A Kolmogorov spectrum for strongly vibrating plates
Gustavo Düring, Christophe Josserand, Giorgio Krstulovic, Sergio Rica
IIs turbulence universal ? A Kolmogorov spectrum for strongly vibrating plates.
Gustavo D¨uring
Facultad de F´ısica, PUC, Chile.
Christophe Josserand
LadHyX, CNRS & Ecole Polytechnique, UMR 7646, 91128, Palaiseau, France.
Giorgio Krstulovic
Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur, CNRS,Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice cedex 4, France.
Sergio Rica
Facultad de Ingenier´ıa y Ciencias, Universidad Adolfo Ib´a˜nez, Santiago, Chile.
In fluid turbulence, energy is transferred from a scale to another by an energy cascade thatdepends only on the energy dissipation rate. It leads by dimensional arguments to the Kolmogorov1941 (K41) spectrum. Remarkably the normal modes of vibrations in elastic plates manifests anenergy cascade with the same K41 spectrum in the fully non-linear regime. Moreover, the elasticdeformations present large “eddies” together with a myriad of small “crumpling eddies”, such thatfolds, developable cones, and more complex stretching structures, in close analogy with spots, swirls,vortices and other structures in hydrodynamic turbulence. We characterize the energy cascade, thevalidity of the constant energy dissipation rate over the scales and the role of intermittency via thecorrelation functions.
Turbulence has remained a central problem in fluid dy-namics since the early experiments of Osborne Reynolds[1]. Perhaps the most salient feature is the setting ofan energy cascade that redistributes the energy amongdifferent Fourier modes of the velocity fluctuations thatare independent of the dissipation caused by viscosity atsmall scales. This energy cascade depends only on theenergy dissipation rate per unit mass, P and the wavenumber k leading to the Kolmogorov K41 [2] spectrum: E k ∼ P / k − / . Despite a century of effort, turbulencestill remains nowadays a major challenge from the exper-imental, theoretical and numerical points of view, andthe very essence of the phenomena has not yet been re-vealed satisfactorily [3–5]. For instance, if the K41 lawcan be deduced using dimensional arguments, the statis-tical description of the turbulence fails to describe cor-rectly the general correlation functions of the velocityfields. In fact, up to date, there is only a single analyt-ical result that maybe derived from the original Navier-Stokes (NS) equations for incompressible fluids, the so-called von K´arm´an-Howarth relation [6], that links thesecond and the third order correlations function of somecomponent of the fluid velocity with the energy dissipa-tion rate. Higher moments of these quantities differ frommean-field based predictions, and experimental as well asnumerical data reveal extreme events, exhibiting heavytailed distributions, that are usually associated with in-termittency . Moreover, one can argue that the difficultyof describing turbulence may come from the fundamen-tal issues raised by the fluid mechanics equations, wit-nessed in particular by the Clay millennium problem onthe regularity of the NS equation in three space dimen- sions. Similarly, fluid turbulence is a hard problem be-cause it is in the same footing that the Euler equationfor perfect fluids which is ab-initio a nonlinear partialdifferential equation without any small parameter thatmay justify an asymptotic scheme nor a rational closurefor the statistical correlations [7]. Since fluid turbulenceis studied in the framework of the NS equations, it isimportant to question whether such cascade dynamicstransporting the energy from the large injection scalesto the small dissipative ones can be observed in differentcontexts and whether the turbulence features describedin fluids are universal or dependent on the dynamicalequations. In particular, while the quadratic nonlinearterm of the NS equation is responsible of the mixing be-tween scales at the heart of the cascade process, noth-ing in the theoretical description of turbulence takes intoaccount the details of the nonlinearity itself. In fact,turbulence has been observed in different contexts suchas Magneto-Hydrodynamics where the magnetic field iscoupled with the flow or viscoelastic turbulence in poly-mer solutions at low Reynolds numbers for instance. Inthese cases even though the nonlinearity differs from theone of NS equations and different spectra can be mea-sured, they have similar properties than the original Kol-mogorov phenomenology.On the other hand, a different class of turbulence ex-ists when a linear term is present in the dynamics, theso-called wave turbulence. Wave propagation dominatesat linear order and the waves interact through the non-linear terms. The dynamics is described