E. J. Skudrzyk 203 
of the turbulent velocities is found to vary in exactly the same manner, and is 
explained by the Kolmogorov equilibrium law [3,5] for the spectral space dis- 
tribution of the turbulent velocity fluctuations. This law is of the form 
indyakie 8 (1) 
where K is a constant and x is the space wave number. Such a power spectrum 
leads to the cube-root distance law, <(As)?> is equal to constant times p”, where 
p is the distance; and, vice versa, a cube-root distance law leads to such a 
power spectrum. 
The Kolmogorov equilibrium law then turns out to be a general law of physics 
that applies not only to turbulence, but to many different phenomena, This law 
applies whenever the following sequence takes place: energy is introduced into 
a fluid at a constant rate, is redistributed until a state of equilibrium is attained, 
and is eventually dissipated by friction or heat conduction. The only other as- 
sumption necessary is that the fluid be infinitely extended. Dimensional con- 
sideration then leads directly to this law in the following manner: the energy 
Dp introduced per unit mass per unit time has the dimensions 
Dyas (2) 
where the expression derivable from the kinetic energy (mv”) has been used to 
arrive at the dimensions of energy. The magnitude of interest is the power 
spectrum E(x) of the phenomenon to be investigated. The power spectrum has 
the dimensions of energy per unit mass per unit space wave number, or 
2 2 3 
BGS) HRS UE 
ee UI oe (3) 
where x = 27/\sp (Asp is the space wavelength and has the dimension/~*). The power 
spectrum must be a function of the parameters that are available for the de- 
scription of the phenomenon. Since the fluid has been assumed to be infinitely 
extended, the only available parameters are Do andx. Hence, E(x) must be a func- 
tion of D, and x: 
E(x) = f(Do, k) (4) 
which, in the simplest case, when written as a power product, is of the following 
form: 
E(k) = DOK" (5) 
Since the dimension of the right hand sides of Eq. (5) and Eq. (3) must be the 
same, the following relation must hold for m and n: 
3 _ pmyn 12% jon 
E(x) =“g = Dox" = (6) 
When we equate the exponents 
3=2m-—n 2=3m (7) 
