Elastic turbulence in two-dimensional cross-slot viscoelastic flows
EElastic turbulence in two-dimensional cross-slot viscoelastic flows
D. O. Canossi, G. Mompean, and S. Berti
Univ. Lille, ULR 7512 - Unit´e de M´ecanique de Lille Joseph Boussinesq (UML), F-59000 Lille, France
We report evidence of irregular unsteady flow of two-dimensional polymer solutions in the absenceof inertia in cross-slot geometry using numerical simulations of Oldroyd-B model. By exploring thetransition to time-dependent flow versus both the fluid elasticity and the polymer concentration,we find periodic behaviour close to the instability threshold and more complex flows at largerelasticity, in agreement with experimental findings. For high enough elasticity we obtain dynamicspointing to elastic turbulence, with temporal spectra of velocity fluctuations showing a power-lawdecay, of exponent in between − −
2, and probability density functions of velocity fluctuationsthat weakly deviate from Gaussianity while high non-Gaussian tails characterise those of localaccelerations.
INTRODUCTION
The rheological behaviour of viscoelastic flows at van-ishingly small inertia can be related to strongly non-linear phenomena and includes an association of viscousand elastic effects, with the latter being typically due tothe presence of flexible long-chain polymers in the solu-tion. The elasticity of the flow can give rise to complexdynamics that are relevant for both fundamental studiesand industrial applications, as e.g. efficient mixing andheat transfer in microdevices [1], or painting and coatingprocesses [2–4].The purely elastic instabilities marking the transitionsbetween different flow regimes have been documented ina variety of geometrical configurations [3, 5–7], includingcomplex ones, such as the abrupt axisymmetric contrac-tion [8] and the lid-driven cavity [9]. The cross-slot setup,made of two perpendicularly intersecting channels withtwo inlets and two outlets, is, in this sense, no excep-tion. Due to its relevance for mixing and rheology, ithas been the subject of extensive studies. Indeed, exper-imental [10–12], theoretical [13, 14] and numerical [15–17] investigations have reported about the existence ofinstabilities solely driven by elasticity in this setup. It isnow known that low-Reynolds-number polymeric flowsin this geometry can display two types of instabilities: afirst one, corresponding to a supercritical pitchfork bifur-cation, to steady asymmetric flow [16, 17], and a secondone leading to unsteady oscillatory behaviour [11, 16, 18].Concerning the latter, it is interesting to recall that it hasbeen provided numerical evidence, in a two-dimensional(2D) flow, that it occurs via a supercritical Hopf bifur-cation [19]; a mechanism relying on the role of stressgradients and the existence of a stagnation point at thecentre of the setup was also proposed [19].Above a critical Weissenberg number (
W i ), meaningfor elasticity larger than a threshold, purely elastic in-stabilities can lead to the appearance of disordered flowscorresponding to the dynamical regime known as elasticturbulence [6, 20]. As shown in the seminal work [6],where a swirling flow between counter-rotating paralleldisks was considered, and in subsequent ones also em-ploying different geometries [21, 22], such flows are remi- niscent of the turbulent ones occurring in Newtonian flu-ids. In particular, they are characterised by a whole rangeof active scales, irregular temporal behaviour, growthof flow resistance and enhanced mixing properties [21].Interestingly, however, the spectrum of velocity fluctu-ations displays power-law behaviours, in both the tem-poral ( E ( f ) ∼ f − δ ) and spatial ( E ( k ) ∼ k − δ ) domains,with an exponent (in absolute value) δ ≈ . >