by the weakor wave turbulence theory (WTT), developed originallyin the sixties [8–11]. WTT is deduced in the limit of a r X i v : . [ n li n . C D ] A ug weak wave amplitudes, so that the nonlinear terms canbe treated perturbatively, by contrast with fluid turbu-lence where the nonlinearity is present at first order inthe dynamics [10, 12, 13]. Based on the long-time statis-tics of randomly fluctuating interacting waves, the WTTdeduces a kinetic equation for the distribution of spectraldensities. Beside thermodynamical equilibrium solution,the key-feature of this equation is that non-equilibriumturbulent stationary solutions also arise, called in generalKolmogorov-Zakharov spectra (KZ). Indeed, as shownby Zakharov [10], they describe a constant flux transfer(or cascade) of conserved quantities ( e.g. energy) be-tween large and small length scales. Examples of waveturbulence go much beyond the cases of surface gravityor capillary waves and concerns systems as diverse asplasma waves [10], nonlinear optics [14], vibrating elas-tic plates [15] and gravitational waves in general rela-tivity [16], among others. Although the analogy withturbulent flows is natural since it describes a flux ofthe energy between scales, the links within the physi-cal mechanisms is less straightforward: in particular, theWTT is based on the resonant interactions between thewaves prescribed by the nonlinearity so that the powerof the nonlinearity is crucial in the process. For instanceif the pertinent nonlinearity is quadratic, the dynamicsis determined by three waves interactions such that boththe wave-numbers and the frequencies of the three wavesform resonant triads. This mechanism implies a specificdependence of the KZ spectrum with the energy flux P as P /N where N is the power of the pertinent nonlinearityof the dynamics ( N = 2 for quadratic interactions, 3 forcubic, etc ). This represents a crucial difference betweenwave turbulence and hydrodynamic turbulence, becausein the last one, the flux of energy appears in the secondorder correlation with the power 2 /
3, which cannot beobtained as a 1 /N power.The turbulent asymptotic expansion of wave systemsin the strongly nonlinear limit is, by construction, sin-gular, because the linear term becomes then subdomi-nant. However, a turbulent fully nonlinear dynamics isstill expected to exist. The dependence of the statis-tical properties with the potentially different nonlinearterms and the connexions with the NS fluid turbulenceis then worth to be investigated. In this paper, we pre-cisely address such singular limit in the case of vibratingelastic plates. This system has indeed recently revealeda strong analogy with hydrodynamic turbulence throughthe derivation of an exact law for a two-point correlationthat is the equivalent of the Kolmogorov’s 4/5-law [17].The weakly nonlinear turbulence regime in vibratingelastic plates has been established ten years ago by threeof us [15]. Based on the dynamical F¨oppl–von K´arm´anequations [18] (described below), we have shown usingthe WTT the existence of a spectrum of direct energycascade. The rather simple comparison between the-ory, numerics and experiments has led to a large num- ber of studies in vibrating plates (for a review see [19]).Experiments performed soon after the theoretical pre-dictions have shown slightly different spectrum powerlaws [20, 21], that have been later explained by the spe-cific features of the dissipation [22]. Furthermore, waveturbulence of plates has shown to be a perfect systemfor investigating different concepts such as inverse cas-cade [23] and transitory dynamics [24]. More recently,thin plates with a high forcing [25, 26], the breakdownof the WTT and the onset of intermittency [27] havebeen extensively studied. In these high forcing regimes,a wave turbulence spectrum is still observed at smallscales, while strong nonlinear regimes appear at largescales, whose nature remain unclear, suggesting the exis-tence of a strong turbulent dynamics.In the following, to investigate this strongly nonlinearregime and the analogy with hydrodynamic turbulence,we will consider a thin elastic plate in the fully nonlinearcase which corresponds to the formal limit with no bend-ing and where linear waves are absent. With no surprise,the WTT spectrum vanishes in this limit and a differentturbulent spectrum is expected to appear. The goal ofthe paper is to characterize such energy repartition andhighlight similarities with hydrodynamic turbulence. THEORETICAL MODEL
