Polymeric stresses, wall vortices and drag reduction

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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
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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).

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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
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Practical Applications

• Trans-Alaska Pipeline System
• The 800-mile-long Trans Alaska Pipeline System
  (TAPS)
• is one of the largest pipeline systems in the
  world.
• An increase in throughput was attributed by drag
  reduction agent (DRA) which is a long chain
  hydrocarbon polymer.
• 1.44 million bbl./day ? 2.136 million bbl./day

www.alyeska-pipe.com
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Practical Applications

• Firefighting Hoses
• Polyethylene oxide (PEO) was shown in the 1960s
  to be very effective in fire hose streams,
  providing dramatic increases in hose
  stream pressure, reach, and volume.
• Possible Medical Applications
• Kamenva et al. 2004 "Blood soluble drag-reducing
  polymers prevent lethality from hemorrhagic shock
  in acute animal experiments," Biorheology vol 41
  p.53-64
• Unthank et al. 1992 "Improvement of flow through
  arterial stenoses by drag reducing agents," J.
  Surg. Res. vol 53 , p. 625630

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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.
• Lumleys time criterion for onset of DR
• Existence of Maximum Drag Reduction
• Virks asymptote
• Turbulence is still sustained in MDR limit.

Time scale of near-wall turbulence
Polymer relaxation time
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Eddies in Wall Turbulence

• Near-wall vortical structures are closely related
  with production of Reynolds shear stress.
  (Quasi-streamwise vortices, low-speed streaks,
  hairpin vortices, vortex packets, etc)

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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

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Motivations

• Much of the past research focus has been on
  accurate characterization of the influence of
  polymer additives on turbulence statistics and
  mechanistic details of DR have been generally
  inferred indirectly from global statistics.
• To elucidate the polymer DR mechanism addressing
  directly
• the influence of polymers on the structure for
  the Reynolds
• stress producing eddies in turbulent wall flow
• Direct Numerical Simulation (DNS) of fully
  developed turbulent channel flows with polymer
  stresses (FENE-P model)
• Three-dimensional eddy structures by conditional
  statistics
• Kim, et al. Effects of polymer stresses on eddy
  structures in drag-reduced turbulent channel
  flow, JFM (2007)

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Polymer Models

• Freely Jointed Bead-Rod Chain Model
• Probability of finding ith link in a small
  range around ?i and ?i
• Probability density for configuration of
  entire chain

Polymer chain
Bead-rod chain
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Average Tension

• Probability of the end-to-end vector R
• if Nk is large and R lt 0.5L
• The average tension
• For an isothermal process the change of the
  Helmholtz free energy of the chain A ( U - TS
  - kT lnZ) is related to the tension F(c)
  in chain by dA ( F(c) ? dR )
• Hookean spring or entropic spring

Bead-rod chain
Elastic Dumbbell
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Constitutive Equation

• Force balance for each bead (Neglecting the
  inertial term)
• Spring force
• Hydrodynamic drag force
• Brownian force
• The equation of change for the configuration
  tensor ?qq?c
• ? ?c average over all the configuration space

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Governing Equations
Continuity Eq.
Polymer stress
Viscous stress
Momentum Eq.
Constitutive Eq.
Reynolds number
Weissenberg number
FENE-P model
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Computational Details
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Mean Velocity

• LDR Upward shifted profile
• HDR Increase in the slope of log-law

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Reynolds and Polymer Stresses

• Mean momentum equation in the streamwise direction

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Near-Wall Vortical Structures

• Vortical structures in polymer solutions are
• Weaker
• Thicker
• Longer
• Fewer

?ci Swirling strength
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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

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Polymer Work on Turbulent Energy

• Turbulent energy equation (no summation on i)

Polymer work
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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

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Polymer Forces around Vortices

• Polymer force inhibits the Q2 pumping of the
• hairpin vortex

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

• Most natural way to describe the average effect
  of polymers on vortices

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Polymer Counter-torque
Strong streamwise polymer torques oppose the
rotation of both legs of the primary hairpin
vortex.
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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.
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Polymer Torque

• Two-point correlation between streamwise
  vorticity and polymer torque

DR18
Colored contour
Line contours
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Axisymmetric Vortex
z
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Model vortex (axisymmetric)

• Burgers -like vortex
• No strain field (simplify problem)

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Configuration tensors around an
axisymmetric vortex

• Substitution of velocity field into the
  constitutive eqns. gives
• Assuming axi-symmetry

Oldroyd-B model

No azimuthal var- iation, so no azimuthal force
FENE-P model
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Polymer forces and torque

• ?-direction polymer force
• Polymer torque in z-direction

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Polymer torque and axisymmetry
Velocity around QSV
LSE of quasi-streamwise vortex at y20

• Polymer torque

Symbols LSE results Line vortex model with
?0.058 b11
???? QSV ???? axisymmetric vortex (d/dtheta0)
Polymer torque
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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

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Viscoelastic counter-torques and axisymmetry

• Axisymmetric Burgers 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.

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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.

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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 ym50,
where um and vm are selected as the most
contributing Q2 event to ttthe mean Reynolds
shear stress.
Newtonian flow
DR18 flow
DR61 flow
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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 auto-generation
for any of the various initial conditions tested.
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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.
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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.

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Evolutions of initial vortical structure
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Growth rate of volume-averaged Reynolds shear
stress
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Effects of Weissenberg No.
These behaviors are consistent with the onset of
DR and the existence of maximum DR limit in the
fully turbulent polymer DR flows, respectively.
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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?

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