epl draft

Lattice Boltzmann spray-like fluids

G. Falcucci1, S. Chibbaro2,3 S. Succi2, X. Shan4 and H. Chen4

1 Dept. of Mechanical and Industrial Engineering, University of “Roma Tre”, Via della Vasca Navale 79, Rome, Italy2 Istituto Applicazioni Calcolo “Mauro Picone”, CNR, V.le del Policlinico 137, Rome, Italy3 Dept. of Mechanical Engineering, University of “Tor Vergata”, Viale Politecnico, Rome, Italy4 EXA Corporation, 450 Bedford Street, Lexington, Massachussetts 02173

PACS 47.11.-j – First pacs descriptionPACS 47.55.Ca – Second pacs descriptionPACS 47.55.dk – Third pacs description

Abstract. - The effects of the competition between short-range attraction and mid-range repulsionin lattice Boltzmann models of single-component non-ideal fluids is investigated. It is shown thatthe presence of repulsive interactions gives rise to long-lived metastable states in the form of multi-droplet spray-like density configurations, whose size can be adjusted by fine-tuning the strength ofthe repulsive versus attractive coupling. This opens up the possibility of using single-componentlattice kinetic models to study a new class of complex flow applications, involving atomization,spray formation, micro-emulsions and possibly, glassy-like phenomena as well.

The dynamics of multi-phase flows with complex trans-lational/rotational order is a rich branch of modern sci-ence, lying at the interface between fluid dynamics andcondensed matter, with many applications in physics,chemistry, engineering and life-sciences [1, 2]. Althoughphase-transitions from isotropic, homogeneous fluids toanisotropic/ordered ones can be traced to a large varietyof underlying microscopic conditions, a common featureof many microscopic scenarios is the competition betweenshort-range attraction and mid/long-range repulsion [3].Short-range attraction typically drives phase-separationthrough dynamical instabilities towards density perturba-tions in the fluid, whereas mid/long-range repulsion frus-trates this tendency, thus giving rise to a variety of com-plex symmetry-breaking structures within the flow. Theanalytical study of these phenomena is obviously a difficultone, mainly due to the need of accounting for the simulta-neous interaction of many disparate scales. Consequently,the investigation of multiphase flows, both classical andquantum, is in a constant need of flexible and efficient com-putational methods. The task of simulating the behaviourof multiphase flows is however a very challenging one, dueto the emergence of moving interfaces with complex topol-ogy [7]. In the last decade, a new class of mesoscopicmethods, based on minimal lattice formulations of Boltz-mann’s kinetic equation, have captured significant inter-est as an efficient and flexible alternative to continuum

methods based on the discretization of the Navier-Stokesequations for non-ideal fluids [8–12]. To date, a very popu-lar mesoscopic approach is the so-called pseudo-potential-Lattice-Boltzmann (LB) method, developed over a decadeago by Shan and Chen (SC) [13]. This method hasgained increasing popularity on account of its simplicityand computational efficiency. In the SC method, poten-tial energy interactions are represented through a density-dependent mean-field pseudo-potential, ψ(ρ), and phaseseparation is achieved by imposing a short-range attrac-tion between the light and dense phases. To date, the SCpseudo-potential has only been used in its original ver-sion, where only nearest-neighbour attractive interactionsare included. This implies that, in the absence of exter-nal constraints, the time-asymptotic state of the phase-separated fluid corresponds to a single droplet configura-tion. Very recently, however, it has been pointed out thatthe inclusion of mid-ranged potentials, extending beyondnearest-neighbour interactions, may open up new perspec-tives and applications of the SC model. In particular, ithas been conjectured that mid-range repulsion should beable to interrupt the phase-separation induced by short-range attraction, thereby providing a physical mechanismpromoting the onset of metastable states in the form oflong-lived droplets of different size [14]. This idea hasbeen pursued before in many endeavors [4–6], includinglattice-gas cellular automata [15]. However, to the best of

