of the percentage drag or dissipation reduction
for decaying homogeneous, isotropic turbulence:
DR ≡
(
ǫf,m − ǫp,m
ǫf,m
)
× 100; (5)
here (and henceforth) the superscripts f and p stand, re-
spectively, for the fluid without and with polymers and
the superscript m indicates the time tm. Figure (1b)
shows plots of DR versus c, for the Weissenberg num-
ber We ≡ τP
√
ǫf,m/ν ≃ 0.35, and versus We, for
c = 1/11 ≃ 0.1. DR increases with c in qualitative accord
with experiments on channel flows (where DR is defined
via a normalized pressure difference); but it drops gently
asWe increases, in contrast to the behavior seen in chan-
nel flows (in which τP is varied by changing the polymer).
In decaying turbulence, the total kinetic energy E(t)
of the fluid falls as t increases; the rate at which it falls
increases with c [Fig. (1c)], which suggests that the ad-
dition of polymers increases the effective viscosity of the
solution. This is not at odds with the decrease of ǫ with
increasing c since the effective viscosity because of poly-
mers turns out to be scale-dependent. We confirm this
by obtaining the kinetic-energy spectrum Ep,m(k) for the
fluid in the presence of polymers at t = tm. For small
concentrations (c ≃ 0.1) the spectra with and without
polymers differ substantially only in the deep dissipa-
tion range, where Ef,m(k) ≪ Ep,m(k). As c increases,
to say c ≃ 0.4, Ep,m(k) is reduced relative to Ef,m(k)
at intermediate values of k [Fig. (2a)]; however, deep in
the dissipation range Ef,m(k) ≪ Ep,m(k). We now de-
fine [12] 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′)], where (∇ · J )k is the Fourier
transform of ∇ · J . The inset of Fig. (2a) shows that
∆ν(k) > 0 for k < 15, but ∆ν(k) < 0 around k =
20. This explains why Ep,m(k) is suppressed relative to
Ef,m(k) at small k, rises above it in the deep-dissipation
range, and crosses over from its small-k to large-k be-
haviors around the value of k where ∆ν(k) goes through
zero.
Given the resolution of our DNS, inertial-range inter-
mittency can be studied only by using extended self sim-
3ilarity [24] as we will report elsewhere. However, we
explore dissipation-range statistics further by calculat-
ing the hyperflatness F6(r) [Fig. (2b)]. The addition of
polymers slows down the growth of F6(r), as r → 0,
which signals the reduction of small-scale intermittency.
This is further supported by the iso-|ω| surfaces shown
in Fig. (3). If no polymers are present, these iso-|ω| sur-
faces are filamentary [25] for large |ω|; polymers suppress
a significant fraction of these filaments.
We use a rank-order method [26] to obtain PC(r2P /L
2)
and find that, as c increases [Fig. (2c)], the extension of
the polymers decreases. We have checked that, in the
passive-polymer version of Eqs.(1) and (2), the extension
of polymers is much more than in Fig. (2c).
Our study contrasts clearly drag reduction in homoge-
neous, isotropic, turbulence and in wall-bounded flows.
In both these cases the polymers increase the overall vis-
cosity of the solution (see, e.g., Fig. (1c) and Ref.[12]).
In wall-bounded flows the presence of polymers inhibits
the flow of the stream-wise component of the momentum
into the wall, which, in turn, increases the net through-
put of the fluid and thus results in drag reduction, a
mechanism that can have no analog in homogeneous,
isotropic turbulence. However, the decrease of ǫ(t) with
increasing c [Fig. (1b)] yields a natural definition of DR
[Eq.(5)] for this case [27]. Thus, if the term drag reduc-
tion must be reserved for wall-bounded flows, then we
suggest the expression dissipation reduction for homo-
geneous, isotropic, turbulence. We have shown that νe
must be scale-dependent; its counterpart in wall-bounded
flows is the position-dependent viscosity of Refs. [4, 7].
Furthermore, as in wall-bounded flows, an increase in c
leads to an increase in DR [Fig. (1b)]. In channel flows
an increase inWe leads to an increase in DR, but we find
that DR falls marginally as We increases [Fig. (1b)].
Our DNS of the Navier-Stokes equation with poly-
mer additives [Eqs. (1) and (2)] resolves the controversy
about drag reduction in decaying homogeneous, isotropic
turbulence and shows clearly that Eq. (5) offers a natu-
ral definition of DR for this case in a far more realistic
