Renormalization group in the theory of fully developed turbulence. The problem of infrared-essential corrections to the Navier-Stokes equation

Using the RG approach to the theory of fully developed turbulence, we consider the problem of possible IR-essential corrections to the Navier-Stokes equation. We formulate an exact criterion for the actual IR-essentiality of the corrections. In accordance with this criterion. we check whether certain classes of composite operators are IR-essential. All of these operators turn out to be actually IR-inessential for arbitrary values of the RG expansion parameter . This confirms the absence of the crossover and enables the RG results obtained for asymptotically small values of to be extrapolated to the physical range >2.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 1, pp. 47–63, April, 1996.Translated by M. V. Chekhova.     

Renormalization group in the theory of fully developed turbulence. Composite operators of canonical dimension 8

Renormalization and critical dimensions of the family of Galilean invariant scalar composite operators of canonical dimension eight are considered within the framework of the renormalization group approach to the stochastic theory of fully developed turbulence.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 92–101, January, 1996.     

Renormalization group in the theory of developed turbulence. The problem of justifying the Kolmogorov hypotheses for composite operators

In this paper, the stochastic theory of developed turbulence is considered within the framework of the quantum field renormalization group and operator expansions. The problem of justifying the Kolmogorov-Obukhov theorem in application to the correlation functions of composite operators is discussed. An explicit expression is found for the critical dimension of a general-type composite operator. For an arbitrary UV-finite composite operator, the second Kolmogorov hypothesis (the viscosity-independence of the correlator) is proved and the dependence of various correlators on the external turbulence scale is determined. It is shown that the problem involves an infinite number of Galilean-invariant scalar operators with negative critical dimensions. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 122–136, January, 1997.     

Diagram equations of the theory of fully developed turbulence

Skeleton diagram equations of turbulence theory — the Dyson equations and the equations for vertices of three types — are obtained nonperturbatively. Their derivation is based on the use of an equation in functional derivatives for the characteristic functional of a hydrodynamic system described by Navier-Stokes equations in the presence of an external random force. The iterative solution of these equations reproduces the perturbation series for second moments that is usually obtained in a more complicated way and also the series for the third moments.Institute of Problems in Mechanics, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 1, pp. 28–37, October, 1994.     

Renormalization group approach and short-distance expansion in theory of developed turbulence: Asymptotics of the triplex equal-time correlation function

Asymptotics of the triplex equal-time correlation function for the turbulence developed in incompressible fluids in the region of widely separated wave vector values is investigated using the renormalization group approach and short-distance expansion. The problem of the most essential composite operators contributing to these asymptotics is examined. For this purpose, the critical dimensions of a family of composite quadratic tensor operators in the velocity gradient are found. Considered in the one-loop approximation, the contribution of these operators turns out to be less substantial (although not significantly) than the contribution of the linear term. The derived asymptotics of the triplex correlator coincide in form with that predicted by the EDQNM approximation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 3, pp. 450–461, December, 1995.     

Renormalization group in turbulence theory: Exactly solvable Heisenberg model

An exactly solvable Heisenberg model describing the spectral balance conditions for the energy of a turbulent liquid is investigated in the renormalization group (RG) framework. The model has RG symmetry with the exact RG functions (the β-function and the anomalous dimension γ) found in two different renormalization schemes. The solution to the RG equations coincides with the known exact solution of the Heisenberg model and is compared with the results from the ε expansion, which is the only tool for describing more complex models of developed turbulence (the formal small parameter ε of the RG expansion is introduced by replacing a δ-function-like pumping function in the random force correlator by a powerlike function). The results, which are valid for asymptotically small ε, can be extrapolated to the actual value ε=2, and the few first terms of the ε expansion already yield a reasonable numerical estimate for the Kolmogorov constant in the turbulence energy spectrum. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 2, pp. 245–262 May. 1998.     

