Improvedεexpansion for three-dimensional turbulence: Two-loop renormalization near two dimensions

Abstract

An improved ε expansion in the d-dimensional (d > 2) stochastic theory of turbulence is constructed at two-loop order which incorporates the effect of pole singularities at d → 2 in coefficients of the ε expansion of universal quantities. For a proper account of the effect of these singularities two different approaches to the renormalization of the powerlike correlation function of the random force are analyzed near two dimensions. By direct calculation it is shown that the approach based on the mere renormalization of the nonlocal correlation function leads to contradictions at two-loop order. On the other hand, a two-loop calculation in the renormalization scheme with the addition to the force correlation function of a local term to be renormalized instead of the nonlocal one yields consistent results in accordance with the UV renormalization theory. The latter renormalization prescription is used for the two-loop renormalization-group analysis amended with partial resummation of the pole singularities near two dimensions leading to a significant improvement of the agreement with experimental results for the Kolmogorov constant.

Nonperturbative renormalization group study of the stochastic Navier-Stokes equation

“…Perturbative renormalization in two dimensions [23,38] also predicts the attainment of this fixed point.…”

“…Locality entitles us to interpret this term as a renormalization counterterm in the sense of Refs. [23,37,38]. Again, we will use the extra freedom introduced by F m r to impose a renormalization condition on the flow.…”

“…In the context of perturbative renormalization the pair (58) specifies the running coupling constant of the model [17,37,38]. In (58) we denoted…”

“…Finally, solving (43) exactly is equivalent to solving an infinite nonclose hierarchy of equations. Perturbative renormalization tells us, however, that there are only a finite number of relevant couplings, at most two for ε 1 and d 2 [17,37,38], determining the scaling properties of the stochastic Navier-Stokes (1). Based on this observation, we now turn to the derivation of a truncation of the right-hand side of (43) in order to derive explicit scaling predictions.…”

“…Since perturbative ultraviolet renormalization forbids nonlocal counterterms [37,38], these approximations are adapted only to the case d > 2. As a consequence, we expect (68) to converge to the fixed point of Sec.…”