Anomalous scaling of passive scalar fields advected by the Navier-Stokes velocity ensemble: Effects of strong compressibility and large-scale anisotropy

Abstract

The field theoretic renormalization group and the operator product expansion are applied to two models of passive scalar quantities (the density and the tracer fields) advected by a random turbulent velocity field. The latter is governed by the Navier-Stokes equation for compressible fluid, subject to external random force with the covariance ∝δ(t-t')k(4-d-y), where d is the dimension of space and y is an arbitrary exponent. The original stochastic problems are reformulated as multiplicatively renormalizable field theoretic models; the corresponding renormalization group equations possess infrared attractive fixed points. It is shown that various correlation functions of the scalar field, its powers and gradients, demonstrate anomalous scaling behavior in the inertial-convective range already for small values of y. The corresponding anomalous exponents, identified with scaling (critical) dimensions of certain composite fields ("operators" in the quantum-field terminology), can be systematically calculated as series in y. The practical calculation is performed in the leading one-loop approximation, including exponents in anisotropic contributions. It should be emphasized that, in contrast to Gaussian ensembles with finite correlation time, the model and the perturbation theory presented here are manifestly Galilean covariant. The validity of the one-loop approximation and comparison with Gaussian models are briefly discussed.

Stochastic Navier-Stokes equation and advection of a tracer field: One-loop renormalization neard= 4

“…Fortunately, despite the difference between the formalism for density and tracer fields in the case of a compressible fluid, the critical dimensions for these two fields have the same form; for detailed analysis see [21]. Thus, expressions (9) and (10) remain true for tracer field as well.…”

“…Usually d plays a passive role -see [20] for Navie-Stokes equation itself and [21,22] for advection of scalar fields -but if d = 4, the situation is similar to the analysis of the incompressible Navier-Stokes equation near space dimension d = 2. In this case an additional divergence appears in the 1-irreducible Green's function �v ′ v ′ � 1-ir , see [34][35][36].…”

“…Further studies are connected with its more realistic generalizations [9][10][11][12][13][14][15][16][17][18]; see also review paper [19]. The RG+OPE technique were also applied to more complicated models, in particular, to the compressible case [20][21][22][23][24][25][26][27][28][29][30][31][32][33].…”

Renormalization group analysis of a turbulent compressible fluid neard= 4: Crossover between local and non-local scaling regimes

“…As it has been shown in [1] and discussed in [14], the model (5) - (6) is invariant with respect to the Galilean symmetry, which results to the UV finitness of the two Green's functions:…”

“…They can be absorbed by a suitable local counterterm v i ∇ 2 v i , and a regular expansion in both y and ε = d − 2 was constructed. Up to now the present model (5) has been investigated at the fixed space dimension d = 3, for which the action (5) contains all terms that can be generated during the renormalization procedure [1,[13][14][15]. However, using the dimensional analysis it can be shown that at d = 4 there appears an additional divergence, also in the Green's function v v .…”