Drag Reduction by Polymers in Wall Bounded Turbulence

0.15 MB·English

by  Victor S. L'vov

Drag Reduction by Polymers in Wall Bounded Turbulence
Victor S. L’vov, Anna Pomyalov, Itamar Procaccia and Vasil Tiberkevich
Dept. of Chemical Physics, The Weizmann Institute of Science, Rehovot, 76100 Israel
Weaddressthemechanismofdragreductionbypolymersinturbulentwallboundedflows. Onthe
basisoftheequationsoffluidmechanicswepresentaquantitativederivationofthe“maximumdrag
reduction (MDR) asymptote” which is the maximum drag reduction attained by polymers. Based
onNewtonianinformationonlyweprovetheexistenceofdragreduction,andwithoneexperimental
parameter we reach a quantitativeagreement with theexperimental measurements.
4
0 The addition of few tens of parts per million (by
0 weight) of long-chain polymers to turbulent fluid flows
2 in channels or pipes can bring about a reduction of the 50 Newtonian, Warholic, 1999
n frictiondragby up to 80%[1, 2, 3, 4]. This phenomenon MDR, Rollin, 1972
a of “drag reduction” is well documented and is used in Rudd, 1969
J 40 Rollin, 1972
technological applications from fire engines (allowing a
DNS, De Angelis, 2003
4 water jet to reach high floors) to oil pipes. In spite of a
MDR, our theory
1
large amountof experimentaland simulationaldata, the 30 Newtonian, our theory
] fundamentalmechanismhasremainedunderdebatefora + V Newtonian plugs
D long time [4, 5, 6]. In such wall-boundedturbulence, the
C dragiscausedbymomentumdissipationatthewalls. For 20
. Newtonianflows(inwhichthekinematicviscosityiscon-
n
stant) the momentum flux is dominated by the so-called
i 10
l Reynolds stress, leading to a logarithmic (von-Karman)
n
[ dependence of the mean velocity on the distance from
the wall[7]. However,withpolymers,the dragreduction 0
2
entails a change in the von-Karman log law such that a 1 10 100
v
muchhighermeanvelocityisachieved. Inparticular,for +
4 y
3 high concentrations of polymers, a regime of maximum
FIG. 1: Mean normalized velocity profiles as a function of
0 drag reduction is attained (the “MDR asymptote”), in-
thenormalized distancefrom thewallduringdragreduction.
7 dependent of the chemical identity of the polymer [2],
The data points from numerical simulations (green circles)
0
3 see Fig. 1. In this Letter we elucidate the fundamen- [11] and theexperimental points(open circles) [12] represent
0 talmechanismforthisphenomenon: whilemomentumis the Newtonian results. The black solid line is our theory
/ producedat a fixed rate by the forcing,polymer stretch- Eq.(13), which for large y+ agrees with von-Karman’s loga-
n ing results in a suppression of the momentum flux from rithmiclawofthewall(2). Thereddatapoints(squares)[13]
i
l the bulk to the wall. Accordingly the mean velocity in represent the Maximum Drag Reduction (MDR) asymptote.
n Thedashedredcurverepresentsourtheoryfortheprofile(19)
v: the channel must increase. We derive a new logarithmic which for large y+ agrees with the universal law (16). The
law for the mean velocity with a slope that fits existing
i arrowmarksthecrossoverfromtheviscouslinearlaw(12)to
X numerical and experimental data. The law is universal, theasymptoticlogarithmiclaw(16). Thebluefilledtriangles
r thus explaining the MDR asymptote. [13]andgreenopentriangles[14]representthecrossover,for
a Turbulentflowsinachannelareconvenientlydiscussed intermediate concentrations of the polymer, from the MDR
[7] for fixed pressure gradients p′ ∂p/∂x where x, y asymptotetotheNewtonianplug. Ourtheoryisnotdetailed
≡ − enough to capture thiscross over properly.