The principle of maximum randomness in the theory of fully developed turbulence. II. Isotropic decaying turbulence

A statistical model for describing the decay of developed isotropic turbulence of an incompressible fluid is proposed. The model uses the distribution function of the velocity pulsations introduced earlier by the authors on the basis of the principle of maximum randomness of the velocity field for a given spectral energy flux. The renormalization-group technique and expansion are used to calculate the correlation functions of the velocity that occur in the equation of spectral energy balance. This leads to a closed equation for the dependence of the energy spectrum on the integral turbulence scaler _c(t). In the inertial interval, this equation gives the Kolmogorov asymptotic spectrum, while for the time dependence ofr _c(t) and the pulsation energye(t) it predicts the power lawsr _c(t)t^2/5 andr(t)t ^–6/5.Physics Research Institute of the St Petersburg University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 1, pp. 150–159, July, 1993.     

Directed-bond percolation subjected to synthetic compressible velocity fluctuations: Renormalization group approach

We study the directed-bond percolation process (sometimes called the Gribov process because it formally resembles Reggeon field theory) in the presence of irrotational velocity fluctuations with long-range correlations. We use the renormalization group method to investigate the phase transition between an active and an absorbing state. All calculations are in the one-loop approximation. We calculate stable fixed points of the renormalization group and their regions of stability in the form of expansions in three parameters (ε, y, η). We consider different regimes corresponding to the Kraichnan rapid-change model and a frozen velocity field.     

Renormalization group analysis for singularities in the wave beam self-focusing problem

A singular solution of the boundary value problem for the system of equations describing wave beam self-focusing is investigated by constructing renormalization group symmetries. New analytic expressions are found that characterize the spatial evolution of a beam with an arbitrary initial profile in a medium with cubic nonlinearity. The behavior of a Gaussian beam is thoroughly analyzed up to the moment the solution singularity is formed, and a hypothesis is proposed for describing the solution structure after the singularity occurs. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp. 405–418, June, 1999.     

On a fully coupled nonlinear parabolic problem modelling miscible compressible displacement in porous media

We study a model describing a compressible and miscible displacement in a porous medium. It consists of a coupled system of nonlinear parabolic partial differential equations. Using nonclassical estimates and renormalization tools, we prove the existence of relevant weak solutions for the problem. This is the first existence result obtained for a transport model containing both the coupling due to the compressibility assumption and the coupling due to the concentration dependent viscosity.     

Structure of the linearized problem for compressible parallel fluid flows

We consider the linearized compressible Navier-Stokes equation near a parallel flow in a cylindrical domain restricting our study to perturbations periodic in the generatrix direction. For any parameter values, we show that the initial value linear evolution problem is solved by the direct sum of a (strictly) contraction semi-group and an analytic semi-group. Any unbounded in time solution of this linear problem comes from isolated eigenvalues with finite multiplicities, which have non negative real part, and whose imaginary part is bounded. In addition, we precise the structure of the spectrum of the generator of the semi-group, locating the essential spectrum stricly on the left side of the complex plane.

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Sunto Si linearizzano le equazioni di Navier-Stokes comprimibili attorno ad un moto lineare in un cilindro. Senza imporre restrizioni sul moto di base, si studia il problema lineare per perturbazioni periodiche nella direzione della generatrice del cilindro. Si prova che il problema di evoluzione lineare è dato dalla somma diretta di un semigruppo di stretta contrazione ed un semigruppo analitico. Ogni soluzione temporalmente non limitata deriva da autovalori isolati di molteplicità finita, aventi parte reale positiva e parte immaginaria limitata. Si studia anche la struttura dello spettro del generatore del semigruppo, costituito nel lato destro del piano complesso da autovalori di molteplicità finita, e nel lato sinistro di autovalori e da uno spettro essenziale situato su di una retta parallela all’asse immaginario.

