Turbulent compressible fluid: Renormalization group analysis, scaling regimes, and anomalous scaling of advected scalar fields

Abstract

We study a model of fully developed turbulence of a compressible fluid, based on the stochastic Navier-Stokes equation, by means of the field theoretic renormalization group. In this approach, scaling properties are related to the fixed points of the renormalization group equations. Previous analysis of this model near the real-world space dimension 3 identified some scaling regime [Theor. Math. Phys., 110, 3 (1997)]. The aim of the present paper is to explore the existence of additional regimes, that could not be found using the direct perturbative approach of the previous work, and to analyze the crossover between different regimes. It seems possible to determine them near the special value of space dimension 4 in the framework of double y and ε expansion, where y is the exponent associated with the random force and ε = 4 − d is the deviation from the space dimension 4. Our calculations show that there exists an additional fixed point that governs scaling behavior. Turbulent advection of a passive scalar (density) field by this velocity ensemble is considered as well. We demonstrate that various correlation functions of the scalar field exhibit anomalous scaling behavior in the inertial-convective range. The corresponding anomalous exponents, identified as scaling dimensions of certain composite fields, can be systematically calculated as a series in y and ε. All calculations are performed in the leading one-loop approximation.

Passive Advection of a Vector Field by Compressible Turbulent Flow: Renormalizations Group Analysis near d = 4

“…In what follows, we heavily rely on a powerful methods of RG framework. The local term proportional to g 20 in (6) is not dictated by the physical considerations, but by a proper renormalization treatment [45]. Let us briefly describe this subtle point.…”

“…where a new field φ = φ(x) has been introduced and it is related to the density fluctuations via the relation φ = c 2 0 ln(ρ/ρ) [45]. Here, a parameter c 0 denotes the adiabatic speed of sound, ρ is the mean value of ρ, and f i = f i (x) is the external force normalized per unit mass.…”

“…Because the calculation of the divergent parts of Feynman diagrams is rather straightforward and proceeds in the usual fashion [21,22,24], we refrain from mentioning here all the technical details. For the latter, we recommend an interested reader to consult our previous works [40,42,43,45].…”

“…In work [45] the velocity part (without β w ) of the system (17) was analyzed. Altogether, three IR attractive fixed points, which define possible scaling regimes of the system, were found.…”

“…This paper is a continuation of our previous works [42][43][44][45] and is organized as follows. In the introductory Section 2 we give a brief overview of the model.…”