C • STATISTICAL THEORIES OF TURBULENCE 



results. Some theoretical speculation and assumptions will therefore be 

 introduced in the following discussions for the purpose of reaching definite 

 conclusions. We shall begin by considering the large scale structure of 

 turbulence, which is associated with small values of k in the spectral 

 representation and large values of r in the correlation representation. 

 Let us now consider the second equation in Eq. 8-1, 



F^{k) = ^ / /(r) cos {Kr)dr (12-1) 



TT Jo 



and expand cos (kt) into a power series. We obtain 



2! ^4! 



^i« = ^' («^o - ^; K^ + ^; .^ - • • • ) (12-2) 



where 



Jn = jj f{r)r-dr (12-3) 



Such a step is justified only when the function /(r) vanishes sufficiently 

 rapidly at infinity (e.g. as a negative exponential function) so that the 

 integrals J„ are convergent. In that case, one may derive from Eq. 12-2 

 a power series expansion for the three-dimensional spectrum F(k) by 

 using Eq. 10-3. This gives 



FiK) =^^Jj,.^ . . )j (12-4) 



Similarly, assuming that h{r) also vanishes sufficiently rapidly at infinity, 

 one can show that the transfer function W{k, t) behaves as k® for small 

 values of k. The spectral equation (Eq. 8-7) then shows that 



dt (^""^^^ = dt 



y? I f{r)rHr 







= (12-5) 



It then follows that 



Vi^ j j{r)r^dr = J, a. constant (12-6) 



Thus, the large scale motions are permanent in the sense that the princi- 

 pal part of F{k) for small values of k remains unchanged. 



The above derivation (including exphcit statements of the necessary 

 convergence assumptions) was given by Lin [S8] for the spectral interpre- 

 tation of the parameter J, which was first obtained by Loitsiansky [£9] 

 from the Kdrmd.n-Howarth equation. Indeed, if one multiplies that equa- 

 tion by r^ and then integrates it with respect to r from zero to infinity, 



one obtains 



J r /"to 



■j; u^ / f{r)r'dr = 2m^ lim {r'h) (12-7) 



"f I Jo J r-^oo 



( 220 ) 