     

Corrections to fully developed turbulent spectra due to compressibility of the fluid

We construct the regular expansion at small compressibilities for the theory of fully developed turbulence of an isotropic homogeneous compressible fluid with MSR-type action. The parameter of the expansion is the Mach numberMa. For the inertial range of a compressible fluid, we study the infrared singularities determined by the transverse fields, which are used in the theory of incompressible fluids. These singularities are connected with the composite operators of transverse fields that are investigated by the quantum field renormalization group method. As a result, it is shown that the transverse fields induce scaling behavior with theMa scaling dimension equal to 1/3 (i.e.,Ma k^–1/3 is the dimensionless scaling parameter of the correlation functions in the inertial range).Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 3, pp. 375–389, March, 1996.Translated by L. O. Chekhov.     

On the nature of turbulence in a problem on the motion of a fluid behind a ledge

In the present paper, we suggest a numerical method for the analysis of the motion of a viscous incompressible fluid under the transition to the turbulent mode for an example of the numerical solution of a three-dimensional space problem on the fluid flow behind a ledge for various values of the Reynolds number. We show that, at the initial stages, the turbulence in the problem is developed via successive bifurcations of generation of a stable cycle, two-dimensional tori, and then three-dimensional tori in the infinite-dimensional phase space of the system.     

Solvability of the initial-boundary-value problem for the equations of motion of a viscous compressible fluid

It is proved that the initial-boundary-value problem for the system of equations describing the motion of a compressible fluid with a constant viscosity is locally solvable with respect to time. The heat conductivity is not taken into account. The solution is found in the class W _q ^2.1 , q>3.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 128–142, 1976.     

Renormalization group in a fermionic hierarchical model

A study is made of a hierarchical model with spin values in a Grassmann algebra defined by a potential of general form. The action of the spin-block renormalization group in the space of Hamiltonians is reduced to a rational mapping of the space of coupling constants into itself. The methods of the theory of bifurcations are used to investigate the nontrivial fixed points of this mapping. A theorem establishing the existence of a thermodynamic limit of the model at these points in a certain neighborhood of a bifurcation value is proved.This work was done with financial support of the Russian Foundation for Fundamental Research (Grant 93-011-16099).State University, Kazan. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 2, pp. 282–293, November, 1994.     

On some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a local in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the \(L_p\) in time and \(L_q\) in space framework with \(2 < p < \infty \) and \(N < q < \infty \) under the assumption that the initial domain is a uniform \(W^{2-1/q}_q\) one in \({\mathbb {R}}^{N}\, (N \ge 2\) ). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve problem by the Banach contraction mapping principle based on the maximal \(L_p\) – \(L_q\) regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key issue for the linear theorem is the existence of \({\mathcal {R}}\) -bounded solution operator in a sector, which combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal \(L_p\) – \(L_q\) regularity theorem. The nonlinear problem we studied here was already investigated by several authors (Denisova and Solonnikov, St. Petersburg Math J 14:1–22, 2003; J Math Sci 115:2753–2765, 2003; Secchi, Commun PDE 1:185–204, 1990; Math Method Appl Sci 13:391–404, 1990; Secchi and Valli, J Reine Angew Math 341:1–31, 1983; Solonnikov and Tani, Constantin carathéodory: an international tribute, vols 1, 2, pp 1270–1303, World Scientific Publishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin, 1992; Tani, J Math Kyoto Univ 21:839–859, 1981; Zajaczkowski, SIAM J Math Anal 25:1–84, 1994) in the \(L_2\) framework and Hölder spaces, but our approach is different from them.     

Study of the higher-order asymptotic behavior of quantum field expansions in the theory of two-dimensional fully developed turbulence

We use a simplified (0+1)-dimensional theory to develop approaches for studying the higher-order asymptotic behavior of quantum field expansions in the two-dimensional theory of fully developed turbulence. We consider the asymptotic behavior of the correlation function in the small-time limit in the theory of fully developed turbulence and derive and investigate the stationarity equation. We show that the perturbation series in this limit has a finite convergence radius.