elan egl Luc p he ovi y tr to al p t o d i of os

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elan egl Luc 3; p he ovi y tr to al p t o d i of os. suppression experimenta However, th depletion of two-dimens serve a stron velocity pro tistics, we s exponents l which theref effects are e mer mole- elongation r relaxation dom, zero- , solenoidal incompres- elocity gra- unit tensor. is the zero- olution vis- u models P H Y S I C A L R E V I E W L E T T E R S week ending18 JULY 2003VOLUME 91, NUMBER 3 mensionality, and thus relevant to three-dimensional tur- bulence as well. We also investigate the limit of vanishingly small polymer concentrations, in which the polymer molecules have no influence on the advecting flow. In this case, the t the mechanical friction between the soap film and the surrounding air [15], and plays a prominent role in the energy budget of Newtonian two-dimensional turbulence [16]. It should be remarked that a model that describes more accurately the polymer dynamics is the FENE-P 034501-1 of large-scale velocity fluctuations observed lly has a simple theoretical explanation. e influence of polymers is not limited to the mean square velocity, which is a genuinely ional effect. In the viscoelastic case, we ob- g intermittency, with exponential tails of the bability density.

  • can there

  • polymer additives

  • sufficiently large

  • polymers reac

  • density function

  • ity fluctuations

  • direct numerical

  • square elongation

  • passive polymers

  • dimensional turbulence


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VOLUME91, NUMBER3
P H Y S I C A LR E V I E WL E T T E R S
week ending 18 JULY 2003
TwoDimensional Turbulence of Dilute Polymer Solutions 1 21 Guido Boffetta,Antonio Celani,and Stefano Musacchio 1 Dipartimento di Fisica Generale and INFM, Universita` degli Studi di Torino,Via Pietro Giuria 1, 10125,Torino, Italy 2 CNRS, INLN, 1361 Route des Lucioles, 06560 Valbonne, France (Received 5 March 2003; published 15 July 2003) We investigate theoretically and numerically the effect of polymer additives on two-dimensional turbulence by means of a viscoelastic model. We provide compelling evidence that, at vanishingly small concentrations, such that the polymers are passively transported, the probability distribution of polymer elongation has a power law tail: Its slope is related to the statistics of finite-time Lyapunov exponents of the flow, in quantitative agreement with theoretical predictions. We show that at finite concentrations and sufficiently large elasticity the polymers react on the flow with manifold consequences: Velocity fluctuations are drastically depleted, as observed in soap film experiments; the velocity statistics becomes strongly intermittent; the distribution of finite-time Lyapunov exponents shifts to lower values, signaling the reduction of Lagrangian chaos. DOI: 10.1103/PhysRevLett.91.034501PACS numbers: 47.27.–i Since the discovery of the conspicuous drag reductionvelocity field evolves according to the two-dimensional obtained by dissolving minute amounts oflong chainNavier-Stokes equation with friction, and is therefore molecules in a liquid, turbulence of dilute polymersmooth at scales smaller than the injection length scale solutions has attracted a lot of attention in view of its[9,10]. For passive polymers, space dimensionality plays industrial applications (see, e.g., Refs. [1–3]). The fluidonly a minor role, and our system is an instance of a mechanics of polymer solutions is appropriately de-generic random smooth flow to which the theory of pas-scribed by viscoelastic models that are able to reproducesive polymers developed by Chertkov [11] and Balkovsky the rheological behavior and many other experimentalet al.[12,13] applies. We check this theory against our observations [4]. For example, it has been shown bynumerical results, and find an excellent quantitative Sureshkumaret al.agreement.that the drag reduction effect can be captured by numerical simulations of the channel flow ofTo describe the dynamics of a dilute polymer solution, viscoelastic fluids [5]. Although the parameters used inwe adopt the linear viscoelastic model (Oldroyd-B), those simulations do not match the experimental ones, the 2 qualitative agreement is remarkable, and all the hall-@tu uru rpuruf; marks of the turbulent flow of polymer solutions are (1) recovered in numerical experiments. Following this premise, it is natural to ask whether a two-dimensional viscoelastic model can reproduce the 1T @t ur ru  ru 2: recent results by Amarouchene and Kellay [6] showing that the turbulent flow of soap films is spectacularly (2) affected by polymer additives (see also Refs. [7,8]). Here we show that this is indeed the case, and that theThe velocity fielduis incompressible, the symmetric suppression oflarge-scale velocity fluctuations observedmatrixis the conformation tensor of polymer mole-experimentally has a simple theoretical explanation.cules, and its tracetris a measure of their elongation However, the influence of polymers is not limited to the[14]. The parameteris the (slowest) polymer relaxation depletion of mean square velocity, which is a genuinelytime. The energy sourcefis a large-scale random, zero-two-dimensional effect. In the viscoelastic case, we ob-mean, statistically homogeneous and isotropic, solenoidal serve a strong intermittency, with exponential tails of thevector field. The pressure termrpensures incompres-velocity probability density. As for the Lagrangian sta-sibility of the velocity field. The matrix of velocity gra-tistics, we show that the values of finite-time Lyapunovdients is defined asruij@iujand1is the unit tensor. exponents lower significantly upon polymer addition,The solvent viscosity is denoted byandis the zero-which therefore reduces the chaoticity of the flow. Theseshear contribution of polymers to the total solution vis-effects are expected to be independent of the space di-cosityt1. The dissipative termumodels mensionality, and thus relevant to three-dimensional tur-the mechanical friction between the soap film and the bulence as well.surrounding air [15], and plays a prominent role in the We also investigate the limit of vanishingly smallenergy budget of Newtonian two-dimensional turbulence polymer concentrations, in which the polymer molecules[16]. It should be remarked that a model that describes have no influence on the advecting flow. In this case, themore accurately the polymer dynamics is the FENE-P
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