Table Of ContentDrag Reduction by a Linear Viscosity Profile
Elisabetta De Angelis1,2, Carlo Casciola1, Victor S. L’vov2,
Anna Pomyalov2, Itamar Procaccia2 and Vasil Tiberkevich2
1 Dip. Mecc. Aeron., Universit`a di Roma “La Sapienza”, Via Eudossiana 18, 00184, Roma, Italy
2Dept. of Chemical Physics, The Weizmann Institute of Science, Rehovot, 76100 Israel
Drag reduction by polymers in turbulent flows raises an apparent contradiction: the stretching
of the polymers must increase theviscosity, so why is thedrag reduced? A recent theory proposed
thatdragreductioninagreementwithexperimentsisconsistentwiththeeffectiveviscositygrowing
linearly with the distance from the wall. With this self consistent solution the reduction in the
Reynolds stress overwhelms the increase in viscous drag. In this Letter we show, using Direct
4
NumericalSimulations,thatalinearviscosityprofileindeedreducesthedraginagreementwiththe
0
theory and in close correspondence with direct simulations of the FENE-P model at the same flow
0
conditions.
2
n
a The addition of few tens of parts per million (by
J weight) of long-chain polymers to turbulent fluid flows
7 in channels or pipes can bring about a reduction of the 50 Newtonian [12]
friction drag by up to 80% [1, 2, 3, 4]. In spite of a Newtonian, DNS [11]
] MDR [2]
D large amountof experimentaland simulationaldata, the 40 MDR asymptote [7]
fundamental mechanism has remained under debate for
C
a long time [4, 5, 6]. Since polymers tend to stretch in
.
in a turbulent flow, increasing thus the bulk viscosity, it 3+0V
l appears contradictory that they should reduce the drag.
n
There must exist a mechanism that compensates for the
[ 20
increased viscosity. Indeed, drag is caused by two rea-
1 sons, one viscous, and the other inertial, related to the
v momentum flux from the bulk to the wall. For a fixed 10
5
rate(perunitmass)ofmomentumgeneratedbythepres-
0
0 sure gradient, reducing the momentum flux can reduce 0
1 the drag. In a recent theory of drag reduction in wall 1 10 y+ 100
0 turbulence [7] it was proposed that the polymer stretch-
4 FIG. 1: Mean velocity profiles as a function of the distance
ing gives rise to a self-consistent effective viscosity that
0 from the wall [in “wall” units cf. Eq. (5)]. The solid
increases with the distance from the wall. Such a pro-
/ line(numericalsimulations [11])andtheexperimentalpoints
n file reduces the Reynolds stress (i.e. the momentum flux (open circles) [12] represent the Newtonian results with von-
li to the wall) more than it increases the viscous drag; the Karman’s log law of the wall observed for y+ >20. The red
n
resultisdragreduction. TheaimofthisLetteristosub- datapoints(squares)[2]representtheMDRasymptote. The
:
v stantiate this mechanism for drag reduction on the basis dashed red curve represents the theoretical universal MDR
i asymptote, Eq. (4) [7].
X of Direct Numerical Simulations.
r The onset of turbulence in channel or pipe flows in-
a
creases the drag dramatically. For Newtonian flows (in
the bulk to the wall. Accordingly the mean velocity in
which the kinematic viscosity is constant) the momen-
the channel must increase. It was shown that when the
tum flux is dominated by the so-called Reynolds stress,
concentrationofthepolymersislargeenoughthereexists
leadingtoalogarithmic(von-Karman)dependenceofthe
anew logarithmiclawfor the meanvelocitywith aslope
meanvelocityonthedistancefromthewall[8]. However,
that fits existing numerical and experimental data. The
with polymers, the drag reduction entails a change in
law is universal, thus explaining the MDR asymptote.
