By Lesieur M., et al. (eds.)
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Example text
Van de Water (Phys. Rev. Lett. 14 (1995) 4651-4654). The first analytical models of the scaling-exponent function ζn = ζ(n) was proposed by Kolmogorov in 1962 (see Eq. (16)); its agreement with the subsequently found values of the exponents ζn proved to be quite poor. Note that according to equation (16) ξ2 is positive and apparently small (µ is positive by definition but hardly large), ξ3 = 0, and ξn are negative for n > 3 and |ξn | grow very quickly with n. The available data shows that the signs of corrections ξn were predicted by equation (16) correctly (but ξ2 is so small, that it is sometimes assumed to be zero), but for higher-order corrections with n > 3 values of |ξn | are always much smaller than they must be according to equation (16).
The data by Mydlarski and Warhaft corresponded to a limited range of not too large Reynolds numbers; therefore even the existence here of the intermediate range of wave numbers k where E11 (k) ∝ k−α , α > 0, was somewhat unexpected. Note also that in this case the found corrections which must be added to the “Kolmogorov exponent” −5/3 prove to be positive while according to K62 the intermittency corrections of the spectral exponent are always negative (equal to −ξ2 ). For this reason the results of this work cannot be compared with the results discussed above which were relating to flows with much higher values of Re.
Org/ prize problems). Clay Institute Problems were considered by their authors as the continuation of the famous “Hilbert’s Problems” – a list of 23 then unsolved problems set up by the famous German mathematician Hilbert at the International Mathematical Congress of 1900 in Paris for solution in the 20th century. For the subject discussed here it is only of importance that seven Clay Institute Prize Problems include a problem called “Navier–Stokes Equations”. A short explanation accompanying the problem title at the internet announcement states that “Our understanding of the Navier–Stokes equations remains minimal.