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

TL;DR

This paper uses field theoretic methods to analyze how passive scalar fields behave under turbulent flows governed by the Navier-Stokes equations, revealing anomalous scaling effects influenced by compressibility and anisotropy.

Contribution

It introduces a systematic calculation of anomalous scaling exponents for passive scalars in compressible turbulence using renormalization group and operator product expansion techniques.

Findings

Anomalous scaling occurs even at small values of the parameter y.

The exponents can be calculated as series in y, including anisotropic effects.

The model maintains Galilean covariance unlike Gaussian models.

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 $\propto \delta(t-t') k^{4-d-y}$, 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.