C • STATISTICAL THEORIES OF TURBULENCE 



relating the pressure to the instantaneous velocity. One may then ob- 

 tain, for example, Apu',^ul' in terms of velocity correlations and attempt 

 to calculate pu'^u'/ by integration. 



Thus, one may expect that, in general, a system of differential equa- 

 tions can be obtained for the correlation coefficients of different orders, 

 each involving velocity correlations of one order higher. To obtain a de- 

 ductive theory, it would be necessary to interrupt this process by some 

 judicious assumption (suggested by experimental information or other 

 theoretical considerations) connecting higher order correlations with ones 

 of lower orders. (See Art. 16 and 17.) 



Many of the existing investigations involve only the dynamical equa- 

 tion (Eq. 7-2) for double correlations. The equations for higher corre- 

 lations have been exploited only recently. Now, Eq. 7-2 represents a sys- 

 tem of six equations. However, since Rik and Tij,k are each determined by 

 a single scalar function (Eq. 6-3 and 6-10), it should be possible to re- 

 duce Eq. 7-2 to a single equation. In fact, von Karman and Howarth [17] 

 found it to be 



6t ^^^^ + 2^ U + yj = 2.u^ (^^ + - /^j (7-4) 



A direct experimental verification of this equation has been made by 

 Stewart [11]. 



If one expands both / and h as power series of r, one obtains a series 

 of relations among the derivatives of those functions. The first of these is 

 commonly written in the form 



f=-104' (7-5) 



where X is Taylor's vorticity scale defined by 



p - - (f L 



The relation (Eq. 7-5) essentially gives the rate of decrease of kinetic 

 energy. It was first established by Taylor [16], both theoretically and 

 experimentally. The equations corresponding to the higher powers of r 

 will be discussed in connection with the small scale structure of turbu- 

 lence (Art. 13). 



C,8. The Spectral Theory of Isotropic Turbulence. The early 

 adoption of statistical correlations for the description of isotropic turbu- 

 lence is at least partly due to the fact that they are relatively easy to 

 measure. Another powerful method for describing a fluctuating field is to 

 analyze it into Fourier components, i.e. to adopt the spectral approach. 

 It is well known that the spectral theory and the correlation theory are 



(210 > 