and z are the lengthwise, wall-normal and spanwise di-
rections respectively. The length and width of the chan-
nel are usually taken much larger than the mid-channel
famousuniversalaspectsofNewtonianturbulentchannel
height L, making the latter a natural re-scaling length
flows is the “logarithmic law of the wall” which in these
for the introduction of dimensionless (similarity) vari-
coordinates is expressed as
ables. Thus the Reynolds number e, the normalized
distance from the wall y+ and the noRrmalized mean ve- V+(y+)=κ−1lny++B, for y+ >30, (2)
K
locity V+(y+) (which is in the x direction with a depen- ∼
where the von-Karman constant κ 0.436 and inter-
dence on y only) are defined by cept B 6.13 [10]. For y+ < 10 onKe≈observes a viscous
e L p′L/ν0 , y+ y e/L , V+ V/ p′L , (1) sub-laye≈r, V+(y+) = y+, Fig. 1. The riddle of drag re-
R ≡ ≡ R ≡ duction is then introduced in relation to this universal
p p
where ν0 is the kinematic viscosity. One of the most law: inthepresenceoflongchainpolymersthe meanve-
2
locity profileV+(y+) (for a fixed value of p′ and channel Armed with the effective equation we establish the
geometry) changes dramatically. For sufficiently large mechanism of drag reduction following the standard
concentration of polymers V+(y+) saturates to a new strategyofReynolds,consideringthe fluid velocityU(r)
(universal, polymer independent) “law of the wall” [2], asasumofitsaverage(overtime)andafluctuatingpart:
V+(y+)=11.7lny+ 17, for y+ >12 . (3)
− ∼ U(r,t)=V(y)+u(r,t) , V(y) U(r,t) . (8)
≡h i
For smaller concentrationof polymers the situation is as
For a channel of large length and width all the averages,
showninFig. 1. TheNewtonianlawofthewall(2)isthe
and in particular V(y) V(y), are functions of y only.
black solid line for y+ >30. The MDR asymptote (3) is ⇒
The objects that enter the theory are the mean shear
the dashedred line. Fo∼r intermediate concentrationsthe
S(y), the Reynolds stress W(y) and the kinetic energy
meanvelocityprofilestartsalongtheasymptoticlaw(3),
K(y); these are defined respectively as
and then crosses over to the so called “Newtonian plug”
with a Newtonian logarithmic slope identical to the in- S(y) dV(y)/dy , W(y) uxuy , K(y)= u2 /2.
≡ ≡−h i h| | i
verse of von-Karman’s constant. The region of values of
A well known exact relation [7] between these objects is
y+ in which the asymptotic law (3) prevails was termed
thepoint-wisebalanceequationforthefluxofmechanical
“the elastic sub-layer” [2]. The relative increase of the
momentum; near the wall (for y L) it reads :
′
mean velocity (for a given p) due to the existence of ≪
′
the new law of the wall (3) is the phenomenon of drag ν(y)S(y)+W(y)=pL . (9)
reduction. Thus the main theoretical challenge is to un-
OntheRHSofthisequationweseetheproductionofmo-
derstand the originof the new law (3), and in particular
mentum flux due to the pressure gradient; on the LHS
its universality, or independence of the polymer used. A
we have the Reynolds stress and the viscous contribu-
secondary challenge is to understand the concentration
tiontothemomentumflux,withthelatterbeingusually
dependentcrossoverbacktotheNewtonianplug. Inthis
negligible (in Newtonian turbulence ν = ν0) everywhere
Letterwearguethatthephenomenoncanbeunderstood
except in the viscous boundary layer. The y dependence
mainly by the influence of the polymer stretching on the
of the effective viscosity ν(y) in the elastic layer will be
y+-dependent effective viscosity. The latter becomes a
shown to be crucial for drag reduction.
crucial agent in carrying the momentum flux from the
A second relation between S(y), W(y) and K(y) is
bulkofthechanneltothewalls(wherethemomentumis
obtainedfromthe energybalance. The energyis created
dissipatedbyfriction). IntheNewtoniancasetheviscos-
by the large scale motions at a rate of W(y)S(y). It is
ity has a negligible role in carrying the momentum flux;
cascadeddownthescalesbyafluxofenergy,andisfinally
this differencegivesriseto the changeofEq. (2)infavor dissipated at a rate ǫ, where ǫ = ν ω2 . We cannot
of Eq. (3) which we derive below. h iji
calculate ǫ exactly, but we can estimate it rather well at
The equations of motion of polymer solutions can be
a point y away from the wall. When viscous effects are
written as [8, 9] : dominant, this term is estimated as ν(a/y)2K(y) (the
∂U/∂t+U ∇U = ∇p+∇ T +ν0 2U , (4) velocity is then rather smooth, the gradient exists and
· − · ∇ can be estimated by the typical velocity at y over the
whereT istheextrastresstensorthatisduetothepoly- distancefromthewall). Hereaisaconstantofthe order
mer. Denoting the polymer end-to-end vector distance ofunity. WhentheReynoldsnumberislarge,theviscous
(normalized by its equilibrium value) as r, the average dissipationisthesameastheturbulentenergyfluxdown
dimensionless extension tensor R is r r , and the scales, which can be estimated as K(y)/τ(y) where
ij i j
R ≡ h i
the extra stress tensor is (with ω ∂U /∂x ), τ(y) is the typical eddy turn over time at y. The latter
ij i j
≡
is estimated as y/b K(y) where b is another constant
T =νp ω R+R ωT ∂R/∂t U ∇R . (5) of the order of unity. We can thus write the balance
· · − − · p
equation at point y as
Here ν (p(cid:0)roportional to the polymer concentr(cid:1)ation) is
p
the polymeric contribution to the viscosity in the limit ν(y)(a/y)2K(y)+bK3/2(y)/y =W(y)S(y), (10)
of zero shear. In order to develop a transparent the-
wherethebiggerofthetwotermsontheLHSshouldwin.
