Polymeric stresses, wall vortices and drag reduction

Ronald J. Adrian

Mechanical and Aerospace Engineering

Arizona State University-Tempe

“High Reynolds Number Turbulence”, Isaac Newton Institute, Sept. 8-12, 2008

Co-workers

Kim, K, Li, C.-F., Sureshkumar, R. Balachandar, S. and Adrian, R. J., “Effects of polymer stresses on eddy structures in drag-reduced turbulent channel flow,” J. Fluid Mech. 584, 281 (2007).

Kim, K, Adrian, R, Balachandar, S, Sureshkumar, R., "Dynamics of HairpinVortices and Polymer- Induced Turbulent Drag Reduction," Phys.Rev.Lett. 100 (2008).

Toms’ Phenomenon• Toms discovered the phenomenon of turbulent drag reduction

by polymer additives by chance in the summer of 1946, when he was actually investigating the mechanical degradation of polymer molecules using a simple pipe flow apparatus.

• By dissolving a minute amount of long-chained polymer mole-cules in water, the frictional drag of turbulent flow could be reduced dramatically. In pipe flows, for example, the drag could be reduced up to 70 % by adding just a few parts per million (ppm) of polymer.

Toms (1949) Proc. Intl. Congress on Rheology, Sec. II, p. 135

Toms (1977) Phys. Fluids *Address at the Banquet of the IUTAM Symposium

on Structure of Turbulence and Drag Reduction

Main Features of Polymer DR• Onset of Drag Reduction

– There exist critical values of parameters (e.g. polymer re-laxation time, concentration..) above which there is onset of DR.

– Lumley’s time criterion for onset of DR

• Existence of Maximum Drag Reduction

– Virk’s asymptote

– Turbulence is still sustained in MDR limit.

2uτ

νλ >Polymer relaxation time

Time scale of near-wall turbulence

After λ

Stretched polymer

Coiled polymer

Structural changes found in experiments

– Increased spacing and coarsening of streamwise streaks– Damping of small spatial scales– Reduced streamwise vorticity– Enhanced streamwise velocity fluctuations– Reduced vertical and spanwise velocity fluctuations and Reynolds stresses– Parallel shift of mean velocity profile in low DR– Increase in the slope of log-law in high DR

Governing Equations

ˆ ˆij i j cc q q=

0i

i

u

x

∂=

∂Continuity Eq.

Momentum Eq.

Constitutive Eq.

00

Reu hτ

τ ν=

/We

h uτ

λ=

0

sμβμ

=

20

/

qb

kT H=

Polymer stressViscous stress

q

Reynolds number Weissenberg number

FENE-P model

ijτ

Near-Wall Vortical Structures• Vortical structures in polymer solutions are:

• Weaker

• Thicker

• Longer

• Fewer

λci: Swirling strength

Conditional Averaged Flow Field•

– Flow structures associated with the event which most contribute the Reynolds stress– Counter-rotating pair of quasi-streamwise vortex– Hairpin vortex

( , ,0)E m mu v=u

( , , 2 )E m m mu v v=u

0( ) | ( ) E=u x u x u

Polymer Work on Turbulent Energy• Turbulent energy equation (no summation on i)

iE

D 12u

i' 2

Dt=− ui

'u2' dU i

dy−

βReτ 0

∂ui'

∂xk

∂ui'

∂xk

+ ui' fi

'

+ddy

− 12ui

'2u2' +

βReτ 0

ddy

12ui

'2⎛

⎝⎜

⎞

⎠⎟ + ui

' ∂p'

∂xi

Polymer work

Conditional Averaged Flow Field•

– Flow structures associated with the event contributing most to the polymer work– Nearly the same as those associated with large Q2 event at similar y-locations

Largest contribution on Ex>0 Largest contribution on Ex<0

Largest contribution on Ey<0 Largest contribution on Ez<0

0 , 0 ,( ) | ( ) & ( )i i m i i mu u f f= =u x x x

Polymer Forces around Vortices•

– Polymer force inhibits the Q2 pumping of the

hairpin vortex

0'( ) | ( ) E=f x u x u

Velocity

Polymer

force

(u,v) (w,v)

(fx,fy) (fz,fy)

DR=18%

See also De Angelis et al. 2002, Dubief, et al. 2005, Stone, et al. 2002 (ECS laminar)

Polymer Counter-torque

DωDt

= ω • ∇( )u+βRe

∇2ω +∇×1−βRe

∇• τ p

⎛

⎝⎜⎞

⎠⎟

Torquedueto polymer stress1 24 44 34 4 4

Red and blue surfaces denotes a positive and negative polymer torques , respectively. 2 2/ 20ih uτΤ =±

Strong streamwise polymer torques oppose the rotation of both legs of the primary hairpin vortex.

Polymer Counter-torque (cont)

Red and blue surfaces denote positive and negative

polymer torques, respectively. 2 2/ 20ih uτΤ =±

Large positive spanwise polymer torques act against rotation at the heads of downstream and secondary hairpin vortices.

Negative torques are exerted on the primary vortex in a direction such that they reduce vortex curvature and thus the inclination angle of the primary hairpin head.