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G. Falcucci1 S. Chibbaro2,3 S. Succi 2 X. Shan 4 H. Chen4

our knowledge, this is the first time that the idea is ap-plied to the Lattice-Boltzmann framework. In this Letter,we provide numerical evidence that a mesoscopic latticeBoltzmann model with mid-range repulsion is indeed ca-pable of sustaining long-lived metastable states in the formof multi-droplet spray-like configurations, whose size andnumber can be controlled by tuning the strength of repul-sive interactions. This opens up the possibility of deploy-ing the computational power of lattice kinetic methods tostudy a variety of very complex fluids flows such sprays,emulsions and glassy-like materials, which are hard to sim-ulate with the standard Shan-Chen model. The kineticlattice Boltzmann equation writes as follows [16, 17]:

fi(~r + ~ci∆t, t+ ∆t) − fi(~r, t) = (1)

−1

τ[fi(~r, t) − f

(eq)i (~r, t)]∆t+ Fi∆t

where fi is the probability of finding a particle at site~r at time t, moving along the ith lattice direction de-fined by the discrete speeds ~ci with i = 1, ..., b. Theleft hand-side of (2) stands for molecular free-streaming,whereas the right-hand side represents the time relax-ation (due to collisions) towards local Maxwellian equi-librium. Finally, Fi represents the total volumetric bodyforce. In particular, we shall use a dynamic mean-field

term connected with bulk particle-particle interactions.The pseudo-potential force consists of two separate com-ponents ~F (~r, t) = ~F1(~r, t) + ~F2(~r, t), defined as follows:

~F1(~r, t) = G1ψ(~r; t)

b1∑

i=0

wiψ(~r1i, t)~c1i∆t

~F2(~r, t) = G2ψ(~r; t)

b1∑

i=0

p1iψ(~r1i, t)~c2i∆t (2)

+ G2ψ(~r; t)

b2∑

i=0

p2iψ(~r2i, t)~c2i∆t

In the above, the indices k = 1, 2 refer to the first andsecond Brillouin zones in the lattice (belts, for simplicity),and ~cki, pki are the corresponding discrete speeds and as-sociated weights (see Figure 1). Finally ~rki ≡ ~r + ~cki∆tare the displacements along the i-the direction in the k-thbelt. Note that the second force ~F2 acts on both belts,so that the short (S) and mid-range (M) components of

the force read as follows ~FS(~r, t) = ~F1(~r, t) + ~F21(~r, t),

and ~FM (~r, t) = ~F22(~r, t), where subscripts 21 and 22 in-

dicate the component of ~F2 acting on the first and sec-ond belts respectively. Note that positive(negative) G cor-respond to repulsion(attraction) respectively and that Gis a measure of potential to thermal energy ratio. Thefirst belt is discretized with 9 speeds (b1 = 8), whilethe second with 16 (b2 = 16) and the weights are cho-sen in such a way as to fulfil the following relations

[18]:∑b1

i=0 wi =∑b1

i=0 pi1 +∑b2

i=0 pi2 = 1;∑b1

i=0 wic2i =

∑b1i=0 pi1c

2i1 +

∑b2i=0 pi1c

2i2 = c2s, c

2s = 1/3 being the lat-

tice sound speed. The numerical values of the weights aregiven in Table .

i |ci|2 wi pi1 pi2

1 − 4 1 1/9 4/635 − 8 2 1/36 4/1359 − 12 4 1/18013 − 20 5 2/94521 − 24 8 1/15120

Table 1: Links and weights of the two-belt, 24-speed lattice.

The pseudo-potential ψ(~r) is taken in the form first sug-gested by Shan and Chen [13], ψ(ρ) =

√ρ0(1 − e−ρ/ρ0)

where ρ0 marks the density value at which non ideal-effects come into play and it is fixed to ρo = 1 in lat-tice units. Taylor expansion of (2) to second-order de-livers the following non-ideal equation of state (EOS)

p ≡ P/c2s = ρ+ (g1+g2)2 ψ2(~r, t), where gk ≡ Gk/c

2s are nor-

malised coupling strengths. Further expansion to fourth-order provides the following expressions for the surfacetension

γ = − (G1 + ξG2)

2c4s

∫ ∞

∞

|∂yψ|2dy (3)

where y runs across the phase interface. Note that thedouble potential allows to change the equation of stateand the surface tension independently. In particular, sinceξ = 12/7 > 1 the effect of the repulsive belt is to lower thesurface tension at a given value ofG1+G2. Thus, the mid-range potential is expected to act as a “surfactant” [1].

