Elastic stresses in turbulent flow with polymers: their role in turbulent drag reduction and ways to measure

Victor SteinbergDepartment of Physics of Complex Systems,

The Weizmann Institute of Science

May 8th, 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 10-100 ppm)

710

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 → Wi4. Since L<η-diss length, polymer “feels” just small

scales→ spatially-smooth, random in time flow

De Gennes (~1990) and further refinements

1/ turbp

!

?

!

!

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)

)(Re 3/23/1 WiFf

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:

/ρτp/ρV) V(tV ~

Constitutive equation for Oldroyd-B modelFull stress tensor-

For solvent part-

For polymer part

PS τττ ~~~

TSS )V(Vητ

~

TP

PP )V(Vη

DtτDλτ

~~

Linear relax

linear

PT

PPPP τ)V()V(ττ) V(t

τ

Dt

τD ~~~~~

 nonlinearity relaxation L/VλWi

and convective derivative:

elastic nonlinearity

Weissenberg number

VLRe

Reynolds number

nonlinearity dissipation

Three limiting cases:

1;1Re.3

0Re;1.2

0;1Re.1

Wi

Wi

Wi 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

Drag Reduction

Wi

Re

Re c

Wic

Elastic Turbulence

Inertial Turbulence

Wi

ReUL

rateshear :

time relaxationpolymer :

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 > cWi Swirling and channel flows

0.980 0.985 0.990 0.995 1.0000

5

10

15

20

25

30

35

40

Boundary Layer - 300 ppm

(dv

/dr)

rms

rppm 100 300 500 700 900 1600 2300 3000 Symbols

Wi

40 24.6 20.6 18 17.1 30.4 78.6 91 Square50 32.9 27.6 24 23.0 45.6 89.8 104 Circle60 41.2 34.4 30 28.6 60.8 134.7 114.4 Triangle70 49.4 41.4 36 34.2 76 179.6 156 Triangle-down80 57.7 48.2 42 40.0 95 224.5 208 Diamond90 65.9 55.2 48 45.6 114 269.4 312 Triangle-left100 74.1 62 54 51.5 95 314.3 416 Triangle-right110 82.3 69 60 57.1 152 359.2 520 Hexagon

↓

rms

bulk

rms

r

V

r

V

100

max

bulkbl WiWi 100max

Local Weissenberg number

0 200 400 600 800 10000

5

10

15

Wi lo

cal

C (ppm)

0 200 400 600 800 1000

5

10

15

20

Wi*

loca

l

C (ppm)

From vertical measurement

rms

=<(dv/dz)2>

Concentration dependence of localWeisseberg number from horizontal plane velocity measurements

Concentration dependence of localWeisseberg number from verticalplane velocity measurements

Theory of polymer dynamics in turbulent or chaotic flow

R< dissipation length

Dynamics of stretching is governed by the velocity gradient only

rtr

VtrV

j

i

)(),(

Dynamics of stretching of a fluid element in random flow is described as

)exp()0()( tRtR

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)

rms

j

i

r

V

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 anddynamics of stretching of a fluid element in a random flow

0R R<< maxR

i

ijji

RVRR

dt

dDynamic equation for the end-to-end vector R=Rn:

10)(

ii RRR

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<1 α>0; at γλ>1 α<0.

)( 1 where

Thus, Wi’=λ plays a role of a local Weissenberg number for a random flowand 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

r1 = 2.25 mmr2 = 6 mmd = 675 md/r1 = 0.3300 m from center100 m above cover glassPolymers: 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:100.64 μg/mL of T4DNAwith 35% sucrose;Rotation speed 120 rpm;Wi=45.8, local Wi=13.4

)t,(Plnt)(S ii 1

Criterion of the coil-stretch transition:

015.007.0 cr 1/sec

1.011 sec

The experimental value for the critical Weissenberg number

2.077.0~

criWand should be compared with theoretical prediction:

1iW~

1 10 100

0.01

0.1

1

10

0 1 2 3 4

-6

-4

-2

0

2

4

6

(d

V/

dr)

rms

Wi

coil-stretch transitionWi

cr=0.7(+/-0.3)

1/Wiloc

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

crWi

Elastic instabilityonset

Force-Extension Relationship of WLC

0.0 0.2 0.4 0.6 0.8 1.00.1

1

10

100

Marko - Siggia (approximation) Exact WLC

FA

/kBT

R/L

L

R

LRkT

FA

4

1

)/1(4

12

C. Bustamante, J.F. Marko & E.D. Siggia, Science 258, 1126 (1992)

0 5 10 15 200.0

0.2

0.4

0.6

0.8

1.0

6.29 g/mL 12.58 25.16 50.32 100.64

x p/L

Wi/Wic

Polymer stretching in elastic turbulence at large Wi>>1

Polymer Solution

T4 DNA: 1.076108 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

33.5 58.0 12.5

14.5 50.0 25

12.6 45.0 50

7.8 35.0 100

PDF of Polymer Extension at large Wi>>1

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

0.25

0.30

Wi = 0 2.7 4.0 5.4 6.7 8.1 10.1 20.2 40.5 60.7 101.2

Pro

bab

ilit

y

Fractional extension

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)

1/ avVVrms(at )

PDF of x/L in elastic turbulence at the highest Wi for five polymer concentrationsLeft inset: peak and average polymer stretching versus concentrationRight 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 concentrationsInset: PDF of inclination angle at Wi/Wi*=7.5 and c≈20ppm

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

Pas

p 5

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

locp Wi130)//(

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 turbulence3. Elastic stresses are about proportional to local Weissenberg number in elastic turbulence4. 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 experiment1. Distribution of polymer stretching on Lagrangian trajectories in 3D

and elastic stresses in 3D in elastic turbulence2. 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