Making use of a simple mathematical model concerning the form of motion associated with each turbulent component, the relation between the space-correlation function and the Eulerian time-correlation function is theoretically studied. While the former correlation R(r) is given by 1-R(r)_??_r2/3, the latter R(t) is given by 1-R(t)_??_tm and the value of m is a function of U/√u2, where U and √u2 are the velocity of the mean flow and the turbulent velocity, respectively. The values of m lie between 2/3 and 1. The minimum value m=2/3 occurs when U_??_√u2, and the maximum value m=1 when U_??_√u2 It is further stressed that the functional form of Lagrangian correlation is exactly the same as that of Eulerian correlation in the case of U_??_√u2, from the view point that the decay-and rebuilding-process of turbulence plays a predominant rôle in both the two correlations.
Harmonic analyses for many terms are troublesome to perform by the ordinal scheme, and in this paper we showed that they are easily obtained by combining sets of simpler analyses. When integers m and n are incommensurable, we represent m×n terms y0, y1, y2, ……, yp, ……, ymn-1 as follows. where m is even. (i) Calculate the Fourier coefficients of the j-th line of (I) using the analyses for m orders whose results are (ii) Calculate the Fourier coefficients of the i-th column of (II) using the analyses for n orders whose result are (iii) The Fourier coefficients of the entire series P (ν), and Q (ν) are shown by, and where As examples we showed schematically the analyses for 36 (4×9), 60 (4×15), 72 (8×9), 120 (8×15) and 180 (5×36) orders, the results of which were given by the tables (IV. 36), (IV. 60), (IV. 72), (IV. 120), and (IV. 180) in the text. In each case mentioned above m is even, but in the case when both m and n are odd the similar scheme can be used.