the von-Karman log law such that a much higher mean
To see how this mechanism works, consider the mod-
velocity is achieved. In particular, for high concentra-
ified Navier-Stokes (NS) equation for the polymer solu-
tions of polymers, a regime of maximum drag reduction
tions [9, 10] :
is attained (the “MDR asymptote”), independent of the
chemical identity of the polymer [2], see Fig. 1. In a ∂U/∂t+U ∇U = ∇p+∇ T +ν0 2U , (1)
· − · ∇
recent theoretical paper [7] the fundamental mechanism
forthisphenomenonwaselucidated: whilemomentumis where ν0 is the kinematic viscosity of the carrier fluid
producedat a fixed rate by the forcing,polymer stretch- and T is the extra stress tensor that is due to the poly-
ing results in a suppression of the momentum flux from mer. Denoting the polymer end-to-end vector distance
2
(normalized by its equilibrium value) as r, the average (y)isself-adjustedinordertoprovideauniversallinear
dimensionless extension tensor R is r r , and Rprofile of the effective viscosity
ij i j
R ≡ h i
the extra stress tensor is (with ω ∂U /∂x ),
ij i j
≡ ν (y)=κ p′L y . (6)
T =ν (ω R+R ωT ∂R/∂t U ∇R) . (2) MDR Vp
p
· · − − ·
Needless to say, in the viscous sublayer the viscosity is
Here νp (proportional to the polymer concentration) is Newtonian: ν =ν0. Itshouldbe notedthatthe possibil-
thepolymericcontributiontotheviscosityinthelimitof ity of drag reduction by increasing of the viscosity looks
zero shear. Note that expression (2) for the extra stress somewhatparadoxical: intheusualNewtoniancasewith
tensor is valid for any dumbbell model of the polymeric constant viscosity, the drag is monotonically increasing
molecule (i.e., FENE-P, Hookean, etc.). with the viscosity. The point is that for the polymer so-
In [7], two simplifying approximations led to a trans- lutions the effective viscosity is not constant anymore,it
parent semi-quantitative theory of drag reduction. The increaseslinearlywiththedistancefromthewallaccord-
first approximation is to ignore the fluctuations of R as ing to Eq. (6). This point is the essential difference of
compared to its mean and to take R R in Eq. (2). thetheory[7]fromallprevious“viscous”theoriesofdrag
Thenthe maincontributionofthe stre≈sshtenisorT inthe reduction (see, e.g., [5]). Therefore a crucial test of the
modified NS equation (1) can be formally written in the theory[7]istointroducesuchalinearviscosityprofileto
form of some effective, R-dependent tensorial viscosity. theNSEq. (3)byhand,andseewhetherweobservedrag
Thetensorialstructureoftheeffectiveviscositydescribes reduction together with its various statistical aspects.
differentdamping ofdifferentcomponentsofthevelocity To this aim we simulate the effective NS Eq. (3) with
fluctuations due to the preferred orientation of polymer properviscosityprofiles(discussedbelow)andshowthat
molecules along the mean shear. The second approxi- the results are in semi-quantitative agreement with the
mation is to ignore the preferential orientation and to corresponding full FENE-P DNS. All simulations were
replace ij δij. These two approximations allow doneinadomain2πL 2L 1.2πL,withperiodicbound-
R ⇒ R × ×
onetosimplify the dumbbellmodel(1)toamodifiedNS aryconditionsinthestreamwiseandspanwisedirections,
equation with an effective viscosity, ν0 ν, and pres- and with no slip conditions on the walls that were sep-
⇒
sure, p P: arated by 2L in the wall-normal direction. An imposed
⇒
massfluxandthesameNewtonianinitialconditionswere
∂U /∂t+U U = P + [ν(ω +ω )] ,
ν =i ν0+νpj∇,j iP =−p∇+i ∂ν/∇∂tj+Uij∇νj.i (3) tuesreldin.eTvheeloRcietyyn)owldassn6u0m00beirnRaell(tchoemrpuuntse.dTwhitehgtrhiedcreens--
R ·
olution is 96 129 96 for the linear viscosity profile
Obviously, the polymer elongation, , depends on the runs and 96 ×193 ×96 for the FENE-P run. The lat-
distance from the wall, leading to theRcorresponding de- ter was done×with D×e = 52.7, η = 0.1, 2 = 1000.
pendenceoftheeffectiveviscosity. Noticethattheabove For a definition of theτse paramepters andRdmetaaxils of the
approximations are uncontrolled. Their verification is numerical procedure see Ref. [14].
