Table Of Content1
7 Lyapunov dimension of elastic turbulence
1
0
2 Emmanuel Lance Christopher VI Medillo Plan1, Anupam Gupta2,
n Dario Vincenzi1 and John D. Gibbon3
a
J 1Universit´e Cˆote d’Azur,CNRS, LJAD,France
6 2FERMaT, Universit´ede Toulouse, CNRS,INPT, INSA,UPS, Toulouse, France
3Department of Mathematics, Imperial College London, London SW7 2AZ, UK
]
n
y Low-Reynolds-number polymer solutions exhibit a chaotic behaviour known as ‘elastic
d
turbulence’ when the Weissenbergnumber exceeds a critical value.The two-dimensional
-
u Oldroyd-Bmodelisthesimplestconstitutivemodelthatreproducesthisphenomenon.To
l makea practicalestimate ofthe resolutionscaleofthe dynamics requiresanassumption
f
. that an attractor of the Oldroyd-B model exists: numerical simulations show that the
s
c quantitiesonwhichthisassumptionisbasedarebounded.WeestimatetheLyapunovdi-
i mensionofthisassumedattractorasafunctionoftheWeissenbergnumberbycombining
s
y a mathematical analysis of the model with direct numerical simulations.
h
p
[ 1. Introduction
1 One of the most remarkable properties of viscoelastic fluids is the formation of insta-
v
bilities at very low Reynolds numbers (Larson 1992; Shaqfeh 1996). Such instabilities
6
are of a purely elastic nature; they occur when inertial forces are negligible and elastic
6
3 forces are strong. In polymer solutions, elastic instabilities eventually lead to a chaotic
2 regimeknownaselasticturbulence (Groisman& Steinberg2000).The emergenceofthis
0 regimeischaracterizedbyafastgrowthoftheLyapunovexponentoftheflowaspolymer
.
1 elasticity exceeds a critical value (Burghelea, Segre & Steinberg 2004). In addition, the
0 kinetic-energyspatialandtemporalspectrahaveapower-lawbehaviour,whichindicates
7 thepresenceofalargenumberofactivescalesintheflow.Thespatialspectrum,however,
1
is steeper than for Newtonian turbulence, i.e. velocity fluctuations are concentrated at
:
v small wave numbers (Groisman & Steinberg 2000; Burghelea, Segre & Steinberg 2007).
i
X Elastic turbulence has important applications in microfluidics in view of the fact
that it strongly enhances mixing in devices that, owing to their microscopic size, are
r
a characterized by a low Reynolds number (Groisman & Steinberg 2001). Moreover, the
potential use of elastic turbulence in the oil industry has recently emerged as a very
promisingapplication.Aqueouspolymersolutionsareindeedusedtorecovertheoilthat
remains trapped inside the pores of reservoir rocks after an initial water flooding, and
elastic turbulence has been proposed as a mechanism to explain the unexpectedly high
efficiency of this oil recovery method (Mitchell et al. 2016).
ThesimplestconstitutivemodelofpolymersolutionsistheOldroyd-Bmodel(Oldroyd
1950) in which the dissolved polymer phase is described by a symmetric tensor field,
termed the polymer conformation tensor, which represents the moment of inertia of
polymers averaged over thermal fluctuations. The Oldroyd-B model is thus a system of
partialdifferentialequations(PDEs)thatdescribesthejointevolutionofthevelocityand
thepolymerconformationtensor.Inparticular,thepolymerfeedbackontheflowisgiven
by a stress term proportional to the conformation tensor. The relevant dimensionless
number isthe Weissenbergnumber;i.e.the ratioofthe polymerrelaxationtime andthe
typicaltime scaleofthe flow.The mainlimitationofthis modelis thatit assumeslinear
2 E. L. C. VI M. Plan, A. Gupta, D. Vincenzi and J. D. Gibbon
polymerelasticitywhich,inextensionalflowsandintheabsenceofpolymerfeedback,can
lead to an unbounded growth of the conformation tensor and hence of polymer stresses.
