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

Von-Karman’s “logarithmic law of the wall”

where

For one observes a viscous sub layer

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

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.

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

The visco-elastic caseThe effect of the polymer enters in the form of a conformation tensor

The derivation of the MDR (for large concentration)

For the derivation of the MDR we consider the limit of large Deborah number

Consequence

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?

The ab-initio calculation of the MDR