one of the main goals of this Letter. We demonstrate The y dependence of the scalar effective viscosity was
that the dynamical model Eq. (3) contains the essential close to being piece-wise linear along the channelheight,
properties of the full FENE-P model Eqs. (1), (2). namely ν = ν0 for y y1, a linear portion with a pre-
In Ref. [7] we modified the Reynolds closureapproach scribedslopefory1 <≤y y2,andagainaconstantvalue
to the situation in which the scalar viscosity cannot be for y2 < y < L. For num≤erical stability this profile was
neglected. Thisapproach,thatwasjustified byconsider- smoothed out according to the differential equation
ingareasonablemodelofthecoil-stretchtransitionofthe
polymers,resultedintheanalyticalformofthe universal d2ν Cν0 (y y1)2 (y y2)2
= exp − exp − ,
MDR logarithmic profile dy2 √2πσLn h− 2σ2 i− h− 2σ2 io
V+(y+)= 1 ln eκ y+ , κ 0.09 , (4) integratedwith initial conditionsν(0)=ν0, and ν′(L)=
κV (cid:0) V (cid:1) V ≈ 0. We chose σ = 0.04L, y1 = L/C, y2 = 3L/4, while C
is the dimensionless value of the slope. Examples of four
presented in the wall units
such profiles are shown in Fig. 2. The slope of the lin-
Reτ ≡Lpp′L/ν0, y+ ≡yReτ/L, V+ ≡V/pp′L. (5) ewaarspvaarrtieodftfhoervdiisffceorseitnytprruonfislea[ni.de.,wpaasrasmmaeltleerrκtVhainn(t6h)e]
′
Here p ∂p/∂x is fixed pressure gradients in the theoretically predicted value for the MDR regime. The
≡ −
streamwisedirectionx,Listhehalf-widthofthechannel simulations with ν (y) require larger e numbers to
MDR
R
(in the wall-normal direction y), and e is the friction sustain the turbulence. Included in the figure is the flat
τ
R
Reynoldsnumber. Equation(4)isinexcellentagreement viscosityprofileofthestandardNewtonianflow. Sincewe
withthe MDRasymptote,seeFig.1. Anotherconclusion keep the throughput constant, the runs differ in the val-
of [7] is that in the MDR regime the polymer extension ues of friction Reynolds numbers eτ √τwL/ν0 where
R ≡
3
10 25
8 20
6 15
ν0 + 0
ν/ V
4 10
2 5
0 0
0 0.2 0.4 y/L 0.6 0.8 1 100 101 y+ 102
FIG. 2: The Newtonian viscosity profile and four examples FIG. 3: The reduced mean velocity as a function of the re-
of close tolinear viscosity profiles employed in the numerical duced distance from the wall. The line types correspond to
simulations. Solid black line: run N, solid blue: run R, red: those used in Fig. 2.
run S, green: T, violet: run U.
τw istheaveragefrictionatthewallτw ν0dU/dy. The
≡
decreased value of Re is a manifestation of the drag re-
τ
ductionmeasuredinpercentageasDR%=(τN τE)/τN. 20
w− w w
Thenormalizedslopes,thevalueof e andthepercent-
τ
R
age of drag reduction for these runs are summarized in
15
Table I.
In Fig. 3 we show the resulting profiles of V0+(y) vs. +V0
y+. Thelinetypesarechosentocorrespondtothoseused 10
in Fig. 2. The decrease of the drag with the increase of
the slope of the viscosity profiles is obvious. Since the
5
slopesofthe viscosityprofilesaresmallerthanneededto
achieve the MDR asymptote for the corresponding e ,
τ
R
the drag reduction occurs only in the near-wall region 0
and the Newtonian plugs are clearly visible. 100 101 y+ 102
It is most interesting to compare the effect of the lin-
ear viscosity profile to the simulation of the FENE-P
model in which both the throughput and e are the
Rτ 1
same. Sucha comparisonwasperformedfor the “S” vis-
cous run, for which e = 214 and the FENE-P run
τ
R 0.8
with e = 212.5. The results are presented in the
τ
R
two panels of Fig. 4. In the upper panel the mean ve-
locity profiles (symbols for FENE-P and dashed line for 0.6
the linear viscosity profile) are seen to correspond very + 0
W
closely. The region with increased slope of the mean ve-
0.4
0.2
TABLE I:DNS parameters for effectiveviscosity runs.
0
Case C Reτ DR% κV 0.2 0.4 y/L 0.6 0.8 1
N 0 245 – –
R 8 227 13.8 0.035
FIG.4: Upperpanel: thereducedmeanvelocityasafunction
S 9 214 21.6 0.042
ofthereduceddistancefromthewall. Lowerpanel: Reynolds
T 10 197 36.9 0.051
stresses across the channel. Continuous line: Newtonian.
U 12 185 42.0 0.065 Dashed line: linear viscosity profile. Symbols: FENE-P.
4
locity fluctuations. The increase in r.m.s. streamwise
(V+) and decrease in r.m.s. wall-normal (V+) velocity
x y
3 fluctuations were observedin many experiments and nu-
merical simulations of drag-reducing flows. In Fig. 5
we compare these quantities for FENE-P and “S” runs.
