1 Studies in the History of Statistics and Probability ... - Sheynin, Oscar

After reducing everything to one space or temporal function, that is,to an ordinary process with stationary increments, we may expectsomething. Still, for determination by experiment we need thestructural function, which is too much. We need it in a parameter formD(r) where r is the distance between the points where the componentof the velocity is measured.The most important considerations are here due to Kolmogorov.According to them, D(r) can only depend on the viscosity of the liquidwhich is responsible for the dissipation, the conversion of the energyof the turbulent heterogeneities into heat (and thus reducingturbulence) and on the amount of energy that being adopted from themain current is gradually passed from large to small whirls (and thussupporting turbulence). The energy is certainly considered for a unitmass of the liquid and unit time. ThereforeD( r) = φ( r, v, ε)where φ is some universal function, v and ε , parameters. Viscosity vis known, and the amount of energy ε is the only parameter changingfrom one experiment to another.If the distance r is sufficiently small as compared with the size ofthe current, for the model of isotropic turbulence to be applicable butlarge enough so that viscosity is not yet essential for whirls of size r,then D(r) does not depend on v. In this case the consideration ofsimilarity leads toD( r) Cεr2/3 2/3= (3.9)where C is a universal constant.For lesser r when viscosity is essential, a formula is not foundalthough it is known thatD rr( v ε)1/2( ) = ( vε) β[ ]3 1/4where β is some universal function of one variable. The dependence(3.9) is called the two thirds Kolmogorov law.There exist spectral analogues of all those statements concerning thestructural function. These conclusions were published in 1940 – 1941and all of them were hypothetical. Intense experimental checks hadbegun after the war [in 1945]. Structural functions are very similar tocorrelation functions so that their estimates have the same unpleasantproperties and it was more convenient to carry out the check byempirically measuring the spectra. No one certainly calculatedsmoothed periodograms, filters were used, see Monin & Jaglom(1967).For my part, I will just say that the measurements had confirmedeverything, the two thirds law for a sufficiently wide interval of thevalues of r, the universality of the constant C and the universaldependence of D(r) expressed through function β for small values of r.80