Polymeric stresses, wall vortices and drag reduction Ronald J. Adrian Mechanical and Aerospace...
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Transcript of Polymeric stresses, wall vortices and drag reduction Ronald J. Adrian Mechanical and Aerospace...
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?