Direct numerical simulations of statistically steady, homogeneous, isotropic fluid turbulence with polymer additives

0.87 MB·English

by  Prasad Perlekar

Direct numerical simulations of statistically steady, homogeneous, isotropic fluid
turbulence with polymer additives
Prasad Perlekar,1,∗ Dhrubaditya Mitra,2,† and Rahul Pandit3,‡
1Department of Mathematics and Computer Science, Eindhoven University of Technology,
P.O. Box 513, 5600 MB Eindhoven, The Netherlands
2NORDITA, Roslagstullsbacken 23, 106 91 Stockholm, Sweden
3Centre for Condensed Matter Theory, Department of Physics,
Indian Institute of Science, Bangalore 560012, India.
We carry out a direct numerical simulation (DNS) study that reveals the effects of polymers on
1
statistically steady, forced, homogeneous, isotropic fluid turbulence. We find clear manifestations
1
of dissipation-reduction phenomena: On the addition of polymers to the turbulentfluid, we obtain
0
a reduction in the energy dissipation rate, a significant modification of the fluid energy spectrum,
2
especially in the deep-dissipation range, signatures of the suppression of small-scale structures,

n including a decrease in small-scale vorticity filaments. We also compare our results with recent
a experimentsand earlier DNS studies of decaying fluid turbulencewith polymer additives.
J

9 PACSnumbers:
2

] I. INTRODUCTION DNS studies of the three-dimensional Navier-Stokes
n
equation coupled to an equation for the polymer con-
y
formation tensor, which describes the polymer additives
d
The addition of small amounts of polymers to a tur-
- at the level of the FENE-P [2, 3] model. Before we give
u bulentfluidleads todramaticchangesthatinclude mod- the details of our study, it is useful to summarise our
l ifications of the small-scale properties of the flow [1–3]
f principal results; these are in two parts.
. and, in wall-bounded flows, the phenomenon of drag re-
s ThefirstpartcontainsresultsfromourDNSofforced,
c duction[4–6],inwhichpolymeradditivesallowthemain-
statistically steady, fluid turbulence, with polymer addi-
i tenance of a given flow rate at a lower pressure gradient
s tives, at moderate Reynolds numbers (Re 80). The
y than is required without these additives. Several exper- λ ≃
forcing is chosen such that the energy injected into the
h imental, numerical, and analytical studies have investi-
fluid remains fixed [19], both with and without poly-
p gateddragreduction[5–13]inwall-boundedflows. These
[ studies have shown that the addition of polymers mod- mers; this mimics the forcing scheme used in the exper-
2 ifies the turbulence significantly in the region near the iments of Ref. [15, 16]. We find that, on the addition
of polymers, the energy in the statistically steady state
v wall; and this leads to an increase in the mean veloc-
and the energy-dissipation rate are reduced. This dissi-
1 ity in the bulk. By contrast, there have been only some
5 investigations of the effects of polymer additives on ho- pation reduction increases with an increase in the poly-
0 mer concentration c at fixed Weissenberg number We,
mogeneous, isotropic turbulence. Examples include re-
0 the ratio (see Table I) of the polymer time scale τ to
cent experiments [14–16], which have been designed to P
. a shearing time scale in the turbulent fluid. The dis-
8 obtain a high degree of isotropy in the turbulent flow,
0 andshell-modelstudies anddirectnumericalsimulations sipation reduction also increases with We if we hold c
0 fixed. The dissipation reduction seen in our simulations
(DNS) [2, 3, 12, 17, 18]. These studies have shown that
1 should not be confused with the phenomenon of drag
the addition of polymers to a turbulent flow leads to
: reduction seen in wall-bounded flows. In the fluid en-
v a considerable reduction in small-scale structures; and
Xi they have alsodiscoveredthe phenomenonof dissipation ergy spectrum we find that the energy content increases
marginallyat small wavevectorson the addition of poly-
reduction, namely, a reduction in the energy dissipation
r mers, but it decreases for intermediate wavevectors. In
a rateǫ,indecayingturbulence[2,3]. Inthispaperwefirst
this partofour study weuse 2563 collocationpointsand
elucidatethephenomenonofdissipationreductionforthe
attain a moderate Reynolds numbers Re 80; but we
case of statistically steady, homogeneous, isotropic tur- λ ≃
do not resolve the deep-dissipation range. (We consider
bulencewithpolymeradditives;wethenstudythesmall-
the deep dissipation range in the following paragraph.)
scale properties of such flows.