2 Clearly, the correct trend of the r.m.s velocity fluctua-
+x tions is observed, indicating that the important features
V
of the mechanism of drag reduction are correctly cap-
tured by the model. An almost quantitative agreement
1
is reached in the region where drag reduction actually
occurs ( y/L 0.1 0.3).
≈ −
0
Inconclusion,weshowedthatthesimple linearviscos-
0 0.2 0.4 0.6 0.8 1
y/L
ity model (3), (6) faithfully demonstrates drag-reducing
properties and, surprisingly enough, the amount of drag
reduction increases with the increase of the slope of the
viscosity profile. Even more interestingly, the behavior
of objects like the Reynolds stress or the velocity fluctu-
ations in the elastic sublayer are in close correspondence
0.8
with the full FENE-P model, indicating that the mecha-
nismsofdragreductionproposedin[7]operatessimilarly
0.6
in both cases.
+
y
V
ThisworkwassupportedinpartbytheUS-IsraelBSF,
0.4
TheISFadministeredbytheIsraeliAcademyofScience,
the European Commission under a TMR grant and the
0.2
Minerva Foundation, Munich, Germany.
0
0.2 0.4 0.6 0.8 1
y/L
FIG. 5: The rms streamwise ( upper panel) and wall-normal [1] B.A. Toms, in Proceedings of the International Congress
(lower panel) velocity fluctuations across the channel. The ofRheologyAmsterdamVol2.0.135-141(NorthHolland,
line typescorrespond to those used in Fig. 4 1949).
[2] P.S. Virk, AIChEJ. 21, 625 (1975)
[3] P.S. Virk, D.C. Sherma and D.L. Wagger, AIChE Jour-
nal, 43, 3257 (1997).
locity (e.g., the region where the drag reduction occurs)
islimitedtoapproximatelyy+ 50,or,innaturalunits, [4] K.R.SreenivasanandC.M.White,J.FluidMech.409,
≤ 149 (2000).
tyo+y/L50≤o0r.2y5/.LIn t0h.2e5o)uttheer preoglyiomne(rNmewolteocnuilaens ipnlutghse, [[56]] JP..-LG..LduemlGeye,nAnensn.IRntervo.dFuclutiiodnMteochP.o1l,ym36e7r(1D9y6n9a)m. ics,
≥ ≥
FENE-Pmodelaresupposedlyunstretchedandthepoly- (Cambridge, 1990).
meric contribution to the effective viscosity is small. In [7] V.S. L’vov, A. Pomyalov, I. Procaccia and V.
contrast, in the viscous model the effective viscosity is Tiberkevich, Phys. Rev. Lett., submitted. Also:
maximum in this region and may be much larger than nlin.CD/0307034.
[8] S. B. Pope, Turbulent Flows(Cambridge, 2000).
ν0. Therefore, if the mechanism of the drag reduction
[9] R.B. Bird, C.F. Curtiss, R.C. Armstrong and O. Has-
in viscous model is the same as in full FENE-P model,
sager, Dynamics of Polymeric Fluids Vol.2 (Wiley, NY
we expect that all statistical quantities qualitatively co-
1987)
incide for both models in the elastic region y/L 0.25, [10] A.N.BerisandB.J.Edwards,Thermodynamics of Flow-
≤
but may differ in the Newtonian plugs y/L 0.25. ing Systems with Internal Microstructure (Oxford Uni-
≥
In Fig. 4, lower panel, we show the normalized versity Press, NY 1994).
Reynolds stresses. Clearly, in the elastic region y/L < [11] E.DeAngelis,C.M. Casciola, V.S.L’vov,R.PivaandI.
Procaccia, Phys.Rev.E., 67, 056312 (2003).
0.25both drag-reducingmodels coincide; this is a strong
[12] M.D. Warholic, H.Massah, T.J. Hanratty, Experiments
evidence that the reduced model (3) captures all essen- in Fluids, 27, 461 (1999).
tial properties of the full model (1) and the mechanisms [13] A. Rollin and F.A. Seyer, Can. J. Chem. Eng.,50, 714-
of drag reduction are the same in both cases. 718 (1972).
Another important characteristic of drag-reducing [14] E.DeAngelis,C.M.Casciola,andR.Piva,Comput.Flu-
flows is behavior of the root-mean-square (r.m.s.) ve- ids, 31, 495 (2002).
