Space-Time Correlations in Stationary Isotropic Turbulence

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Abstract
The correlation between Fourier velocity components at the same wave number k and two instants separated by a time τ is investigated for a homogeneous, isotropic, and stationary turbulent field. Ensemble averages of four velocity components are related to second‐order averages as in a normal distribution. A power series in τ for the time‐displaced kinetic energy spectrum E(k, τ) is derived and the coefficient of τ2 evaluated explicitly using the Heisenberg spectrum for τ = 0. The asymptotic behavior of this term for an infinite Reynolds number R shows that the relaxation time associated with eddies of wave number k varies as k−1. These results were verified by numerical integrations for finite large R. The resulting discrepancy with the dimensional result τkαk−2/3 is ascribed to the nonuniversality of the method of analysis followed. A general integrodifferential equation for E(k, τ) is then derived and shown to admit solutions in similarity variables of the form ξ = k τ1/β. The case β = 2/3 corresponds to the Kolmogoroff spectral law. Recent objections to the quasi‐normal approximation are criticized briefly in an appendix.