Kolmogorov similarity in freely decaying turbulence

Abstract

A direct numerical simulation of the three‐dimensional Navier–Stokes equation at high Reynolds numbers is performed by the spectral method with 3×340³ effective modes (85³ independent degrees of freedom) starting from a high‐symmetric flow. Kolmogorov’s [C. R. (Dokl.) Acad. Sci. URSS 3 0, 301 (1941)] similarity forms of the energy spectra (the one‐dimensional longitudinal and lateral energy spectra as well as the three‐dimensional one) in the universal range are observed in a decaying period after the enstrophy takes the maximum value. During this period the energy decays exponentially in time and the microscale Reynolds number changes from 100 to 60. At the lower part of the universal range Kolmogorov’s inertial range spectrum E(k,t)=Aε(t)^2/3k^−5/3 is observed over nearly one decade of the wavenumber, where the Kolmogorov constant A≊1.8. At the higher part of the universal range, on the other hand, it has an exponential tail with an algebraic correction, E(k,t)/[ε(t)ν⁵]^1/4 =B(k/k_d(t))^m exp[−β(k/k_d(t))], R)], where B≊8.4, m≊−1.6, β≊4.9, and k_d(t)=[ε(t)/ν³]^1/4 is the energy dissipation wavenumber.