Inspite ofitssimplicity,the modelsuccessfullyreproducesthe mainfeaturesofelastic
turbulence(Fouxon&Lebedev2003;Bertiet al.2008;Berti&Boffetta2010;Thomases,
Shelley & Thiffeault 2011; Grilli, Va´zquez-Quesada & Ellero 2013). In particular, nu-
merical simulations show that elastic turbulence is observed also in two-dimensional
settings. It has attracted much attention in the mathematical community over the last
two decades in which analysts have focused on the existence, uniqueness and regularity
of solutions (see, e.g., Lei, Masmoudi & Zhou 2010; Barrett & Su¨li 2011; Constantin &
Kliegl 2012; Elgindi & Rousset 2015; Hu & Lin 2016, for some recent studies). To our
knowledgetherearefewtheoreticalresultsonthepropertiesofelasticturbulentsolutions.
A mathematicaldefinition ofthe number ofdegreesoffreedomofasystemofPDEs is
given by the dimension of its globalattractor (Robinson 2001).For the two-dimensional
case the bounds found by Constantin & Kliegl (2012) on the L2-norms of the vorticity
ω and the polymer conformationtensor σ, ω and σ , are exponential in time (and
2 2
double exponential for ∇σ ). Thus, nokbokundedklonkg-time averages have been
2
k k h·i
found to exist and therefore, in a strictly rigorous sense, no global attractor is known
to exist.Mathematically we cango no further.However,numericalsimulations ofelastic
turbulence (see 4) suggest that ω , σ and ∇σ are indeed bounded in time for
2 2 2
§ k k k k k k
various values of Wi. One practical way of progressing is to work under the following
technical assumption with a subsequent strategy:
(i) Given that numerical calculations of ω , σ and ∇σ are finite in time, as
2 2 2
k k k k k k
suggested in 4, we assume that a global attractor exists;
§ A
(ii) Based on (i), we estimate the Lyapunov dimension of , which will include the
long-time averages ω , σ and ∇σ : the numericAal bounds found for these
2 2 2
hk k i hk k i hk k i
quantities in 4 in terms of Wi are used in our estimates.
§
A connection between the system dynamics and the attractor dimension is provided
by the notion of the Lyapunov exponents via the Kaplan–Yorke formula. For ordinary
differentialequations(ODEs),theLyapunovexponentscontroltheexponentialgrowthor
contractionofvolume elements in phase space,andthe Kaplan–Yorkeformula expresses
the balance between volume growth and contraction realized on the attractor. The
Kaplan–Yorke formula is used to define the Lyapunov dimension of the attractor and
is the following: for Lyapunov exponents µ > µ > > µ > , the Lyapunov
1 2 n
··· ···
dimension d ( ) is given by
L
A
µ1+...+µN−1
d ( )=N 1+ , (1.1)
L
A − µ
N
−
where the number N of µ is chosen to satisfy
n
N−1 N
µ >0 but µ <0. (1.2)
n n
nX=1 nX=1
Note that according to the definition of N, the ratio of exponents in (1.1) satisfies
06 µ1+...+µN−1 <1. (1.3)
µ
N
−
Insimpleterms,thevalueofN thatturnsthesignofthesumoftheLyapunovexponents
Lyapunov dimension of elastic turbulence 3
as in (1.2) is that value that bounds above d ( ) such that
L
A
N 16d ( )<N. (1.4)
L
− A
The non-integer Lyapunov dimension d ( ) generally bounds above the fractal and
L
A
Hausdorff dimensions d ( ) and d ( ). The Kaplan–Yorke formula originated as a
F H
A A
phase-space argument for ODEs but has been rigorously applied to global attractors in
PDEsbyConstantin&Foias(1985)(seealso Gibbon&Titi1997).Tousethemethodfor
PDEs to find an estimate for d ( ), it is necessary to extend the idea of the Lyapunov
L
A
exponents to global Lyapunov exponents via a trace formula for the two-dimensional
Oldroyd-B model that is explained in 3. This uses the methods of Constantin & Foias
§
(1985),Constantin,Foias&Temam(1988),Doering&Gibbon(1991)and,inparticular,
∞
the L -estimates of Constantin (1987). For a two-dimensional system of side L, the
resolution length ℓ of the smallest feature in the dynamics is connected to d ( ) by
res L
A
d ( ) (L/ℓ )2.