Wealsoobtainthe structuralpropertiesofthe fluidwith
We do this by conducting a series of high-resolution 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
Ref. [15, 16]. Furthermore, we find, as in our study of
∗Electronicaddress: [email protected]
decaying turbulence [3], that the polymer extension in-
†Electronicaddress: [email protected]
creases with an increase in the polymer relaxation time
‡Electronicaddress: [email protected];
τ . We compare our results, e.g., those for the energy
alsoatJawaharlalNehruCentreForAdvancedScientificResearch, P
Jakkur,Bangalore,India spectrum, with their counterparts in our earlier study of
2
decaying fluid turbulence with polymer additives [3]. In figurations), the identity tensor with elements δ ,
αβ
such comparisons, we use averages over the statistically f(r ) (L2 I3)/(L2 r2) the FENE-P potential that
P ≡ − − P
steady state of our system here; and, for the case of de- ensures finite extensibility, r Tr( ) and L the
P
caying turbulence, we use data obtained at the cascade- lengthandthemaximumpossible≡extpensionC,respectively,
completiontime,atwhichaplotoftheenergydissipation ofthe polymers,andc µ/(ν+µ)adimensionless mea-
rate versus time displays a maximum. sure of the polymer co≡ncentration [18]; c = 0.1 corre-
In the second part of our study we carry out the sponds, roughly, to 100ppm for polyethylene oxide [6].
highest-resolution DNS, attempted so far, of forced, sta- Table I lists the parameters of our simulations.
tisticallysteady,fluidturbulencewithpolymeradditives;
we drive the fluid by an external, stochastic force as in
N δt L ν τP c We
Ref. [20]. This part of our study has been designed to
NSP-256A 256 5.0 10−4 100 5 10−3 0.5 0.1 3.5
uncover the effects of polymers on the deep-dissipation × ×
NSP-256B 256 5.0 10−4 100 5 10−3 1.0 0.1 7.1
range, so the Reynolds numbers is small (Reλ 16). × ×
By comparing fluid energy spectra, with and w≃ithout NSP-512 512 10−3 100 5 10−2 1.0 0.1 0.9
×
polymers, we find that the polymers suppress the en-
ergy in the dissipation range but increase it in the deep-
TABLE I: The cube root N of the number of collocation
dissipation range. Finally, we calculate the second-order
points, the time step δt, the maximum possible polymer ex-
velocity structure function S (r) directly from the en-
2 tension L, the kinematic viscosity ν, the polymer-relaxation
ergy spectrum via a Fourier transformation; this shows time τP, and the polymer concentration parameter c for our
that S2(r) with polymers is smaller than S2(r) without four runs NSP 256A, NSP 256B and NSP 512. We also
− − −
polymers in this range. carry out DNSstudies of theNS equation with thesame nu-
The remaining part of this paper is organised as fol- mericalresolutionsasinourNSPruns. TheTaylor-microscale
lows. In Section II we present the equations we use for ReynoldsnumberRe √20 f/p3νǫf andtheWeissenberg
λ ≡ E ν
tthhee mpoeltyhmoderwseoulusteiofonrathnednduemscerriibcea,liinntesgurbasteiocntioonftIhIeAse, NnSuPmbe2r56WB:eR≡eλτPp8ǫ0fν/aνndaNreSPas 5fo1l2lo:wRs:eλNSP1−6;25th6eAKanold-
esuqlutas;tiaosnws.eSheacvtieomnIeInItiiosndeedvoatbeodvteo,tahdeissecuasrseiodnivoidfeodurinrteo- mNSoPg−oro2v56dAissipBa,tηio≃nl1e.n0g7tδhx;scaanldeηf−o≡rr(uνn3/NǫSfνP)1/≃45.1F2,orηour1r9uδnxs,
wher−eδx −L/N is≃thegridresolutionofours−imulation≃s. The
tItwIaIionCspaaanrctdso;nthtcheluesdfieicrnosgtnddpiasirnctusisussbicosonenc.ttaioinnedIIIinDs.uSbescetcitoionnIsVIIcIoAn-- liTninetdtedgyra≡1l.3lue≡arnmngsdt/hlTinestdcdaaylreelain1st.2f≡o;lal(on3wdπs,/:f4oN)rSPNPS−kP−215E561(Ak2a),/nl(idnPtNSEP2(−k.0)5)25aa6nnBdd:
≃ ≃ − ≃
T 4.0.
eddy
≃
II. EQUATIONS AND NUMERICAL METHODS
We model a polymeric fluid solution by using the A. Numerical Methods
three-dimensional, Navier-Stokes (NS) equations for the
fluid coupled with the Finitely Extensible Nonlinear
Weconsiderhomogeneous,isotropic,turbulence,sowe
Elastic-Peterlin (FENE-P) equation for the polymer ad-
use periodic boundary conditions and solve Eq. (1) by
ditives[3]. Thepolymercontributiontothefluidismod-
using a pseudospectral method [21, 22]. We use N3
elled by an extra stress term in the NS equations. The collocation points in a cubic domain (side L = 2π).
