Statistical mechanics and large-scale velocity fluctuations of turbulence

Abstract

Turbulence exhibits significant velocity fluctuations even if the scale is much larger than the scale of the energy supply. Since any spatial correlation is negligible, these large-scale fluctuations have many degrees of freedom and are thereby analogous to thermal fluctuations studied in the statistical mechanics. By using this analogy, we describe the large-scale fluctuations of turbulence in a formalism that has the same mathematical structure as used for canonical ensembles in the statistical mechanics. The formalism yields a universal law for the energy distribution of the fluctuations, which is confirmed with experiments of a variety of turbulent flows. Thus, through the large-scale fluctuations, turbulence is related to the statistical mechanics.

Log-normal distribution from a process that is not multiplicative but is additive

“…The functional form of the correlation only affects details of the approach to the log-normal distribution. To normalize the scale R, we generalize the definition of the correlation length L v 2 [20], Figure 4 shows thatv 2 R at R ≫ L v 2 in a variety of turbulent flows is log-normal [6] and is consistent with our model [20], regardless of the correlation functions that depend on the configuration of the flow. Not so con- Fig.…”

“…This term is dominant over the second term in the thermodynamic limit of N ≫ 1, where Eq. (A.3) reproduces formulae of the thermodynamics [20].…”

“…Since its fluctuations exist at any small scale [23], the calculations are made with different resolutions if necessary. The one-dimensional field is divided into segments with length R [6,20]. Over each of the segments, the center of which is tentatively defined as x * , we average the energy v 2 asv…”

“…This is the case even in turbulence [20]. We have defined the number of the subsegments with length 4L v 2 as N = R/4L v 2 in Eq.…”

“…(6),f (E) = E C−1 exp(−E/T ) Γ(C)T C . (A.4a)Especially in the thermodynamic limit, the distribution is reduced to our past model[20],f (E) = E ζN −1 exp(−E/T ) Γ(ζN )T ζN for N ≫ 1. (A.4b)This model corresponds to the exact Eq.…”