Depletion of nonlinearity in two-dimensional turbulence. Abstract

Transcription

1 Depleion of nonlineariy in wo-dimensional urbulence Andrey V. Pushkarev and Wouer J.T. Bos LMFA, CNRS, Ecole Cenrale de Lyon, Universié de Lyon, 6934 Ecully, France arxiv:42.535v [physics.flu-dyn] 6 Dec 24 Absrac The srengh of he nonlineariy is measured in decaying wo-dimensional urbulence, by comparing is value o ha found in a Gaussian field. I is shown how he nonlineariy drops following a wo-sep process. Firs a fas relaxaion is observed on a imescale comparable o he ime of formaion of vorical srucures, hen a long imes he nonlineariy relaxes furher during he phase when he eddies merge o form he final dynamic sae of decay. Boh processes seem roughly independen of he value of he Reynolds number. PACS numbers: eb, Gs, Jv

2 I. SELF-ORGANIZATION, COHERENCE AND DEPLETION OF NONLINEAR- ITY Turbulen flows display he endency o generae flow srucures, saring from an unorganized iniial sae. This srucuring of urbulen flows is perhaps mos impressive in freely evolving wo-dimensional urbulence, where large scale vorices are generaed hrough nonlinear ineracion []. Even hough inuiively one easily appreciaes from visualisaions ha his self-organizaion of he flow increases is spaio-emporal coherence, i is no sraighforward o give a quaniaive measure of his coherence. In he presen invesigaion we sudy he link beween flow coherence and he srengh of he nonlineariy during he selforganizaion process of wo-dimensional urbulence saring from an iniial sae consising of saisically independen modes. We will firs illusrae how hese quaniies are relaed in nonlinear sysems in general. To illusrae our approach we consider a nonlinear sysem of he form ψ = N(ψ 2 )+L(ψ), () where ψ(x,) is a funcion of space and ime, N(ψ 2 ) is he nonlinear erm and L(ψ) represens he linear erms. As a measure of he emporal coherence creaed by he selforganizaion process we can evaluae he rae a which on average he sysem evolves in ime. Such a measure is ψ 2, where he brackes denoe a space or ensemble average. If he mean-square ime derivaive decreases, i indicaes ha he emporal coherence of he considered field increases. Squaring boh sides of equaion () ψ 2 = N(ψ 2 )+L(ψ) 2, (2) we direcly see ha he coherence will be deermined by an inerplay of linear and nonlinear effecs. A purely linear sysem will no display an ineresing self-organizaion in he sense ha he differen modes are no coupled in he absence of nonlineariy. To simplify he problem we could herefore consider a purely nonlinear sysem in he absence of linear erms. In ha case we have ψ 2 = N(ψ 2 ) 2. (3) This shows ha in he mos elemenary case of he nonlinear evoluion of a sysem in he absence of any linear effecs, he average emporal coherence is compleely deermined by 2

3 he mean-square nonlineariy. We will consider he evoluion of he laer quaniy in wodimensional urbulen flows. I is well known for freely decaying wo-dimensional urbulence in a periodic domain ha he final dynamic sae consiss of wo slowly evolving coheren vorices. The voriciy of he resuling flow-field shows a monoonic correlaion wih he sream-funcion [2], corresponding o an equilibrium-sae where he nonlinear ineracions are small. Since coheren srucures are also observed a shorer imes, local equilibrium saes should reduce he nonlocal ineracions already a shor and inermediae imes and we expec hus he mean-square nonlineariy o be small compared o he quaniy in a noncoheren velociy field even a hese shorer imes. The definiion of a non-coheren field is an ineresing quesion iself. We will follow he idea of Kraichnan and Panda [3] and compare o a field wih he same energy disribuion, consising of independen Fourier modes. This choice is no unique and does no allow o meaningfully quanify all possible non-gaussian feaures of urbulence, as for insance ensrophy producion [4]. For he presen purpose, which is he analysis of he depleion of he mean-square nonlinear erm, i seems however a good choice. The mean-square nonlineariy as compared o is value in a Gaussian field was firs invesigaed by Kraichnan and coworkers [3, 5, 6] in hree dimensional decaying urbulence. Furher sudies [4, 7 9] aimed a indenifying mechanisms which can explain he depleion of nonlineariy hrough local flow srucuring. One such mechanism, he alignmen of velociy and voriciy, was suggesed o play an imporan role in he dynamics of urbulence []. The sudy of Kraichnan and Panda[3], pu forward he idea ha velociy-voriciy alignmen migh be one manisfesaion of a more generic feaure of urbulen flows, which is is endency o diminish he srengh of he nonlineariy. This seems indeed o be he case since he depleion of nonlineariy is also observed in magneohydrodynamic urbulence [] and he advecion erm in he advecion of a passive scalar also shows his endency [2, 3]. Our sudy is a logical exension of hese works o he case of wo-dimensional urbulence. One firs lesson ha we can learn from he oucome of he presen work is ha, if he nonlineariy is reduced in wo-dimensional urbulence, i is no he velociy-voriciy alignmen which is he underlying mechanism which explains he depleion of nonlineariy in boh woand hree-dimensional urbulence, since he voriciy is always perpendicular o he velociy in wo-dimensional flow. Alignmen, in general, can however play an imporan role, as was shown in [4], where a preferenial alignmen of he voriciy-gradien and sream-funcion 3

