Fabula 



viscosity increase. Similar behavior was found for the lower P301B concentra- 

 tions, except that the viscosity spectral shifts was not so large. The theoretical 

 prediction for the absence of any non -Newtonian effects is given in the following 

 section. 



DISCUSSION 



A theoretical estimate of the requirements for viscoelastic effects on the 

 grid-turbulence spectrum can be obtained as follows. With the molecular theo- 

 ries of viscoelasticity of dilute polymer solutions, a spectrum of relaxation 

 times is associated with the solvated macromolecular coils. When the theoreti- 

 cal results for infinitesimal oscillatory shear are generalized in order to obtain 

 materially objective forms, as in the work of Spriggs and Bird (11), the same 

 characteristic times appear, as long as one assumes that the coils are subject 

 only to small perturbations. We will use the molecular theory value of the ter- 

 minal (largest) molecular-coil relaxation time t^ as the best available estimate 

 for the largest viscoelastic time of the dilute solutions. 



A variety of characteristic times can be associated with turbulent flows. 

 With respect to the effects of viscoelasticity upon grid turbulence, the time 

 characteristic of the spectral transfer of energy from larger to smaller eddies 

 would seem to be appropriate. This is the Onsager time of inertial spectral 

 transfer, as used by Corrsin (12): 



where E(/<) is the three-dimensional energy spectrum function and k is the 

 wavenumber vector magnitude, Nondimensionalizing with dissipation variables, 

 we obtain, for k^ = (e/v^)^""^ , 



t^(e/v)^^^ ^ [k/kj)^ E(k)/( £1^5)1/4]- 1/2 ^ (7) 



where (e/V)^/^ is the rms vorticity in isotropic turbulence. Figure 23 gives 

 the plot of Eq. (7) using a Heisenberg spectrum for E(k), The curve is taken 

 from Corrsin's report (12), Also shown are points based on measurements of 

 E(/<) in grid turbulence by Uberoi (13), The Lagrangian integral time scale of 

 strain-rate fluctuations would seem to be necessarily of the same order or 

 larger than t^, which can be interpreted as the time for energy associated with 

 wavenumber k to be transferred to wavenumber 2k . 



Corrsin also considers two other characteristic times: t^^ for energy ex- 

 change due to pressure-velocity correlations and t^ for viscous dissipation, 

 t^ has to be assumed to be approximately equal to t^, t^ becomes appreciably 

 smaller than t^ only for k/k^ > 1. In this region, we can expect that the effects 

 of viscoelasticity to be difficult to separate experimentally from purely viscous 

 non-Newtonian effects. 



Thus as the viscoelastic relaxation time, t , is increased from values far 

 smaller than the minimum characteristic times associated with the turbulence, 



66 