We have simulated droplet formation by integratingthe LBE Eq. (2) in a 2D lattice using the nine-speed2DQ9 model [12, 17], out of a noisy density backgroundwith initial density ρin = ρ0ln2 + δρ in a periodic do-main. The force, however, is evaluated on a larger 24-speed stencil (see Fig. 1). In all simulations, τ = 1 andGeff = G1 + G2 = −4.9. This choice yields a surface

tension γ ≈ 0.5 and a density ratio ∆ρρ ≈ 10 between liq-

uid and vapour phases, respectively. In this configuration,we have studied the changes in the qualitative behaviourof the droplet formation as a function of the strength ofthe repulsive force. We have performed a systematic scanover the force strength, while keeping Geff = −4.9 in the(G1, G2) plane, with a standard resolution of 1282 gridpoints, and with final simulation time t = 50000. Bydefining a characteristic time-scale tcap = Hµ

γ , µ being thedynamic viscosity of the fluid and H the dimension of thedomain, the final time corresponds to about 50tcap. Ithas been checked that the the same qualitative behaviouris reproduced at higher resolution, 2562 and 5122 latticesites. In analogy with droplet theory and kinetic Isingsimulations [19,20], we identify the following four regions:as follows; i) |Geff | ≡ |G1 +G2| < |Gc| = 4: homogeneous

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Lattice Boltzmann spray-like fluids

fluid (zero droplets); ii) |Geff | > |Gc|, G2 = 0: non-homogeneous fluid, one droplet (standard Shan-Chen);iii) |Geff | > |Gc|, 0 < G2 < G2c: non-homogeneousfluid (several metastable droplets); iv) |Geff | > |Gc|,G2 > G2c: quasi-ordered fluid (many stable droplets,”emulsion”);

An example of fluid behaviour in the metastable regioniii) is given in Fig. 2, corresponding to G2 = 10. From thisfigure, we observe that as many as 30 droplets are formedat time t = 10000 ∼ 10tcap, while at time t = 30000 andt = 50000, this number falls down to 11 and 7, respec-tively. A typical fluid configuration in the stable regioniv) is given in Figure 3, corresponding to G2 = 13 andt = 50000. From this figure, nucleation of quasi-orderedliquid droplets is apparent. It is interesting to observe thatin this regime, the fourth order expression eq. (3) wouldyield an unphysical negative surface tension, indicatingthat higher order terms can no longer be neglected. Nu-merical exploration of this region shows however that themesoscopic model continues to provide physically plausi-ble results, with the surface tension smoothly decreasingby about a factor two with increasing G2. This is consis-tent with the physical behaviour of surfactants, where thesurface tension decreases by a factor of 2− 3 without evervanishing [1].

The physical scenario is summarised in figure 4, wherethe interfacial area is plotted as a function of time. Fromthis figure, we observe that the standard Shan-Chen fluidattains its single-droplet steady-state on a time scale ofa few ’capillary’ times, 5tcap ∼ 5000 time-steps. Directinspection (not displayed for space limitation) shows thatthe one-droplet configuration attains the lowest value ofthe pseudo-potential energy associated with the forces de-fined in eq. (2). By raising G2, multi-droplet configura-tions are excited, as clearly displayed by the plateaux inthe time evolution of the interfacial area A(t) at differentvalues of G2. Each plateau associates with a given numberof droplets, the transition between two different plateauxmarking the discrete change in the number of droplets.From Figure 4 it is apparent that, even though in thelimit t→ ∞ all configurations are expected to decay to thesingle-droplet ground state, the time it takes to reach thisasymptotic state becomes increasingly larger as G2 ap-proaches the critical value G2c. In fact, as G2 > G2c, thisrelaxation time is virtually infinite, corresponding to theattainment of a new stable phase, i.e. the quasi-orderedstate represented in Figure 3. Remarkably, the presentLB method permits to straddle across the entire region ofmetastability, connecting the one-droplet Shan-Chen stateall the way to the quasi-ordered state. It is instructive toinspect the number of droplets at a given time as a func-tion of the repulsive strength G2. Such information is dis-played in Figure 4b at times t = 5, 10tcap and t = 50tcap.From this figure, we see that the number of droplets as afunction of time is a monotonically non-decreasing func-tion of G2, with the smallest values of G2 showing a cleartendency to relax to the same asymptotic value n = 1.