3, corre-sponding to a smooth flow essentially dominated by thelargest spatial scales. It is worth to remark that such ex-perimental findings are supported by theoretical predic-tions based on a simplified uniaxial model of viscoelasticfluid dynamics in the absence of walls and in homoge-neous isotropic conditions [23]. At the same time, it wasrecently pointed out in [24] that numerical simulationsbased on standard constitutive models may be dramat-ically affected by the polymer-stress diffusivity typicallyadded to the evolution equations to ensure numerical sta-bility, and that this particularly applies to flows charac-terised by regions of pure strain. Notably, using a cellularforcing in two dimensions, it was shown that kinetic en-ergy spectra are considerably flatter in the absence ofartificial polymeric diffusion and scale as k − . [24].The elasticity-driven transition to turbulent-like stateswas experimentally investigated in cross-slot devices ofdifferent aspect ratio (vertical size over channel width),for more and less concentrated polymer solutions [25].Independently of the aspect ratio, it was found that themore concentrated solution undergoes a transition to un-steady flows that become progressively more irregularwhen the Weissenberg number is increased. The powerspectra of velocity fluctuations, obtained from single-point time series of the streamwise component measuredin the outlet channel at midway from the lateral walls(both in the horizontal and vertical directions), werecharacterised by the presence of marked peaks (a funda-mental frequency plus some harmonics), and by a power-law behaviour of exponent smaller than −
3, at smalland large
W i values, respectively. In particular, for thesmaller aspect ratio, continuous spectra and features typ-ical of elastic turbulence were observed when
W i (cid:38)
W i did not show similar spectral proper- a r X i v : . [ c ond - m a t . s o f t ] F e b ties.In this letter we explore the unsteady viscoelastic flowregime occurring in a 2D cross-slot geometry at high elas-ticities and vanishing Reynolds number ( Re ) by meansof extensive numerical simulations, for different polymerconcentrations. For this purpose we adopt Oldroyd-Bmodel, i.e. the simplest possible one, to describe thedynamics of the viscoelastic fluid. As in [26], whereelastic turbulence was simulated in a 2D Taylor-Couettesystem, we integrate the model evolution equations us-ing the open-source code OpenFOAM ® [27, 28], whichallows control of the numerical instabilities associatedwith large Weissenberg numbers [29]. We provide nu-merical evidence of the emergence of turbulent-like fea-tures for quite concentrated solutions when W i is largeenough. We analyse the transition to irregular dynamicsand we characterise the statistical properties of the high-
W i flows, discussing the similarities and differences withexperimental results.
MODEL AND METHODS
We consider an isothermal, incompressible, inertialess,2D viscoelastic fluid flow in a cross-slot geometry. Thelatter consists of two perpendicular and bisecting chan-nels of identical width d , with opposing inlets (here,along the x direction) and outlets (along the y direc-tion), as shown schematically in fig. 1. The velocity field u ( x , t ) = ( u x ( x , t ) , u y ( x , t )) at position x and time t evolves according to the momentum conservation equa-tion ρ (cid:20) ∂ u ∂t + ( u · ∇ ) u (cid:21) = ∇ · T − ∇ p (1)and the incompressibility condition ∇ · u = 0, where T is the total (viscous plus elastic) stress tensor, p thepressure and ρ the density.In the framework of Oldroyd-B model [30, 31], thestress tensor T is the sum of a viscous component σ = η s ˙ γ , with η s the zero-shear dynamic viscosity of the sol-vent and ˙ γ = ∇ u +( ∇ u ) T the strain-rate tensor, and anelastic one τ due to polymers. The constitutive equationfor the extra-stress tensor τ reads: τ + λ (cid:20) ∂ τ ∂t + ∇ · ( uτ ) − ( ∇ u ) T · τ − τ · ∇ u (cid:21) = η p ˙ γ , (2)where λ represents the largest polymer relaxation timeand η p the polymer contribution to viscosity. An impor-tant parameter is the viscosity ratio β = η s / ( η s + η p ),which is inversely proportional to the polymer concen-tration. Let us remark that in the limit β →
