Two-parameter expansion in the renormalization-group analysis of turbulence

The renormalization of the solution of the Navier-Stokes equation for randomly stirred fluid with long-range correlations of the driving force is analysed near two dimensions. It is shown that a local term must be added to the correlation function of the random force for the correct renormalization of the model at two dimensions. The interplay of the short-range and long-range terms in the large-scale behaviour of the model is analysed near two dimensions by the field-theoretic renormalization group. A regular expansion in 2ε=d-2 and δ=2-λ is constructed, whered is the space dimension and λ the exponent of the powerlike correlation function of the driving force. It is shown that in spite of the additional divergences, the asymptotic behaviour of the model near two dimensions is the same as in higher dimensions, contrary to recent conjectures based on an incorrect renormalization procedure.     

Helical structures,fractal dimensions and renormalization-group approach in homogeneous turbulence

The concept of helicity-fluctuation hierarchy as generating the fractal structure of turbulence in conjunction with the renormalization-group theory are utilized to renormalize the iterative solution of the Navier-Stokes equation in all orders of perturbation theory. This solution produces the value of the intermittency exponent μ = 0.4 for both the lognormal and the β model of homogeneous turbulence.     

On the theory of turbulence: A non eulerian renormalized expansion

A non Eulerian framework for a renormalized theory of isotropic homogeneous steady state turbulence at high Reynold's numbers is developed. By construction it is invariant under random Galilei transformations. A direct interaction factorization is free of infrared singularities and yields Kolmogorov scaling for the static as well as for the dynamic correlation and response functions.     

Statistics of long-wavelength fluctuations and the expansion rate of Richtmyer-Meshkov turbulence zone

The important problem of Richtmyer-Meshkov turbulence (RMT) is solved. Much work has been devoted to the experimental, numerical (direct numerical simulation), and semiphenomenological (turbulent diffusion and bubble envelope models) analysis of RMT.All of them were of approximate character. They considered the evolution of a mixing layer, and its average thickness h(t) was found. Then, the approximate value of the most important exponent θ (h ∝ t ^θ) was judged from the slope of the h(t) curve in the lnt-lnh coordinates. In this work, the theoretical approach for the exact determination of θ is developed.     

Scalar and tensor spherical harmonics expansion of the velocity correlation in homogeneous anisotropic turbulence

The representation theory of the rotation group is applied to construct a series expansion of the correlation tensor in homogeneous anisotropic turbulence. The resolution of angular dependence is the main analytical difficulty posed by anisotropic turbulence; representation theory parametrises this dependence by a tensor analogue of the standard spherical harmonics expansion of a scalar. The series expansion is formulated in terms of explicitly constructed tensor bases with scalar coefficients determined by angular moments of the correlation tensor.     

Probability density,diagrammatic technique,and epsilon expansion in the theory of wave turbulence

We apply the methods of field theory to study the turbulent regimes of statistical systems. First we show how one can find their probability densities. For the case of the theory of wave turbulence with four-wave interaction we calculate them explicitly and study their properties. Using those densities we show how one can in principle calculate any correlation function in this theory by means of direct perturbative expansion in powers of the interaction. Then we give the general form of the corrections to the kinetic equation and develop an appropriate diagrammatic technique. This technique, while resembling that of ϕ⁴ theory, has many new distinctive features. The role of the ϵ = d − 4 parameter of ϕ⁴ theory is played here by the parameter κ = β + d − α − γ where β is the dimension of the interaction, d is the space dimension, α is the dimension of the energy spectrum and γ is the “classical” wave density dimension. If κ > 0 then the Kolmogorov index is exact, and if κ < 0 then we expect it to be modified by the interaction. For κ a small negative number, α < 1 and a special form of the interaction we compute this modification explicitly. We neglect the encountered IR divergencies with the intend to study them in a later publication.     

Express analysis of turbulence parameters

A method utilizing a Hartmann sensor for obtaining estimates of the external scale L ₀ of turbulent fluctuations and the structural constant C _n ² during the variation of the parameters of the wavefront of a light beam that has passed through turbulence is suggested. The method is based on Zernike polynomial decomposition of phase fluctuations at the given aperture and on analysis of this expansion coefficient’s statistics. Application of the method for making an estimate of the turbulence characteristics in a water cell yielded results that are in good agreement with the estimates obtained by other methods.     

On the character of peculiarities in the position-space renormalization-group transformations

The image magnetic field is shown to be a discontinuous function of the object magnetic field near the zero-field point for T < T_c in the thermodynamic limit in the position-space renormalization-group procedure as applied to an infinite-range interaction model.     

A renormalization-group analysis for two unidirectionally coupled Feigenbaum systems at the hyperchaos threshold

Saratov Branch, Radio Engineering and Electronics Institute, USSR Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofisika, Vol. 33, No. 7, pp. 788–792, July, 1990.