model than those of Refs. [11, 13]. We also find a non-
trivial modification of the fluid kinetic-energy spectrum
especially in the deep-dissipation range [Fig. (2b)] that
can be explained in terms of a polymer-induced, scale-
dependent viscosity. Experiments [16, 17] do not resolve
the dissipation range as clearly as we do, so the experi-
mental verification of the deep-dissipation-range behav-
ior of Fig. (2a) remains a challenge. Earlier theoretical
studies [10, 11] have also not concentrated on this dis-
sipation range. The reduction in the small-scale inter-
mittency [Fig. (2b)] and in the constant-|ω| isosurfaces
[Fig. (3)] is in qualitative agreement with channel-flow
studies [2], where a decrease in the turbulent volume frac-
tion is seen on the addition of the polymers, and water-jet
studies [21], where the addition of the polymers leads to
a decrease in small-scale structures. We hope our work
N δt L ν τP c
NSP-96 96 1.0× 10−2 100 10−2 0.1− 3 0.1, 0.2, 0.3, 0.4
NSP-192 192 1.0× 10−2 100 10−2 1 0.1, 0.4
NSP-256A 256 1.0× 10−2 100 10−2 1 0.1, 0.4
NSP-256 256 4.0× 10−3 100 10−3 1 0.1, 0.4
TABLE I: The parameters N , δt, L, ν, τP and c for our
four runs NSP− 96, NSP− 192, NSP− 256A, and NSP− 256.
NSP− 96, NSP− 192, NSP− 256A use type I initial conditions;
NSP− 256 uses an initial condition of type II . We also carry
out DNS studies of the NS equation with the same numerical
resolutions as our NSP runs. Re ≡ √20Ef,m/
√
3νǫf,m
and We ≡ τP
√
ǫf,m/ν; NSP-96: Re = 47.1 and We =
0.03, 0.17, 0.24, 0.28, 0.31, 0.41, 0.48, 0.55, 0.62, 0.68, 1.03;
NSP-192 and NSP-256A: Re = 47.1 and We = 0.35; NSP-256:
Re = 126.6 and We = 0.76.
will stimulate more experimental studies of drag or dis-
sipation reduction in homogeneous, isotropic turbulence.
We thank C. Kalelkar, R. Govindarajan, V. Kumar,
S. Ramaswamy, L. Collins, and A. Celani for discus-
sions, CSIR, DST, and UGC(India) for financial support,
and SERC(IISc) for computational facilities. DM is sup-
ported by the Henri Poincare´ Postdoctoral Fellowship.
∗ Electronic address: perlekar@physics.iisc.ernet.in
† Electronic address: Dhrubaditya.MITRA@obs-nice.fr
‡ Electronic address: rahul@physics.iisc.ernet.in;
also at Jawaharlal Nehru Centre For Advanced Scientific
Research, Jakkur, Bangalore, India.
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40.5 1 1.5
2
3
4
5
x 10−4
t/τ
ǫ
(a)
0.1 0.2 0.3 0.4
5
10
15
20
25
c
D
R 0.1 0.8
3.6
4
4.4
4.8
We
D
R (b)
0.2 0.4 0.6 0.8 1 1.20.03
0.04
0.05
0.06
t/τ
E
(c)
FIG. 1: (Color online) (a) Temporal evolution of the energy dissipation rate ǫ (run NSP-256) for concentrations c = 0.1(−−) and
c = 0.4(solid line), with τ ≡
√
2E(t = 0)/3L2; (b) percentage drag-reduction DR versus c (run NSP-192); the inset shows the
mild variation in DR with We (runs NSP-96); (c) temporal evolution of the total fluid energy E for concentrations c = 0.1(−−)
and c = 0.4(solid line) (runs NSP-256). In (a) and (c) the plots for c = 0 (o−) are shown for comparison.
100 101
10−30
10−20
10−10
k
E
p
,m
(k
)
o
r
E
f
,m
(k
)
20 40 60
−1
2
5
8
x 10−4
k
∆
ν
(k
)
(a)
0.5 1 1.5 2
50
100
150
200
r
F
6
(r
)
(b)
10−3 10−2
10−6
10−4
10−2
r2P /L
2
P
C
(r
2 P
/
L
2
)
(c )
FIG. 2: (Color online) (a) Plots of the energy spectra Ep,m(k) or Ef,m(k) versus k (run NSP-192) for c = 0.1(−−) and
c = 0.4(solid line) [Ep,m(k) is unchanged if we use N = 256, with all other parameters the same (run NSP-256A)]; inset:
polymer contribution to the scale-dependent viscosity ∆ν(k) versus k for c = 0.1(−−); ∆ν(k) = 0 (solid line) is also shown
for reference; (b) the hyper-flatness F6(r) as a function of r (run NSP-256) and concentration c = 0.4(solid line). In (a) and
(b) the corresponding plots with c = 0 (o−) are shown for comparison. (c) The cumulative PDF PC(r2P /L2) versus r2P/L2 for
c = 0.1(−−) and c = 0.4(solid line) (run NSP-256).
FIG. 3: (Color online) Constant-|ω| isosurfaces for |ω| =
〈|ω|〉 + 2σ at tm without (a) and with (b) polymers, (run
NSP-256) and c = 0.4; 〈|ω|〉 is the mean and σ the standard
deviation of |ω|.
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[27] In some steady-state simulations [11, 22] DR is associated
with Ep(k) > Ef (k), for small k. We obtain this for type
II , but not type I , initial conditions; but Eq.(5) yields
drag reduction for both of these initial conditions.


