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 ${\mathbf{u}}_a\left(x\right)$, 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 ${\mathbf{u}}_a\left(x\right)$, we have shown the velocity field correlators $\langle {\mathbf{u}}_{a_1}\left(x_1\right)...{\mathbf{u}}_{a_n}\left(x_n\right)\rangle$ 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 $\left(\mathbf{u}\mathbf{\nabla }\right)\mathbf{u}$, 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.

Figure 1

One-loop contribution to the Green function. Primed lines denote the $u^{\prime }$ fields incident to the vertices. Crossed line denotes the stirring force correlators. Latin letters indicate vector indices, Greek letters stand for the scale arguments of external lines.

Figure 2

The dependence of turbulent viscosity ${\nu }_A\left(k\right)$ on the observation scale $\xi =A/L$ and the dimensionless momentum $x=kL$. Calculations were performed in the IR region using the equation (30) at the assumption of invariant $g_A{\nu }_A^3=g_L{\nu }_L^3$, i.e., small $K\left(\xi \right)$, and normalization momentum $x_{*}=4\pi$ used to evaluate $g_L$, in accordance to $k=2$ energy injection limit in simulated turbulence from John Hopkins Turbulence Database.

Figure 3

Feynman diagrams used for evaluation of the pair correlator of velocity field.

Figure 4

One-dimensional energy spectra. Calculated for the set of parameters of the John Hopkins Turbulence Database for the isotropic turbulence grid simulations. The curves show the spectra for observation scales $A=0.1,0.2,0.5,1.0$. The $E\left(k\right)$ numerical energy spectrum, obtained for the ${128}^3$ decimation of the original JHTDB ${1024}^3$ data, is shown by dashed line. The normalization scale $x_{*}=4\pi$ corresponds to the $k=2$ energy injection limit. All quantities are normalized to ${\left(2\pi \right)}^3$ volume.

Figure 5

One-loop contribution to the correlation function. Greek letters $\alpha$ and $\beta$ denote scale arguments.