Analytical theories of turbulence and the ε expansion

Abstract

The fixed-point form of hydrodynamic equations emerging from renormalization group analysis of strong turbulence is analyzed using perturbation expansion in powers of the renormalized coupling constant (Reynolds number) Re*∝ε1/2. In the second order of ε expansion, transport equations are derived for energy and variance of a passive scalar which are identical to the inertial-range form of the eddy-damped quasinormal Markovian approximation (EDQNM), but without adjustable parameters which must be determined from experiment. The energy and scalar variance spectra are predicted to be Ek=1.617ε̄2/3k−5/3, and Θk=1.146N̄ ε̄−1/3k−5/3, where ε̄ and N̄ are, respectively, the dissipation rate of energy and scalar variance. This level of agreement with experiment has in the past proved difficult to obtain from renormalized perturbation theory applied to the bare hydrodynamic equations, without introduction of parameters that must be adjusted to different values for different dynamical fields. The fidelity of the present method, which emphasizes in a particular way the distant interactions in k space, may be related to the degree of nonlocality of the bare dynamics as manifested by possible generic Beltramization tendencies in strong turbulence.