Manifestations of drag reduction by polymer additives in decaying, homogeneous, isotropic turbulence

Perlekar, P.

Mitra, D.

Pandit, R.

NARCIS 

Mitra, Dhrubaditya

Name not available

Manifestations of Drag Reduction by Polymer Additives in Decaying, Homogeneous, Isotropic Turbulence

Pure OAI Repository

Indian Academy of Sciences

Crossref

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06
Manifestations of Drag Reduction by Polymer Additives
in Decaying, Homogeneous, Isotropic Turbulence
Prasad Perlekar,1, ∗ Dhrubaditya Mitra,2, † and Rahul Pandit1, ‡
1Centre for Condensed Matter Theory, Department of Physics,
Indian Institute of Science, Bangalore 560012, India.
2Observatoire de la Côte d‘Azur, BP 4229, 06304 Nice Cedex 4, France.
The existence of drag reduction by polymer additives, well established for wall-bounded turbulent
flows, is controversial in homogeneous, isotropic turbulence. To settle this controversy we carry out
a high-resolution direct numerical simulation (DNS) of decaying, homogeneous, isotropic turbulence
with polymer additives. Our study reveals clear manifestations of drag-reduction-type phenomena:
On the addition of polymers to the turbulent fluid we obtain a reduction in the energy dissipation
rate, a significant modification of the fluid energy spectrum especially in the deep-dissipation range,
a suppression of small-scale intermittency, and a decrease in small-scale vorticity filaments.
PACS numbers: 47.27.Gs, 47.27.Ak
The dramatic reduction of drag by the addition of
small concentrations of polymers to a turbulent fluid con-
tinues to engage the attention of engineers and physicists.
Significant advances have been made in understanding
drag reduction both experimentally [1, 2, 3] and theo-
retically [4, 5, 6, 7] in channel flows or the Kolmogorov
flow [8]. However, the existence of drag-reduction-type
phenomena in turbulent flows that are homogeneous and
isotropic [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] remains
controversial. Some experimental [15, 16, 17, 18], numer-
ical [11, 12, 13, 14], and theoretical [9, 10] studies have
suggested that drag reduction should occur even in ho-
mogeneous, isotropic turbulence; but other studies have
refuted this claim [19].
To settle this controversy we have initiated an exten-
sive direct numerical simulation (DNS) of decaying, ho-
mogeneous, isotropic turbulence in the presence of poly-
mer additives. We monitor the decay of turbulence from
initial states in which the kinetic energy of the fluid is
concentrated at small wave vectors; this energy then cas-
cades down to large wave vectors where it is dissipated
by viscous effects; the energy-dissipation rate ǫ attains
a maximum at tm, roughly the time at which the cas-
cade is completed. A recent shell-model study [13] has
suggested that this peak in ǫ can be used to quantify
drag reduction by polymer additives. Since shell mod-
els are far too simple to capture the complexities of real
flows, we have studied decaying turbulence in the Navier-
Stokes (NS) equation coupled to the Finitely Extensible
Nonlinear Elastic Peterlin (FENE-P) model [20] for poly-
mers. Our study, designed specifically to uncover drag-
reduction-type phenomena, shows that the position of
the maximum in ǫ depends only mildly on the polymer
concentration c; however, the value of ǫ at this maxi-
mum falls as c increases. We use this decrease of ǫ to de-
fine the percentage drag (or dissipation) reduction DR in
