Turbulence in Viscoelastic Fluids 



as noted in Fig. 3. Evidently the major effects of interest do not occur in this 

 low-wavenumber region. 



At high wavenumbers, a much more difficult analysis is necessary in view 

 of the much greater mathematical complexity of Eq. (1) as compared to Eq. (2) 

 or other expressions of this kind. However, a beginning may be made in the 

 following manner. It has been noted (7,47,56) that the predominant kinematic 

 process at large wavenumbers is the "stretching" of fluid elements. This proc- 

 ess may be characterized by defining, at a given wavenumber, a stretch rate as 



, = X (6) 



in which V denotes the relative elongational velocity and x the length of the fluid 

 element under consideration. Batchelor (7) has shown that the inverse of the 

 characteristic stretch rate may be identified with a time scale characteristic of 

 the dissipative eddies in a turbulent flow, so that one may define a Deborah 

 number characterizing the dissipative process as 



Separately it has been shown analytically (2,13,34) and in part confirmed 

 experimentally (6) that the resistance offered to stretching by a fluid element of 

 viscoelastic material is much larger than that of a Newtonian fluid and is de- 

 pendent exponentially on the dimensionless grouping defined in Eq. (7). For ex- 

 ample, in the case of a deformation of a two-dimensional area element described 

 by Eq. (1) it may be shown readily that the ratio of stresses developed in a 

 viscoelastic fluid to those in a Newtonian fluid is of the form 



(t ) - 



'"' viscoelastic 1 /q\ 



Thus, it is noted viscoelastic materials possess a maximum stretch rate at 

 which the tensile stresses would increase without limit. This conclusion has 

 been shown to be independent of the form of the constitutive equation used to 

 portray the fluid properties (2). 



K we now consider a given turbulent flow field, the energy levels available 

 to deform a fluid element are fixed at some finite value by inertial effects, from 

 which it follows, in view of Eq. (8), that the rates of stretching will be far lower 

 than in a Newtonian fluid. Equivalently, since the stretching processes for a 

 Newtonian fluid are primarily associated with the dissipative eddies, one may 

 predict the increased resistance to stretching to inhibit much of the small-scale 

 high-frequency turbulence. In view of the form of Eq. (8) it would appear that 

 the elastic effects would serve to cut off rather suddenly the high-wavenumber 

 end of the spectrum, starting at eddies with time scales of the order of the fluid 

 relaxation time. Several qualitative observations based on observing the spread 

 of a dye filament (25,50,52) have shown that the structure of the turbulent flow 

 is grossly altered in drag reducing systems in just this way: the dye filaments 

 spread as large discrete "lumps" of material, whereas in a Newtonian fluid the 



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