Grid Turbulence in Dilute High-Polymer Solutions 



Spectral Shift Due to Viscosity Increase 



The comparison of the 4j{k) spectrum for the solution with the correspond- 

 ing spectrum for water must be made with allowance for viscosity difference. 

 Non-Newtonian behavior is indicated only if the spectral changes cannot be cor- 

 related with the predictions of the Newtonian-fluid scaling laws for the effects 

 of a simple viscosity increase. 



As shown in Ref. 1, the good agreement of the normalized dissipation spec- 

 tra measured in air and water, with R,^ varying from 40,000 to 68,000, indicates 

 that the effect upon 0(k) in the dissipation range of a change in v should be pre- 

 dicted by assuming invariance of the function* 0(k)/(ei/5)^'"' measured at a 

 particular x/m for water and considered as a function of k/k^. Thus the spec- 

 tral shift in 0(k) due to a change in v arises only from the changes in k^^ and 

 {ev^)^'"^ . Furthermore because the wind-tunnel data also imply that eM/Ug^ for 

 a given x/M is essentially independent of R^, then e will be assumed to be fixed 

 for a given x/m. This assumption should be valid as long as the energy - 

 containing scales are largely separate from the dissipation scales, as discussed 

 in Ref. 1. With 



5(k/kj) = 0(k, x/M, v)/{ev^)^'"^ , 



where 0(k, x/m, v) has been measured for a particular grid, x/M and v , the ef- 

 fect of changing v to v' in the dissipation range should be given by 



0'(k, x/M, v') = ^Ck/k^Xev'S)^^" , (3) 



where k^ = {e/v'^) '' and where e remains constant. For conciseness, this 

 scaling law will be referred to as Kolmogoroff similarity with constant dissipa- 

 tion. If m is the logarithmic derivative of ^(k/kj) with respect to k/k^, it can 

 be shown that 



4>'/4>^ {v'/v)^'''''^'\ (4) 



Then the spectral level shift in db is approximately 



y (m + I) log (.Vv). (5) 



Thus if there is an inertial subrange with m = -5/3, the spectral level shift due 

 to viscosity in that inertially dominated range is zero as expected. Of course, 

 viscosity increase would not affect the even lower wavenumber range which is 

 also inertially dominated. Thus Eq. (5) applies only where m < -5/3. The upper 

 wavenumber for which spectral measurements were made was typically of the 

 order of k^. Since m ~ -1 for k k^ = 1, the factor (m + 5/3) varies from to 

 roughly -16/3 over the applicable range in this work. Thus for a 10% increase 

 in V , the spectral shift will be about -1.6 db. Since the viscosity increases due 



''Because of equipment limitations, e was not determined but was replaced by 

 the isotropic viscous dissipation rate calculated from 0(k). 



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