4 gradien was observed in decaying wo-dimensional urbulence. Wih respec o his laer work, he originaliy of he presen approach is ha we will compare saisics in a decaying urbulen flow wih a Gaussian reference field having he same energy specrum a every ime-insan, as proposed in [3]. I allows o show how he emporal coherence of he flow evolves in ime. We do no focus on he precise local relaxaion mechanism and argue, following [3], ha i is perhaps no he deailed opological mechanism which is universal, bu he endency o creae coherence by decreasing, saisically, he norm of he nonlinear erm. II. THE DEPLETION OF NONLINEARITY IN TWO-DIMENSIONAL TURBU- LENCE For an incompressible fluid flow, equaion () is given by he Navier-Sokes equaions wih he nonlinear erm given by u = N +ν u, u =, (4) N = u u+ p, (5) wih p he pressure divided by he fluid densiy, which is assumed homogeneous and consan, and ν he kinemaic viscosiy. In wo-dimensions i is convenien o consider he evoluion of ω = e u, he componen of he voriciy normal o he plane of he velociy, wih ω = N ω +ν ω, (6) N ω = e N = u ω. (7) In he absence of linear effecs, i will be he quaniy Nω 2, which measures he emporal Eulerian coherence of he voriciy field, as suggesed by equaion(3). For a srongly coheren field his quaniy will be small compared o he quaniy in a non-coheren velociy field. As explained above, we will compare o a field wih he same energy disribuion, consising of independen Fourier modes, by measuring Nω 2 (8), N 2 ω,g 4

5 Resoluion ν R L R λ 4 4 TABLE I: Deails of he simulaions. We define R L = UL/ν, wih U he rms-value of one of he velociy componens and L he box-size. The Taylorscale Reynolds number is defined as R λ = Uλ/ν, wih λ = U 5ν/ǫ. where Nω,G 2 is he mean-square voriciy advecion erm measured in a mulivariae Gaussian field wih he same energy specrum. The mean-square nonlinear erm is no invarian under Galilean ransformaions, i.e. adding a spaially uniform velociy field o he flow will change is value. Adding such a field will also change he esimae in he Gaussian field so ha measure (8) compensaes, a leas in par, for such effecs. In he presen work we will se he mean flow u = and only consider he saisically isoropic case. The evoluion of equaion (6) was compued using a wo dimensional pseudo-specral Fourier code in sream funcion-voriciy formulaion. The compuaion was performed on a (2π) 2 square domain wih periodic boundary condiions. The resoluion was varied beween 256 and 248 Fourier modes in each direcion. The iniial velociy field is defined in Fourier space and corresponds o a field of incompressible saisically independen Fourier modes, given by u i (k) = E(k) πk P ij(k)a j (k), (9) where a i (k) is a Gaussian whie noise process and E(k) k 4 exp( k 2 /k 2 ) is he kineic energy specrum. The value of k ses he dominan correlaion wavenumber of he iniial energy disribuion and is k =. The Riesz-operaor, P ij (k) = δ ij k i k j /k 2, () projecs he random noise on he plane perpendicular o he wavevecor o ensure incompressibiliy of he iniial velociy field. The kinemaic viscosiy ν and he corresponding values for he iniial Reynolds numbers based on he box-size and on he Taylor lenghscale, respecively, are summarized in able I. 5