0, one re-covers the upper-convected Maxwell (UCM) model [31],accounting for the dynamics of very concentrated solu-tions. At fixed β , the control parameters of the dy-namics specified by eqs. (1) and (2) are the Reynolds d inletinlet outletoutlet xy d 12 FIG. 1. (Colour online) Schematic of the cross-slot geome-try. The dotted square is the area where the analyses wereconducted, with the two dots indicating the positions wheretime series were recorded: probe 1 (entrance, red), probe 2(exit, blue). Inset: zoom of the central area and typical meshrefining towards the centre of the setup; note that simulationswere performed with at least twice finer meshes. Re = ρU b d/ ( η s + η p ) and Weissenberg W i = λU b /d num-bers, where U b is the (uniform) velocity at the inlet.In spite of important limitations, such as the infiniteextensibility of polymers - and the consequent unboundednature of extensional viscosity at strain rates ≥ / (2 λ )- or the absence of shear-dependent viscosity, Oldroyd-Bmodel corresponds to the simplest differential constitu-tive equation for viscoelastic fluids, and it exhibits nor-mal stress differences. Furthermore, it has been success-fully employed to numerically reproduce the basic phe-nomenology of elastic turbulence in different 2D config-urations [7, 26, 32]. Numerical simulations
Equations (1) and (2) are integrated by means of theopen-source numerical solver rheoTool ® , which wasdeveloped in the framework of the OpenFOAM ® simula-tion code [28]. This solver is based on a finite-volume dis-cretisation and makes use of the log-conformation tech-nique [33] to control the numerical instabilities appearingat high W i values. We remark that no polymer-stressdiffusion is included.The cross-slot configuration has recently been pro-posed as a benchmark problem [34], for its geometricalcharacteristics and the existence of the instability lead-ing to asymmetric flow at appropriate β , Re and W i values. Similarly to the reference studies with this setup,here we set a length to width ratio of 10 : 1 for eachof the four “arms”, which was previously shown to beenough to ensure a fully developed flow away from theinlet in a channel [35]. The global mesh adopted for thenumerical integration is composed of four blocks, each ofwhich corresponds to an arm, with increased density ofgrid points when approaching the centre of the system,plus a central square block with the smallest (uniform)grid size. The results presented in the following wereobtained with a total of 12801 computational cells, cor-responding to 51 ×
51 cells and a minimal grid spacing∆ x min = ∆ y min ≈ . d in the central region. The meshrefinement towards the centre in each arm is realised viaa geometric progression relation with a stretching factor f s = 0 . U b is applied atboth inlets, where a homogeneous Neumann (zero gradi-ent) boundary condition is specified for the pressure field,whereas polymeric extra-stresses are set to zero. At theoutlets, a homogeneous Dirichlet (zero value) boundarycondition is imposed for pressure, as well as zero-gradientones for velocity and extra-stress fields. At the walls, no-slip conditions ( u = 0) are applied to the velocity fieldand a linear extrapolation technique is adopted for theextra-stresses [28]. The velocity and stress initial condi-tion corresponds to no flow.The Weissenberg number was varied by changing thepolymer relaxation time λ only; the polymer concentra-tion was set by choosing η s and η p such that their sumis constant. The Reynolds number, accounting for therelative strength of the non-linear inertial term to theviscous one in eq. (1), was kept fixed at Re = 0 by ne-glecting the term ( u · ∇ ) u in eq. (1) [17], but we checkedthat including the latter (and setting Re = 0 .
1) did notstrongly affect the results on the instability critical pa-rameters. Further, the dynamics appear not to be verysensitive to the presence of the term ρ∂ t u in eq. (1). RESULTS
When increasing the elasticity of the solution, whilekeeping β fixed, in our numerical integrations, we observea destabilisation of the flow, in agreement with previousstudies [11, 16]. The sequence of flow states that areselected depends on the polymer concentration, however,and here we provide a full picture of the stability portraitof the system as a function of both β and W i . Let uspreliminarily remark that below the onset of purely elas-tic instabilities the flow coming from each of the inletssplits into two streams of equal flow rate, a symmetricstate, at the outlets (see fig. 2a). For concentrated so-lutions ( β (cid:46) . β = 1 / (a) Wi = 0 .
