Renormalization of viscosity in wavelet-based model of turbulence

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 $⟨{\mathbf{u}}_{a_1}(x_1)\dots {\mathbf{u}}_{a_n}(x_n)⟩$ 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.

20 pages, revtex, 5 eps figures

References in corpus (3)

• A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence
• Second order structure function in fully developed turbulence
• Influence of helicity on scaling regimes in the extended Kraichnan model

Cited by in corpus (3)

• Fluctuating relativistic dissipative hydrodynamics as a gauge theory
• Multiresolution quantum field theory in light-front coordinates
• Are there any Landau poles in wavelet-based quantum field theory?