Periodically kicked turbulence is theoretically analyzed within a mean field
theory. For large enough kicking strength A and kicking frequency f the
Reynolds number grows exponentially and then runs into some saturation. The
saturation level can be calculated analytically; different regimes can be
observed. For large enough Re we find the saturation level to be proportional
to A*f, but intermittency can modify this scaling law. We suggest an
experimental realization of periodically kicked turbulence to study the
different regimes we theoretically predict and thus to better understand the
effect of forcing on fully developed turbulence.Comment: 4 pages, 3 figures. Phys. Rev. E., in pres

A. Pumir

A. S. Monin

B. P. Barber

D. Lohse

Detlef Lohse

G. S. Lewis

G. Stolovitzky

G. Zocchi

H. Effinger

K. R. Sreenivasan

L. A. Crum

M. Faraday

M. Smith

P. J. Zandbergen

R. Camussi

R. Labbe

S. Grossmann

Th. Gebhardt

U. Frisch

English

arXiv

Crossref

Periodically kicked turbulence

Periodically kicked turbulence is theoretically analyzed within a mean-field theory. For large enough kicking strength A and kicking frequency f the Reynolds number grows exponentially and then runs into some saturation. The saturation level Resat can be calculated analytically; different regimes can be observed. For large enough Re we find Resat¿Af, but intermittency can modify this scaling law. We suggest an experimental realization of periodically kicked turbulence to study the different regimes we theoretically predict and thus to better understand the effect of forcing on fully developed turbulence

