Some Progress in Turbulence Theory 



pqdpdq I a^p^ G (k;t',s) U (p;t,s) U (q;t,s) ds 



A 



f 







(15) 



Kpd ^ (•<:*', s) G (p;t,s) U (q;t,s) ds 



H(k;t,t') = - ^^J j pqdpdq b^p^ J^ G(k;s,t') G (p; t , s) U (q; t , s) ds . (16) 

 A ^' 



Here ■, . . ; , ' . ' ■,'""';', „■",'" 



ai,^^ = — (l-xyz-2y2 z^), b. =— (xy+z^), ''■■.-- 



kpq 2 kpq k ..-^ 



where x, y, z are the cosines of the interior angles opposite k, p, q, respec- 

 tively, in a triangle with the latter numbers as sides. The integration /J^ ex- 

 tends over all regions of the (p,q) plane where this triangle can be formed. 

 Kinematic viscosity is t^ . These equations are a complete set which determines 

 E(k,t), u(k; t, t'), and G(k; t, t') for t > 0, t' > 0, if the initial spectrum 

 E(k, 0) is given. The spectrum and correlation functions are related by E(k, t) = 

 27rk^ u(k; t, t). The kinetic energy per unit mass at time t is 



■ ■' E (k,t) dk . - 



■■ ■■■ ■ ■■' ■ % . ^ ,:. : : 



Also, u(k; t, t') = u(k; t', t), while G(k; t, t') vanishes for t < t'. 



The equations above share, with the direct- interaction equations for dis- 

 persion, the property of nonlocalness, both in the present Fourier representa- 

 tion and if they are transformed back into physical space. As before, this 

 expresses the finite length and time scales of the eddy -transport process. 



The direct- interaction equations preserve some important properties of 

 the exact turbulence dynamics. Equations (12) and (15) yield 



/ 



T(k,t) dk = , (17) 







which expresses conservation of kinetic energy by the nonlinear processes. 

 Moreover, and equally important, the equations guarantee the realizability of 

 u(k ; t, t'). This means that a vector random process can be constructed for 

 which u(k; t, t'), as found from the direct-interaction equations, actually is 

 the covariance scalar. This implies an infinite set of realizability inequalities, 

 the simplest and most important of which is 



E(k,t) > 0, |U(k; t,t')| 2 < U(k;t,t) U(k;t',t') . '' (18) 



793 



