Renormalization of viscosity in wavelet-based model of turbulence

TL;DR

This paper reformulates turbulence theory using wavelet-based scale-dependent fields, demonstrating finite correlators without divergences and deriving scale-dependent viscosity through a renormalization group approach in a simplified 3D model.

Contribution

It introduces a wavelet-based reformulation of turbulence that avoids divergences and derives scale-dependent viscosity using a renormalization group framework in a simplified setting.

Findings

Correlators are finite by construction without regularization.

Viscosity depends on the observation scale within the Kolmogorov range.

One-loop corrections describe the scale dependence of velocity correlations.

Abstract

Statistical theory of turbulence in viscid incompressible fluid, described by the Navier-Stokes equation driven by random force, is reformulated in terms of scale-dependent fields $\mathbf{u}_a(x)$, defined as wavelet-coefficients of the velocity field $\mathbf{u}$ taken at point $x$ with the resolution $a$. Applying quantum field theory approach of stochastic hydrodynamics to the generating functional of random fields $\mathbf{u}_a(x)$, we have shown the velocity field correlators $\langle \mathbf{u}_{a_1}(x_1)\ldots \mathbf{u}_{a_n}(x_n)\rangle$ to be finite by construction for the random stirring force acting at prescribed large scale $L$. The study is performed in $d=3$ dimension. Since there are no divergences, regularization is not required, and the renormalization group invariance becomes merely a symmetry that relates velocity fluctuations of different scales in terms of the Kolmogorov-Richardson picture of turbulence development. The integration over the scale arguments is performed from the external scale $L$ down to the observation scale $A$, which lies in Kolmogorov range $l \ll A \ll L$. Our oversimplified model is full dissipative: interaction between scales is provided only locally by the gradient vertex $(\mathbf{u}\nabla) \mathbf{u}$, neglecting any effects or parity violation that might be responsible for energy backscatter. The corrections to viscosity and the pair velocity correlator are calculated in one-loop approximation. This gives the dependence of turbulent viscosity on observation scale and describes the scale dependence of the velocity field correlations.