Polymer Torque• Two-point correlation between streamwise vorticity and polymer torque

DR=18%

' 'x x

Rω ωColored contour

' 'x x

Rω Τ

Line contours

Axisymmetric Vortex z

Model vortex (axisymmetric)

• Burgers ”-like” vortex

No strain field (simplify problem)

vθ =Ωb2

2r1−e

−rb

⎛

⎝⎜⎞

⎠⎟

2⎛

⎝⎜⎜

⎞

⎠⎟⎟

vr=0

vz=0

ω(r) =

1r

∂∂r

(rvθ ) −1r∂vr

∂θ=Ωe

−rb

⎛

⎝⎜⎞

⎠⎟

2

Configuration tensors around an axisymmetric vortex

• Substitution of velocity field into the constitutive eqns. gives

– Assuming axi-symmetry

Oldroyd-B model

FENE-P model

…

No azimuthal var- iation, so no

azimuthal force

1rrc =

…

Polymer forces and torque θ-direction polymer force

• Polymer torque in z-direction

fθ =1−βRe

1r2

∂∂r

r2τ rθ( ) +1r∂τθθ

∂θ+

∂τθz

∂z+

τθr −τ rθ

r

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

=1−βRe

−2Ω rb2

e−

rb

⎛

⎝⎜⎞

⎠⎟

2⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

Τz =1−βRe

1r

∂∂r

(rfθ ) −1r∂fr

∂θ⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

=1−βRe

4Ωr2 −b2

b4

⎛

⎝⎜⎞

⎠⎟e

−rb

⎛

⎝⎜⎞

⎠⎟

2⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

Polymer torque and axisymmetryVelocity around QSV

Symbols: LSE results

Line: vortex model with Ω=0.058 & b=11

QSV

axisymmetric

vortex (d/dtheta=0)

LSE of quasi-streamwise vortex at y+=20Polymer torque

Polymer torque

Viscoelastic Drag Reduction Principle•Drag is reduced by intrinsic viscoelastic counter-torques that

retard the rotation of turbulent vortices

•Counter-torques exist around the vortices only if the flow is non-

axi-symmetric

•Deviations from axi-symmetry occur when the vortex is

embedded in a strain field, e.g.

– Quasi-streamwise wall vortices imbedded in the strain

field of its image vortex

– Bent vortices, i.e. heads of hairpins

Viscoelastic counter-torques and axisymmetry

• Axisymmetric Burger’s vortex

generates zero azimuthal net

force, and hence zero counter

torque.

• Quasi-streamwise vorticies near the wall are not axi-symmetric, so a net torque can be developed.

• The core of the vortex in the head region is not axisymmetric because the flow is faster under the arch of the head than above it. Hence non-zero counter torque also occurs around the arch.

∂τθz

∂θ= 0

θ

∂τθz

∂θ≠ 0

Conclusions

• In fully turbulent flow polymer forces are associated with

the Q2 pumping of the hairpin vortex and the ejection/

sweep motions at the flanks of streamwise vortices in a

that opposes the motion. • They apply counter-torques to the rotation of the

vortices, Within the validity of the FENE-P model, this is

the fundamental mechanism for reducing turbulent

stresses and drag.

Evolution of initial vortical structuresThe initial structure is the conditionally averaged flow field with Q2 event vector, (um,vm,0) of strength =2.0 specified at ym

+=50, where um and vm are selected as the most contributing Q2 event to ttthe mean Reynolds shear stress.

DR=18% flow

DR=61% flow

Newtonian flow

Threshold for the auto-generation

In low DR flow, the threshold kinetic energy for the generation of secondary vortices increases, especially in the buffer layer. For the high-DR simulations we did not observe auto-generation for any of the various initial conditions tested.

Autogeneration occursAutogeneration fails

Newtonian flow

Low DR flow

Effects of polymer stress on auto-generation

To see suppression of the auto-generation by the polymer stresses more directly, we compared the evolution in the absence of the polymer stress from the same initial velocity fields as one of the LDR simulations.

Reynolds shear stress more rapidly increases in the absence of the polymer stress.

2nd Simulation• In the dynamical simulations presented so far, the polymers

were initially stretched or compressed according to the

straining of the conditionally averaged velocity field extracted from a turbulent flow that was already drag-reduced. The

behavior we have found does not necessarily explain the mechanisms that lead up to the occurrence of drag reduction.

• To determine how polymer stresses act to modify turbulence

in Newtonian fluids we imagine creating a fully turbulent flow without polymers, and then abruptly turning the polymer

stresses on.

Evolutions of initial vortical structure

Growth rate of volume-averaged Reynolds shear stress

−u'v'

vol(t) =

1V

−u'v' dV∫

−u'v'vol

(t)

−u'v'vol

(t=0)

Effects of Weissenberg No.

' ' ( 300)

' ' ( 0)vol

vol

u v t

u v t

+

+

− =− =

Onset of reduction on Reynolds shear stress

Asymptotic behavior

These behaviors are consistent with the onset of DR and the existence of maximum DR limit in the fully turbulent polymer DR flows, respectively.

Conclusions

• Polymers cut-off the autogeneration of hairpin eddies, thereby– reducing the number of vortices– inhibiting drag by reducing the coherent stress

associated with hairpin packets.Kim, Adrian Balachandar and Sureshkumar, PRL (2008)

• Future WorkLarge-scale and very-large scale motions account for over half of the Reynolds shear stress in Newtonian flow. How do polymers influence them?