35 (b) Wi = 1 . Wi = 10 (d) Wi = 20 FIG. 2. (Colour online) Snapshots of the magnitude of thevelocity field (colour) and flow streamlines (black lines) for β = 1 / Re = 0. Increasing W i , different regimes areobserved: steady symmetric (a), steady asymmetric (b), un-steady disordered flow (c, d). the excess flow rate in a stream, as a function of
W i we verified (results not shown) that this transition isa supercritical pitchfork bifurcation. Our values of thecritical Weissenberg number are in good agreement withthose reported in previous benchmark studies [34] (rela-tive difference of less than 0 .
05) both for β = 1 / β = 0. In this range of low β values, a sec-ond instability manifests when W i is further increasedbeyond a second threshold value close to 1, leading totime-dependent behaviour in the form of regular oscilla-tions of the asymmetric flow pattern (which stays similarto that of fig. 2b). The situation changes for more di-luted solutions ( i.e. when β (cid:38) . β , an increase of W i eventually gives rise tospatially and temporally more complex flows akin to elas-tic turbulence ones. Two illustrative examples at fixedtime are shown in fig. 2c,d for β = 1 / W i .The complete stability portrait, obtained by spanningthe ( β, W i ) plane with a large number of simulations,is shown in fig. 3, where the different point types corre-spond to the different dynamical regimes observed; herewe only show a limited subset of the results from thesimulations performed. By measuring the amplitude andfrequency of the time series of | u ( x (2) ∗ , t ) | at the fixed lo-cation x (2) ∗ (corresponding to point 2 in fig. 1) for W i close to the onset of the unsteady regime and for differ-ent concentration values, we could assess that the secondinstability is a supercritical Hopf bifurcation (see insetof fig. 3 for β = 1 / W i (cid:1) A m p li t ude F r equen cy Wi FIG. 3. (Colour online) Stability diagram in the ( β, W i ) planeat Re = 0. The green squares, blue diamonds and red dotsrespectively correspond to steady symmetric, steady asym-metric and unsteady flow. The dashed ( W i (I) c ) and contin-uous ( W i (II) c ) lines are fits using eq. (3); here a (I)0 (cid:39) . a (I) − (cid:39) − . a (II)0 (cid:39) . a (II) − (cid:39) .
05. Inset: amplitude andfrequency of | u ( x (2) ∗ , t ) | vs W i at the onset of unsteady flow,for β = 1 / FENE-P model at non-zero Re and large β . Indeed, thevelocity signal displays a growth of its amplitude thatis fairly well described by ( W i − W i (II) c ) / , with W i (II) c the critical Weissenberg number, and an approximatelylinear decrease of its frequency with W i . For both thefirst and the second instability, the critical Weissenbergnumber,
W i (I) c and W i (II) c respectively, grows with grow-ing β , which is reasonable since increasing β correspondsto decreasing polymer concentration. The faster growthof W i (I) c ( β ) causes the shrinking of the region of steadyasymmetric flow. Determining the functional dependen-cies W i ( i ) c ( β ) (with i = I , II) from stability analysis is notan easy task, due to the formation of a birefringent strandand a diverging base state associated with the infinite ex-tensibility of polymers [14]. Since here we are mainly in-terested in characterising the boundaries, in the ( β, W i )plane, of the regions where elastic turbulence could beexcited, we proceed heuristically, especially focusing on
W i (II) c ( β ). In order to account for non-zero β effects, weconjecture that W i (II) c ( β ) = W i (II) c (0) f ( β ), where f ( β )is a positive analytic function, except for β → f (0) = 1. Our numerical results suggest that the dataare compatible with a Laurent expansion at second orderaround the point β = 1. Somehow more surprisingly, wefind that the same functional shape can also be used tofit the W i (I) c ( β ) data, indicating that: W i ( i ) c = W i ( i ) c (0) (cid:34) a ( i )0 + a ( i ) − − β + a ( i ) − (1 − β ) (cid:35) , (3)where a ( i ) − = 1 − a ( i )0 − a ( i ) − using the constraint f (0) = 1, and i = I , II. In fig. 3 we report a comparison between afit with function (3) (dashed and continuous lines for i =I , II, respectively) and the numerical data; the agreementis rather good for both instability types, confirming ourconjecture.To conclude this discussion, we mention that in ourcalculations with a more refined grid or at Re = 0 . β and W i .We now consider the transition to turbulent-like flow.In the following we will present the results of the anal-ysis performed for increasing
