Elastic stresses in turbulent flow with polymers: their role in turbulent drag reduction and ways to measure Victor Steinberg Department of Physics of Complex Systems, The Weizmann Institute of Science May 8 th, 2009

Turbulent drag reduction is characterized by the friction factor that can be reduced up to 5-6 times by adding high molecular weight polymer molecules (usually M= and concentration of ppm)

Two basic theories and their misconceptions Lumley (~1970) and further refinements 1.Coil-stretch transition does occur in random flow-only then polymer can modify flow 2.Buffer layer, where polymer mostly stretched→highest extension strain rate→ highest viscosity 3.Time criterion → Wi 4.Since L<η-diss length, polymer “feels” just small scales→ spatially-smooth, random in time flow De Gennes (~1990) and further refinements ! ? ! ! 1.No coil-stretch transition in random flow 2.Even partially stretched polymer stores elastic energy 3.Elastic energy~visc diss gives r* at which spectra is modified 4.Based on (3) and also el energy~turb energy gives explanation for onset drag reduction and maximum drag asymptote ? ?

Numerical simulations of the last decade give a great insight into the problem: (i)observation of drag reduction in full scale numerical simulations in a turbulent channel flow of viscoelstic fluid (Beris et al 1997 and later on) (ii)role of elastic stresses and coil-stretch transition in drag reduction in turbulent Kolmogorov flow, which reflects in universal function for the friction coefficient (Boffetta, Celani, Mazzino 2004) (iii)Stretching of polymer in turbulent shear f low (Davoudi,Schumacher 2006) Theory: first quantitative theory, which take into account the elastic stresses, makes predictions about elastic waves and spectra and links together elastic turbulence and mhd (Lebedev et al 2000, 2001, 2003) Conclusion: measurements and characterization of polymer stretching and elastic stresses in turbulent flow are the key issues to solve turbulent drag reduction problem

Hydrodynamics of polymer solutions Equation of motion: Constitutive equation for Oldroyd-B model  Full stress tensor-  For solvent part-  For polymer part Linear relax linear

nonlinearity relaxation and convective derivative: elastic nonlinearity Weissenberg number Reynolds number nonlinearity dissipation

Three limiting cases: Hydrodynamic turbulence Elastic turbulence Turbulent drag reduction Elastic stress is the main source of nonlinearity at large Wi and low Re. Elastic turbulence is a random flow solely driven by the nonlinear elastic stresses generated by stretched polymers in the flow of a dilute polymer solution. It can be excited at arbitrarily small Re (being independent of it), if Wi exceeds a critical value. Elastic turbulence is a model system to understand more complicated phenomenon- turbulent drag reduction

Elastic Turbulence

Elastic Turbulence A. Groisman & V. Steinberg, Nature 405, 53 (2000) Randomness (in time), spatially smooth flow fields Sharp growth of the flow resistance Algebraic decay of power spectra Efficient mixing Random in time, spatially smooth flow field Thus, elastic turbulence is a dynamic state solely driven by the nonlinear elastic stresses at small Re (being independent of it) and at Wi > Swirling and channel flows

ppm Symbols Wi Square Circle Triangle Triangle-down Diamond Triangle-left Triangle-right Hexagon ↓ Local Weissenberg number

Concentration dependence of local Weisseberg number from horizontal plane velocity measurements Concentration dependence of local Weisseberg number from vertical plane velocity measurements

Theory of polymer dynamics in turbulent or chaotic flow R< dissipation length Dynamics of stretching is governed by the velocity gradient only Dynamics of stretching of a fluid element in random flow is described as Since polymer molecule follows deformation of a fluid element, it can be stretched significantly even in a random flow, if the velocity correlation time is longer than the relaxation time, λ ( Lumley(1970 ) E. Balkovsky, A. Fouxon, V. Lebedev, PRL 84, 4765 (2000) M. Chertkov, PRL 84, 4761 (2000) where γ is the Lyapunov exponent.( γ≈ )

Statistics of polymer molecule extensions: Coupling between statistics of polymer stretching and dynamics of stretching of a fluid element in a random flow  R<< Dynamic equation for the end-to-end vector R=Rn: Stationary probability distribution function of the end-to-end length R: At α  of right tail the majority of molecules is strongly stretched. At α>  of left tail the majority of molecules has nearly equilibrium size. α changes sign at γλ=1. At γλ 0; at γλ>1  α<0. where Thus, Wi ’ =  λ plays a role of a local Weissenberg number for a random flow and the condition  =0 can be interpreted as the criterion for the coil-stretch transition in a random flow that occurs at Wi’=1.