decaying homogeneous, isotropic turbulence; we also ex-
plore other accompanying physical effects and show that
they are in qualitative accord with drag-reduction exper-
iments [18, 21]: In particular, DR increases with c (upto
25% in one of our simulations). For small values of c
the energy spectrum of the fluid is modified appreciably
only in the dissipation range; however, this suffices to
yield significant drag reduction. We show that vorticity
filaments and intermittency are reduced at small spatial
scales and that the extension of the polymers decreases
as c increases.
The NS and FENE-P (henceforth NSP) equations are
Dtu = ν∇
2
u +
µ
τP
∇.[f(rP )C] −∇p; (1)
DtC = C.(∇u) + (∇u)
T .C −
f(rP )C − I
τP
. (2)
Here u(x, t) is the fluid velocity at point x and time t,
incompressibility is enforced by ∇.u = 0, Dt = ∂t +u.∇,
ν is the kinematic viscosity of the fluid, µ the viscos-
ity parameter for the solute (FENE-P), τP the polymer
relaxation time, ρ the solvent density (set to 1), p the
pressure, (∇u)T the transpose of (∇u), Cαβ ≡ 〈RαRβ〉
the elements of the polymer-conformation tensor C (an-
gular brackets indicate an average over polymer con-
figurations), I the identity tensor with elements δαβ ,
f(rP ) ≡ (L
2 − 3)/(L2 − r2P ) the FENE-P potential that
ensures finite extensibility, rP ≡
√
Tr(C) and L the
length and the maximum possible extension, respectively,
of the polymers, and c ≡ µ/(ν + µ) a dimensionless mea-
sure of the polymer concentration [22]. c = 0.1 corre-
sponds, roughly, to 100ppm for polyethylene oxide [1].
We consider homogeneous, isotropic, turbulence, so we
use periodic boundary conditions and solve Eq. (1) by
using a massively parallel pseudospectral code [23] with
N3 collocation points in a cubic domain (side L = 2π).
We eliminate aliasing errors [23] by the 2/3 rule, to
obtain reliable data at small length scales, and use a
second-order, slaved Adams-Bashforth scheme for time
marching. For Eq. (2) we use an explicit sixth-order
central-finite-difference scheme in space and a second-
order Adams-Bashforth method for temporal evolution.
2
The numerical error in rP must be controlled by choosing
a small time step δt, 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 pseudospec-
tral DNS of the NS equation alone. Table I lists the
parameters we use. We preserve the symmetric-positive-
definite (SPD) nature of C at all times by using[22] the
following Cholesky-decomposition scheme: If we define
J ≡ f(rP )C, Eq. (2) becomes
DtJ = J .(∇u) + (∇u)
T .J − s(J − I) + qJ , (3)
where s = (L2 − 3 + j2)/(τP L
2), q = [d/(L2 − 3)− (L2 −
3 + j2)(j2 − 3)/(τP L
2(L2 − 3))], j2 ≡ Tr(J ), and d =
Tr[J .(∇u) + (∇u)T .J ]. Since C and hence J are SPD
matrices, we can write J = LLT , where L is a lower-
triangular matrix with elements ℓij , such that ℓij = 0 for
j > i. Thus Eq.(3) now yields (1 ≤ i ≤ 3 and Γij = ∂iuj)
Dtℓi1 =
∑
k
Γkiℓk1 +
1
2
[
(q − s)ℓi1 + (−1)
(i mod 1) sℓi1
ℓ211
]
+ (δi3 + δi2)
ℓi2
ℓ11
∑
m>1
Γm1ℓm2
+ δi3Γi1
ℓ233
ℓ11
, for i ≥ 1;
Dtℓi2 =
∑
m>2
Γmiℓm2 −
ℓi1
ℓ11
∑
m>2
Γm1ℓm2
+
1
2
[
(q − s)ℓi2 + (−1)
(i+2)s
ℓi2
ℓ222
(
1 +
ℓ221
ℓ211
)]
+ δi3
[ℓ233
ℓ22
(
Γ32 − Γ31
ℓ21
ℓ11
)
+ s
ℓ21ℓ31
ℓ211ℓ22
]
, for i ≥ 2;
Dtℓ33 = Γ33ℓ33 − ℓ33
[
∑
m<3
Γ3mℓ3m
ℓmm
]
+
Γ31ℓ32ℓ21ℓ33
ℓ11ℓ22
− s
ℓ21ℓ31ℓ32
ℓ211ℓ22ℓ33
+
1
2
[
(q − s)ℓ33
+
s
ℓ33
(
1 +
∑
m<3
ℓ23m
ℓ2mm
)
+
sℓ221ℓ
2
32
ℓ211ℓ
2
22ℓ33
]
. (4)
The SPD nature of C is preserved by Eq.(4) if ℓii > 0,
which we enforce explicitly [22] by considering the evolu-