W i at β = 1 /
9. Notwith-standing some quantitative differences, the phenomenol-ogy holds similar in the whole range ( β (cid:46) .
56) of con-centrated solutions, including for UCM ( β = 0). In thecase of more diluted solutions, while we observed somehints of the onset of irregular flow, we could not reacha fully developed regime and we cannot conclude aboutthe emergence of elastic turbulence. Notice that for suchlarge values of β , the critical Weissenberg number W i (II) c grows very rapidly, making the simulations more andmore delicate. t/λ −2.0−0.51.02.54.0 u y / (cid:1) u y (cid:0) t FIG. 4. (Colour online) Temporal evolution (subset of thetotal data set, see text) of the y -component of velocity atthe outlet (probe 2), normalised by its time average over thewhole time series, after the transient, for W i = 1 . , , , Re = 0 and β = 1 / Our analysis is based on the measurement of time se-ries of the velocity components at two different positionsmarked as probe 1 ( x (1) ∗ , entrance) and probe 2 ( x (2) ∗ ,exit) (see fig. 1), over long durations corresponding toat least 800 λ , and up to 1000 λ . As for the experimentsreported in [25], we choose to focus on the axial com-ponent u y ( x (2) ∗ , t ) at the exit probe, whose behaviour ispresented in fig. 4 for several values of W i . Remark thatin this figure the initial transient was removed and onlya subset of the data record is shown.The spectra of u y ( x (2) ∗ , t ) are shown in fig. 5. All thosecorresponding to the developed regime are averages overten spectra computed from consecutive subintervals of -1 f λ -4 -3 -2 -1 E y ( f ) / E t o t y Wi = 25Wi = 20Wi = 12Wi = 6 f λ E y ( f ) / E t o t y Wi = 3Wi = 1.55
FIG. 5. (Colour online) Temporal spectra of fluctuations ofthe axial velocity at the outlet u y ( x (2) ∗ , t ), normalised by theirintegral E toty in the elastic turbulence regime for Re = 0 and β = 1 /
9; the curves have been vertically shifted to ease read-ability. The dashed black curves stand for E y ( f ) ∼ f − δ ,the fitted values of δ are δ (cid:39) (2 . , . , . , . ± . W i = 6 , , ,
25, respectively. Inset: similar spectra at lowerelasticity. For
W i = 1 . (cid:38) W i (II) c , a single frequency peak isfound; at larger W i = 3 more discrete frequencies are present. the velocity time series obtained for a given value of
W i (after the transient). For
W i (cid:38)
W i (II) c , time dependencymanifests in the form of regular oscillations with a singlefrequency close to 0 . /λ (see inset of fig. 5). At slightlyhigher Weissenberg number ( W i = 3 in fig. 4) the flow isstill periodic but it is now characterised by more discretefrequencies; correspondingly, the spectrum shows severaldistinct peaks associated with a fundamental frequencyand some harmonics (inset of fig. 5). The occurrence ofa transitional periodic regime was also found in differentsetups [36, 37]. Above
W i ≈
5, the flow loses periodic-ity and the velocity spectra become continuous. Indeed,starting from 5 (cid:46)
W i (cid:46)
10 they result to be quite well de-scribed by a power-law function (fig. 5). When elasticityis increased in the range
W i >
10, the faster fluctuatingbehaviour of the flow is accompanied by quite wide andirregular oscillations, over longer durations. The flownow loses its spatial asymmetry to alternatively selectthe outlet in the positive/negative y -direction. Such aphenomenon has a strong impact on the statistics of thetransversal velocity component u x ( x (2) ∗ , t ) at the outlet(and similarly on u y ( x (1) ∗ , t ) at the inlet), whose fluctu-ations are accompanied by irregular jumps between twomean values of opposite sign (see fig. 6), thus compli-cating their analysis. A detailed