C • STATISTICAL THEORIES OF TURBULENCE 



relaxed the requirement and assumed that the similarity spectrum might 

 be adequate only for the calculation of higher moments of F{k). If the 

 similarity spectrum is accurate only for the calculation of ]QK^F{K)dK and 

 higher moments, Goldstein shows that the law of decay becomes 



u\t -u) = a + hit - to) + c{t - toy (14-23) 



This includes one more constant than Eq. 14-17. Further generalization 

 involving higher powers of t — to is immediate. 



Comparison of the laws of decay with experiments will be made in the 

 next article. 



C,15. The Process of Decay. We shall now examine the whole 

 process of decay and compare the above theoretical laws with experi- 

 ments, whenever such evidence is available. 



0.20 



0.16 



0.12 



0.08 



0.04 



200 



400 600 



x/M 



800 



1000 



Fig. C,15a. Change of vorticity scale during a 

 decay process at low Reynolds numbers. 



For small Reynolds numbers of turbulence, the nonviscous range can- 

 not be expected to occur. The process of decay may be described by 

 adopting the law (Eq. 14-17) for the first part of the decay process and 

 the law (Eq. 14-14) for the last part. This is shown by the experimental 

 results^^ of Batchelor and Townsend (Fig. C,15a). The slope of the curve 

 (X^, vi) begins with a value 10 and ends with a value 4. 



15 The experimental agreement in this case should be accepted with some reserva- 

 tion, since so little data are available. See \S0,31\ for detailed discussions. 



< 230 ) 



