Transcript Active and hibernating turbulence in channel flow of Newtonian and viscoelastic fluids Li Xi and Michael D.

Active and hibernating turbulence in channel flow of Newtonian and
viscoelastic fluids
Li Xi and Michael D. Graham
University of Wisconsin-Madison
Abstract
Channel flow: near-wall structure
Re=3600
Wi=0 (Newtonian)
Hibernating
(only bottom half
shown)
Minimal spanwise
size vs. Wi
Wi = 80, b=0.97, Ex=103
57% DR
Virk, P. S., AIChE J. 1975
Simulations in minimal flow units
MDR (maximum drag reduction): insensitive to
polymer-related properties.
FENE-P dumbbell model
b = maximum extensibility
Maximum drag reduction
Newtonian
38%
DR
55%-69%
DR
Virk, P. S., AIChE J. 1975
Warholic, M. D. et al. Exp.
Fluids, 1999
• Minimal periodic box that
sustains turbulence.
• Isolates the dynamics of
individual coherent
structures.
• Applied in Newtonian flow by
Jiménez and Moin (J. Fluid
Mech., 1991), and Hamilton,
Kim and Waleffe (J. Fluid
Mech., 1995)
Time scale for convection of momentum
• Universal mean velocity profile: the Virk profile
• Features of structure in MDR regime:
• Reynolds shear stress is greatly reduced across the
channel,
• Weak streaks and streamwise vortices
• Weak streamwise dependence
• Spatiotemporal intermittency
Time scale for diffusion of momentum
Time scale for polymer relaxation
Time scale for polymer deformation
Solvent shear viscosity
Top
wall
Time series of mean velocity and wall shear rate
Re=3600
Wi=0
Bottom
wall
Experiments
Active and hibernating turbulence
Top wall
• Red:
constant
vortex
strength Q.
• Green:
constant vx.
• What sustains turbulence at MDR?
• Why is there an upper limit of DR?
• Why is MDR universal (w.r.t. polymer properties)?
• What causes the completely different turbulent
spatiotemporal dynamics in the MDR regime?
• Can we achieve high drag reduction without polymer
via active control?
Near laminar-turbulence
transition.
Mean flow rate
vs. Wi
Active
Drag reduction by polymers-phenomena
Schematic
Flow structure in hibernation
Selected snapshots during a hibernation
period (Wi=29):
Wi=29
b=0.97
Ex=103
Bottom wall
Turbulent channel flow of drag-reducing polymer
solutions is simulated in minimal flow geometries.
Even in the Newtonian limit, we find intervals of
“hibernating” turbulence that display many features of
the universal maximum drag reduction (MDR)
asymptote observed in polymer solutions: weak
streamwise vortices, nearly nonexistent streamwise
variations and a mean velocity gradient that
quantitatively matches experiments. As viscoelasticity
increases, the frequency of these intervals also
increases, while the intervals themselves are
unchanged, leading to flows that increasingly resemble
MDR.
MFU results: overview
Observation: Normal “active” turbulence is
punctuated by long periods of low wall shear rate
accompanied by increasing mean velocity and
followed by a burst in shear rate
• “Hibernation”
• Found in both Newtonian and viscoelastic flows
• Quantifiable with identification criteria:
Hibernation:
• Very weak vortices and streaks,
• Very weak streamwise dependence,
• “Near-Virk” mean velocity profile -- even in
Newtonian flow!
Hypothesis 1: on intermittency and MDR
Polymer
stretches
Active turbulence: substantial
stretching of polymer
molecules
New turbulent
fluctuations grow,
destabilizing
hibernation and
generating active
turbulence
Prediction: hibernation
dominates as MDR is
approached.
Hibernating flow: weak vortices,
Reynolds stresses and
streamwise waviness.
Polymers relax.
Polymer stretching
suppresses active
turbulence and causes
hibernation.
Increased
Frequency
as Wi ↑.
Polymer
relaxes
Times scales of hibernation
As Wi increases:
• Lifetime of active
turbulence is
reduced.
• Duration of
hibernation is
almost invariant.
• Hibernation
takes a larger
fraction of time.
average
duration of
active
turbulence
Hypothesis 2: hibernation is a saddle point
Exact coherent states
(ECS) in plane
Couette flow
fraction of
time taken
by
hibernation
Upper branch
?
average
duration of
hibernation
Lower branch:
very weak
streamwise
dependence
Waleffe, F., Phys. Fluids, 2003
Total shear viscosity
Polymer extensional viscosity
Solvent extensional viscosity
Viscoelasticity compresses the lifetime of active
turbulence intervals while having little effect on
hibernation itself.
Acknowledgments
Fabian Waleffe, John Gibson, NSF-CBET