FENE-P equation approximates a polymer molecule by
We eliminate aliasing errors by the 2/3 rule [21, 22],
a nonlinear dumbbell, which has a single relaxationtime
to obtain reliable data at small length scales; and we
and an upper bound on the maximum extension. The
use a second-order, slaved, Adams-Bashforth scheme for
NS and FENE-P (henceforth NSP) equations are
time marching. In earlier numerical studies of homo-
µ geneous, isotropic turbulence with polymer additives it
D u = ν 2u+ .[f(r ) ] p+f; (1)
t ∇ τ ∇ P C −∇ has been shown that sharp gradients are formed dur-
P
f(r ) ing the time evolution of the polymer conformation ten-
Dt = .( u)+( u)T. P C−I. (2) sor; this can lead to dispersion errors [18, 23]. To
C C ∇ ∇ C− τ
P avoid these dispersion errors, shock-capturing schemes
Here u(x,t) is the fluid velocity at point x and time t, have been used to evaluate the polymer-advection term
incompressibilityis enforcedby .u=0, D =∂ +u. , [(u ) ] in Ref. [23]. In our simulations we have mod-
t t
∇ ∇ ·∇ C
ν is the kinematic viscosity of the fluid, µ the viscos- ified the Cholesky-decomposition scheme of Ref. [18],
ity parameter for the solute (FENE-P), τ the poly- which preserves the symmetric positive definite nature
P
mer relaxation time, ρ the solvent density (set to 1), p of the tensor . We incorporate the large gradients
the pressure, f(x,t) the external force at point x and of the polymerCconformation tensor by evaluating the
time t, ( u)T the transpose of ( u), R R polymer-advection term [(u )ℓ] via the Kurganov-
αβ α β
∇ ∇ C ≡ h i ·∇
the elements of the polymer-conformation tensor (an- Tadmorshock-capturingscheme[24]. Forthederivatives
C
gular brackets indicate an average over polymer con- on the right-hand side of Eq. (2) we use an explicit,
3
fourth-order, central-finite-difference scheme in space; The SPD nature of is preserved by Eqs. (6) if ℓ > 0,
ii
C
and the temporal evolution is carried out by using an whichweenforceexplicitly [18]byconsideringtheevolu-
Adams-Bashforth scheme. The numerical error in r tion of ln(ℓ ) instead of ℓ .
P ii ii
must be controlled by choosing a small time step δt, We resolve the sharp gradients in the polymer con-
otherwise r can become larger than L, which leads to formation tensor by discretizing the polymer advection
P
a numerical instability; this time step is much smaller termbyusing the Kurganov-Tadmorscheme[24]. Below
than what is necessary for a pseudospectral DNS of the we show the discretization of the advection term u∂ ℓ,
x
NS equation alone. Table I lists the parameters we use. where u (u,v,w) and ℓ is one of the components of
≡
We preserve the symmetric-positive-definite (SPD) na- the ℓ ; the discretization of the other advection terms
αβ
tureof atalltimesbyusing[18]thefollowingCholesky- in Eq. (6) is similar.
C
decomposition scheme: If we define
H H
i+1/2,j,k i−1/2,j,k
f(r ) , (3) u∂ ℓ = − ,
J ≡ P C x δx
Eq. (2) becomes u [ℓ+ +ℓ− ]
i+1/2,j,k i+1/2,j,k i+1/2,j,k
H =
D = .( u)+( u)T. s( )+q , (4) i+1/2,j,k 2
t
J J ∇ ∇ J − J −I J a [ℓ+ ℓ− ]
where i+1/2,j,k i+1/2,j,k− i+1/2,j,k ,
− 2
L2 3+j2
s = −τPL2 , ℓ±i+1/2,j,k = ℓi+1,j,k∓ δ2x(∂xℓ)i+1/2±1/2,j,k ,
d/(L2 3) (L2 3+j2)(j2 3)
a u , (7)
q = − − − − , i+1/2,j,k i+1/2,j,k
(τ L2(L2 3)) ≡ | |
P
−
j2 Tr( ),and where i,j,k = 0,...(N 1) denote the grid points and
−
≡ J δx = δy = δz is the grid spacing along the three direc-
d = Tr[ .( u)+( u)T. ].