6 E()/E().6.4 ǫ()/ǫ() (a) (b) FIG. : Time evoluion of he energy (a) and energy dissipaion (b). The emporal evoluion of he kineic energy E = u i u i /2 and ensrophy Z = ω 2 /2 is given by E = ǫ Z = η, () wih ǫ and η he energy and ensrophy dissipaion rae. In a periodic wo-dimensional decaying flow wihou energy and ensrophy sources, boh he kineic energy and ensrophy are monoonically decaying funcions of ime. Since he energy dissipaion ǫ = 2νZ, as soon as he ensrophy has decayed considerably, he energy will remain approximaely consan a high Reynolds numbers. This behaviour is shown in figure, which illusraes he ime evoluion of he kineic energy and energy dissipaion rae, respecively. Figures 2 depic snapshos of he voriciy fields of he 24 2 run a 3 (a) and a 85(b). The condensaion of all he voriciy ino wo counerroaing vorices is observed for long imes. I will be shown now ha his sae corresponds o a flow wih a very small magniude of he nonlineariy. Infigure 3 we show heevoluion of Nω 2 / N2 ω,g. As we can see fromhe figure, his raio very rapidly decreases a he iniial ime. Afer his momen, during he furher evoluion of he flow, peaks are observed which correspond o he ineracion beween differen vorices. The opposie, local minima correspond o a quie sae where here is no ineracion beween 6

7 Ly Lx 4 (a) Ly Lx (b) FIG. 2: Snapshos of he voriciy field a 3 (a) and a 85 (b). differen vorices and all vorices are well separaed. This is illusraed by he snapshos which are presen on he figure. The momens of ime which correspond o he snapshos are noed by arrows. We have checked ha he same qualiaive picure is observed during he oher peaks and local minima of he ime-evoluion. The fac ha vorex merging increases he value of he nonlineariy can be undersood inuiively since when wo vorices approach, he relaive velociy beween he cores of he vorices is parallel o heir radial voriciy gradiens so ha he value of (u ω) 2 is high during hese evens. We have checked ha he order uniy peaks are no an arefac of he Gaussian random 7

8 N 2 ω / N 2 ω,g (a) N 2 ω / N 2 ω,g (b) FIG. 3: Evoluion of he raio of he oal mean square nonlinear erm in urbulen flow o he equivalen value in he Gaussian flow for shor imes (a) and long imes (b). Snapshos of he voriciy field are shown for four differen ime insans. In he op figure hese snapshos only show a quarer of he domain o more clearly idenify he flow srucures. Resuls correspond o he 24 2 run. fields we used for comparison. In figure 4 we show he resuls for he mean-square nonlineariy compared o wo independenly generaed Gaussian fields for he 52 2 run. I is observed ha he low frequency srucure of he ime evoluion is no significanly affeced when changing he reference field. Only a zoom a he fine ime-srucure shows a difference 8

9 N ω / Nω,G N 2 2 / N ω ω,g* FIG. 4: Evoluion of he raio of he oal mean square nonlinear erm in urbulen flow o he equivalen value in he Gaussian flow. The urbulen saisics for he 52 2 run are compared o wo independenly generaed Gaussian fields Nω,G 2 and N2 ω,g o check he influence of he choice of he Gaussian fields on he resuls. in behaviour. A he end of he simulaion only wo weakly ineracing vorices are observed, figure 2(b). A his ime i is observed ha he mean square nonlineariy is reduced o only a few percen of is Gaussian value. I seems ha in he wo-dimensional case here are wo imescales over which he nonlineariy is reduced: a very shor one a he beginning of he simulaion of he order of ime over which he iniial vorices form, and a long one, of he order of he ime i akes o condense all he energy in he wo counerroaing vorices in figure 2. To give a saisical descripion of he muliscale characer of he depleion of nonlineariy in 2D urbulence we define he nonlineariy specrum of he Navier-Sokes equaions as W(k)dk = N 2. (2) The voriciy nonlineariy specrum is given by W ω (k)dk = N 2 ω (3) and he relaion beween he wo specra is W ω (k) = k 2 W(k). (4) 9

10 N 2 N G 2 N 2 ω N 2 ω,g FIG. 5: Time evoluion of he oal mean square nonlinear erm in urbulen flow o he equivalen value in he Gaussian flow for he values N 2 / N G 2 and N 2 ω / N 2 ω,g. Therefore, he raio of he specrum of he mean square nonlinear erm in he urbulen flow o he equivalen value in he Gaussian flow does no depend on which nonlineariy we consider: W ω (k) W ω,g (k) = W(k) W G (k). (5) However, he raios of he inegraed quaniies are no necessarily he same, which is illusraed in figure 5, where we compare N 2 / N G 2 and N 2 ω / N2 ω,g for he 242 run. I is shown, however, ha he wo raios are very close. We show W(k) and W G (k) a = 4 in figure 6 (a). I is shown ha he specra are an approximaely consan funcion of he wavenumber in he inermediae range of wavenumbers. The raio W(k)/W G (k) is shown in figure 6 (b). I is shown ha he depleion akes place in almos he complee specrum. In hree dimensional urbulence, according o closure heory, he depleion was consan hroughou he inerial range of scales [5]. Noe ha he highes wavenumbers, beyond k = 3, are affeced by he wavenumber runcaion and he srong flucuaions a hese scales are raher numerical arefacs han par of he urbulen dynamics. An obvious quesion is of course how he above resuls depend on he iniial value of he Reynolds number. Are he resuls essenially inviscid, so ha we can expec hem o hold in he limi of infiniely high Reynolds numbers, or do hey depend on he relaive