The vibration of a bending-free elastic plate comesfrom the usual dynamical version of the F¨oppl–vonK´arm´an equations [28] for the vertical amplitude of thedeformation ζ ( x, y, t ) and for the Airy stress function χ ( x, y, t ) that describes the three in–plane stresses: ρ ∂ ζ∂t = ζ xx χ yy + ζ yy χ xx − ζ xy χ xy + F + D ; (1)1 E ∆ χ = − (cid:0) ζ xx ζ yy − ζ xy (cid:1) . (2)Here ∆ = ∂ xx + ∂ yy is the usual Laplacian and ρ and E are, respectively, the mass density and the Young modu-lus E of the plate. F and D are the forcing and dissipa-tion respectively.When F = D = 0, the equations (1) and (2) derivefrom a Hamiltonian principle [15, 19]. More important,it is a well posed system, in the sense that the energyis compound by two positive (hence bound from below)quantities, namely the kinetic energy and the stretchingenergy per unit mass: E kin = 12 S (cid:90) ˙ ζ d r , E stret = E ρS (cid:90) (cid:2) ∆ − (cid:0) ζ xx ζ yy − ζ xy (cid:1)(cid:3) d r , (3)here S is the area plate. For dimensional purposes, weuse everywhere the energies per unit mass, E , E kin , and E stret , with dimensions of the square of a speed. RESULTSTurbulent behavior
We performed numerical simulations of equations (1)and (2) by using a standard pseudo-spectral method in aperiodic domain with an additive random forcing at largescales and a viscous damping acting at small scales. Theexplicit form of forcing and dissipation are discussed inMethods. Turbulent states should not depend explicitlyon their details, provided that these mechanisms are wellseparated in the wavenumber space.Figures 1 a-b show the surface plate deformation ζ ( x, y ) and the plate vertical speed ˙ ζ ( x, y ) respectively.The displacement field displays a coarse scale, super-imposed with a fine scale fluctuations dominated by alarge number of wrinkles of various size. Such multi-scalestates are the most prominent characteristic of turbulentsystems. The local speed of the plate in Fig.1-b demon-strates that the elastic plate is not at equilibrium as itdisplays a myriad of excited modes. When bending is ab-sent, it is natural to expect the appearance of highly non-linear geometrical structures. Such structures are visiblein Fig. 1-c that shows a close-up of the plate deflection.The small scale wrinkles consist in a random assemblyof moving ridges and conical points. Ridges and conicalpoints are in fact the fundamental equilibrium configu-rations of elastic plates in the bending-free limit. Thesestructures have been vastly studied since the pioneeringworks performed in the 90s [29–31]. In the bending-freelimit, the elastic deformations favor the bending modesbecause they cost no energy. Then, the deformations ofthe plates are controlled by the stretching that modifiesthe locally plane metric of the sheet. Because of geomet-rical constrains, it is not always possible to have a fullydevelopable surface everywhere. The system thus createssingular structures: linear ridges [29] and punctual de-velopable cones or more commonly named D-cones [30],which concentrate the plate curvature and the stretchingenergy. Therefore, the dynamics of a bending-free elas-tic plate corresponds to a myriad of singularities moving(apparently) randomly over the sheet. In Supporting In-formation one can see a movie of these defect induceddynamics.To catch precisely these singular structures of dimen-sion 1 (riges) and 2 (D-cones) we have plotted in Fig. 1-dthe first order correction to the instantaneous Gaussiancurvature: G ( x, y, t ) = ζ xx ζ yy − ζ xy (4)which, after (2), is the source of the in-plane stressesvia the Airy function χ ( x, y, t ). The Gaussian curvatureshows the complex network of ridges and D-cones.An immediate analogy can be established with hydro-dynamic turbulence. Fully nonlinear elastic plate turbu-lence seems to be characterized by a “crumpling cascade” where wrinkles play the role of whirls and instead of vor-tex filaments and sheets, the localized singularities comein the form of ridges and D-cones. To illustrate this ideafurther, we integrate the system (1) and (2) starting forma flat state, with no forcing ( F = 0), but with an initialvelocity at large-scales and let the system decay. Snap-shots of the velocity field, shown on Fig. 2, exhibit thedynamics of this decaying turbulence configuration. Ini-tially, all the velocity fluctuations are contained at largescales Fig. 2-a. Then, due to the non-linear mode inter-actions, instabilities appear creating smaller and smallerstructures Fig. 2-b. After this transient, a turbulentstate is observed where fluctuations at all scales coexistFig. 2-c. Finally, at large times, the dissipative termsacting at small scales kill the turbulent state, smoothingout the fluctuations (see Fig. 2-d). Kolmogorov spectrum