This is consistent with the theoretical expectation of asingle-droplet steady-state in the long-time limit in theregion G2 < G2c.

Our numerical results are supported by the followingtheoretical considerations. In the single-droplet region,mass conservation delivers ρlVl + ρgVg = ρiV = M , whereindex i stands for initial conditions, Vl,g denotes the to-tal volume of the liquid/gas phase, and A is the interfa-cial area. The droplet radius, R, follows from a simple

rearrangement: πR2

H2 = ρi−ρl

ρl−ρg, where ρl,g can be com-

puted a priori by combining mechanical stability of theinterface (zero divergence of momentum-flux tensor), with

Maxwell’s area rule [21],∫ ρ−1

g

ρ−1

l

(p(ρ)−p0)dρ = 0, where p0 is

the constant bulk pressure. With our choice Geff = −4.9,ρi = ρ0ln2 = ln2, we obtain ρg ≈ 0.17 and ρl ≈ 1.85,so that the liquid area covers approximately one third ofthe simulation box. This is confirmed by numerical ex-periments, which yield ρg = 0.172 and ρl = 1.865. Thecomputed density predicts R ≈ 40, in good agreementwith the numerical value provided by the simulations. Inour simulations, R = 40, Laplace’s law R = γ

∆p yieldsγ = 0.0457, in good match with the value γ ≈ 0.04 ob-tained by numerically integrating the Eq.(3).

With two parameters at our disposal, G1 and G2, thepresent model allows a separate control of the equation ofstate and surface tension, respectively. In particular, thenon-ideal part of the equation of state depends only onA1 = G1 + G2, whereas surface tension effects are con-trolled by the combination G1 + 12

7 G2. Since in the vicin-ity of γ → 0 higher order terms come into play, it provesexpedient to define a new coefficient

A2 = G1 + λG2 (4)

where the numerical factor λ plays the role of a renor-malisation parameter, whose departure from zeroth-ordervalue 12

7 is a measure of the influence of the higher-orderterms. We found that λ ≈ 3/2 provides satisfactory agree-ment with the numerical results. Given this renormalisa-tion, the numerical value of G2c can be estimated by not-ing that, to fourth-order, the total force, ~Ftot = ~F1 + ~F2,in the present LB model is given by:

~Ftot = −(

c2sA1ψ~∇ψ +A2c

4s

2ψ~∇∆ψ

)

. (5)

A dimensional argument 1, gives l2 =c2

s

2A2

A1

= 16

A2

A1

, thisyields

l ∼ 1√6

√

A2

A1→ l

l1=

√

1 − G2

2|Geff ||(6)

where l is the typical size of a nucleus and l1 is the typicalsingle-droplet dimension. The resulting spinodal value, at

1Strictly speaking, this estimate is only valid for droplets with

radius significantly larger than the lattice spacing, ∆x/l ≪ 1.

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G. Falcucci1 and S. Chibbaro2,3 S. Succi 2 and X. Shan 4 and H. Chen4

which l → 0, turns out to be G2c = 9.8 . For this value,the second term in Eq. (5) diverges, thus signaling a phasetransition. This value is found to be in good agreementwith the numerical simulations, which indicate completenucleation starting around a value of G2 ≈ 10, as shownin fig. 4.