J ∇ ∇ J tions.
andhence areSPDmatrices;wecan,therefore,write We use the following initial conditions (superscript
JeClem=enLtsLTℓi,j,JwshuechrethLatisℓiaj =low0efro-rtrjia>ngiu,laanrdmatrix with 0P)m:n(kC)m0vn0n((kx))exp=(ιθnδ(mkn)), fworithalml ,xn; =anxd,y,uz0m, (Pkm)n ==
(δ k k /k2) the transverse projection operator, k
mn m n
ℓ211 ℓ11ℓ21 ℓ11ℓ31 the w−ave vector with components km =( N/2, N/2+
J ≡ ℓℓ1111ℓℓ3211 ℓ21ℓℓ32211++ℓℓ22222ℓ32 ℓℓ22311ℓ+31ℓ+232ℓ+22ℓℓ23332 . (5) 1cbh,e.or.ss.ed,niNsst/ur2icb)huattnehddatmuntaihgfeonriimtnuildtyieablkekt=wine|eektn|i,c0-θenna(nekrd)−gy2rπasn,pdaeonc−mtdruvnmn0u(mkis)-
Equation (4) now yields (1 i 3 and Γij = ∂iuj) the E0(k) = k4exp( 2.0k2). This initial condition corre-
≤ ≤ −
following set of equations: sponds to a state in which the fluid energy is concen-
trated, to begin with, at small k (large length scales);
1 sℓ
D ℓ = Γ ℓ + (q s)ℓ +( 1)(imod1) i1 and the polymers are in a coiled state. Our simula-
t i1 Xk ki k1 2h − i1 − ℓ211 i tions are run for 45Teddy and a statistically steady state
ℓ is reached in roughly 10T , where the integral-scale,
i2 eddy
+(δ +δ ) Γ ℓ
i3 i2 ℓ11 mX>1 m1 m2 emdedayn--tsuqrunaorveervteilmoceitTyedadnyd≡lurms/lint,kw−it1hEu(krm)/stherEo(okt)-
+δi3Γi1ℓℓ233, for i≥1; tahnedinNtSePgral2l5en6gBthwsecaalles.oAclaoirnnrtgy≡woiuPtthkopuurrer-uflnusidN,SPPN−Sk2s5im6A-
11
−
ℓ ulations till a statistically steady state is reached; this
i1
D ℓ = Γ ℓ Γ ℓ
t i2 mi m2 m1 m2 takes about 10 15T . Once this pure-fluid simula-
− ℓ eddy
mX>2 11 mX>2 tionreachesast−atisticallysteadystate,weaddpolymers
1 ℓ ℓ2 to the fluid at 27T ; i.e., beyond this time we solve
+ (q s)ℓ +( 1)(i+2)s i2 1+ 21 eddy
2h − i2 − ℓ222(cid:16) ℓ211(cid:17)i the coupled NSP equations 1 and 2 by using the meth-
ods given above. We then allow 5 6T to elapse, so
+δ ℓ233 Γ Γ ℓ21 +sℓ21ℓ31 , for i 2; that transients die down, and the−n weedcdoyllect data for
i3hℓ22(cid:16) 32− 31ℓ11(cid:17) ℓ211ℓ22i ≥ fluid and polymer statistics for another 25Teddy for our
Γ3mℓ3m Γ31ℓ32ℓ21ℓ33 runs NSP 256A and NSP 256B.