By forcing at large scales and dissipating at smallscales, a turbulent out-of-equilibrium steady state is ob-tained, such as the one observed in Fig. 1. A quantitativemeasurement of these steady turbulent states is obtainedthrough the spectral densities of the plate velocity. Asin fluid turbulence, here we compute the average kineticenergy spectrum and the kinetic energy flux for differ-ent wave numbers. The kinetic energy spectrum per unitmass, E kin ( k ), is defined through the kinetic energy perunit mass by E kin = (cid:90) E kin ( k ) dk, where E kin ( k ) = 2 πk (cid:68) | ˙ ζ k | (cid:69) , the velocity field ˙ ζ k ( t ) = (cid:82) ˙ ζ ( r , t ) e i k · r d r and isotropy is assumed. In addition, thekinetic energy flux P kin ( k ) is defined through the transferequation: ∂∂t E kin ( k ) = − ∂∂k P kin ( k ) . (5)Since E kin ( k ) has dimensions of L /T the dimensionsof P kin ( k ) is L /T . Notice that the stretching energy(3) is a quadratic quantity, therefore analogous defini-tions can be given for its spectrum and flux (See Meth-ods for explicit definitions). We have measured directlythe energy fluxes for various turbulent runs with differentforcing amplitudes. Figure 3-a shows the time-averagedkinetic (solid lines) energy fluxes, normalized by theirmean value ¯ P kin in the well defined transparency win-dow where they are flat. This transparency window isusually called the inertial-range in the context of hydro-dynamic turbulence. At small scales, dissipation takesplace and produces a bottleneck that invades the iner-tial range as the mean energy flux increases, although aninertial range is still clearly present. This bottleneck is a)b) c)d) FIG. 1. Turbulent states of a bending-free plate. a) Surface plot of the plate deformation ζ ( x, y ). b) Surface plot of the platevertical velocity ˙ ζ ( x, y ). c) Detail of the wrinkles in a zoom of the surface plot of the plate deformation ζ ( x, y ). d) Snapshotof the gaussian curvature (4) of the respective area. The density plot displays the values of log | G | . Simulations were madeat resolution of 4096 with an additive random forcing, see Methods for more details.a) b)c) d)FIG. 2. Simulations for decaying turbulence. Snapshots of ˙ ζ .Time goes as: a) t = 0, b) t = 0 .
5, c) t = 65 , and d) t = 100(in time units of (1) and (2)). related to dissipative effects, that have been shown to benon-trivial [22]. Figure 3-a also displays the stretchingenergy fluxes (dashed lines), which are notably smallerthan the kinetic energy fluxes. Therefore, numerical sim-ulations indicate that the pertinent energy flux for thecascade is the kinetic energy one, because it shows a con-stant energy flux along scales in an inertial range. Aconstant energy flux is usually associated with a turbu-lent regime that leads to a power law behavior for the corresponding energy spectrum. This is indeed the casefor the kinetic energy spectrum, as shown in Fig. 3-bwhere the k − / Kolmogorov law is apparent.The kinetic turbulent spectrum can be discussed on di-mensional grounds in the footprint of Kolmogorov. How-ever, here in addition to the constant energy flux per unitmass P and the wavenumber k (with dimension inverseof a length), the spectrum should depend a priori on theadditional material parameter E/ρ . Since a characteris-tic length can be defined, namely λ = ( E/ρ ) / /P , onegets generically E kin ( k ) = P / k / Φ (cid:18) k ( E/ρ ) / P (cid:19) , (6)where Φ( · ) is an arbitrary function of the dimensionlessargument kλ . Contrary to fluid turbulence, the existenceof this extra length λ , does not allow to uniquely de-termine a fixed exponent for the power law spectrum.Remarkably, numerics suggest that the kinetic and thestretching energy spectra (13) can be treated indepen-dently. Considering that for the kinetic energy the elasticproperty of the material does not intervene, the kineticenergy spectrum should follow E kin ( k ) = C kin P / k − / , (7)with C kin = Φ(0) (cid:54) = 0 a constant. The numerical simula-tions show indeed a good agreement with this predictedscalings (7) for both k and P .Because the energy (3) combines two different contri-butions we address the role of the stretching energy. Weshow (See methods for the proof), via a simple argument -8 -6 -4 -2 -4 -3 -2 FIG. 3. a) Time averaged kinetic energy flux (solid lines)and the stretching energy flux (dashed lines). Different mark-ers (and colors) are for different runs with increasing forcingamplitude or fluxes (see arrow in figure b). The fluxes havebeen normalized by their total mean flux value in the inertialrange (see methods for values). The inset displays the sameplot but in log-log. b) Time averaged kinetic energy spectranormalized by P / . The dashed line displays the Kolmogorovscaling. Inset: Time averaged stretching energy. The dashedline displays the thermalization energy scaling. The arrowindicates the different runs with increasing values of the flux P . The value of ¯ P kin varies in the range (0 . , .