The spinodal point can also be estimated by resortingto classical homogeneous nucleation theory [22]. On thebasis of this theory, the equilibrium cluster distributionfor a cluster of size n is given by:

pe(n) = pe(1) exp

[

−2πR(γn√n− γ1)

KT

]

(7)

where γn < γ1 is the value of the surface tension suchthat the equilibrium distribution would correspond to an-droplet configuration. The expression (7) builds on twoassumptions: i) the equilibrium cluster distribution hasthe form pe(n) ∼ exp [−W (n)/KT ] and ii) W (n) is equalto the product of the area of the cluster and the surfacetension of a flat liquid surface with the same tempera-ture and composition of the nucleus and (i.e. W (n) =γA(n)). Moreover, the volume of a droplet is given byπR2, while with n droplets, the same volume is πnR2

n,so that Rn ≈ R/

√n. Finally, all nuclei are assumed to

be spherical and with the same radius. Based on the ex-pression (7), the surface tension γ = γ1/

√n would vanish

in the limit n → ∞, corresponding to l → 0. By in-specting the expression (3), and taking into account thedefinition of Geff , γ = 0 implies G2c = 2|Geff |, con-sistently with the expression (6). Let us define G2(n) asthe value of G2 such that the n-droplet configuration is al-lowed to survive the longest before decaying into the n = 1ground state, i.e. G1 + 3

2Gn = 1/√n. By recalling that,

by definition, G1 + 32G2c = 0, the following scaling rela-

tion is readily obtained: n = (1− ξ)−2, where we have setξ ≡ G2/G2c. Interestingly, a dynamic extension of suchscaling law, namely n(t) = (1 − ξ)−p(t) provides a reason-able fit of the numerical data, with p(t) ∼ 2 at small t,and p(t) → ∞ (step-function) at t→ ∞ .Larger simulations (5122) confirm the basic picture de-scribed in this paper, namely a seamless transition froma homogeneous gas to a quasi-ordered configuration ofdroplets, whose size (number) is controlled by the relativestrength of repulsive and attractive interactions. As an ex-ample, in Figure 5 the spectrum of density fluctuations isshown in the initial and final stage of the evolution. Thetypical size of the droplets is identified with the inversewave-number 2π/kmax, where kmax is the wave-numberat which the peak of the spectrum is attained (k = 1 cor-responds to a droplet radius R = L). From this figurewe see that with G1 = −15 and G2 = 10.1, the peak isaround k ∼ 4, while the SC case gives k ∼ 1, correspond-ing to about 16 droplets in the same volume.

Summarising, we have shown that the inclusion ofa mid-range repulsive potential within the Shan-Chenformulation of non-ideal lattice fluids, discloses a very richphysical picture. In particular, it opens up the possibility

Fig. 1: Sketch of the discrete-velocity stencil used in the ex-tended Lattice Boltzmann scheme. The integers denote themagnitude of the corresponding discrete speed.

of realising multiphase flows with long-lived metastablestates characterised by multi-droplet configurations. Themodel allows the independent tuning of the size of thedroplet and of the density ratio between the liquid andthe gas phases. This should permit the efficient handlingof complex applications, such as micro-emulsions [23],multiphase sprays, globular protein crystallisation [24]and, possibly, glassy-like states as well.

SS wishes to acknowledge financial support from theproject INFLUS (NMP3-CT-2006-031980) and SC fromthe PI2S2 COMETA project also supported by a Marie-Curie UE grant. Fruitful discussions with M. Sbragaglia,R. Benzi, L. Biferale, I. Procaccia and F. Toschi are kindlyacknowledged.

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Lattice Boltzmann spray-like fluids

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 20

40 60

80 100

120 140 0

20 40

60 80

100 120

140

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

0 20 40 60 80 100 120 140 0

20

40

60

80

100

120

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2 2.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 20 40 60 80 100 120 140 0

20

40

60

80

100

120

140

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2

Fig. 2: Spatial distribution of the fluid density. The formationof a large number of droplets is well visible. Configurationparameters: Geff/Gc = −4.9, nx = ny = 128; 2a: Singledroplet (standard Shan-Chen), t = 50tcap, 2b: Multi-droplet,G2 = 10 , t = 10tcap. 2c: Multi-droplet, G2 = 10 , t = 50tcap.

[8] G. R. Mc Namara, G. Zanetti, Phys. Rev. Lett., 61, 2332,(1988).

[9] F. Higuera, S. Succi, R. Benzi, Europhys. Lett., 9, 345,(1989)

[10] S. Ansumali, and Iliya V. Karlin Phys. Rev. Lett. 95,260605 (2005).

[11] R. Benzi, S. Succi, M. Vergassola, Phys. Rep., 222, 145,(1992).

[12] S. Chen , and G.D. Doolen, Annual Rev. Fluid Mech., 30,329 (1998).