L res
A ∼
2. The Oldroyd-B model for polymer solutions
On the periodic unit square Ω = [0, 1]2, the dimensionless form of the Oldroyd-B
model is
∂ u+u·∇u= ∇p+Re−1∆u+β(WiRe)−1∇·σ+F , (2.1a)
t
−
∂ σ+u·∇σ =(∇u)σ+σ(∇u)⊤ Wi−1(σ I)+Pe−1∆σ, (2.1b)
t
− −
whereuistheincompressiblevelocityfield,(∇u) =∂ u ,σisthepolymerconformation
ij j i
tensor,pispressure,andwhereRe,Pe andWi aretheReynolds,P´ecletandWeissenberg
numbersrespectively.Thepositiveconstantβ dependsonthepolymerconcentrationand
equilibrium extension. The forcing F is time-independent, periodic and divergence-free
(∇·F =0).
The Laplacian term in (2.1b) originates from the diffusion of the centre of mass
of polymers (El-Kareh & Leal 1989), which ensures the global regularity of the two-
dimensional Oldroyd-B model (Constantin & Kliegl 2012) and improves the stability of
numerical simulations, even though the values of Pe used in practice are considerably
lower than its realistic values (Sureshkumar & Beris 1995; Thomases 2011).
Intwodimensions,(2.1)canberewrittenintermsofthescalarvorticityω =zˆ·(∇ u):
×
∂ ω+u·∇ω =Re−1∆ω+β(WiRe)−1zˆ·∇ (∇·σ)+f, (2.2a)
t
×
∂ σ+u·∇σ =(∇u)σ+σ(∇u)⊤ Wi−1(σ I)+Pe−1∆σ, (2.2b)
t
− −
where f =zˆ·(∇ F)andu=∇⊥∆−1ω with∇⊥ ( ∂ ,∂ ). SinceF isperiodic,the
y x
× ≡ −
spatial average of ω is zero, and the inverse Laplacian of ω is properly defined. Figure
1 shows snapshots of the ω and Trσ fields from a numerical simulation of (2.2) in the
elastic-turbulence regime. Note that Trσ is concentrated over very thin regions, which
is associated with large gradients in the polymer-conformation-tensor field.
3. Lyapunov dimension
In the PDE case the phase space is now infinite dimensional. Define q = (ω,σ) and
denotebyδq =(δω,δσ)theinfinitesimaldisplacementaboutq;δqsatisfiesthelinearized
set of equations from (2.2):
δq˙ =A(t)δq (3.1)
4 E. L. C. VI M. Plan, A. Gupta, D. Vincenzi and J. D. Gibbon
(a) (b)
Figure 1: Pseudocolor plot of: (a) ω and (b) log Trσ for Re = Re /√2, Wi = 20,
10 c
β =0.2; and a cellular forcing with n=4 and f =0.16 (see 4 for the details).
