The energy spectrum in the universal range of two-dimensional turbulence

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Abstract
Direct numerical simulation of two-dimensional Navier-Stokes equations at large Reynolds numbers is made by the spectral method with 13642 modes starting from a high-symmetric random initial velocity field. Two wavenumber ranges, governed by different similarity laws are observed after the enstrophy dissipation rate η(t) takes the maximum value. At small wavenumbers the energy spectrum is stationary in time, while at larger wavenumbers it decays according to the similarity law predicted by the enstrophy cascade theory, and the shape of the energy spectrum E(k,t) is expressed by E(k,t) = Aη(t)1/6v3/2(k/kd)-3exp[ –√2A(k/kd)], where k is the wavenumber, t the time, v the kinematic viscosity of fluid, kd = η(t)1/6v1/2 the dissipation wavenumber, and A ≈ 1.6. Concerning the enstrophy dissipation rate the following properties are observed: (i) As the Reynolds number R increases, the time of maximum enstrophy dissipation rate is delayed, probably in proportion to ln R. (ii) It approaches finite positive values in the inviscid limit if the above-mentioned time-lag is taken into account, (iii) It decays inversely proportionally to the cubic of time, so that the enstrophy is expressed as a sum of a constant term and a term which decays inversely proportionally to the square of time. This paper discusses why power laws of the energy spectrum observed in most of previously reported direct numerical simulations of two-dimensional periodic flows were steeper than k-3.