3G (k 



Some Progress in Turbulence Theory 



t) k^ f' r r'''^ 



= - — J cis j pdp J E (q) sin 



^t 2 |^_p| 



(q,k) 



dq 

 G (p,t-s) G (k,s) — i + ... , G (k,0) = 1 



q 



(10) 



X (t) 



J dq J E (q) G (q,s) ds , (11) 



where sin (q, k) is the sine of the interior angle between q and k in a triangle 

 with sides k, p, q, and G (k , t - t ') is the Fourier transform of G(x, t ; x', t') 

 with respect to x - x'. Equation (10) was discretized and truncated in k, dis- 

 cretized in time, and integrated forward by a simple predictor-corrector 

 scheme. Finally, k (t) was computed from Eq. (11). Variation of step sizes 

 and truncated limits, interpreted by extrapolation techniques, was used to 

 verify that numerical- integration errors were negligible. 



Figure 1 compares k (t) as found [llj from the numerical experiment and 

 from the direct-interaction approximation for the spectrum choice in Eq. 

 (9b), wherein the accuracy of the direct- interaction approximation was found 

 to be poorer. In the present case of dispersion by a statistically stationary 

 velocity field, the Lagrangian velocity correlation (correlation of a fluid parti- 

 cle's current velocity with its initial velocity) is easily shown to be u^ (t) =- 

 1/3 (v-(0) v.(t)> = dK(t)'dt. Figure 2 compares the curves of u^ (t) obtained 

 from the numerical experiment and from the direct-interaction results. 



ISOTROPIC TURBULENCE DECAY 



The direct-interaction equations for the decay of isotropic turbulence are 

 similar in general appearance to those for turbulent dispersion; but they are 

 also more complicated because the Navier- Stokes equation is a vector equation, 

 nonlinear in the unknown dynamical variables, in contrast to the linear scalar 

 equation for T (x, t). The final equations of the approximation comprise an 

 energy balance equation which determines the spectrum function E(k, t), an 

 equation for the time correlation u(k; t, t') of the Fourier amplitude at wave- 

 number k, and an equation for the mean response G(k; t, t') of Fourier ampli- 

 tude k to infinitesimal perturbations. These equations are (g and T are 

 unrelated to previous G and T) [l2j: 



f— + 2 ukA E (k,t) = T (k,t), T (k,t) = 4 TTk^ S (k;t,t) (12) 



— + i^k^j U (k;t,t') = S (k;t,t') (13) 



— + i^k^l G (k;t,t') = H (k;t,t'), G (k;t',t') = 1 (14) 



3t 



791 