Lohse, Detlef

NARCIS 

University of Twente Research Information

rlandsPHYSICAL REVIEW E OCTOBER 2000VOLUME 62, NUMBER 4Periodically kicked turbulenceDetlef Lohse*Department of Applied Physics and J. M. Burgers Centre for Fluid Dynamics, University of Twente, 7500 AE Enschede, Nethe~Received 23 May 2000!Periodically kicked turbulence is theoretically analyzed within a mean-field theory. For large enough kickingstrengthA and kicking frequencyf the Reynolds number grows exponentially and then runs into some satu-ration. The saturation level Resat can be calculated analytically; different regimes can be observed. For largeenough Re we find Resat}A f , but intermittency can modify this scaling law. We suggest an experimentalrealization of periodically kicked turbulence to study the different regimes we theoretically predict and thus tobetter understand the effect of forcing on fully developed turbulence.PACS number~s!: 47.27.2iri-xeehees-diayri-umreoteocageinf-inhraisiders.reror-tndncebeal-.Periodically driven flow is ubiquitous. Faraday’s expement @1# is an idealized version thereof; more relevant eamples are the earth’s atmosphere, driven by periodical hing of the sun, or the blood flow in veins, driven by thbeating heart. Another example is the gas flow insidesonoluminescing bubble that is periodically kicked by tcollapsing bubble wall@2#. Another example of periodicallykicked turbulence is the numerical realization of homogneous shear flow@3# where periodical remeshing is necesary.In this paper we set up a mean-field theory for periocally kicked flow, based on the mean-field theory for decing turbulence@4#, which was able to describe the expementally measured energy decay in turbulent liquid-heliflow with fixed external length scale@5#. The goal of thispaper is to theoretically understand the different flowgimes, which are to be expected, to explore the effectintermittency corrections on these regimes, and to ultimainitiate experiments.Another motivation for the paper is to study the effecta specified type of forcing on turbulence. In most theoretistudies on turbulence, a Gaussian random noise, actinthe largest length scales, is assumed. Only recently expmentalists started to systematically vary the type of forc@6,7#. This paper is a further step towards the analysis omore specific type of forcing.To define the model, we have to~i! calculate the energyinput during the kick and~ii ! know how the energy is dissipated in the timeDt between successive kicks.~i! Kick: As an illustration, consider plane shear flowthe 1 direction; the flow is sheared in the 3 direction. Twidth of the channel isL, the velocity of the upper plate isU,the lower one is at rest. The average energy dissipationcan be calculated to be~see, e.g., Ref.@8#!e52UL~^u3u1&A2n]3^u1&A!, ~1!where^&A denotes the average over thex-y plane andn isthe viscosity. For laminar flow the first term in the bracketzero and the second one is2nU/L. In turbulent flow in the*Email address: lohse@tn.utwente.nlPRE 621063-651X/2000/62~4!/4946~4!/$15.00-at-a----flyflonri-gaetemiddle of the channel the second term on the right-hand swill hardly contribute for large enough Reynolds numbeThe first term represents the total turbulent flow energyE,order of magnitude wise. Therefore, in general,Ė~ t !52e~ t !;ULE~ t !1U2L2n. ~2!Imagine now a short intense kick of timeDtkick!Dt on theflow by rapidly moving the upper plate withU. After thiskick the initial energyE0 increases according to Eq.~2!,E15E01S E01 UnL D UL Dtkick , ~3!where we have assumedDtkick!L/U. Assume isotropic tur-bulence in the flow center and define a Reynolds numbe1Re~ t !5Lu1,rms~ t !n5A23LAE~ t !n. ~4!Then Eq.~3! translates toRe15Re0A112A1 Relam2Re02~5!with the dimensionless kicking strengthA5 12 DtkickU/L!1and the ‘‘laminar’’ Reynolds number Relam523 DtkickU2/n.We choose this name because for very small Re0 w haveRe15Relam . For very large Re0@Relam we have Re15(11A)Re0. Shear flow and the ‘‘derivation’’ of Eq.~5! areonly thought of as a motivation; there will be many othexperimental situations where the energy input roughly cresponds to a law of type~5!.~ii ! Decay: In Ref.@4# we calculated how the turbulenactivity decays within a timet for flow with fixed externallength scaleL. The calculation was based on Effinger aGrossmann’s variable range mean-field theory of turbule@9# in which viscous subrange and inertial subrange can1Note that this is not the standard definition and gives lower vues for the laminar-turbulent transition than what one is used to4946 ©2000 The American Physical Societynyre---inhk.qve,tionire-PRE 62 4947PERIODICALLY KICKED TURBULENCEtreated equally well. The result of Ref.