[13] X. Shan, and H. Chen Phys Rev E 47, 1815 (1993).[14] Falcucci G., Bella G., Chiatti , Chibbaro S., Sbragaglia

M., and Succi S., Communications in ComputationalPhysics, Vol. 2, No. 6, pp. 1071-1084.

[15] J.P. Boon, C. Bodenstein, D. Hanon J. Mod. Phys. C 9,1559 (1998).

[16] S. Succi Lattice Boltzmann equation for Fluid Dynamics

0

0.5

1

1.5

2

2.5

3

3.5

0 20 40 60 80 100 120 140 0

20

40

60

80

100

120

140

Fig. 3: Isocontours of the density distribution at time t =10000 ∼ 10tcap, Geff/Gc = −4.9, nx = ny = 128, G2 = 19.The configuration contains approximately 100 droplets. Notethat the droplet radius is about 128/15/2 ∼ 4 lattice units, stillsufficiently larger than the lattice thermal length ∆tcs ∼ 1/

√3.

This is usually sufficient to rule out serious non-hydrodynamiceffects, although this issue is surely worth a detailed futureinvestigation.

and beyond (Oxford U.P. 2001).[17] D.A. Wolf-Gladrow Lattice-gas Cellular Automata and

Lattice Boltzmann Models (Springer, Berlin, 2000).[18] M. Sbragaglia, R. Benzi, L. Biferale, S. Succi, K.

Sugiyama, and F. Toschi, Phys. Rev. E 75, 026702 (2007).[19] P. A. Rikvold, H. Tomita, S. Miyashita, and Scott W

Sides, Phys. Rev. E 49, 5080, (1994).[20] V.I. Kalikmanov , Statistical Physics of fluids (Springer,

Berlin, 2001).[21] R. Benzi, L. Biferale, M. Sbragaglia, S. Succi, and F.

Toschi, Phys. Rev. E 74, 021509 (2006).[22] J. L. Katz, Pure & Appl. Chem. 64, 1661 (1992).[23] M. Nekovee, P. V. Coveney, H. Chen, and B.M. Boghosian

Phys. Rev. E 62, 8282, (2000).[24] S. Brandon, P. Katsonis, and P.G. Vekilov, Phys. Rev. E

73, 061917, (2006).

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G. Falcucci1 S. Chibbaro2,3 S. Succi 2 X. Shan 4 H. Chen4

0 10000 20000 30000 40000 50000Time

0

1000

2000

3000

Surf

ace

Are

a

1 10G

2

1

10

100

n dr

ople

ts

Simul. t=5tcap

Simul. t=10tcap

Simul. t=50tcap

analytical t=0analytical t=5t

cap

analytical t=10tcap

analytical t=50tcap

Multi droplets

"emulsion"

Fig. 4: Fig 4a: Total interfacial area (perimeter in 2D)as a function of time at increasing values of G2 =0, 4.1, 6.1, 7.6, 9.1, 10, 13. The final time, t = 50000 time-steps,corresponds to 50tcap. The arrows indicate times at whichdensity snapshots are shown in the previous figures. Fig 4b:Number of droplets versus G2 at t = 5, 10, 50tcap. The verticalline denotes the theoretical spinodal point G2c. The single-droplet region is associated with G2 → 0. The analytical fitn(t) = (1 − ξ)−p(t), with p(t) = 2

1+t/10, is also shown.

0 4 8 12 16 20k

0

0,2

0,4

0,6

0,8

1

S(k)

/Sm

axShan-Chen t = 5e5G1 = -15 G2 = 10.1 t = 5e5t = 0

Fig. 5: Spectrum of density fluctuations, normalized to its peakvalue, for the standard Shan-Chen (dashed) and for the caseG1 = −15 and G2 = 10.1 (solid line) after 5 × 105 time-steps.The latter case shows a clear shift in the spectrum, peakedaround k = 4. This indicates a reduction of about a factor 4 indroplet size, corresponding to approximately 16 droplets in thecomputational box. The dotted line corresponds to spectrumat t = 0, with fluctuations of the order of 10−5, not visible onthe scale of the figure. Grid size 5122.

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