tion of ln(ℓii) instead of ℓii.
We use the following initial conditions (superscript
0): C0mn(x) = δmn for all x; and u
0
m(k) =
Pmn(k)v
0
n(k) exp(iθn(k)), with m, n = x, y, z, Pmn =
(δmn − kmkn/k
2) the transverse projection operator, k
the wave-vector with components km = (−N/2,−N/2 +
1, . . . , N/2), k = |k|, θn(k) random numbers distributed
uniformly between 0 and 2π, and v0n(k) chosen such
that the initial kinetic-energy spectra are either of type
I, with EI(k) = k2 exp(−2k4), or of type II, with
EII(k) = k4 exp(−2k2).
In addition to u(x, t), its Fourier transform uk(t), and
C(x, t) we monitor the vorticity ω ≡ ∇ × u, the kinetic-
energy spectrum E(k, t) ≡
∑
k−1/2<k′≤k+1/2 |u
2
k′
(t)|, the
total kinetic energy E(t) ≡
∑
k E(k, t), the energy-
dissipation-rate ǫ(t) ≡ ν
∑
k k
2E(k, t), the cumulative
probability distribution of scaled polymer extensions
PC(r2P /L
2), and the hyperflatness F6(r) ≡ S6(r)/S
3
2 (r),
where Sp(r) ≡ 〈{[u(x + r) − u(x)] · r/r}
p〉 is the order-p
longitudinal velocity structure function and the angular
brackets denote an average over our simulation domain
at tm. For notational convenience, we do not display the
dependence on c explicitly.
Figure (1a) shows that ǫ first increases with time,
reaches a peak, and then decreases; for c = 0 this peak
occurs at t = tm. The position of this peak changes
mildly with c but its height goes down significantly as
c increases. This suggests the following natural defini-
tion [13] of the percentage drag or dissipation reduction
for decaying homogeneous, isotropic turbulence:
DR ≡
(
ǫf,m − ǫp,m
ǫf,m
)
× 100; (5)
here (and henceforth) the superscripts f and p stand, re-
spectively, for the fluid without and with polymers and
the superscript m indicates the time tm. Figure (1b)
shows plots of DR versus c, for the Weissenberg num-
ber We ≡ τP
√
ǫf,m/ν ≃ 0.35, and versus We, for
c = 1/11 ≃ 0.1. DR increases with c in qualitative accord
with experiments on channel flows (where DR is defined
via a normalized pressure difference); but it drops gently
as We increases, in contrast to the behavior seen in chan-
nel flows (in which τP is varied by changing the polymer).
In decaying turbulence, the total kinetic energy E(t)
of the fluid falls as t increases; the rate at which it falls
increases with c [Fig. (1c)], which suggests that the ad-
dition of polymers increases the effective viscosity of the
solution. This is not at odds with the decrease of ǫ with
increasing c since the effective viscosity because of poly-
mers turns out to be scale-dependent. We confirm this
by obtaining the kinetic-energy spectrum Ep,m(k) for the
fluid in the presence of polymers at t = tm. For small
concentrations (c ≃ 0.1) the spectra with and without
polymers differ substantially only in the deep dissipa-
tion range, where Ef,m(k) ≪ Ep,m(k). As c increases,
to say c ≃ 0.4, Ep,m(k) is reduced relative to Ef,m(k)
at intermediate values of k [Fig. (2a)]; however, deep in
the dissipation range Ef,m(k) ≪ Ep,m(k). We now de-
fine [12] the effective scale-dependent viscosity νe(k) ≡
ν + ∆ν(k), with ∆ν(k) ≡ −µ
∑
k−1/2<k′≤k+1/2 uk′ · (∇ ·
J )−k′/[τP k
′2Ep,m(k′)], where (∇ · J )k is the Fourier
transform of ∇ · J . The inset of Fig. (2a) shows that
∆ν(k) > 0 for k < 15, but ∆ν(k) < 0 around k =
20. This explains why Ep,m(k) is suppressed relative to
Ef,m(k) at small k, rises above it in the deep-dissipation
range, and crosses over from its small-k to large-k be-