investigation of the be-haviour of such a two-state system goes beyond the scopeof the present work.In the turbulent-like regime ( W i > E y ( f ) ∼ f − δ beyond a frequency that, as in experimen-tal studies [25], slightly increases with W i . The absolutevalue of the exponent is found to be in the range 2 (cid:46) δ (cid:46) W i ; −12−60612−40−2002040−80−4004080100 200 300 400 500 600 700 t/λ −40−2002040 u x / (cid:1) u x (cid:0) t FIG. 6. (Colour online) Temporal evolution (subset of thetotal record) of u x ( x (2) ∗ , t ), normalised by its time averageover the whole time series, after the transient, for W i =11 , , ,
20 (from top to bottom), Re = 0 and β = 1 / the latter feature is also detected in experiments [25, 38].In particular, we find δ (cid:39) (2 . , . , . , . ± . W i = 6 , , ,
25, respectively. The spectra are thusoverall less steep than those previously found in exper-iments [6, 25] and those theoretically predicted assum-ing homogeneity and isotropy [23], pointing to more en-ergetic small scales, as e.g. the quite localised ones(fig. 2c,d) stemming from intense polymer stretching, andless smooth flow. However, they bear an interesting sim-ilarity with those obtained in 2D numerical simulations,without artificial polymer-stress diffusion, of Oldroyd-Bmodel in the presence of a cellular forcing generating dis-tinct regions of strain and vorticity [24]. A possible rea-son for the difference with the prediction of [23] is the lackof the statistical symmetries assumed by the theory in thepresent case. Indeed, our flow is neither homogeneous(due to the presence of the walls, but also of the high-strain region close to the centre of the setup), nor fullyisotropic, as we typically observe that u rmsy > u rmsx forthe root-mean-square (rms) velocity components. More-over, the turbulent intensity u rms /u , here defined as theratio of the rms to the mean value of the full velocitymodulus u ≡ | u | (with the overbar denoting a temporalaverage), can quite easily exceed 0 .
5, and be as high as ≈ . δ for W i ≥
25 shouldbe taken with caution, as they may also likely depend onthe length of the inlet/outlet channels.To further characterise the statistical properties of −4 −3 −2 −1 0 1 2 3 4 (u y − ¯u y )/σ u y -4 -3 -2 -1 P (cid:0) ( u y − ¯ u y ) / σ u y (cid:1) (a) Wi = 6Wi = 12Wi = 15Wi = 20Wi = 25 −4 −3 −2 −1 0 1 2 3 4 (u x − ¯u x )/σ u x -4 -3 -2 -1 P (cid:0) ( u x − ¯ u x ) / σ u x (cid:1) Wi = 11Wi = 12Wi = 15Wi = 20 −6 −4 −2 0 2 4 6 w y10 -4 -3 -2 -1 P [ w y ] (b) Wi = 6Wi = 12Wi = 15Wi = 20Wi = 25 −6 −4 −2 0 2 4 6 w x -4 -3 -2 -1 P [ w x ] FIG. 7. (Colour online) Probability density functions of nor-malised velocity fluctuations u (cid:48) y = ( u y − u y ) /σ u y (a) andtemporal increments w y = ( ∂ t u y − ∂ t u y ) /σ ∂ t u y (b), where u y ≡ u y ( x (2) ∗ , t ), the overbar denotes the temporal averageand σ the standard deviation, for different values of W i , Re = 0 and β = 1 /
9. The insets show the pdf’s of thesame quantities along x − direction (velocity and temporal-increment fluctuations, in (a) and (b), respectively). In allpanels the solid black lines are standard Gaussian pdf’s. our elastic turbulent flows, we computed the probabil-ity density functions (pdf’s) of the fluctuations of thevelocities u x,y ( x (2) ∗ , t ), as well as of the local accelera-tions ∂ t u x,y ( x (2) ∗ , t ), obtained from the temporal signalsat probe 2. The results are presented in fig. 7, whereall variables are rescaled with the corresponding stan-dard deviation σ . The statistics of u y fluctuations arenot far from Gaussian, but negatively skewed for largeenough W i (fig. 7a), probably due to the establishmentof a transversal flow component via intermittent bursts[42]. Those of u x are less so (inset of fig. 7a), instead, andshow a bimodal shape for 10 (cid:46) W i (cid:46)