0
§
1 β
u·∇δω+ ∆δω δu·∇ω+ zˆ·∇ (∇·δσ)
= − Re − WiRe × ,
δσ 1
(∇δu)σ+σ(∇δu)⊤ δu·∇σ u·∇δσ+(∇u)δσ+δσ(∇u)⊤ + ∆δσ
− − − Wi Pe
where the explicit form of the operator A(t) is obtained by using u = ∇⊥∆−1ω to
expressuandδuintermsofωandδω,respectively.FollowingConstantin&Foias(1985),
take different sets of initial conditions q(0)+δq (0) which evolve into q(t)+δq (t) for
i i
i=1,...,N.Iftheyarechosentobelinearlyindependent,theseδq formanN-volumeor
i
parallelepipedofvolumeV (t)= δq δq ... δq ,whichchangesalongthesolution
N 1 2 N
q(t), so we need to find its time e|volut∧ion. This∧is giv|en by V˙ =V Tr[AP ], which is
N N N
easily solved to give
t
V (t)=V (0)exp Tr[AP ](τ)dτ , (3.2)
N N N
(cid:26)Z (cid:27)
0
where P (t) is an L2-orthogonal projection onto the finite dimensional subspace
N
span δq ,δq ,...,δq . In terms of the time average , the sum of the first N global
1 2 N
{ } h·i
Lyapunov exponents is taken to be (Constantin & Foias 1985)
N
µ = Tr[AP ] . (3.3)
n N
h i
nX=1
To estimate the Lyapunovdimension, we wish to find the value of N that turns the sign
of Tr[AP ] . This value of N bounds above d ( ) as in (1.4). As we are interested in
N L
h i A
elastic turbulence, we assume Re =O(1), Pe 1 and Wi 1.The derivation has been
≫ ≫
relegatedtotheAppendix.Thefinalresultisthatthefollowinginequalitiesaresufficient
conditions on N for the growthrate Tr[AP ] to be negative and the N-volume V to
N N
h i
contract:
N >cRe σ 3 1/3 1+lnRe+ln σ 3 1/3 1/2, (3.4a)
hk k2i hk k2i
N >cRe2/3 ∇σ (cid:0)2 1/3 1+lnRe+ln ∇σ(cid:1) 2 1/2 1/3, (3.4b)
hk k2i hk k2i
(cid:0) (cid:1)
where 2 = 2dx, σ 2 = σ 2, ∇σ 2 = ∂ σ 2 and c is regarded
k·k2 Ω|·| | | i,j| ij| | | i,j,k| k ij|
as a generic, pRositive, dimensionlPess constant. These cPonditions depend on estimates
for σ 3 and ∇σ 2 . It has been shown in Constantin & Kliegl (2012) that these
k k2 k k2
(cid:10) (cid:11) (cid:10) (cid:11)
Lyapunov dimension of elastic turbulence 5
are exponential in time and thus cannot be used. Instead we turn to direct numerical
simulations to find their behaviour in terms of Wi.
4. Direct numerical simulations of elastic turbulence
To simulate solutions of (2.2), we use the approach described in Gupta, Perlekar &
Pandit (2015) and Gupta & Pandit (2016). For the time integration, we use the fourth-
orderRunge–Kuttaschemewithtimestepδt=10−4,whilethefourth-ordercentral-finite-
difference scheme with 10242 collocation points is employed for the spatial derivatives.
To accurately resolve the steep gradients of σ, we apply the Kurganov–Tadmor shock-
capturing scheme to the advection term in (2.2b) (Kurganov & Tadmor 2000), which
allows us to set Pe = in (2.2). The velocity is calculated from the vorticity via the
Poissonequation∆ψ =∞ω,whereψisthestreamfunction:u=∇⊥ψ.Thepseudospectral
method is used to solve the Poisson equation in Fourier space. The simulations were
performed on [0,2π]2 and the solutions were rescaled appropriately.
Three kinds of forcing are considered: a cellular forcing f = f n[cos(nx)+cos(ny)]
0
−
with n =4, a cellular forcing with n=10, and a Kolmogorov forcing f = f ncos(nx)
0
−
with n=8. In all simulations Re =Re /√2, where Re is the critical Reynolds number
c c
above which instabilities occur; hence in the absence of polymers (β =0) the stationary
vorticity field shows alternating vortices and antivortices for the cellular forcing and a
parallel sinusoidal flow for the Kolmogorovforcing. The parameter β is set to β =0.2.
The numerical simulations show that σ 2 1/3 is greater than σ 3 1/3 and, at
large Wi, σ 2 1/3 Wiα with α h0k.∇7 fokr2ithe three different fohrkcinkg2si considered
hk∇ k2i ∼ ≈
here (see figure 2). This result is in agreementwith the observationof large gradients in
thepolymerconformationfieldinfigure1aswellinpreviousnumericalsimulations(Berti
et al. 2008; Berti & Boffetta 2010). It follows that the value of N such that N-volumes
contract is determined by inequality (3.4b). We conclude that, in the elastic-turbulence
regime and under the specified assumptions, the two-dimensional Oldroyd-B model has
a finite-dimensional global attractor with Lyapunov dimension
d ( )6 , (4.1)
L
A N
where istheminimumvalueofN satisfying(3.4b).Thus,uptologarithmiccorrections,
N
d ( ).CWiα, (4.2)
L
A
where α 0.7 and C depends on Re.