Dtℓ33 = Γ33ℓ33 ℓ33 + − −
− hmX<3 ℓmm i ℓ11ℓ22
ℓ ℓ ℓ 1
21 31 32
s + (q s)ℓ III. RESULTS
− ℓ211ℓ22ℓ33 2h − 33
s ℓ2 sℓ2 ℓ2
+ 1+ 3m + 21 32 . (6) We now present the results that we have obtained
ℓ33(cid:16) mX<3ℓ2mm(cid:17) ℓ211ℓ222ℓ33i from our DNS. In addition to u(x,t), its Fourier
4
transform uk(t), and (x,t), we monitor the vor- tives:
ticity ω u, Cthe kinetic-energy spectrum
E(k,t) ≡ ≡ k−∇1/2×<k′≤k+1/2|u2k′(t)|, the total kinetic ddEt = ǫν +ǫpoly+ǫinj, (8)
energy (t)P E(k,t), the energy-dissipation-rate
ǫscνa(tle)d≡poEνlyPmke≡rk2eEPx(tkken,ts)i,ontshePp(rrPo2b/aLb2i)l,itythdeisPtrDibFutoiofnthoef ǫν = −νV1 Z u·∇2u,
strain and the modulus of the vorticity, and the eigen-
µ 1
values of the strain tensor. For notational convenience, ǫ = ( ) u [f(r ) ] ,
poly P
wedonotdisplaythedependence onc explicitly. Insub- τP (cid:26)V Z ·∇ C (cid:27)
section IIIA we present the time evolution of E and ǫ 1
ν ǫ = f u. (9)
and provide evidence for dissipation reduction by poly- inj V Z ·
mer additives. This is followed by subsections IIIB and
IIIC that deal, respectively, with the effects of polymers Inthestatisticallysteadystate dE =0andtheenergyin-
dt
onfluid energyspectra andsmall-scalestructuresin tur- jectedisbalancedbythefluiddissipationrateǫ andthe
ν
bulent flows. In subsection IIID we examine the modi- polymer dissipation ǫ . Our simulations are designed
poly
fication, by polymer additives, of fluid-energy spectra in to keep the energy injection fixed. Therefore, we can
the deep dissipation range. determine how the dissipation gets distributed between
the fluid and polymer subsystems in forced, statistically
steady turbulence.
Beforewepresentourresultsforthekinetic-energydis-
A. The Energy and its Dissipation Rate sipationrate,wefirstcalculatethesecondorder-structure
function S (r) via the following exact relation [25],
2
We first consider the effects of polymer additives on
∞ sin(kr)
the time evolution of the fluid energy E for our runs S (r)= 1 E(k)dk. (10)
2
NSP 256AandNSP 256B; this is showninFig. 1. The Z0 (cid:20) − kr (cid:21)
− −
polymersareaddedtothefluidatt=27T . Theaddi-
eddy
In Fig.2 we give a log-logplotof S (r), compensatedby
tionofpolymersleadstoanewstatisticallysteadystate; 2
(r/L)−2, as a function of r/L. We find that, for small
specifically, we find for We = 3.5 and We = 7.1, that
r, S (r) r2, which implies that our DNS resolves the
the average energy of the fluid with polymers is reduced 2 ∼
analytic range, which follows from a Taylor expansion,
in comparison to the average energy of the fluid with-
ofS (r)[26];thisguaranteesthatenergy-dissipationrate
out polymers. By using Eq. (1), we obtain the following 2
hasbeencalculatedaccurately. InFig.3wepresentplots
3.4
2.5
3.2 )]
r
(
2
S
3.0
2
2.0
)(cid:3)
E
2.8 /(cid:2)
r
(
[0
2.6 1
og 1.5
l
2.4
2.2
20 25 30 35 40 45 50 55 60 1.0
2.0 1.5 1.0 0.5
t/Teddy (cid:0) (cid:0)log (r/ )(cid:0) (cid:0)
10 (cid:1)
FIG.1: (Coloronline)AplotofthefluidenergyE versusthe FIG. 2: (Color online) Log-log (base 10) plots of the second-
dimensionlesstimet/Teddy (runsNSP-256AandNSP-256B)for orderstructurefunctionS2(r),compensatedby(r/L)−2,ver-
Weissenberg numbers We=3.5 (blue circles) and We=7.1 susr/L,forourrunNSP-256B(bluesquare)andforthepure-
(blackdashedline). Thecorrespondingplotforthepure-fluid fluid case (red circle). The regions in which the horizonal
case is also shown for comparison (red line). The polymers black lines overlap with the points indicate the r2 scaling
are added to thefluid at t=27Teddy. ranges.
energy-balanceequationfor the fluid withpolymer addi- of ǫ (t) versus t/T for We=3.5 and We=7.1 with
ν eddy
5
the polymer concentration c = 0.1. We find that the
average value of ǫ decreases as we increase We. This
ν
1.2
1.0
0.8
(cid:4)
0.6
0.4
FIG. 4: (Color online) Log-log (base 10) plots of the cumu-
0.2
20 25 30 35 40 45 50 55 60 lative PDF PC(r2/L2) versus the scaled polymer extension
P
t/T r2/L2 for We=3.5 (bluedashed line for run NSP-256A) and
eddy P
We=7.1(fullblacklineforrunNSP-256B).NotethatasWe
increases so does the extension of the polymers. These plots
FIG.3: (Coloronline)Plotsoftheenergydissipationrateǫ
ν
versust/T (runsNSP-256AandNSP-256B)forWeissenberg are obtained from polymer configurations at t=60Teddy.