5) for allthe runs (see Methods). that the stretching energy flux related to the stretch-ing energy must vanish in a statistically steady state un-der an additive forcing and a viscous-like dissipation, asthe one considered in this work. This result follows di-rectly from the observation that the stretching energyflux appears to be the time derivative of the stress corre-lation function which must be zero in steady state turbu-lent regime. Therefore, one would expect the stretchingmodes to eventually thermalize. Since the stretching en-ergy is quadratic in γ = ∆ − (cid:0) ζ xx ζ yy − ζ xy (cid:1) (see (3)),the equilibrium distribution corresponds to equipartitionof the Fourier modes of γ , leading to the equipartitionspectrum of the form: E stret ( k ) = C stret k, (8) with C stret a constant proportional to the mean energy,that plays the role of an effective temperature. In theinset of Fig. 3 the stretching spectra for different forcingamplitude are shown to be consistent with the equipar-tition law 8, for the scales where the dissipation is negli-gible. Intermittency and beyond Kolmogorovphenomenology
The Kolmogorov phenomenology discussed in the pre-vious section is based on dimensional analysis and mean-field assumptions that neglect the existence of extremefluctuations. Nevertheless, it is well known that devia-tions exist to such predictions and they become impor-tant when looking at higher order statistics [7]. These de-viations are associated with the intermittent statistics ofthe fields and are somehow inherent to fluid turbulence.A complete understanding of such fluctuations is stillmissing. Unlike hydrodynamic turbulence, the theory ofweak wave-turbulence predicts Gaussian statistics for thedistribution of wave amplitudes, therefore no intermit-tency can be observed within the range of validity of thetheory. However, when the non-linearities become of thesame order than the linear dispersive terms, the wave tur-bulence theory breaks down and an intermittent statisticscan manifest [32]. Such intermittencies have been experi-mentally observed in gravity-capillary waves [33] and sug-gested for thin elastic plates [34]. Moreover, numericalsimulations of elastic plates at strong forcing have shownimportant deviations from the WTT predicted scalingsfor the plate deformation [26, 27], leading to intermit-tency signatures [27, 34].Intermittency is usually addressed looking at the mo-ments of the so-called structure functions of the fields,that provide information of the variation of the fields ata given scale [7]. For elastic plates, because of the fastdecay of the deformation spectrum, a second order differ-ence is needed to observe an intermittent scaling [27]. Inour system, where bending waves are absent and the dy-namics is thus fully non-linear, the intermittent behavioris expected to occur at any forcing. Following [27], we in-troduce the (second variation) increments δ (cid:96) ζ of the platedeformation and their corresponding structure functions S p ( (cid:96) ), namely S p ( (cid:96) ) = (cid:104)| δ (cid:96) ζ | p (cid:105) with δ (cid:96) ζ = ζ ( x + (cid:96) ) − ζ ( x ) + ζ ( x − (cid:96) ) . (9)At very small scales, the regularity of ζ ( x ) impliesthe scaling S p ( (cid:96) ) ∼ (cid:96) p . However, in the inertial rangea non-trivial scaling can appear. We define then theanomalous exponents (in the spirit of hydrodynamic tur-bulence) ξ p , from the moments following S p ( (cid:96) ) ∼ (cid:96) ξ p . From our numerics we measure that ξ ≈ .