20, which reflectsthe importance of the flow-asymmetry alternation eventsin this range of elasticities. A qualitatively similar phe-nomenology is found at the entrance probe 1, provided x and y indices are exchanged. The statistics of fluctua-tions of the accelerations are remarkably less dependenton the Weissenberg number (and the probe location),suggesting a faster (with W i ) onset of scaling propertiesat small scales. As it can be seen in fig. 7b, the corre-sponding pdf’s display high tails that are indicative of non-Gaussian statistics, as is typical in turbulent flowsand as observed in elastic turbulence experiments [42].This finding highlights the intermittent behaviour of localaccelerations, likely due to the passage through the sys-tem of transient intense filamentary structures (fig. 2c,d,but see also [37, 43, 44] about the role of elastic propa-gating wavy patterns).
CONCLUSIONS
We investigated numerically the dynamics of Oldroyd-B fluids in a 2D cross-slot geometry for broad ranges ofthe Weissenberg number and the polymer concentration,focusing on the possibility to obtain elastic turbulence.We detected two instabilities: the first one, present onlyfor rather concentrated solutions (see also [12]), leadsto steady asymmetric flow; the second one, less docu-mented, manifests for all viscosity ratios β <
W i (II) c on the viscosity ratio, we found a heuristicexpression that allows to quantitatively delimit the re-gions W i > W i (II) c ( β ) where elastic turbulence may beexcited.Close to the onset of the second instability, the flow ofquite concentrated solutions displays regular oscillationsin time, while at larger elasticities its dynamics appearmore irregular. The frequency spectra measured in one ofthe outlets and far from the walls show distinct peaks for W i (cid:38)
W i (II) c , while for W i (cid:38) − δ , point-ing to elastic turbulence. As in experiments [25, 38], thescaling range occurs beyond a frequency that moderatelyincreases with W i , and δ decreases with W i . However,we obtain values 2 < δ <
3, somehow smaller than the ex-perimental ones and the theoretical prediction for the ho-mogeneous isotropic case [23]. While we cannot excludean impact of the 2D nature of our flow here, and we recallthe influence of the inlets/outlets’ length on the resultsfor
W i ≥
25, we remark that the symmetries assumed inthe theory clearly do not hold for our setup. Similarly en-ergetic spectra have been recently found in simulationsof 2D Oldroyd-B cellular flows without polymer-stressdiffusion [24].Further, the statistics of axial velocity componentsare found to be weakly non-Gaussian in the developedregime, while those of transversal ones also exhibit a bi-modal pdf for 10 < W i <
20 due to the alternationsof the spatial flow asymmetry occurring in this range of
W i . The pdf’s of both components of the local accelera-tions, instead, present high non-Gaussian tails indicativeof intermittency. Such a phenomenology agrees with thatobserved in experiments (see e.g. [42]).In summary, we reproduced the different dynamicalregimes experimentally observed in cross-slot devices,and we obtained turbulent-like states bearing good sta-tistical resemblance with elastic turbulence. The quanti-tative differences highlighted call for further theoreticaland numerical developments. In the future it would beinteresting to explore such dynamics in three-dimensionalflows.
ACKNOWLEDGMENTS
D. O. Canossi acknowledges financial support froma PhD grant funded by the Brazilian agency CNPq(
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