eddy
numbersWe=3.5 (bluecircles) and We=7.1(black dashed
line). The corresponding plot for the pure fluid case is also
shown for comparison (red line). The polymers are added to B. Energy spectra
thefluid at t=27T .
eddy
Inthissubsectionwestudyfluid-energyspectraEp(k),
in the presence of polymer additives, for two different
suggeststhefollowingnaturaldefinitionofthepercentage valuesoftheWeissenbergnumberWeandfixedpolymer
dissipation reduction for forced, homogeneous, isotropic concentration c = 0.1 (Fig. 5). We find that the energy
turbulence: content at intermediate wave-vectors decreases with an
increase in We. At small wave-vector magnitudes k, we
observea smallincrease in the spectrum on the addition
ǫf ǫp
DR h νi−h νi 100%; (11) of the polymers, but this increase is within our numer-
≡(cid:18) hǫfνi (cid:19)× ical, two-standard-deviation error bars. Because of the
moderate resolution of our simulations we are not able
toresolvethedissipationrangefullyinthesesimulations.
here (and henceforth) the superscripts f and p stand, re-
Weaddressthisissuebyconductinghigh-resolution,low-
spectively, for the fluid without and with polymers; and
Reynolds-number simulations in Sec. IIID.
the angular brackets denote an average over the statis-
tically steady state. Percentage dissipation reduction,
DR, rises with We; this indicates that ǫp increases with
ν
C. Small-scale structures
We. [34] Thus, in contrast to the trend we observed
in our decaying-turbulence DNS [3], DR increases with
We: For We = 3.5, DR 30% and, for We = 7.1, We now investigate how polymers affect small-scale
≃
DR 50%. Our interpretation is that this increase of structures in homogeneous, isotropic, fluid turbulence;
≃
DR with We arises because the polymer extensions and, and we make specific comparisons with experiments [15,
therefore, the polymer stresses are much stronger in our 16]. We begin by plotting the PDFs of the modulus of
forced-turbulence DNS than in our decaying-turbulence the vorticity ω and the local energy dissipation rate
DNS(atleastfortheReynoldsnumbersthatweachieved ǫ = (∂|u |+∂ u )2/2 in Figs. 6. We find that the
loc i,j i j j i
in Ref. [3]). In Fig. 4 we show the cumulative PDF of additioPnofpolymersreducesregionsofhighvorticityand
the scaledpolymerextension; this showsclearlythatthe high dissipation [Figs. (6)]. Furthermore, we find that,
extensionofthepolymersincreaseswithWe. Ingeneral, onnormalising ω orǫ bytheirrespectivestandardde-
loc
| |
the calculation of PDFs from numerical data is plagued viations,the PDFs ofthese normalisedquantities forthe
byerrorsoriginatingfromthebinningofthedatatomake fluidwithandwithoutpolymerscollapseontoeachother
histograms. Here instead we have used the rank-order (within our numerical error bars) as shown in Figs. 7.
method to calculate the corresponding cumulative PDF Our results for these PDFs are in qualitative agreement
which is free of binning errors [28]. withtheresultsofRefs.[15,16](seeFig.2ofRef.[15]and
6
]
k) 0
(
p
E
[
0
1
g −2
o
l
d
n
a
] −4
)
k
(
fE
[
0
1−6
g
o
l 0 0.5 1 1.5 101
log (k)
10 100
10-1
FIG. 5: (Color online) Log-log plots (base 10) of the energy
csp=ect0r.a1Eanpd(k)Wveer=sus3.k5 ((brulunessNqSuPa-r2e5s)6AoranWdeNS=P-72.516(Bb)lafcokr )loc10-2
P((cid:6) 10-3
stars); we give two-standard-deviation error bars. The corre-
spondingpure-fluidspectrumEf(k) (redcircles) isshownfor 10-4
comparison.
10-5
10-6
0 2 4 6 8 10 12
(cid:5)loc
Fig. 3 of Ref. [16]). Earlier high-resolution, large-Re ,
λ
DNS studies of homogeneous, isotropic fluid turbulence
without polymer additives (see,e.g., Ref.[29,30]andref- FIG. 6: (Color online) Semilog plots (base 10) of the PDFs
erencestherein)haveestablishedthatiso-ω surfacesare P(ω)versus ω (toppanel)andP(ǫloc)versusǫloc (bottom
filamentaryforlargevaluesof ω . InFig.|8w| eshowhow pan|el|), for our|ru|n NSP-256B, with [c=0.1, We=7.1 (blue,
such iso-ω surfaces change on| t|he addition of polymers dashed line)] and without [c = 0 (full, red line)] polymer
| | additives.
