Log-Stable Distribution and Intermittency of Turbulence

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  • Kida Shigeo
    Research Institute for Mathematical Sciences, Kyoto University

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  • Log Stable Distribution and Intermitten

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The logarithm of the breakdown coefficient εr⁄εl, εr being the mean energy dissipation rate averaged over a sphere of radius r is shown, under a similarity assumption, to obey a stable distribution, the characteristic function of which is given by \varphi(z|r⁄l)=(r⁄l)(μ⁄2<SUP>α−2)[iz−(zeiπ⁄2)α]</SUP>, where μ>0 and 0<α≤2. The scaling exponent of the p-th order moment of the energy dissipation rate is calculated to be μp=μ(pα−p)⁄(2α−2), which is in excellent agreement with the experiments (Anselmet et al. 1984) when the intermittency parameter is μ=0.20 and the characteristic exponent of the distribution is α=1.65. The probability density function of εr diverges as 1/εr(−ln εr)α+1 at the origin and decreases as exp [−A(ln εr)α⁄(α−1)], where A>0, as ε→∞. The present results include the log-normal theory for α=2 and coincide with the prediction of μp due to the β-model in the limit α→0.

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