oryweproposetoignorethefluctuationsofRcompared
We notethatcontrarytoEq. (9)whichisexact,Eq.(10)
to its mean. In other words, we will take R R .
≈ h i isnotexact. Weexpectithowevertoyieldgoodorderof
Simplifying further the tensor structure, assuming that
magnitude estimates as is demonstrated below. Finally,
= δ we can rewrite Eq. (4) in the form
Rij R ij we quote the experimental fact [2, 16] that outside the
viscous boundary layer
∂U /∂t+U U = P + ν(ω +ω ) , (6)
i j j i i j ij ji
ν =ν0+νp ∇, P =−p∇+∂ν/∇∂t+U ∇ν , (7) W(y) c2N, for Newtonian flow,
R · = (11)
with a scalar (y-dependent) effective viscosity ν. K(y) ( c2 , for viscoelastic flow.
V
3
The coefficients c and c are bounded from above by neglecting the second term on the LHS, using Eq. (11),
N V
unity. (The proof is c W /K 2 u u / u2 + and finally re-scaling the variables results in
u2 1, because (u | |u≡)2| |0). W≤hi|lhe cx yiis|khnoxwn
yi ≤ x ± y ≥ N ∂V+/∂y+ =c y+/c y+ . (14)
quite accurately, c 0.5, the actual values of c varies N v V
N ≈ V
somewhatfrom experimentto experiment, such that the
Fromherefollowsimmediately the new “logarithmiclaw
ratioc /c (whichisallthatweneedtousebelow)varies
V N of the wall”
between 0.3 and 0.7. In our estimates we take c /c =
V N
0.5. V+(y+)=κ −1lny++B , κ =c /c y+, (15)
V V V V N v
We show now that Eqs. (9 – 11) are sufficient for
deriving the Newtonian law (2) and the viscoelastic law where the intercept B is still unknown (but will be de-
V
(3)withequalease. BeginwiththeNewtoniancase. The terminedmomentarily). Theslopeofthenewlawisinde-
resultofthisderivationisnotnew-butwewanttostress pendent of the polymer concentration andis greaterthan
thatthe same equationsgiveriseto boththe wellknown the von-Karman slope 1/κ ; the ratio of the slopes is
K
and the new results. Substitute Eq. (11) in Eqs. (9, 10) in fact κ y+c /c , which is about 4.9 according to the
K v N V
withν(y)=ν0,turningthemintoalgebraicequationsfor above estimates cN/cV ≈ 0.5 and y+ ≈ 5.6. Compar-
K(y)andS(y),andeventuallytoafirstorderdifferential ing with the measured ratio of slopes in Eqs. (2) and
equation for V(y). In the viscous sub-layer K(y) = 0, (3) which is about 5.1, we consider our estimates to be
and the solution in re-scaled coordinates is quite on the mark. As said before, this increase in slope
is the phenomenon of drag reduction. We stress that
V+(y+)=y+ , y+ y+ a/c , (12) the information gained from the Newtonian data alone is
≤ v ≡ N
sufficient to predict drag reduction, since c 1.
Outside the viscous sub-layer (y+ >yv+) we find To find BV we match the logarithmic laVw≤(15) to the
viscoussublayersolution(12)bythevalueofV+(y+)and
V+(y+)=κ−1lnY(y+)+B ∆(y+), (13a) its derivative. First, the logarithmic law has slope 1 at
K −
B =2y+ κ−1ln e(1+2κ y+)/4κ , (13b) y+ = κ−1. Next, matching at this point the viscous so-
v − K K v K lumtion VV+(y+)=κ−1 to (15) we find B =ln(eκ )/κ .