2. In Kol-mogorov theory as well as for WWT, a linear relationbetween the different exponent is expected ξ p = ξ p , p ξ p ℓ/L -3 -2 -1 ξ p ( ℓ ) p = 0 . p = 0 . p = 1 p = 2 p = 3 p = 4 p = 5 p = 6 ℓ/L -3 -2 -1 Z ( ℓ ) / ( P ℓ ) -1-0.50 ζ p FIG. 4. Intermittency exponents ξ p as a function of p showingclearly a departure from a linear law, indicating the existenceof anomalous exponents and intermittency. The right downinset shows the structure functions local slopes as a functionof (cid:96) . The inertial range is delimited by the two vertical greenlines. The left-up panel shows evidence of the 1-law (10).The numerics was done under the same conditions as previousfigures. witnessing the self-similar nature of the dynamics. Forvibrating elastic plates however, a deviation from thislinear law has been observed, indicating thus clearly thepresence of intermittency that breaks the self-similarityof crumpling dynamics [27]. In the bottom inset of Fig.4, we present the local slope of the structure functionsdefined as ξ p ( (cid:96) ) = d log S p ( (cid:96) ) d log (cid:96) for different orders p . A rel-atively flat behavior is observed in the inertial range andthe anomalous exponents ξ p are measured by averaging ξ p ( (cid:96) ) in this window. Fig. 4 shows ξ p as a function of p , exhibiting a clear departure from the linear behavior(dashed line) and thus intermittent statistics. Moreover,we have verified the exact result derived by two of us forthin elastic plates that is valid for both, weak and strongwave turbulence [17]. This result is the equivalent to theonly exact result of hydrodynamic turbulence known asthe 4 / Z ( (cid:96) ), thatdepends on the first order variation of the fields χ , ζ and˙ ζ (see Methods). In analogy with the 4 / Z ( (cid:96) ) = − P (cid:96). (10)The structure function Z ( (cid:96) ), normalized by P (cid:96) is dis-played in (top) inset of Fig. 4. An excellent agreementwith the prediction (10) is observed over a decade with noadjustable parameter. Finally, we have also studied thevertical speed structure function (cid:28)(cid:16) ˙ ζ ( r + (cid:96) ) − ˙ ζ ( r ) (cid:17) (cid:29) .Unlike, hydrodynamic turbulence, it does not scales lin- early with the distance (cid:96) . This difference can be ex-plained by the existence of the extra-length λ , suggest-ing that the exponent can not be uniquely determinedby dimensional analysis. This fact is also related to thecomplexity of the non-linear term, that is precisely takeninto account in (10). DISCUSSION
Substantial evidence shows robustness among turbu-lent behavior in ordinary fluids and in the case of crum-pling vibrations of an elastic plate in the zero limit thick-ness. As presented, both manifest an energy cascade withthe well known K41 spectrum E k ∼ P / k − / . Moreimportant, a myriad of randomly interacting highly non-linear crumpling structures (folds, ridges and d-cones) atall relevant scales dominate the dynamics. They induceintermittency that we have quantitatively investigatedby studying high-order correlators that confirm the ap-pearance of an intermittent behavior. Nevertheless, theunderlying plate dynamics differs notably from the one ofordinary fluids: though in incompressible fluids the totalenergy consists purely of kinetic energy, in elastic platesthe energy is compound of two separately (positive) ener-gies: the kinetic and the stretching energy. Therefore, itis expected that two different cascades may exist in theelastic situation, a kinetic energy one and a stretchingenergy one. Because the forcing is additive to the defor-mation acceleration (1), only a constant kinetic energyflux (see Fig. 3-a) exists resulting in a K41 cascade. Onthe other hand, the stretching energy flux vanishes (seeFig. 3-a) and consequently Gibbs equipartition is ob-served for the stretching spectrum, as shown in the insetof Fig. 3-b. Naturally it is expected that other forcingmay display a stretching energy cascade without any ki-netic energy cascade. Or perhaps two distinct cascadesa kinetic energy K41 cascade and a stretching cascadesimultaneously. APPENDIX
In the following we describe the mathematical and nu-merical methods used in the present paper.