haviors around the value of k where ∆ν(k) goes through
zero.
Given the resolution of our DNS, inertial-range inter-
mittency can be studied only by using extended self sim-
3
ilarity [24] as we will report elsewhere. However, we
explore dissipation-range statistics further by calculat-
ing the hyperflatness F6(r) [Fig. (2b)]. The addition of
polymers slows down the growth of F6(r), as r → 0,
which signals the reduction of small-scale intermittency.
This is further supported by the iso-|ω| surfaces shown
in Fig. (3). If no polymers are present, these iso-|ω| sur-
faces are filamentary [25] for large |ω|; polymers suppress
a significant fraction of these filaments.
We use a rank-order method [26] to obtain PC(r2P /L
2)
and find that, as c increases [Fig. (2c)], the extension of
the polymers decreases. We have checked that, in the
passive-polymer version of Eqs.(1) and (2), the extension
of polymers is much more than in Fig. (2c).
Our study contrasts clearly drag reduction in homoge-
neous, isotropic, turbulence and in wall-bounded flows.
In both these cases the polymers increase the overall vis-
cosity of the solution (see, e.g., Fig. (1c) and Ref.[12]).
In wall-bounded flows the presence of polymers inhibits
the flow of the stream-wise component of the momentum
into the wall, which, in turn, increases the net through-
put of the fluid and thus results in drag reduction, a
mechanism that can have no analog in homogeneous,
isotropic turbulence. However, the decrease of ǫ(t) with
increasing c [Fig. (1b)] yields a natural definition of DR
[Eq.(5)] for this case [27]. Thus, if the term drag reduc-
tion must be reserved for wall-bounded flows, then we
suggest the expression dissipation reduction for homo-
geneous, isotropic, turbulence. We have shown that νe
must be scale-dependent; its counterpart in wall-bounded
flows is the position-dependent viscosity of Refs. [4, 7].
Furthermore, as in wall-bounded flows, an increase in c
leads to an increase in DR [Fig. (1b)]. In channel flows
an increase in We leads to an increase in DR, but we find
that DR falls marginally as We increases [Fig. (1b)].
Our DNS of the Navier-Stokes equation with poly-
mer additives [Eqs. (1) and (2)] resolves the controversy
about drag reduction in decaying homogeneous, isotropic
turbulence and shows clearly that Eq. (5) offers a natu-
ral definition of DR for this case in a far more realistic
model than those of Refs. [11, 13]. We also find a non-
trivial modification of the fluid kinetic-energy spectrum
especially in the deep-dissipation range [Fig. (2b)] that
can be explained in terms of a polymer-induced, scale-
dependent viscosity. Experiments [16, 17] do not resolve
the dissipation range as clearly as we do, so the experi-
mental verification of the deep-dissipation-range behav-
ior of Fig. (2a) remains a challenge. Earlier theoretical
studies [10, 11] have also not concentrated on this dis-
sipation range. The reduction in the small-scale inter-
mittency [Fig. (2b)] and in the constant-|ω| isosurfaces
[Fig. (3)] is in qualitative agreement with channel-flow
studies [2], where a decrease in the turbulent volume frac-
tion is seen on the addition of the polymers, and water-jet
studies [21], where the addition of the polymers leads to
a decrease in small-scale structures. We hope our work
N δt L ν τP c
NSP-96 96 1.0 × 10−2 100 10−2 0.1 − 3 0.1, 0.2, 0.3, 0.4
NSP-192 192 1.0 × 10−2 100 10−2 1 0.1, 0.4
NSP-256A 256 1.0 × 10−2 100 10−2 1 0.1, 0.4
NSP-256 256 4.0 × 10−3 100 10−3 1 0.1, 0.4
