Exact theory for the self-similarity and decay of homogeneous turbulence

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Abstract
It is shown that space-time dilatation invariance (x→ξ−1x, t→ξ−2t, with concomitant transformations for dependent variables) and linearity of the Φ-equation engender an exact, time-explicit generic form for the solution applicable to freely-decaying homogeneous incompressible fluid turbulence. This solution features a summation over mutually independent dynamical modes labeled by the dilatation scaling-index n(>1). Without the assumption of isotropy nor introduction of a closure approximation procedure, the theory provides an explanation for the experimentally observed self-similarity of the correlation tensors and the decay laws 〈‖u(x, t)‖2〉∝t−n for the different types and decay stages of homogeneous turbulence.