C • STATISTICAL THEORIES OF TURBULENCE 

 We then obtain a power series for j{r) in the form 



Kr) = 1 +^^ ,2 +/:^) ,4 + . . . (i3_2) 



with 



(_l)„^2y(2n)(0) = 72„ = J^" K2»Fi(/c)d«, W = 0, 1, . . . (13-3) 



In terms of the three-dimensional spectrum, these integrals become 



^2n = ,o^ I iWo^ I o^ / X.''-F{K)dK (13-4) 



(2n-f- l)(2n-|-3) 



Here it is useful to recall that 7o is proportional to the energy, and that 

 li is proportional to the rate of energy dissipation. 



Consider now the dynamical relations in the correlation theory. We 

 expand both /(r, t) and h{r, t) in power series of r, 



(13-5) 



i,( A ^0) 3 , 



and substitute them into the Karman-Howarth equation (Eq. 7-4). As 

 observed before (Art. 7) the terms independent of r give the energy 

 relation. The terms in r^ give the vorticity equation in the form 



^ - 70K"u^ = -10^ (13-6) 



at Xi 



or 



CO 



ir-2"."*aJi=-10j;i (13-7) 



where coi is the vorticity vector, co^ is the mean square value of one com- 

 ponent of the vorticity, and X^ is defined by 



^ = ^xyj., /." = @)^^„ (13-8) 



The second term on the left side of Eq. 13-7 represents the change of 

 vorticity due to stretching or contraction of the vortex tube without the 

 action of viscosity. It is well known that, in a perfect fluid, the circulation 

 around a vortex tube is permanent and hence the vorticity increases at 

 a rate in proportion to its rate of stretching. The right-hand side repre- 

 sents the dissipation of viscosity by viscous forces. 



Taylor [SB] suggested that this relation represents one of the basic 

 mechanisms in the process of turbulent motion. The rotation of the fluid 

 is being slowed down by the effect of viscosity. This loss is partly com- 



( 222 > 



