C,7 • DYNAMICS OF ISOTROPIC TURBULENCE 



where p is the density of the fluid, p is the pressure, v is the kinematic 

 viscosity coefficient, and A is the Laplacian operator. It might be ex- 

 pected that one could, from Eq. 7-1, derive the equations governing the 

 behavior of all the statistical properties of turbulent motion, such as the 

 level of turbulence, the correlation functions, etc. However, as one pro- 

 ceeds to construct the equations for such purposes, it becomes at once 

 clear that we are always faced with the difficulty of having fewer equa- 

 tions than unknowns, caused primarily by the nonlinear terms in the 

 differential equations. Unless additional assumptions are introduced, de- 

 ductions from such an approach are quite limited. In this article, we only 

 discuss the results following the formal construction of the equations for 

 the change of the correlation functions. The necessary additional assump- 

 tions will be taken up later (Art. 16 and 17). 



To obtain the equation for the change of the double correlation func- 

 tion, one may multiply Eq. 6-1 by m^ and add to it a similar equation 

 obtained by the interchange of the role of the points P and P' . The lead- 

 ing term of the combined equation is then 



, dUi . dui d , .. 



Upon averaging, this yields an equation for the time rate of change of the 

 double correlation tensor u^Rij.. 



However, there are clearly terms of other types appearing in the 

 equation. The nonlinear term on the left of Eq. 7-1 gives rise to triple 

 correlations and the pressure term gives rise to a pressure-velocity corre- 

 lation. It can be easily verified by using Eq. 6-13 that the pressure term 

 vanishes identically, and the equation finally reduces to 



^ {u^Biu) - u' ^ ifij,k + fkj,i) = 2pu''ARik (7-2) 



The appearance of the triple correlation in Eq. 7-2 would suggest the 

 attempt to estabUsh a relation governing its time rate of change. This 

 can be done by combining three equations with leading terms 



obtained by multiplying the equations of motion at P, P' , and P" respec- 

 tively by suitable factors. Upon averaging, an equation for {d/dt){u^fi^k,i) 

 is obtained, but this equation also involves correlations of the fourth order 

 and pressure-velocity correlations. The latter can be eliminated in terms 

 of velocity correlations by the following process. From Eq. 7-1, one may 

 obtain, by taking its divergence, a Poisson equation 



< 209 ) 