(cid:2) (cid:3) m V V V V
where κ =c /b and we defined We note that if we substitute the experimental value
K N
κ−1 =11.7 we find B = 17 in perfect agreement with
K V −
Y(y+) = y++ y+2 y+2+ 2κ −2 2, Eq. (3). We thus write the law (15) in its final form
− v K (with just one constant remaining)
h2κ2y+q2+4κ Y(y+)(cid:0) y+(cid:1) +i.1
∆(y+) = K v K − . 1
2κ2y+ V+(y+)= ln eκ y+ . (16)
(cid:2)K (cid:3) κ V
V
aFnodr y∆+(y≫+)yv+ E0q.s. (13) turns into (2) since Y(y+)→y+ ereNnocteettohaatnythmisoudneilveorfsathlerepsuoll(cid:0)tymiseorbdtay(cid:1)innaemdiwcsit,haonudt rtehfe-
Notethat→Eqs(13)pertaintothewholey+domain. By only assumptions are that the polymer viscosity domi-
taking the experimental values of κ and B we compute natesthe momentumtransferintheelasticlayerandthe
K
y+ 5.6 to be compared with the experimental value of logarithmic law (15) is valid all the way to y+. At this
v ≈ m
5.5 0.5, cf. [7]. The resulting Newtonian profile, Eqs. pointwedemonstratethatthesetwoassumptions,which
±
(12) and (13), is shown in Fig. 1 as the black solid line. lead to the universality of (16), are well supported by a
The excellent agreement with the experimental and nu- closer consideration of the polymer dynamics.
mericaldatain the entireregionofy+ indicates thatour The fundamental reason for the appearance of a y-
balance equations are sufficiently accurate and we can dependent viscosity is the coil-stretch transition of the
proceed to the viscoelastic case. polymers under turbulent shear. This transition occurs
To see how the law (3) emerges we consider Eqs. (9) inanyreasonablemodelofthepolymer-turbulenceinter-
and(10) withy-dependenteffective viscosity. We should action, the simplest of which is of the form [15]
warn the reader that this is not fully justified – there
are terms in the full viscoelastic equations which cannot d /dt= /τ˜ +s(y) , s(y) ω2 , (17)
R −R p R ≡ h iji
be simplified to the form of a space dependent viscos-
q
ity. Nevertheless we propose (and justify further below) where τ˜ is the polymer relaxation time. In the elastic
p
thatthetermswitheffectiveviscosityarethemainterms sublayer we estimate s(y)=g K(y)/y, with some con-
that allow the momentum flux to be carried in the elas- stantg O(1),andexpectacoilstretchtransitionwhen
tic sublayer. In other words, when the concentration g K(y≈)/y = 1/τ˜ . Definingpτ gτ˜ we then find in
p p p
of the polymer is large enough we can neglect the sec- the elastic sublayer (where polym≡ers are stretched)
p
ond term in favor of the first in Eq. (9) and estimate
′ 2
ν(y)=Lp/S(y). Substituting this estimate in Eq. (10), K(y)=(y/τ ) . (18)
p
4
Thisequationisimportantsinceitcanbesolvedtogether Navier-Stokes equations with a space dependent viscos-
withEqs. (9)and(10)tofindS+(y+)whenthepolymers ity according to the above theory. We predict that a
are stretched. The answer is viscosity profile that remains constant for 0 y+ y+,
≤ ≤ V
andthengrowslinearlytowardsthecenter,shouldresult
1 y+L
S+(y+) = (19) in drag reduction in much the same way as seen in ex-
κ y+ eℓ
V n(cid:16)R (cid:17) perimentswithpolymers. Suchsimulationswouldrender
+ 1+ 2κ y+c 2 1 y+L 2 , direct support to the views offered in this Letter.
K v V − eℓ
r
(cid:2)(cid:0) (cid:1) (cid:3)(cid:16)R (cid:17) o
where ℓ=2κ y+c τ √Lp′. Equation(14) isrecaptured We acknowledge useful discussions with Rama Govin-
K v V p
from this, more general equation, when e is large and darajan. This work was supported in part by the US-
... 1. Then the identity of the polyRmer is lost, giv- IsraelBSF,TheISFadministeredbytheIsraeliAcademy
{ing r}is→e to the universal drag reduction asymptote (16). of Science, the European Commission under a TMR
It can be also checked that the matching point y+ used grant and the Minerva Foundation, Munich, Germany.
m
above indeed connects smoothly the viscous sublayer to
the asymptotic law (16) (see arrow in Fig. 1). We can
thereforeconcludethatourderivationofEq.(16)isfully
consistent with the polymer dynamics, and we under-
[1] B.A. Toms, in Proceedings of the International Congress
stand how the polymer characteristics drop out, leading
ofRheologyAmsterdamVol2.0.135-141(NorthHolland,
to the universal law (16). 1949).
