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 e
ddy structures in drag-reduced turbulent channel fl
ow,” J. Fluid Mech. 584, 281 (2007).
Kim, K, Adrian, R, Balachandar, S, Sureshkumar, R
., "Dynamics of HairpinVortices and Polymer- Indu
ced Turbulent Drag Reduction," Phys.Rev.Lett. 10
0 (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 po
lymer molecules using a simple pipe flow apparatus.
• By dissolving a minute amount of long-chained polymer molecules in water, the frictional drag of turbulent flow could be re
duced 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 relaxation time, concentration..) above which there is onset
of DR.
– Lumley’s time criterion for onset of DR
Polymer relaxation
time
 

2
u
Time scale of nearwall turbulence
• Existence of Maximum Drag Reduction
– Virk’s asymptote
– Turbulence is still sustained in MDR limit.
Stretched polymer
After 
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
Continuity Eq.
Momentum Eq.
Constitutive Eq.
cij  qˆi qˆ j
c
u i
0
xi
Viscous stress
ui
u
p

uj i  

t
x j
xi x j
cij
cij
Polymer stress
  ui (1   ) 

 ij 


Re

x
Re

o
j

o


u j

ui
1 
1
 uk
 cik

ckj 
cij   ij 

t
xk
xk xk
We  1  ckk / b

 ij
q
Re 0 
u h
0
Reynolds number
FENE-P model
s

0
We 

h / u
Weissenberg number
q02
b
kT / H
•
•
•
•
•
Near-Wall Vortical Structures
Vortical structures in polymer solutions are:
Weaker
Thicker
Longer
Fewer
ci: Swirling strength
Conditional Averaged Flow Field
•
u(x) | u(x0 )  u E
– Flow structures associated with the event which most contribute the Reynolds stress
– Counter-rotating pair of quasi-streamwise vortex
– Hairpin vortex
u E  (um , vm ,0)
u E  (um , vm , 2vm )
Polymer Work on Turbulent Energy
• Turbulent energy equation (no summation on i)
D
1
2
ui' 2
Dt
  ui' u2'
d 
 
dy 
dU i
dy
1
2

'
i
 ui' f i '
Re 0 xk xk
u u 
'2 '
i 2
Polymer work
u u

'
i

Re 0
d
dy
1
2
'2
i
u
'

' p
  ui x

i
Ei
Conditional Averaged Flow Field
•
u(x) | ui (x 0 )  ui ,m & fi (x 0 )  fi ,m
– 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 Ey<0
Largest contribution on Ex<0
Largest contribution on Ez<0
Polymer Forces around Vortices
•
f '(x) | u(x0 )  u E
DR=18%
– Polymer force inhibits the Q2 pumping of the
hairpin vortex
(u,v)
(w,v)
(fx,fy)
(fz,fy)
Velocity
Polymer
force
See also De Angelis et al. 2002, Dubief, et al. 2005, Stone, et al. 2002
(ECS laminar)
Polymer Counter-torque
 1 

D
 2
   u
  
 p
Dt
Re
 Re

1 4 44 2 4 4 43


Torque due to polymer stress
Strong streamwise polymer torques oppose the
rotation of both legs of the primary hairpin vortex.
Red and blue surfaces denotes a positive and negative
polymer torques i h 2 / u2  20 , respectively.
Polymer Counter-torque (cont)
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.
Red and blue surfaces denote positive and negative
i h 2 / u2  20
polymer torques, respectively.
Polymer Torque
• Two-point correlation between streamwise vorticity and polymer torque
DR=18%
Colored contour
R '  '
Line contours
R ' '
x x
x
x
Axisymmetric Vortex
z
Model vortex (axisymmetric)
• Burgers ”-like” vortex
No strain field (simplify problem)
2
 r 


 b 
b


v  
1 e
2r 


vr  0
2
vz  0
1 
1 vr
 (r) 
(rv ) 
 e
r r 
r 
 r
 
 b
2
Configuration tensors around an
axisymmetric vortex
• Substitution of velocity field into the constitutive eqns. gives
– Assuming axi-symmetry
crr  1
…
Oldroyd-B model
…
No azimuthal variation, so no
azimuthal force
FENE-P model
Polymer forces and torque
• -direction polymer force
f 
1   1  2
1     z   r   r 
r




 2

r
Re  r r
r 
z
r

 
2
 r 


1   2 r  b  

 2 e

Re  b


• Polymer torque in z-direction
1   1 
1 f r 
z 
(rf ) 


Re  r r 
r  
2
 r 
  2
2

1  
r  b   b  

e
4 


4
Re   b 


Polymer torque and axisymmetry
LSE of quasi-streamwise vortex at y+=20
Polymer torque
Velocity around QSV
Symbols: LSE results
Line: vortex model
with =0.058 & b=11
 QSV
 axisymmetric
vortex (d/dtheta=0)
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 nonaxi-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.

  z
0

• Quasi-streamwise vorticies near t
he wall are not axi-symmetric, so
a net torque can be developed.
• The core of the vortex in the head
region is not axisymmetric becaus
e the flow is faster under the arch
of the head than above it. Hence
non-zero counter torque also
occurs around the arch.
  z
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 structures
The 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.
Newtonian
flow
DR=18% flow
DR=61% flow
Threshold for the auto-generation
Low DR flow
Newtonian flow
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 autogeneration for any of the
various initial conditions
tested.
Autogeneration occurs
Autogeneration fails
Effects of polymer stress on autogeneration
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
1
u'v ' vol (t) 
u 'v ' vol (t)
u 'v ' vol (t  0)
u'v ' dV

V
Effects of Weissenberg No.
u ' v ' vol (t   300)
u ' v ' vol (t   0)
Asymptotic behavior
Onset of reduction on Reynolds shear stress
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 Work
Large-scale and very-large scale motions account for
over half of the Reynolds shear stress in Newtonian
flow. How do polymers influence them?