≈
5. Conclusions
Through a mathematical and numerical analysis of the two-dimensional Oldroyd-
B model in the elastic-turbulence regime, we have made a practical estimate of the
dimension of its (assumed) global attractor, as suggested by numerical simulations. We
have also shown that the complexity of the attractor is related to the formation of
large gradients in the polymer conformation field. Although the asymptotic power-law
dependence of d ( ) on Wi was found to be the same for different forcings, further
L
A
studies are required to confirm the universality of the exponent α.
As mentioned in the introduction, the main limitation of the Oldroyd-B model is
the absence of a maximum polymer elongation. Other constitutive models of polymer
solutions, such as the FENE-P model, overcome this limitation by introducing a non-
linear elastic force that diverges when Trσ approaches the square of the maximum
elongation (see Thomases et al. 2011; Gupta & Pandit 2016, for the application of the
6 E. L. C. VI M. Plan, A. Gupta, D. Vincenzi and J. D. Gibbon
<||∇ σ||2>1/3 Wi0.7 104 <||∇ σ||22>1/3 Wi0.7
103 <||σ||3>12/3 <||σ||23>1/3
<||ω||22>1/2 103 <||ω||22>1/2
102 2
Wi0.6 102 Wi0.5
101
101
100
10 −1 100 101 10100 −1 100 101
(a) Wi (b) Wi
104 <||∇ σ||2>1/3 Wi0.7
2
<||σ||3>1/3
2
<||ω||2>1/2
2 10−5
102 Wi0.5 E(k)
10−10
100
10 −1 100 101 100 101 102
(c) Wi (d) k
Figure2:Plotof ω 2 1/2 (blacksquares), σ 3 1/3 (bluecircles)and ∇σ 2 1/3 (red
hk k2i hk k2i hk k2i
triangles) as a function of Wi for Re = Re /√2, β = 0.2, and (a) cellular forcing with
c
n=4,f =0.16,(b)cellularforcingwithn=10,f =2.50,(c)Kolmogorovforcingwith
0 0
n = 8, f = 1.28; (d) Kinetic-energy spectrum for Wi = 20 for case (a) in red, case (b)
0
in blue and case (c) in black. The spectrum shows a power-law behaviour E(k) k−a
≃
with a 3.1 for case (a), a 3.0 for case (b) and a 3.7 for case (c).
≈ ≈ ≈
FENE-P model to the study of elastic turbulence). However, using a non-linear elastic
force does not prevent the formation of large gradients in the σ-field (Thomases et al.
2011). In other words, large values of ∇σ seem to occur independently of the form of
theforcethatdescribestheelasticityofpolymers.Ourestimate(4.1)dependson ∇σ
2
k k
rather than σ . This fact suggests that, albeit based on the Oldroyd-B model, (4.1)
2
k k
may also be relevant to other constitutive models.
Elastic turbulence is observed at low Re and high Wi. When both Re and Wi are
high,theadditionofpolymerstoatwo-dimensionalturbulentflowsuppresseslarge-scale
velocity fluctuations (Amarouchene & Kellay 2002). This phenomenon is once again
correctlyreproducedby the Oldroyd-Bmodel(Boffetta, Celani& Musacchio2003).Our
estimate(A16),whichgivesasufficientconditionforthecontractionofN-volumes,holds
forallvaluesofRe.Itwouldbeinterestingtoinvestigatetheimplicationsofthisestimate
for the attractor dimension in the high-Re regime.
