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

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 a scaling regime [N. V. Antonov et al., Theor. Math. Phys. 110, 305 (1997)]. The aim of the present paper is to explore the existence of additional regimes, which 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 $\varepsilon$ expansion, where $y$ is the exponent associated with the random force and $\varepsilon =4-d$ 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 $\varepsilon$. All calculations are performed in the leading one-loop approximation.

Figure 1

Graphical representation of the bare propagators in the model (3.1).

Figure 2

Graphical representation of the interaction vertices in the model (3.1).

Figure 3

Renormalization group flow diagram in the plane $(g_1,g_2)$ at $y=4$ and $\varepsilon =1$, with $\alpha =\infty$ and $u=v=1$. Three fixed points FPI, FPII, and FPIII are marked by an open circle, a closed circle, and a closed rhombus, respectively.

Figure 4

Domains of IR stability of the fixed points for the model (3.1) in the plane $(y,\varepsilon )$.

Figure 5

Graphical representation of the bare propagators ${\langle \theta {\theta }^{\prime }\rangle }_0$ and ${\langle \theta \theta \rangle }_0$.

Figure 6

Graphical representation of the interaction vertex $V_j^3$.