Drag Reduction by Polymers in Wall-bounded Turbulence Itamar Procaccia The Weizmann Institute of...
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Drag Reduction by Polymers Drag Reduction by Polymers in Wall-bounded Turbulencein Wall-bounded Turbulence
Itamar Procaccia
The Weizmann Institute of Science
Work with: V.S. L’vov, A. Pomyalov and V. Tiberkevich(Weizmann Institute of Science)
R. Benzi, E. De Angelis, C. Casciola(Universita di Roma, I, II
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Von-Karman’s “logarithmic law of the wall”
where
For one observes a viscous sub layer
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In the presence of long chain polymers the mean velocity profile changes dramatically. For sufficiently large concentration of polymers
For smaller concentration of polymers there are non-universal crossovers.
The relative increase of the mean velocity (for a fixed p’) due to the existence of the new law of the wall IS the phenomenon of drag reduction
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Derivation of von-Karman’s log-law of the wall
The momentum balance equation
In the viscous sub-layer
Outside the viscous layer W(y)=const, but we need to know more to find the velocity profile.
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The energy balance equationThe energy is created by the large scale motions at a
rate of W(y) S(y)
It is cascaded down the scales by a flux of energy, and is finally dissipated at a rate
Experimentally, outside the viscous sub-layer
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The visco-elastic caseThe effect of the polymer enters in the form of a conformation tensor
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The derivation of the MDR (for large concentration)
For the derivation of the MDR we consider the limit of large Deborah number
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Consequence
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The prediction is that the effective viscosity is linear in the distance from the wall!
Is this blue-sky dreaming, or is a linear viscosity profile Sufficient for drag reduction?
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The ab-initio calculation of the MDR
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