Renormalization of viscosity in wavelet-based model of turbulence

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 $L$. Since there are no divergences, regularization is not required, and renormalization group invariance becomes merely a symmetry that relates velocity fluctuations of different scales. The one-loop corrections to viscosity and to the pair velocity correlator are calculated. This gives the dependence of turbulent viscosity on observation scale and describes the scale dependence of the velocity field correlations.