Application of renormalization group techniques to transport in the presence of nonlinear sources and sinks

The paper considers the problem of spreading of a localized distribution due to diffusion in the presence of nonlinear sources and sinks modeling annihilation and creation of material in chemical reactions. The evolution of the parameters characterizing the problem, namely, the amplitude and radius of the distribution, has been investigated under the assumption that the distribution is self-similar. These parameters have been calculated using the renormalization-group method with renormalization of the diffusion coefficient and the total amount of material. As a result, it is possible to classify various regimes of the asymptotic behavior of the system at long times according to the sign of nonlinear interaction constant and the spatial dimensionality. In particular, the conditions under which the regime of asymptotic freedom and the peaking regime are realized have been found. The renormalization group method not only allows us to calculate the exponents of functions with power-law behavior, but also to track the transition to the asymptotic regime and calculate numerical coefficients. Zh. éksp. Teor. Fiz. 115, 1497–1510 (April 1999)     

Stochastic analysis of a displacement front in a randomly heterogeneous medium

The behavior of the interface in a two-phase immiscible fluid flow in a randomly heterogeneous porous medium is investigated. The medium is described by the permeability distribution which represents a random field with given statistical characteristics. When the approach proposed is used, it turns out to be possible to relate the statistical characteristics of the interface with the statistical characteristics of the permeability field and the properties of the phases. On the basis of this relation an important characteristic of the two-phase flow, namely, the average saturation distribution in the neighborhood of the interface, can be calculated.     

Self-similar solution of the plane problem of the evolution of a hydraulic fracture crack in an elastic medium

The plane problem of the evolution of a hydraulic fracture crack in an elastic medium is considered. It is established that a self-similar solution is only possible at a constant rate of fluid injection. The solution for the value of the crack opening is presented in the form of a series expansion in Chebyshev polynomials of the second kind, and expansion coefficients are obtained as a solution of the algebraic set of equations which arise when projecting the balance equation for injected fluid mass on Chebyshev polynomials. When there is no part of the region unfilled with fluid (a fluid lag), the gradient of the crack opening at the crack tip turns out to be singular when the finiteness of the medium stress intensity factor is taken into account. According to the estimate made, the rate of convergence of the series expansion for the solution in Chebyshev polynomials is fairly rapid for a small injection intensity.     

A Functional Formulation of the Statistical Theory of Turbulence

Doklady Physics - The Navier–Stokes equation in the presence of external regular and random forces is considered. The statistical solution is described in terms of the characteristic...     

Two-dimensional problem for internal waves generated by moving singular sources

When bodies move in a liquid with inhomogeneous density in a gravitational field waves are excited even at low velocities and in the absence of boundaries. They are the so-called internal waves (buoyancy waves), which play an important part in geophysical processes in the ocean and the atmosphere [1–4]. A method based on the replacement of the bodies by systems of point sources is now commonly used to calculate the fields of internal waves generated by moving bodies. However, even so the problems of the generation of waves by a point source and dipole are usually solved approximately or numerically [5–11]. In the present paper, we obtain exact results on the spectral distribution of the emitted waves and the total radiation energy per unit time for some of the simplest sources in the two-dimensional case for an incompressible fluid with exponential density stratification. The wave resistance is obtained simply by dividing the energy loss per unit time by the velocity of the source. In the final section, some results for the three-dimensional case are briefly formulated for comparison.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 77–83, March–April, 1981.     

Role of local and nonlocal interactions in the formation of the developed turbulence regime

The successful use of the renormalization group method for calculating the universal constants of developed turbulence has provoked a discussion on the extent to which the results obtained correspond to the ideas of Kolmogorov's theory of localization of the intermodel couplings, since the computational procedure employed was based on consideration of the essentially nonlocal direct effect of the small-scale on the large-scale modes. Within the framework of a field-theory approach, it is shown that the use of the renormalization group method in conjunction with the -expansion in fact means taking into account the local and filtering out the nonlocal intermodel interactions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 36–42, July–August, 1990.     

Experimental investigation of variability of ultraviolet fluxes at the earth's surface

Some results of two years of regular measurements of UV radiation at the earth's surface in Siberia (56.5^o N, 85^o E) are given. It is shown that the presence of snow cover increases the UV irradiation of the earth by scattered radiation by 9–15%. In cloudless but windy weather, quasi-periodic fluctuations of UV irradiation with periods of 5–15 min and with fluctuation swings of 1–10% are observed. Low-level, heavy cloudiness in summer considerably reduces (by an average of 30–35%) UV irradiation, especially when the sun is low (down to 30^o). Mid-and upper-level clouds increase the UV irradiation of the earth by scattered radiation. When the sun is high (30–50^o) and there is heavy cloudiness, this increase can reach 28% in the UV-A spectral range. Tomsk State University. Academician V. D. Kuznetsov Siberian Physics and Technology Institute, Tomsk State University. Institute of Optical Monitoring, Siberian Branch of the Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 71–74, September, 1998.     

Energy characteristics of harmonic internal wave generators

The propagation of internal waves plays an important role in liquid media with layers that vary according to density (stratified liquids) and are located in a gravitational field, which include the Earth's atmosphere and oceans. Highly controlled experiments are essential for investigating efficient generators of internal waves (in particular, harmonic internal waves). Hence, it is important to compare the efficiences of various types of internal wave generators. This problem is considered for the simplest forms of stratification: discontinuous and uniform (with a constant buoyancy frequency N). Although there are very few studies of oscillations in the case of discontinuous stratification, there are even fewer investigations of uniform stratification (e.g., see [1–4]). A comparison of the efficiences of different types of generators has not been made for the latter case. This is done below on the basis of energy estimates for two types of generators: for objects (a sphere or cylinder) that undergo small harmonic oscillations in a liquid and for objects with pulsating volumes.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 53–59, July–August, 1986.