Numerics
We solve the F¨oppl–von K´arm´an equations (1)-(2) witha standard pseudo-spectral code in a square domain ofsize 2 π with periodic boundary conditions. The forcing F is white-noise in time of variance f and its Fouriermodes are non-zero only for wave-vectors 1 . ≤ | k | ≤ / D = − ν ( − ∆) n ν ˙ ζ − α ( − ∆) − n α ˙ ζ (11)The run presented in Fig. 2 was performed with 1024 collocation points, F = 0, n ν = 2, ν = 4 × − , α = 0and a random initial condition at large scale only on ˙ ζ .All the other runs presented in this work were performedwith 4096 collocation points, n ν = 3, ν = 2 × − , n α = 2, α = 100 and f = 1 , ,
27 and 100. For the differ-ent forcing amplitudes, the measured value of the fluxeswere 0 . , . , . . The energy spectra
The energies in (3) maybe seen as quadratic contri-bution of ˙ ζ and γ ( r ) = − ∆ − (cid:0) ζ xx ζ yy − ζ xy (cid:1) .The finalquadratic contributions read E kin = 12 (cid:90) | ˙ ζ k | d k and E stret = E ρ (cid:90) | γ k | d k . (12)We define the kinetic energy and the stretching spectraby averaging over the angular variables in the wavenum-ber space, hence E kin ( k ) = πk (cid:68) | ˙ ζ k | (cid:69) and E stret ( k ) = π Eρ k (cid:10) | γ k | (cid:11) , (13)where we have defined (cid:104) . . . (cid:105) = π (cid:82) . . . dϕ k by the angu-lar average. The energy fluxes
Because the energy is quadratic in ˙ ζ and γ , the energyflux can be straightforward defined as in hydrodynamicturbulence. By making a scale-by-scale energy budgetthe energy fluxes are: P kin ( k ) = − (cid:90) k ∂E kin ( p ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) dp = − (cid:90) k (cid:68) ˙ ζ p { ζ, χ } − p (cid:69) dp, (14) P stret ( k ) = − (cid:90) k ∂E stret ( p ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) dp = (cid:90) k (cid:68) χ p { ζ, ˙ ζ } − p (cid:69) dp, (15)where the subscript 0 stands for the free of forcingand dissipation time variations of the fields through (1)with F = D = 0. For simplicity, we define { f, g } ≡ f xx g yy + f yy g xx − f xy g xy . We have also used the factthat the angular averages satisfy (cid:10) (∆ − f ) p (∆ − g ) − p (cid:11) = (cid:10) (∆ − f ) p g − p (cid:11) . The total energy flux results from theaddition of these previous expressions. Steady state with zero stretching energy flux
Following [17], we introduce the spatial correlationfunction E stret ( (cid:96) ) = 12 E (cid:104) ∆ r χ ( r )∆ r (cid:48) χ ( r (cid:48) ) (cid:105) , (16)with (cid:96) = r (cid:48) − r , which on the limit of (cid:96) → E stret ( (cid:96) ) = −(cid:104) χ ( r (cid:48) ) { ζ ( r ) , ˙ ζ ( r ) }(cid:105) = 1 V (cid:90) χ ( r + (cid:96) ) { ζ ( r ) , ˙ ζ ( r ) } d r . (17)The last equality considers that for an homogeneous sys-tem the statistical average can be taken as a spatial av-erage. From the convolution theorem the Fourier trans-form ˙ E stret ( p ) = χ p { ζ, ˙ ζ } − p . In a steady state the timederivative of the correlation function must be zero hence χ p { ζ, ˙ ζ } − p = 0 and the stretching energy flux (15) mustbe zero. In Ref. [17] an exact result for turbulence of thin elasticplates was found. It provides a K´arm´an-Howarth-Monintype relationship for the energy flux:12 ∇ (cid:96) · (cid:104) J [ δχ,δζ ] δ ˙ ζ (cid:105) = − P, (18)where δ stands for the first difference of the field, e.g. δζ = ζ ( x + (cid:96) ) − ζ ( x ). The vector J is defined by J [ f,h ] = f y h yx − f x h yy , f x h xy − f y h xx . Under the assumption of isotropy, (18) implies the 1 -law (10) for the structure function Z ( (cid:96) ) ≡ (cid:104) J [ δχ,δζ ] δ ˙ ζ (cid:105) · ˆ (cid:96) = − P (cid:96), (19)where ˆ (cid:96) is the unit vector along (cid:96) .GD, GK and SR thank to FONDECYT grants N ◦ [1] Reynolds O (1883) Phil. Trans. Roy. Soc.
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