Turbulence in Viscoelastic Fluids 



to the impact reading due to time averaging of the fluctuating stresses. The 

 paper by Astarita and Nicodemo (4) includes a careful consideration of this 

 effect. 



At this point it is appropriate to consider the recent velocity profile meas- 

 urements, reported by Elata et al. (18), Ernst (22,23), and Meyer (41), in terms 

 of these probe limitations. The data obtained by Ernst and analyzed by Meyer 

 in terms of an altered velocity profile equation appear to suffer from a 7% dis- 

 crepancy between the integrated flow rate as computed from Pitot tube meas- 

 urements and the recorder flow rate, the recorder flow rate being the lower of 

 the two. Data are analyzed by using a velocity correlation of the type 



4 = A log y^ + B . (9) 



u ^ 



The data suggest the non-Newtonian effect appears as a change in the numerical 

 value of B, while A remains the same. In Eq. (9) the minor term contributing 

 to the value of u/u* is b, and over most of the range of the data it seems that 

 the noted changes in b could possibly be attributed to the flow rate discrepancy. 

 Thus, it is not clear if the apparent shifts in the velocity curves are significant 

 or not. 



Qualitative considerations of the curves predicted by the Meyer equation 

 (41) show that gross changes in the shape of the curves, as compared to those of 

 Newtonian fluids, must take place in the sublayer regions. At equal flow rates 

 the velocity profile in the turbulent core is predicted to be flatter in viscoelastic 

 fluids than in Newtonian, since u* is smaller. At the wall, since the shearing 

 stresses are lower, the drag-reducing fluid must also have the flatter profile. 

 The only way to join these two flatter curves is by means of a very sharp curve 

 in the outer sublayer or buffer zones, implying very sharp changes in the pro- 

 file in the buffer zone. This would appear to be unlikely on physical grounds, as 

 it implies very sudden changes in the radial momentum transport rates. As it 

 is doubtful whether useful probe measurements may be made in this region, 

 other measurements may be necessary to resolve these questions and to infer 

 the detailed shapes of the velocity profiles. One possibility would appear to be 

 the use of heat transfer measurements employing drag-reducing fluids of low 

 Prandtl number. 



Notably the solutions used in the above studies were very dilute and exhib- 

 ited only slight drag reduction effects. Other data (4), using solutions showing 

 major drag reduction effects, indicate that if the impact readings are interpreted 

 directly as velocity, the apparent flowrate may be as much as 30% lower than 

 the true flowrate. As these results appear to have obtained using systems simi- 

 lar to those used by Elata et al. (18), the absence of similar effects in the latter 

 study is surprising. Clearly it would be premature to suppose any validity for 

 reported turbulent velocity profile measurements on systems which show sig- 

 nificant viscoelastic effects, as there is at present no rigorous method for dif- 

 ferentiating between the inertial velocity contributions desired and the effects 

 arising due to viscoelastic (normal stress) terms. 



31 



