Advection of a passive scalar field by turbulent compressible fluid: renormalization group analysis near d = 4

¹ Department of Theoretical Physics, Faculty of Physics, Saint-Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034 Russia
² Faculty of Sciences, P.J. Šafárik University, Moyzesova 16, 040 01 Košice, Slovakia and Fakultät für Physik, Universität Duisburg-Essen, D-47048 Duisburg, Germany

^a e-mail: n.antonov@spbu.ru
^b e-mail: n.gulitskiy@spbu.ru
^c e-mail: kontramot@mail.ru
^d e-mail: tomas.lucivjansky@uni-due.de

Published online: 22 March 2017

Abstract

The field theoretic renormalization group (RG) and the operator product expansion (OPE) are applied to the model of a density field advected by a random turbulent velocity field. The latter is governed by the stochastic Navier-Stokes equation for a compressible fluid. The model is considered near the special space dimension d = 4. It is shown that various correlation functions of the scalar field exhibit anomalous scaling behaviour in the inertial-convective range. The scaling properties in the RG+OPE approach are related to fixed points of the renormalization group equations. In comparison with physically interesting case d = 3, at d = 4 additional Green function has divergences which affect the existence and stability of fixed points. From calculations it follows that a new regime arises there and then by continuity moves into d = 3. The corresponding anomalous exponents are identified with scaling dimensions of certain composite fields and can be systematically calculated as series in y (the exponent, connected with random force) and ε = 4 − d. All calculations are performed in the leading one-loop approximation.