(c = 0.1; We = 3.5 or 7.1). In particular, the addition
of polymers suppresses a significant fraction of these fil-
aments (compare the top and middle panels of Fig. 8);
ical invariants: Q Tr(A2)/2 and R Tr(A3)/3,
and this suppression becomes stronger as We increases where A is the velo≡cit−y-gradient tensor ≡u −[35]. Topo-
(middle and bottom panels of Fig. 8). In addition to ∇
logical properties of such a flow can be classified [31, 32]
suppressingeventswhichcontributetolargefluctuations
by a Q R plot, which is a contour plot of the joint
in the vorticity, the addition of polymers also affects the −
probability distribution function (PDF) of Q and R. In
statistics of the eigenvalues of the rate-of-strain matrix
Fig. 11 we give Q R plots from our DNS studies with
(Sij = (∂iuj +∂jui)/√2), namely, Λn, with n = 1,2,3. and without polym−ers; althoughthe qualitative shape of
They providea measureof the localstretching and com-
these joint PDFs remains the same, the regions of large
pression of the fluid. In our study, these eigenvalues are
R and Q are dramatically reduced on the additions of
arranged in decreasing order, i.e., Λ > Λ > Λ . In-
1 2 3 polymers;thisisyetanotherindicatorofthesuppression
compressibility implies that Λ =0; therefore, for an
i i of small-scale structures.
incompressiblefluid, oneoftPhe eigenvalues(Λ1)mustbe
positive and one (Λ ) negative. The intermediate eigen-
3
value Λ can either be positive or negative. In Figs. (9)
2
and (10) we plot the PDFs of these eigenvalues. The D. Effects of polymer additives on
deep-dissipation-range spectra
tails of these PDFs shrink on the addition of polymers.
This indicates thatthe addition ofthe polymers leads to
a substantial decrease in the regions where there is large Inthe previoussubsectionswehavestudiedthe effects
strain, a result that is in qualitative agreement with the of polymer additives on the structural properties of a
experimentsofRef.[15](seeFig.3(b)ofRef.[15]). Evi- turbulent fluid at moderate Reynolds numbers. We now
denceforthesuppressionofsmall-scalestructuresonthe investigate the effects of polymer additives on the deep-
addition of polymers can also be obtained by examin- dissipation range. To uncover such deep-dissipation-
ing the attendant change in the topologicalproperties of rangeeffects,weconductaveryhigh-resolution,butlow-
a three-dimensional, turbulent flow. For incompressible, Re (= 16) DNS study (NSP 512). The parameters
λ
ideal fluids in three dimensions there are two topolog- used in our run NSP 512 ar−e given in Table I. The
−
7
100
10-1
10-2
)
(cid:12)
/|(cid:11) 10-3
(|(cid:10)
P10-4
10-5
10-6
0 2 4 6 8 10 12 14 16
/
|(cid:7)| (cid:8)(cid:9)
100
10-1
10-2
)
(cid:18)
/loc(cid:17) 10-3
((cid:16)
P10-4
10-5
10-6
0 5 10 15 20 25 30
/
(cid:13)loc (cid:14)(cid:15)
FIG. 7: (Color online) Semilog plots (base 10) of the scaled
PDFsP(ω/σ)versus ω/σ (toppanel)andP(ǫ /σ )ver-
ω loc ǫ
| | | |
susǫ /σ (bottompanel),whereσ andσ arethestandard
loc ǫ ω ǫ
deviationsfor ω andǫ ,respectively,forourrunNSP-256B,
| | loc FIG.8: (Coloronline)Constant-ω isosurfacesfor ω = ω +
with [c = 0.1, We = 7.1 (dashed line)] and without [c = 0 | | | | | |
2σ at t 60T without (top panel) and with polymers
(line)] polymer additives. These plots are normalized such [mωiddlepa≈nelWeded=y 3.5 (NSP-256A) and bottom panelWe=
that the area undereach curveis unity.