Appendix. Derivation of the decay rate of N-dimensional volumes
Here we show the main steps needed to obtain (3.4). The trace Tr[AP ] can be
N
expressed as (e.g. Constantin & Foias 1985; Doering & Gibbon 1995; Robinson 2001)
N
Tr[AP ]= Φ ·AΦ dx, (A1)
N n n
Z
nX=1 Ω
Lyapunov dimension of elastic turbulence 7
where Φ N = (φω,φσ) N with (φσ)⊤ = φσ is an orthonormal set spanning
{ n}n=1 { n n }n=1 n n
the subspace generated by the displacements (δω ,δσ ) N . The orthonormality of
{ n n }n=1
Φ N should be interpreted as follows: φωφωdx+ φσ :φσdx=δ , whence
{ n}n=1 Ω m n Ω m n mn
R R
N N
Φ 2dx= φω 2+ φσ 2 dx=N. (A2)
nX=1ZΩ| n| nX=1ZΩ(cid:16)| n| | n| (cid:17)
The symbol ‘:’ denotes the inner product between matrices, σ : σ′ Tr[σ⊤σ′]. The
≡
following inequality (e.g. Robinson 2001) will also be used when we estimate the norms
of φω,φσ or of their gradients
n n
N
Θ Tr[ ∆P ]= φω∆φω +φσ :∆φσ dx>cN2. (A3)
≡ − N −nX=1ZΩ(cid:16) n n n n(cid:17)
The terms appearing in (A1) can be explicitly listed as (the terms that follow from
u·∇δω and u·∇δσ vanish because of the incompressibility of u)
N N
Tr[AP ]=Re−1 φω∆φωdx+Pe−1 φσ :∆φσdx (A4a)
N Z n n Z n n
nX=1 Ω nX=1 Ω
N N
φωv ·∇ωdx φσ :(v ·∇σ)dx (A4b)
− Z n n − Z n n
nX=1 Ω nX=1 Ω
N N
+β(WiRe)−1 φωzˆ·(∇ ∇·φσ)dx Wi−1 φσ 2dx (A4c)
Z n × n − Z | n|
nX=1 Ω nX=1 Ω
N N
+ φσ :[(∇v )σ+σ(∇v )⊤]dx+ φσ :[(∇u)φσ +φσ(∇u)⊤]dx,(A4d)
Z n n n Z n n n
nX=1 Ω nX=1 Ω
where v = ∇⊥∆−1φω. By using (A3), the Laplacian terms in (A4a) can be shown to
n n
satisfy (under the assumption Pe>Re)
N N
Re−1 φω∆φωdx+Pe−1 φσ :∆φσdx6 Re−1Θ. (A5)
Z n n Z n n −
nX=1 Ω nX=1 Ω
We treat the advective terms (A4b) by using the result of Constantin (1987) and its
∞
subsequent use in Doering & Gibbon (1991): i.e. a Cauchy–Schwarz inequality, an L
bound for the terms involving v , and (A3)
n
N
φωv ·∇ω+φσ :(v ·∇σ) dx 6c(1+lnΘ)1/2Θ1/4( ∇ω + ∇σ ).
(cid:12)(cid:12)(cid:12)nX=1ZΩh n n n n i (cid:12)(cid:12)(cid:12) k k2 k k2
(cid:12) (cid:12) (A6)
(cid:12) (cid:12)
Toestimatethefeedbacktermin(A4c),weuseintegrationbyparts,theCauchy–Schwarz
inequality, the relations ∇ (φωzˆ) = ∇φω and ∇·φσ 2 6 ∇φσ 2, and (A3):
| × n | | n| | n| | n|
N N
φωzˆ·(∇ ∇·φσ)dx 6 ∇φω ∇φσ dx6Θ. (A7)
(cid:12)(cid:12)(cid:12)nX=1ZΩ n × n (cid:12)(cid:12)(cid:12) nX=1ZΩ| n|| n|
(cid:12) (cid:12)
The second t(cid:12)erm in (A4c) satisfies (cid:12)
N
Wi−1 φσ 2dx=bWi−1, 0<b6N 6cΘ1/2. (A8)
Z | n|
nX=1 Ω
8 E. L. C. VI M. Plan, A. Gupta, D. Vincenzi and J. D. Gibbon
Thestretchingtermin(A4d)thatinvolves∇v isfirstintegratedbyparts.TheCauchy-
n
Schwarz inequality is then applied twice to obtain
N N
φσ :[(∇v )σ+σ(∇v )⊤]dx 62 (φσ ∇σ v + ∇φσ σ v )dx.