@4# is that for giveninitial Reynolds number Rei , the time dependence of Re(t)@defined as in Eq.~4!# is determined by the inverse functiooft~Re!t53ce,`@F~Re!2F~Rei !#, ~6!wheret5L2/n andF(Re) is given byF~Re!512 Re2$2g1Ag21Re2%112glogH g1Ag21Re2Re J .~7!ce,`5(6/b)3/2 ~in this theory! is the dimensionless energdissipation in the large Reynolds number limit,g59/ce,` ,and b is the Kolmogorov constant@10#, which is the onlyfree parameter in the theory of Ref.@4#. From experiment@11# b56.0, a value that we take in all calculations heConsequently,ce,`51.0 andg59.0. Rather than the Reynolds number@Eq. ~4!# one could also give the TaylorReynolds number Rel @4#,Rel5A 15 Re2ce,`~g1Ag21Re2!. ~8!Equations~5!–~7! with t5Dt ~the time between successive kicks! fully define the present model. The two maphysical parameters in the model are the kicking strengtAand the kicking frequencyf 51/Dt. The third physical pa-rameter is Relam , the minimal Reynolds number after a kicWe pick Relam51 throughout.Figure 1 shows Re(t) for fixed A and three different kick-ing frequenciesf for the initial Reynolds number Re05Relam51. During each cycle there is a kick Re0→Re1according to Eq.~5! and a subsequent decay according to E~6!. Overall, there is growth up to some saturation leResat(A, f ), achieved aftertsat(A, f ). In this saturation stateFIG. 1. Time series Re(t) for fixed A50.1 and three kickingfrequenciesf 510/t, f 5100/t, and f 5500/t, bottom to top. Thecorresponding ~upper! saturation levels are Reusat51.3, Reusat517.3, and Reusat5136, respectively...lenergy input and loss through decay inDt balance, and thedegree of excitation fluctuates between a lower saturalevel Relsat and an upper saturation levelReusat5RelsatA112A1~Relam /Relsat!2. ~9!The ~lower! level of saturation Relsat(A, f ) is given throughthe implicit equation1t f53ce,`FF~Relsat!2FS RelsatA112A1S RelamRelsatD 2D G .~10!For large Reusat , Relsat@g one has the explicit resultFIG. 2. Saturation level Relsat ~lower curve of pair! and Reusat~upper curve of pair! as a function off for three different kickingstrengthsA50.01 ~solid!, A50.1 ~dashed!, and A50.5 ~dashed-dotted!, bottom to top. Below Reusat;g59.0 the excited state islaminar, above Reusat;9 it is turbulent.FIG. 3. Reusat and Relsat for f 510, 102, 103, 104, bottom totop. The increasing difference between Reusat and Relsat at the rightedge of the figure, has its origin in the breakdown of the requmentA!1.yheerit-int ie-cr-rcg:vsehreotilthtalinanillsecor-ichre-withbestaseaseseceithes-Re.ebtleol-Doin-nd?dt,re-er-der-the62No.al4948 PRE 62DETLEF LOHSEReusat53t fce,`~A112A21!'3tce,`A f . ~11!In Fig. 2 we show a log-log plot of Rel ,usat as a function offfor different kicking strengthA. Two regimes are seen:~i!For small Rel ,usat&g a laminar regime in which after the decathe kick always brings back the level of excitation to tlaminar value Relam51. ~ii ! For large Rel ,usat*g we have aturbulent scaling regime with Reusat} f . The transition fromthe laminar to the turbulence regime takes place aroundf trans53tA. ~12!Similarily, these two regimes are also seen in Fig. 3 whwe plotted Rel ,usat vs A. In the turbulent regime for largeRel ,usat , it is Rel ,usat}A.We now come to the important question of how intermtency effects@12# change the exponents calculated withthis mean-field theory. In Ref.@13# intermittency effectshave been included into the mean-field theory of Ref.@4# ona phenomenological basis. One possibility for their effecthat the dimensionless energy dissipation ratece becomesslightly Reynolds number dependent even in the large Rnolds number limit@13#, ce}Re2k, with k5(9/8)dz2 /(113dz2/8). Here,dz2'0.03 is the experimentally found deviation from the classical scaling exponentz252/3 of thesecond-order velocity structure function. The consequenof this small (k'0.03) scaling correction can straightfowardly be embodied in the present mean-field approachperiodically kicked turbulence. The result is that in the tubulent regime the saturation level now obeysRel ,usat}~A f !1/(12k) ~13!rather than Eq.