Towards a Scaling Theory of Drag Reduction

Flexible polymers in dilute solution enhance the viscosity in slow flows. But in strong, rapidly varying, shear fields, they behave elastically. A turbulent cascade (from large to small scales) should thus be deeply modified when the elastic stresses become comparable to the Reynold's stress. A (tentative) scaling picture for these effects has been proposed by M. Tabor and the present author1): it involves one unknown exponent n relating polymer deformation (~,) and spatial scales (r) in the cascade. We now show that, depending on the control parameters (turbulent power; polymer concentration and molecular weight) the cascade may proceed according to two "scenarios". In the first scenario 1 ) the smallest Kolmogorov eddy occurs when the chains are only partly stretched. In the second scenario, the smallest eddies display nearly full chain extension: polymer degradation is expected to be much more serious in the latter case. We also transpose these ideas to wall turbulence, in the first scenario. At a distance y from the wail, the smallest eddy scale available r**(y) is a decreasing function of y (very different from the classical Lumley picture, where it is an increasing function of y). The overall result is again an increase of the buffer layer, provided that the polymer concentration exceeds a-very low but finitethreshold. We point out finally that the elastic effects discussed here could be present in many other systems: one amusing (although impractical) example is a binary mixture near a consolute point: the concentration fluctuations should be very similar, in this effects, to a polymer solution with coil size (the correlation length), at the overlap concentration c*.