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M. V. Kompaniets

Publications and source records attributed to M. V. Kompaniets.

25 records · Page 2Linked to original sources
arXiv2013

Anomalous scaling of a passive vector field in $d$ dimensions: Higher-order structure functions

The problem of anomalous scaling in the model of a transverse vector field $θ_{i}(t,x)$ passively advected by the non-Gaussian, correlated in time turbulent velocity field governed by the Navier--Stokes equation, is studied by means of the field-theoretic renormalization group and operator product expansion. The anomalous exponents of the $2n$-th order structure function $S_{2n}(r) = <[θ(t,x) - θ(t,x+ r)]^{2n}>$, where $θ$ is the component of the vector field parallel to the separation $r$, are determined by the critical dimensions of the family of composite fields (operators) of the form $(\partialθ\partialθ)^{2n}$, which mix heavily in renormalization. The daunting task of the calculation of the matrices of their critical dimensions (whose eigenvalues determine the anomalous exponents) simplifies drastically in the limit of high spatial dimension, $d\to\infty$. This allowed us to find the leading and correction anomalous exponents for the structure functions up to the order $S_{56}$. They reveal intriguing regularities, which suggest for the anomalous exponents simple "empiric" formulae that become practically exact for $n$ large enough. Along with the explicit results for modest $n$, they provide the full description of the anomalous scaling in the model. Key words: passive vector field, turbulent advection, anomalous scaling, renormalization group, operator product expansion.

cond-mat.stat-mech↗
arXiv2008

Renormalization group in the infinite-dimensional turbulence: third-order results

The field theoretic renormalization group is applied to the stochastic Navier-Stokes equation with the stirring force correlator of the form k^(4-d-2ε) in the d-dimensional space, in connection with the problem of construction of the 1/d expansion for the fully developed fluid turbulence beyond the scope of the standard epsilon expansion. It is shown that in the large-d limit the number of the Feynman diagrams for the Green function (linear response function) decreases drastically, and the technique of their analytical calculation is developed. The main ingredients of the renormalization group approach -- the renormalization constant, beta function and the ultraviolet correction exponent omega, are calculated to order epsilon^3 (three-loop approximation). The two-point velocity-velocity correlation function, the Kolmogorov constant C_K in the spectrum of turbulent energy and the inertial-range skewness factor S are calculated in the large-d limit to third order of the epsilon expansion. Surprisingly enough, our results for C_K are in a reasonable agreement with the existing experimental estimates.

nlin.CD↗
arXiv2004

An improved $\eps$ expansion for three-dimensional turbulence: two-loop renormalization near two dimensions

An improved $\eps$ expansion in the $d$-dimensional ($d > 2$) stochastic theory of turbulence is constructed at two-loop order which incorporates the effect of pole singularities at $d \to 2$ in coefficients of the $\eps$ expansion of universal quantities. For a proper account of the effect of these singularities two different approaches to the renormalization of the powerlike correlation function of the random force are analyzed near two dimensions. By direct calculation it is shown that the approach based on the mere renormalization of the nonlocal correlation function leads to contradictions at two-loop order. On the other hand, a two-loop calculation in the renormalization scheme with the addition to the force correlation function of a local term to be renormalized instead of the nonlocal one yields consistent results in accordance with the UV renormalization theory. The latter renormalization prescription is used for the two-loop renormalization-group analysis amended with partial resummation of the pole singularities near two dimensions leading to a significant improvement of the agreement with experimental results for the Kolmogorov constant.

nlin.CD↗
arXiv2002

Renormalization-group approach to the stochastic Navier--Stokes equation: Two-loop approximation

The field theoretic renormalization group is applied to the stochastic Navier--Stokes equation that describes fully developed fluid turbulence. The complete two-loop calculation of the renormalization constant, the $β$ function, the fixed point and the ultraviolet correction exponent is performed. The Kolmogorov constant and the inertial-range skewness factor, derived to second order of the $\eps$ expansion, are in a good agreement with the experiment. The possibility of the extrapolation of the $\eps$ expansion beyond the threshold where the sweeping effects become important is demonstrated on the example of a Galilean-invariant quantity, the equal-time pair correlation function of the velocity field. The extension to the $d$-dimensional case is briefly discussed.

nlin.CD↗
arXiv2002

Renormalization group in the statistical theory of turbulence: Two-loop approximation

The field theoretic renormalization group is applied to the stochastic Navier--Stokes equation that describes fully developed fluid turbulence. The complete two-loop calculation of the renormalization constant, the beta function and the fixed point is performed. The ultraviolet correction exponent, the Kolmogorov constant and the inertial-range skewness factor are derived to second order of the $ε$ expansion.

nlin.CD↗
arXiv2002

Two-loop calculation of the scaling behavior of two-dimensional forced Navier-Stokes equation

Asymptotic properties of the solution of two-dimensional randomly forced Navier-Stokes equation with long-range correlations of the driving force are analyzed in the two-loop order of perturbation theory with the use of renormalization group. Kolmogorov constant of the energy spectrum is calculated for both the inverse energy cascade and the direct enstrophy cascade in the second order of the $ε$ expansion.

nlin.CD↗