TABLE I: The parameters N , δt, L, ν, τP and c for our
four runs NSP− 96, NSP− 192, NSP− 256A, and NSP− 256.
NSP− 96, NSP− 192, NSP− 256A use type I initial conditions;
NSP− 256 uses an initial condition of type II . We also carry
out DNS studies of the NS equation with the same numerical
resolutions as our NSP runs. Re ≡
√
20Ef,m/
√
3νǫf,m
and We ≡ τP
√
ǫf,m/ν; NSP-96: Re = 47.1 and We =
0.03, 0.17, 0.24, 0.28, 0.31, 0.41, 0.48, 0.55, 0.62, 0.68, 1.03;
NSP-192 and NSP-256A: Re = 47.1 and We = 0.35; NSP-256:
Re = 126.6 and We = 0.76.
will stimulate more experimental studies of drag or dis-
sipation reduction in homogeneous, isotropic turbulence.
We thank C. Kalelkar, R. Govindarajan, V. Kumar,
S. Ramaswamy, L. Collins, and A. Celani for discus-
sions, CSIR, DST, and UGC(India) for financial support,
and SERC(IISc) for computational facilities. DM is sup-
ported by the Henri Poincaré Postdoctoral Fellowship.
∗ Electronic address: perlekar@physics.iisc.ernet.in
† Electronic address: Dhrubaditya.MITRA@obs-nice.fr
‡ Electronic address: rahul@physics.iisc.ernet.in;
also at Jawaharlal Nehru Centre For Advanced Scientific
Research, Jakkur, Bangalore, India.
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0.5 1 1.5
2
3
4
5
x 10
−4
t/τ
ǫ
(a)
0.1 0.2 0.3 0.4
5
10
15
20
25
c
D
R 0.1 0.8
3.6
4
4.4
4.8
We
D
R (b)
0.2 0.4 0.6 0.8 1 1.2
0.03
0.04
0.05
0.06
t/τ
E
(c)
FIG. 1: (Color online) (a) Temporal evolution of the energy dissipation rate ǫ (run NSP-256) for concentrations c = 0.1(−−) and
c = 0.4(solid line), with τ ≡
√
2E(t = 0)/3L2; (b) percentage drag-reduction DR versus c (run NSP-192); the inset shows the
mild variation in DR with We (runs NSP-96); (c) temporal evolution of the total fluid energy E for concentrations c = 0.1(−−)
and c = 0.4(solid line) (runs NSP-256). In (a) and (c) the plots for c = 0 (o−) are shown for comparison.
10
0
10
1
10
−30
10
−20
10
−10
k
E
p
,m
(k
)
o
r
E
f
,m
(k
)
20 40 60
−1
2
5
8
x 10
−4
k
∆
ν
(k
)
(a)
0.5 1 1.5 2
50
100
150
200
r
F
6
(r
)
(b)
10
−3
10
−2
10
−6
10
−4
10
−2
r2P /L
2
P
C
(r
2 P
/
L
2
)
(c )
FIG. 2: (Color online) (a) Plots of the energy spectra Ep,m(k) or Ef,m(k) versus k (run NSP-192) for c = 0.1(−−) and
c = 0.4(solid line) [Ep,m(k) is unchanged if we use N = 256, with all other parameters the same (run NSP-256A)]; inset:
polymer contribution to the scale-dependent viscosity ∆ν(k) versus k for c = 0.1(−−); ∆ν(k) = 0 (solid line) is also shown
for reference; (b) the hyper-flatness F6(r) as a function of r (run NSP-256) and concentration c = 0.4(solid line). In (a) and
(b) the corresponding plots with c = 0 (o−) are shown for comparison. (c) The cumulative PDF P C(r2P /L2) versus r2P /L2 for
c = 0.1(−−) and c = 0.4(solid line) (run NSP-256).
FIG. 3: (Color online) Constant-|ω| isosurfaces for |ω| =
〈|ω|〉 + 2σ at tm without (a) and with (b) polymers, (run
NSP-256) and c = 0.4; 〈|ω|〉 is the mean and σ the standard
deviation of |ω|.
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with Ep(k) > Ef (k), for small k. We obtain this for type
II , but not type I , initial conditions; but Eq.(5) yields
drag reduction for both of these initial conditions.


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Manifestations of Drag Reduction by Polymer Additives in Decaying,
  Homogeneous, Isotropic Turbulence

Manifestations of Drag Reduction by Polymer Additives in Decaying, Homogeneous, Isotropic Turbulence

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