Experimental Setup r 1 = 2.25 mm r 2 = 6 mm d = 675  m d/r 1 =  m from center 100  m above cover glass Polymers: λ-DNA and T4DNA Single Polymer Dynamics: coil-stretch transition in a random flow. S. Geraschenko, C. Chevallard, V. Steinberg, Europhys. Lett. 71, 221 (2005) Y. Liu and V. Steinberg, submitted (2008, 2009) and in preparation

Polymer solution: μg/mL of T4DNA with 35% sucrose; Rotation speed 120 rpm; Wi=45.8, local Wi=13.4

Criterion of the coil-stretch transition: 1/sec sec The experimental value for the critical Weissenberg number and should be compared with theoretical prediction:

Determination of of the coil-stretch transition from velocity field and statistics of polymer stretching measurements Elastic instability onset

Force-Extension Relationship of WLC C. Bustamante, J.F. Marko & E.D. Siggia, Science 258, 1126 (1992) Polymer stretching in elastic turbulence at large Wi>>1

Polymer Solution T4 DNA:  10 8 Daltons (165.6 kbp). T4 DNA solution in a buffer (10 mM Tris-HCl, 2mM EDTA, 10mM NaCl) with sucrose. Fluorescent T4 DNA is also used as probes (10 -3 ppm): contour length 70  m when stained with YoYo-1, which is 1060 persistent length. (s) Sucrose, % T4 DNA,  g/mL

PDF of Polymer Extension at large Wi>>1 PDF of polymer extension at different values of Wi in solution of 25  g/mL T4 DNA with 50% sucrose. The dash line is the calculated distribution of end-to-end distance for a Gaussian chain in coiled state. The PDF is based on the statistics of 700~1100 molecules for each Wi. (typical polymer conformation at Wi = 2.7, 8.1, and 101.2) (at )

PDF of x/L in elastic turbulence at the highest Wi for five polymer concentrations Left inset: peak and average polymer stretching versus concentration Right inset: Scaling exponent of PDFs tails α as function of Wi and concentration

Elastic (full symbols) and viscous (open symbols) stresses versus Wi for five concentrations Inset: PDF of inclination angle at Wi/Wi*=7.5 and c≈20ppm

Elastic stresses and local Weisssenberg as a function of concentration Inset: local Weissenberg versus Wi for five polymer concentrations

Normalized elastic stress versus local Weissenberg number for five polymer concentrations and concentration dependence of the elastic stress saturation value on concentration (Inset)

What can be learnt from elastic turbulence relevant to turbulent drag reduction problem: 1.Polymer can be stretched (and even overstretched-DNA) in random flow! 2. Elastic stresses exceeds viscous stresses up to two orders of magnitude in elastic turbulence 3. Elastic stresses are about proportional to local Weissenberg number in elastic turbulence 4.Local Weisssenberg number( and so elastic stresses?) are up to two orders of magnitude exceed those in a bulk! 5.The latter in turbulence can lead to drag reduction, since most of momentum is stored in the stretched polymers (elastic stresses) rather than in Reynolds stresses and not transfer to the wall (nothing to do with sharply increasing longitudinal viscosity!?)

Conclusions Measurements and characterization of polymer stretching and elastic stresses in turbulent flow are the key issues to solve turbulent drag reduction (two fields should be measured like V and H in MHD) What is the next in experiment 1.Distribution of polymer stretching on Lagrangian trajectories in 3D and elastic stresses in 3D in elastic turbulence 2.Development of a new stress sensor to measure directly elastic stresses locally and globally (image tensometry) 3.Direct measurements of statistics and spatial distribution of elastic stresses in turbulent flow (including elastic stress boundary layer and its relation to buffer layer(?), elastic waves and their spectra) What we expect from theory Quantitative predictions of scaling dependence of friction coefficient on Re and Wi Scaling dependence of elastic stresses with Re and Wi Role of nonlinearity of polymer and elasticity and concentration on drag reduction

Saturation of Polymer Stretching: Nonlinearity PDF of polymer extension at different values of Wi in solution of 12.5  g/mL T4 DNA with 58% sucrose. The dash line is the calculated distribution of end-to-end distance for a Gaussian chain in coiled state. The PDF is based on the statistics of 700~800 molecules for each Wi. saturation

A. Groisman & V.Steinberg, Nature 405, 53 (2000) Swirling flow between two parallel disks Random flow at Wi>>1 & Re→0. S. Gerashchenko, C. Chevallard & V. Steinberg, Europhys. Lett. 71, 221 (2005) DNA (21  m) Coil-stretch transition: Wi local = 0.77±0.20 Flow stretches polymer, it produces elastic stress, which will feed back to the flow and change the flow properties at strong back reaction. Elastic Turbulence & Polymer Stretching

Longest Relaxation Time of DNA For Hookean dumbbell model: σ ~ = Aexp(-t/ )+B Relaxation of 14 individual T4 DNA molecules averaged together in a solution of 25  g/mL T4 DNA with 50.0% sucrose.