~11!. Equation ~13! may offer a new andindependent way to experimentally determine intermittenexponents.Another effect related to intermittency is the followinWe expect the total energy to build up again and again otime scales larger thanDt and then to suddenly drop becauan energy pulse is traveling downscale. Such behaviorbeen observed in numerical simulations of periodicallymeshed homogeneous shear flow@3# and in simulations ofperiodically kicked shell models of turbulence@results to bepublished#. As based on a mean-field theory the modelthis paper is only applicable to the mean energy and nothese fluctuations.We now come back to an experimental realization. It wbe easier to perform experiments in a closed system raysn,esy-esto-yeras-ftolerthan in a channel flow. A particular suited experimensetup for periodically kicked turbulence would be the flowa cylinder between two counter-rotating disks@6,14#. Also,Rayleigh-Benard convection may be well suited. Here, asexample, we take Taylor-Couette flow@15#: If the radii of theinner and outer cylinder are similar, the energy input wstill roughly follow Eq. ~5!. Take water as a fluid that han51026 m2/s and takeL51 cm, thent5100 s. Realisti-cally achievable kick strengths would beA5DtkickU/(2L);(0.1 s)(0.1 m/s)/(231 cm)50.5, e.g., the whole rangeA,1 in which the theory is applicable. For this value thscaling regime sets in atf trans50.06 Hz. Roughly two de-cades of scaling are necessary to explore intermittencyrections and to distinguish between Eqs.~11! and~13!. Thatis, one has to go up to frequencies of around 6 Hz, whshould be achievable. At these relatively large kicking fquencies, measurements can only revealinstantaneousval-ues. To get statements on the averaged quantities dealtin this paper, ensemble averaging is necessary. This isdone by repeatedly probing the flow at some fixed phafter the respective kick. Averaging over the results at ph01 will give Reusat , and averaging over the results at phaDt2 will give Relsat , etc.We suggest to perform a periodically kicked turbulenexperiment and to measure Resat as a function of bothA andf. To our knowledge, it would be one of the first ones wsome active control on the type of forcing. Immediate qutions to ask are: Does the level of saturation for largeindeed only depend on the productA f as suggested by Eqs~11! and ~13! or do boundary effects carry on into thstrongly turbulent central regime and cause a more surelation? If so, an application of the so commonly used vume forcing for turbulence becomes more questionable.the ~scaling! relations between the quantities introducedthis paper, e.g., Rel ,usat( f ,A) offer a new way to measure intermittency effects? What modifications arise if forcing adecay do not decouple as assumed in this simple modelThe author thanks L. Biferale, M. Brenner, B. Eckharand F. Toschi for discussions. This work is part of thesearch program of the Stichting voor Fundamenteel Ondzoek der Materie~FOM!, which is financially supported bythe Nederlandse Organisatie voor Wetenschappelijk Onzoek~NWO!. This research was also supported in part byEuropean Union under Contract No. HPRN-CT-2000-001and by the National Science Foundation under GrantPHY94-07194. We also thank the Institute of TheoreticPhysics in Santa Barabara for its hospitality.tter@1# M. Faraday, Philos. Trans. R. Soc. London52, 319 ~1831!.@2# L. A. Crum, Phys. Today47, 22 ~1994!; B. P. Barber, R. A.Hiller, R. Löfstedt, S. J. Putterman, and K. R. Weninger, PhRep.281, 65 ~1997!.@3# A. Pumir, Phys. Fluids8, 3112~1996!.@4# D. Lohse, Phys. Rev. Lett.73, 3223~1994!.@5# M. Smith, R. J. Donelly, N. Goldenfeld, and W. F. Vine.Phys. Rev. Lett.71, 2583~1993!.@6# R. Labbe, J. F. Pinton, and S. Fauve, J. Phys. II6, 1099~1996!.@7# R. Camussi, S. Ciliberto, and C. Baudet, Phys. Rev. E56, 6181~1997!.@8# Th. Gebhardt, S. Grossmann, M. Holthaus, and M. Lo¨hden,Phys. Rev. E51, 360 ~1995!.@9# H. Effinger and S. Grossmann, Z. Phys. B: Condens. Ma-s.ch.PRE 62 4949PERIODICALLY KICKED TURBULENCE66, 289 ~1987!.@10# A. S. Monin and A. M. Yaglom,Statistical Fluid Mechanics~MIT Press, Cambridge, MA, 1975!.@11# K. R. Sreenivasan, Phys. Fluids7, 2778~1995!.@12# U. Frisch, Turbulence ~Cambridge University Press, Cambridge, 1995!.@13# S. Grossmann, Phys. Rev. E51, 6275 ~1995!; G. Stolovitzkyand K. R. Sreenivasan,ibid. 52, 3242~1995!.@14# G. Zocchi, P. Tabeling, J. Maurer, and H. Willaime, PhyRev. E50, 3693~1994!; for a review on this type of turbulencesee P. J. Zandbergen and D. Dijkstra, Annu. Rev. Fluid Me26, 137 ~1987!.@15# G. S. Lewis and H. L. Swinney, Phys. Rev. E59, 5457~1999!.

Periodically kicked turbulence

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