(cid:12)(cid:12)(cid:12)nX=1ZΩ n n n (cid:12)(cid:12)(cid:12) nX=1ZΩ | n|| || n| | n|| || n|
(cid:12) (cid:12)
W(cid:12)e then use the same techniques emplo(cid:12)yed to get (A6) and obtain:
N
φσ :[(∇v )σ+σ(∇v )⊤]dx 6c(1+lnΘ)1/2 Θ1/2 σ +Θ1/4 ∇σ .
(cid:12)(cid:12)(cid:12)nX=1ZΩ n n n (cid:12)(cid:12)(cid:12) (cid:0) k k2 k k2(cid:1)
(cid:12) (cid:12) (A9)
(cid:12) (cid:12)
The other stretching term in (A4d) is estimated by first applying the Cauchy–Schwarz
inequality and (A3):
1/2
N N
2
φσ :[(∇u)φσ+φσ(∇u)⊤]dx 62 ∇u Φ 2 dx .
(cid:12)(cid:12)(cid:12)nX=1ZΩ n n n (cid:12)(cid:12)(cid:12) k k2(cid:12)(cid:12)(cid:12)ZΩ(cid:16)nX=1| n| (cid:17) (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Wethe(cid:12)nnotethatintwodimensions ∇u =(cid:12) ω andu(cid:12)setheLieb–Thirring(cid:12)inequality
2 2
k k k k
for the set of orthonormalfunctions Φ N (Constantin, Foias & Temam 1988)
{ n}n=1
N
φσ :[(∇u)φσ +φσ(∇u)⊤]dx 6c ω Θ1/2. (A10)
(cid:12)(cid:12)(cid:12)nX=1ZΩ n n n (cid:12)(cid:12)(cid:12) k k2
(cid:12) (cid:12)
Using (A5)–(A10(cid:12)), the trace Tr[AP ] can now be estim(cid:12) ated in terms of Θ as follows
N
Tr[AP ]6 βWi−1 1 Re−1Θ bWi−1+c(1+lnΘ)1/2Θ1/2 σ
N 2
− − k k
(cid:0) (cid:1)
+c(1+lnΘ)1/2Θ1/4( ∇σ + ∇ω )+c ω Θ1/2. (A11)
2 2 2
k k k k k k
In the third term in the right-handside of (A11), it is useful to write Θ1/2 =ΘaΘ1/2−a
with 0<a<1/2. Taking the time average of (A11) and applying the Cauchy–Schwarz
and Ho¨lder’s inequalities on the time variable yields
Tr[AP ] 6 βWi−1 1 Re−1 Θ bWi−1+c ω 2 1/2 Θ 1/2
h N i − h i− hk k2i h i
(cid:0) (cid:1)
+c (1+lnΘ)3/2Θ3a 1/3 Θ3(1/2−a) 1/3 σ 3 1/3
h i h i hk k2i
+c (1+lnΘ)Θ1/2 1/2 ∇σ 2 1/2+ ∇ω 2 1/2 . (A12)
h i hk k2i hk k2i
The bound in (A12) can be improved by estim(cid:0)ating ∇ω 2 . We first m(cid:1)ultiply (2.2a)
hk k2i
byω,integrateoverspace,andtakethetimeaverage(notingthatisnecessarytouseour
assumption that ω is finite and hence ∂ ω 2 =0):
k k2 h tk k2i
Re−1 ∇ω 2 = ωzˆ·∇ fdx +β(WiRe)−1 ωzˆ·∇ (∇·σ)dx . (A13)
hk k2i DZΩ × E DZΩ × E
We then integrate by parts and apply the Cauchy–Schwarzinequality, first in space and
then in time, to obtain
∇ω 2 1/2 6Re f +βWi−1 ∇σ 2 1/2. (A14)
hk k2i k k2 hk k2i
Moreover,Jensen’s inequality allows us to express (A12) as a function of Θ :
h i
Tr[AP ] 6 βWi−1 1 Re−1 Θ bWi−1+c ω 2 1/2 Θ 1/2
h N i − h i− hk k2i h i
(cid:0)+c(1+ln Θ(cid:1))1/2 Θ 1/2 σ 3 1/3+cRe(1+ln Θ )1/2 Θ 1/4 f
h i h i hk k2i h i h i k k2
+c 1+βWi−1 (1+ln Θ )1/2 Θ 1/4 ∇σ 2 1/2. (A15)
h i h i hk k2i
(cid:0) (cid:1)
