1. Elastic Turbulence Theory Misha Chertkov Los Alamos Nat. Lab. Polymer Stretching by Turbulence (Statistics of a Passive Polymer) Pure (Re >1) Elastic Turbulence of dilute polymer solution Inertia-Elastic Turbulence (Re>>1,Wi>>1). Drag reduction. Santa Barbara Santa Barbara 02/10/00 Thanks A. Groisman, V. Steinberg, E. Balkovsky, L. Burakovsky, G. Falkovich, G. Doolen, D. Preston, S. Tretiak, B. Shraiman http:/www.princeton.edu/~chertkov/elturb.tex /polyprl.tex

2. Polymer Stretching by Turbulence MC, chao-dyn/9911011 submitted to PRL Balance of forces Models of Elasticity A B C linear (Hook) dumb-bell nonlinear dumb-bell nonlinear chain Scale Separation smallest scale of the flow equilibrium polymer length stretched polymer length >> The question: to describe statistics of passive polymer ? Advection >> Diffusion Batchelor ‘59 Kraichnan ‘68 Shraiman, Siggia ‘94,’95 MC,Falkovich,Kolokolov,Lebedev ‘95 MC,Gamba,Kolokolov ‘94 Balkovsky,MC,Kolokolov,Lebedev ‘95 Bernard,Gawedzki,Kupianen’98 MC, Falkovich, Kolokolov ‘98 Balkovsky,Fouxon ‘99 Statistics of passive scalar advected by the large scale “Batchelor” velocity is understood Passive = statistics of is given

3. Passive linear polymer A CLT for the Lyapunov exponent statistics at First order transition: the polymer stretches indefinitly if advection exceeds diffusion Lumley ‘72 Balkovsky,Fouxon,Lebedev’99 Nonlinearity beats the stretching !! (saddle point parameter) PDF diss. scale PDF

4. Passive nonlinear polymer B MC ‘99 saddle point parameter Passive nonlinear chain C linear conformations are dominant N (number of segments) >>1 is an additional saddle parameter Notice the nonlinear dependance coming from the “equilibration” of the stretching by the nonlinearity PDF

5. Non-Newtonian hydrodynamics of a dilute polymer solution a dilute polymer solution MC, submitted to PRL Elastic part of the stress tensor in the kinetic theory approximation Hydrod. Inter-polymer Stretched Equilibrium scales distance polymer length polymer length >> Scale separation Navier-Stokes equation Regime of a weak elasticity (linear stretching) =>OldroydB model Regime of a strong elasticity (nonlinear stretching)=> local relation between and : Weissenberg number the maximal tension the largest eigenvalue of the direction of the eigenvector n- is the polymer solution concentration N>>1- is the dimensionless polymer length nondeg. deg. constitutive equation

6. Pure Elastic Turbulence (experiment) Groisman, Steinberg ‘96-’99 Power spectra of velocity fluctuations pure solvent d=10mm d=20mm Wi=13 Re=0.7 Swirling flow between two parallel disks transition to turbulence 80ppm polyacrylamide+ 65% sugar+1% NaCl in water

7. Pure Elastic Turbulence (theory) Elastic dissipation >> Viscous dissipation, Advection + constitutive equation Nonlinear diffusion poor-man scaling K- is the pumping amplitude of

8. Inertia-elastic Turbulence (instead of conclusions) Energy containing scale Viscous (Kolmogorov) scale Increase of n - polymer density Dissipation due to elasticity at the Kolmogorov scale is less then the viscous dissipation The drag reduction (dissipation dominated by the elasticity onset) The energy is dissipated at the elastic scale Polymers start to overlap each other (the kinetic approximation fails) According to Lumley’69 the increase in bulk dissipation (viscosity) is accompanied by a swelling of a boundary layer, that leads to the drag reduction