7.1 (run NSP-256B)]; ω is the mean and σ the standard
ω
deviation of ω. | |
| |
fluid is driven by using the stochastic-forcing scheme of
Ref. [20]. In Fig. 12 we plot fluid-energy spectra with
additionofpolymersleadstoadecreaseinthemagnitude
and without polymer additives. The general behavior of
of S (r). Our plots for S (r) are similar to those found
these energy spectra is similar to that in our decaying- 2 2
intheexperimentsofRef.[14]. Inoursimulationsweare
turbulence study [3]. We find that, on the addition
able to reach much smaller values of r/η than has been
of polymers, the energy content at intermediate wave-
possible in experimental studies on these systems [14];
vectors decreases, whereas the energy content at large
however,wehavenotresolvedtheinertialrangeverywell
wave-vectorsincreasessignificantly. Wehavecheckedex-
in these runs. Note that the spectra Ep(k), with poly-
plicitly that this increase in the energy spectrum in the
mers, and Ef(k), without polymers, cross each other as
deep-dissipation range is not an artifact of aliasing er-
shown in Fig. 12. But such a crossing is not observed in
rors: Note first that this increase starts at wave vectors
the corresponding plots of second-order structure func-
whose magnitude is considerably lower than the dealias-
tions (Fig. 14). This can be understood by noting that
ing cutoff k in our DNS; furthermore, the enstrophy
max S (r)combineslarge-andsmall-kparts[33]oftheenergy
spectrumk2Ep(k),whichweplotversusk inFig.13,de- 2
spectrum.
cays at large k; this indicates that the dissipation range
has been resolved adequately in our DNS.
For homogeneous, isotropic turbulence, the relation-
IV. CONCLUSIONS
shipbetweenthesecond-orderstructurefunctionandthe
energy spectrum is given in Eq. (10) [25]. Using this re-
lationship and the data for the energy spectrum shown Wehavepresentedanextensivenumericalstudyofthe
in Fig. 12, we have obtained the second-order structure effects of polymer additives on statistically steady, ho-
function S (r) for our run NSP 512. We find that the mogeneous, isotropic fluid turbulence. Our study com-
2
−
8
plements, and extends considerably, our earlier work [3].
Furthermore, our results compare favorablywith several
recent experiments.
Our first set of results show that the average viscous
energydissipationratedecreasesontheadditionofpoly-
mers. This allows us to extend the definition of dissipa-
tion reduction, introduced in Ref. [3], to the case of sta-
tisticallysteady,homogeneous,isotropic,fluidturbulence
with polymers. We find that this dissipation reduction
increases with an increase in the Weissenberg number
We, at fixed polymer concentration c. We obtain PDFs
of the modulus of the vorticity, of the eigenvalues Λ of
n
therate-of-straintensorS,andQ Rplots;wefindthat
−
these are in qualitative agreement with the experiments
of Refs. [15, 16].
Our second set of results deal with a high-resolution
DNS that we have carried out to elucidate the deep-
FIG. 9: (Color online) Semilog (base 10) plots of the PDF
dissipation-rangeformsof(a)energyspectraand(b)the
P(Λ1) versus thefirst eigenvalue Λ1 of the strain-rate tensor
S for therun NSP 256B, with [We=7.1 (bluedashed line)] related second-order velocity structure functions. We
− find that this deep-dissipation-range behavior is akin
and without [c = 0 (full red line)] polymer additives. These
plots are normalized such that the area under each curve is to that in our earlier DNS of decaying, homogeneous,
unity. isotropic, fluid turbulence with polymers [3]. Further-
more, the results we obtain for the scaled, second-order,
velocity structure S (r) yield trends that are in qualita-
2
tive agreement with the experiments of Ref. [14].
We hope that the comprehensive study that we have
presentedherewillstimulatefurtherdetailedexperimen-
tal studies of the statistical properties of homogeneous,
isotropic fluid turbulence with polymer additives.
V. ACKNOWLEDGMENTS
We thank J. Bec, E. Bodenschatz, F. Toschi, and
H. Xu for discussions, Leverhume trust, European Re-
searchCouncilundertheAstroDynResearchProjectNo.
FIG. 10: (Color online) Semilog (base 10) plots of the PDF 227952, CSIR, DST, and UGC (India) for support, and
P(Λ2) versusthesecond eigenvalueΛ2 ofthestrain-rateten- SERC(IISc)forcomputationalresources. Twoofus(RP
sor S for the run NSP 256B, with [We = 7.1 (blue dashed andPP)are members of the InternationalCollaboration
−
line)] and without [c = 0 (full red line)] polymer additives. for Turbulence Research and they acknowledge support
These plots are normalised such that the area under each fromtheCOSTActionMP0806. Whilemostofthiswork
curveis unity. is being carried out PP was a student at IISc.
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FIG.11: (Coloronline)Contourplotsofthejoint PDFP(R,Q)from ourDNSstudieswith(left)andwithout(right)polymer
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)
-3 k
k) 10 pE( 10-3
(
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fE fE 10-9
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-12
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k k
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