Lyapunov dimension of elastic turbulence 9
Note that for this last inequality to hold, a must be such that Θ3(1/2−a) and (1 +
lnΘ)3/2Θ3a are concave functions. The choice a = 1/5 guarantees that Θ3(1/2−a) is
concaveforallvalues ofΘ and(1+lnΘ)3/2Θ3a is concavefor Θ>5.This restrictionon
Θ is justified since Θ > cN2 and in elastic turbulence N is large. The trace Tr[AP ]
N
h i
is thus guaranteedto be negative if Θ satisfies
h i
1 βWi−1 Θ >cRe ω 2 1/2 Θ 1/2+cRe2(1+ln Θ )1/2 Θ 1/4 f (A16)
− h i hk k2i h i h i h i k k2
(cid:0) +cRe(cid:1)(1+ln Θ )1/2 Θ 1/2 σ 3 1/3+ 1+βWi−1 Θ 1/4 ∇σ 2 1/2 .
h i nh i hk k2i (cid:0) (cid:1)h i hk k2i o
Under the assumptions that Re = O(1) and Wi 1, it is easy to see that (1
βWi−1) Θ Θ is greater than each of the terms≫in the first line of (A16). As f−
2
h i ≈ h i k k
is independent of Wi, Θ dominates the forcing term. In the elastic-turbulence regime
h i
of the Oldroyd-B model, the kinetic-energy spectrum decays rapidly as a function of
the wave number (Berti et al. 2008; Berti & Boffetta 2010), and ω is expected to be
2
k k
small as Wi increases. Therefore, Θ also controls the enstrophy term in (A16)—this
h i
is confirmed numerically in figure 2.
The number of significant terms in (A16) is thus reduced to two. Direct numerical
simulations of the Oldroyd-B model show that in elastic turbulence Trσ grows as Wi
increases and σ develops strong gradients (Berti et al. 2008; Berti & Boffetta 2010).
Therefore, both σ and ∇σ are expected to increase with Wi. To analyze the σ-
2 2
and ∇σ- terms ikn (kA16), ikt is ukseful to note that for any independent of Θ 1/2 and
M h i
any 0<λ61 the following holds
Θ 1/2 >c 1/(2−λ)(1+ln )1/2(2−λ) = Θ >c Θ λ/2(1+ln Θ )1/2, (A17)
h i M M ⇒h i Mh i h i
which can be proved by generalizing an analogous result by Doering & Gibbon (1991).
The inequalities below are thus sufficient conditions for (A16) to hold:
Θ 1/2 >cRe σ 3 1/3 1+lnRe+ln σ 3 1/3 1/2, (A18a)
h i hk k2i hk k2i
Θ 1/2 >cRe2/3 ∇σ (cid:0)2 1/3 1+lnRe+ln ∇σ(cid:1) 2 1/2 1/3, (A18b)
h i hk k2i hk k2i
(cid:0) (cid:1)
Since Θ >cN2 (see (A3)), these can be converted into sufficient conditions on N, as in
(3.4).
The authors would like to acknowledge useful discussions with S. Musacchio. The
work of ELCMP was supported by EACEA through the EMMA program. ELCMP and
DV acknowledge the support of the EU COST Action MP 1305 ‘Flowing Matter.’ The
authors also acknowledge the Obervatoire de la Cˆote d’Azur for computing resources.
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