Two-field theory of incompressible-fluid turbulence

The turbulent velocity field in wave-number space is decomposed into two distinct fields. One is a purely chaotic field; while the other is a correction field, and carries all the phase information. Application of this decomposition to a thin shell of wave numbers in the dissipation range allows the elimination of modes in that shell, with the usual mode-coupling problems being circumvented by the use of a conditional average. The (conditional) mean effect of the eliminated modes appears as an increment to the viscosity, with terms of order ${\lambda }^2$ being neglected, where λ is a dimensionless measure of bandwidth thickness, such that 0≤λ≤1. An iteration (with appropriate rescaling) to successively lower shells reaches a fixed point, corresponding to a renormalized turbulent viscosity. As previously reported [W. D. McComb and A. G. Watt, Phys. Rev. Lett. 65, 3281 (1990)], the spectrum of the purely chaotic field is found to take the Kolmogorov -5/3 power-law form, with a value for the Kolmogorov spectral constant of α=1.6, independent of λ over the range of bandwidths for which the theory is valid.