The mechanism of drag reduction is then the suppres- [2] P.S. Virk, AIChEJ. 21, 625 (1975)
sion of the Reynolds stress in the elastic sublayer. The [3] P.S. Virk, D.C. Sherma and D.L. Wagger, AIChE Jour-
Reynolds stress is the main agent for momentum flux in nal, 43, 3257 (1997).
the Newtonian case, and its suppressionresults in an in- [4] K.R.SreenivasanandC.M.White,J.FluidMech.409,
149 (2000).
crease of the mean mechanical momentum (velocity) in
[5] J. L. Lumley,Ann.Rev.Fluid Mech. 1, 367 (1969).
the channel. The increase in viscosity of course leads to
[6] P.-G. de Gennes Introduction to Polymer Dynamics,
increased energy dissipation, but this is immaterial for
(Cambridge, 1990).
the phenomenon. This is the main difference between [7] S. B. Pope, Turbulent Flows(Cambridge, 2000).
dragreductioninwall-boundedturbulence andinhomo- [8] R.B. Bird, C.F. Curtiss, R.C. Armstrong and O. Has-
geneousturbulence[18]. Wenotethatthetheorypredicts sager, Dynamics of Polymeric Fluids Vol.2 (Wiley, NY
averylowvalueofK(y)intheelasticlayer. Inrealityin 1987)
[9] A.N.BerisandB.J.Edwards,Thermodynamics of Flow-
everywallboundedturbulentflowstherewillbeejectaof
ing Systems with Internal Microstructure (Oxford Uni-
parcels of fluid with high level of K(y) from the Newto-
versity Press, NY 1994).
nianregiontowardsthe walls. Thesewouldbe pickedup [10] M.V. Zagarola and A.J. Smits, Phys. Rev.Lett. 78, 239
in measurements, and would seem to contradict the low (1997).
value of K(y) predicted above. In fact there is no real [11] E.DeAngelis,C.M. Casciola, V.S.L’vov,R.PivaandI.
contradictionsincesuchrandomejectadonotcontribute Procaccia, Phys.Rev.E., 67, 056312 (2003).
muchtothe momentumflux thatis socrucialtoourdis- [12] M.D. Warholic, H.Massah, T.J. Hanratty, Experiments
in Fluids, 27, 461 (1999).
cussion. We alsonote aterminthe dynamicalequations
[13] A. Rollin and F.A. Seyer, Can. J. Chem. Eng.,50, 714-
fortheviscoelasticflowoftheformd r r /dt,whichcan-
h i ji 718 (1972).
not be written as an effective viscosity. This term van- [14] M.J. Rudd,Nature, 224, 587 (1969)
ishes for uncorrelated rotations of individual polymeric [15] R. B. Bird, R.C. Armstorng and O. Hassager, Dynam-
molecules. However it can contribute to the energy flux ics of Polymeric Liquids, (John Wiley, New York, 1977,
from the mean shear to turbulent fluctuations in wall Vol.2).
boundedflowsduetoapossible“correlationinstability”, [16] P.K. Ptasinski, F.T.M. Nieuwstadt, B.H.A.A. van den
Brule and M.A. Hulsen, Flow, Turbulence and Combus-
whichleadstosynchronizedrotationofneighboringpoly-
tion, 66, 159 (2001).
mers. Amoredetailedconsiderationofsuchtermsshould
[17] Theimportanceofspacedependentviscositywasalready
provide also the cross over behavior seen in Fig. 1.
pointed out, cf. R. Govindarajan, V.S. Lvov and I. Pro-
Finally we propose direct numerical simulations in caccia, Phys.Rev. Lett., 87, 174501 (2001).
channel flows to test our approach. Instead of simulat- [18] R.Benzi,E.DeAngelis,R.GovindarajanandI.Procac-
ingtheviscoelasticequations,weproposetosimulatethe cia, Phys. Rev.E, in press. Also: physics/0304091