Saturation of Polymer Stretching: Nonlinearity & Concentration Peak extension of T4 DNA as a function of Wi.Saturated PDF at different concentration.

Saturation of Polymer Stretching: Nonlinearity PDF of polymer extension at different values of Wi in solution of 25  g/mL T4 DNA with 50% sucrose. The dash line is the calculated distribution of end-to-end distance for a Gaussian chain in coiled state. The PDF is based on the statistics of 700~1100 molecules for each Wi. saturation

Turbulent Drag Reduction by adding flexible polymer molecules For years the experimental and theoretical studies on turbulent drag reduction were concentrated on extracting relevant information from structure and statistics of velocity field measurements. Only recently it was suggested to investigate also an elastic stress field, which has direct relation to changes of turbulent flow by adding polymers. Experimentally we study structure and statistics of the elastic stress field in a related elastic turbulent flow. We found unexpected for this flow boundary layer for elastic stresses and strongly non-uniform distribution of elastic stresses in the flow. We develop currently similar approach for turbulent drag reduction problem

viscous scale pumping scale inertial range viscous scale Balkovsky, Fouxon, Lebedev ’00 Fouxon, Lebedev ‘03 advection relaxation polymer stress relaxation -advection ballance scale Scales smaller than are slaved by the larger scales eddies: a)damped “Alfven”-waves b) smaller scales theory is linear/passive c) emergence of new range – nonuniversal scaling viscous scale pumping scale Turbulence in dilute polymer solution

Theory of Elastic Turbulence E. Balkovsky, A. Fouxon, V. Lebedev, PRE 64, (2001) A. Fouxon, V. Lebedev, Phys. Fluids 15, 2060 (2003)

In the framework of molecular theory polymer stress tensor can be expressed as where n is polymer concentration,  is relaxation time, and end-to-end vector and the average conformation tensor In Hookean approximation of polymer elasticity and by neglecting thermal noise one gets uniaxial tensor Then equations of motion can be rewritten in the form similar to MHD equations with zero magnetic resistance and linear damping at Re >1 (1) (2)

Spatially smooth temporally random flow: (i) in inertial turbulence at scales below dissipation length (ii) in elastic turbulence 1. Velocity power spectrum decays faster than 2. Smallness of higher order space derivatives in velocity field or in cross-correlation function of velocity field. Thus, Batchelor regime Properties of smooth random flow:

Problem of small scale fluctuations v, B’ based on Eqs. (1,2) is reduced to linearly decaying passive field problem advected in large scale fields V,B that leads to power law velocity spectrum (spherically normalized) Velocity Spectrum in Elastic Turbulence from experiment Eqs. (1,2) show instability at Wi>1 that leads to random statistically steady state. The latter is stabilized probably due to back reaction of stretched polymers on velocity field (Eq.(2)). In the case, when viscous and relaxation dissipations of the same order, one gets Then elastic stress can be estimated as and saturates as well as rms of velocity gradients

What is the source of high flow resistance in the elastic turbulence? Momentum balance equation: wall stress solvent shear stress Reynolds stress Polymer stress (mostly elastic stress) Solvent shear stress: the same as in a laminar shear flow, if averaged across the layer. Reynolds’ stress: low fluctuating velocities, low contribution into R. stress  less than 0.5% of in the standard set-up, and 16 times less in the 1:4 set-up. Thus, the whole increase in flow drag is due to polymer stress τ. A. Groisman & V.Steinberg, PRL 86, 934 (2001)

-normalized rms vorticity at different radial locations in the bulk of the cell from r/R=0.2 till 0.66 (80 ppm) Elastic stresses Local Weissenberg number T. Burghelea, E. Segre, V. Steinberg, PRL 96, (2006) 80 ppm PAAm

Rms velocity grad Boundary layer Velocity boundary layer 80 ppm PAAm Local Weissenberg number

Dependence of rms of the velocity gradient in a boundary layer vs Wi (80 ppm PAAm) Local Weissenberg number

New characteristic spatial scale-boundary layer width: So the scale can be related only to scales of elastic stresses: in passive scalar problem one has ; while in elastic turbulence

r = 1mm d = 200  m

Single Polymer Dynamics: coil-stretch transition in a random flow. S. Geraschenko, C. Chevallard, V. Steinberg, Europhys. Lett. 71, 221 (2005) Y. Liu and V. Steinberg, submitted (2008, 2009) and in preparation