11 4 W(k) W G (k) 2 W(k) k (a) 2.5 W (k) WG(k).5 2 k (b) FIG. 6: Top: specrum of he nonlinear erm W(k) and W G (k) evaluaed a = 4. Boom: normalized power specrum W(k)/W G (k) of he nonlinear erm evaluaed a = 4. srengh of he viscous effecs compared o ha of he nonlinear erm? In figure 7 he resuls are shown for he differen simulaions. I is observed ha he rapid relaxaion becomes roughly independen of he Reynolds number for he 52 2, 24 2 and248 2 simulaions. The long-ime evoluions are qualiaively similar. For numerical reasons, we have no carried ou he simulaion for long imes, bu he qualiaive similariy beween he oher hree simulaions indicaes ha he long-ime evoluion, like he shor-ime evoluion, is no qualiaively changed by he value of he Reynolds number. Boh long and shor ime

12 N 2 ω / N 2 ω,g (a) N 2 ω / N 2 ω,g (b) FIG. 7: Evoluion of he raio of he oal mean square nonlinear erm in urbulen flow o he equivalen value in he Gaussian flow for shor imes (a) and long imes (b). processes appear o be inviscid. III. DISCUSSION AND CONCLUSION In he presen invesigaion we have shown how he mean-square nonlineariy evolves in 2D urbulence, compared o a Gaussian reference field. In paricular, we observed a wo-sep regime of he suppression of nonlineariy. The firs sep corresponds o he fas formaion 2

13 of vorices and could be qualified as rapid relaxaion. The second phase corresponds o he evoluion of he sysem hrough he muual ineracions of he vorices. If we consider only a shor ime inerval, as in figure 3 (Top), i seems ha he mean square nonlineariy falls o a value of he order of half he Gaussian value and hen remains consan, as seen qualiaively in 3D urbulence (e.g. [3, 5]). Comparing he srongly flucuaing ime-evoluion of he raio Nω / N 2 ω,g 2 wih flow-visualizaions a differen ime-insans, i was observed ha low values correspond o siuaions where vorical srucures are well separaed in space. Whenever vorices inerac, a srong increase of he raio is observed. The firs, rapid relaxaion process is possibly caused by local alignmen properies of he flow [4]. The fac ha a very long imes he nonlineariy is furher reduced was prediced by an enropy opimizaion argumen by Joyce & Mongomery [2], and more formally explained in [6, 7]. Noe ha his final sae of wo vorices, depleed from nonlinear ineracion was prediced by applying saisical mechanics o he global quaniies (ensrophy, energy averaged over he spaial domain) leading o he predicion of he mos probable final dynamic sae in he limi of small viscosiy. A recen review aricle [8] gives an overview of exising aemps o predic he formaion of coherence using he applicaion of saisical mechanics o wo-dimensional flows. These approaches mosly focus on he long-ime limi of sysems in he inviscid limi. The resuls in he presen work sugges ha also he rapid relaxaion process is of inviscid naure, and migh herefore also be reaable in he framework of Euler s equaions. We have no invesigaed he role of he boundary condiions on he depleion of nonlineariy. I is cerain ha he deailed long-ime dynamics mus be affeced by his. Indeed, i was shown ha he final dynamic sae is differen when an unbounded fluid is considered [9, 2] or when rigid boundaries are aken ino accoun [2, 22]. However, in all hese cases he final sae is characerized by large-scale coheren srucures, ha we can expec o be in a near equilibrium sae wih a low value of he mean-square nonlineariy, compared o a field consising of random independen modes. The long-ime endency of he mean-square nonlineariy will hus no be changed imposing differen boundary condiions. Furhermore, i can be expeced ha he iniial fas relaxaion is relaively independen of he boundary condiions, as long as he iniial correlaion lenghscale is small compared o he domain-size. Transposing he exising heoreical ideas of inviscid global relaxaion o local flow srucuring, migh prove a good saring poin o undersand he rapid relaxaion evens observed 3

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