Fedor V. Shugaev

On the problem of beam focusing in the turbulent atmosphere

Two parts of the problem were analyzed. The first one is the adequate description of turbulence. The result is the simulation of the evolution of the refractive index due to turbulence. The second one is the beam focusing on condition that the refractive index is subject to spatial and temporal variations. The turbulence was simulated with the aid of the solution to the Navier-Stokes equations. Three kinds of initial conditions were used: (i) the vortical field was given, the velocity divergence (dilatation) being zero and the temperature being constant; (ii) the velocity divergence was given, the vorticity being zero and the temperature being constant; (iii) there is a temperature distribution, the vorticity and the dilatation being zero. In all cases, the initial values of the density are constant. The problem is set in the infinite space, the initial data being random functions. The solution of the Navier- Stokes equations was reduced to the solution of integral equations of the Volterra type. The iterative procedure was used. The comparison of the subsequent iterations allows to conclude that the convergence takes place. The problem of compensation for turbulent distortions of a laser beam was solved. The resolving function determines the necessary deformation of the mirror. The knowledge of the resolving function indicates the way to the beam focusing in the turbulent atmosphere.

Acoustic radiation frequency of a cylindrical vortex

Acoustic radiation frequency of a cylindrical vortex in the air has been calculated. Calculations are based on the Navier-Stokes equations using the expansion of the functions in powers of small parameter characterizing the initial vorticity. Non-uniform system of parabolic differential equations with constant coefficients is obtained. The initial radius of the cylinder varies over a wide range. The problem is considered for the case of plane flow. In contrast to previously results, it is shown that at small values of vorticity, the frequency of the acoustic radiation depends only on the geometric size (the initial radius of the vortex).

Modeling of laser beam propagation through turbulence

Our approach for modeling laser beam propagation through turbulence involves parabolic equation method and results of experimental investigation in laboratory. The analytic solution to the problem of the Gaussian beam propagation through non-uniform gas has been derived. The solution depends on the refracted index, i.e. on the gas density. The density distribution can be found from the Navier-Stokes system. The appropriate solution may be constructed by two ways : (i) as a series in powers of vorticity which is supposed to be small; (ii) with the aid of the parametrix method which includes an iterative procedure. It follows from the solution that acoustic radiation of vortex rings arises. Statistical properties of the propagating beam were found from the solution to the parabolic equation as average over time. In experiments the propagation path was equal to 7 m. The laser beam propagation was accompanied by convection and lateral wind. The frequency of turbulent fluctuations was equal to 2-10 Hz. Phase trajectories were found as well as statistical properties of the beam intensity in turbulent gas flow. The conclusion is as follows. Statistical characteristics traditionally used for the estimation of the laser beam special distortions in the open space transmission channels are to be complemented by the dynamic parameters such as the space of embeddings dimension, characteristic frequencies for the phase trajectories and so on.

Laser beam propagation through an ensemble of vortex rings in air

The problem of the evolution of an ensemble of vortex rings in air has been solved. The full system of the Navier -Stokes equations was used. The parametrix method was applied. The calculations were performed for a wide range of the ring parameters (circular and elliptic cross-sections, various diameters of the rings, their different orientation in the space etc.). The initial value problem is as follows. The vorticity has non-zero value only inside the rings at initial instant, the density and the temperature being constant everywhere at t=0. If the density is known, then it is possible to find the refractive index. The solution to the Navier-Stokes equations is an oscillating one. Thus the refractive index is an oscillating function with respect to time. These results enable to model turbulence in an adequate way without using the Taylor frozen turbulence hypothesis. The evolution of the frequency spectrum of the density fluctuations was obtained. These results were compared with Tatarskiis data. The intensity of a laser beam propagating through the ensemble of vortex rings in air was found with the aid of the parabolic equation method. A numerical procedure is set forth which allows to solve the problems of superresolution without using regularization methods. The task is as follows. There is a set of experimental data and an instrument function (with some error). We change the domain in such a manner that the corresponding MTF has nowhere zero values. The procedure enables to solve problems of focusing in the turbulent atmosphere.

Local-linear method of super-resolution for compensation of image distortions using new model of turbulence

Atmospheric turbulence is one of the important factors that influence on scene spatial resolution. In order to restore an image with minimum distortions one must know the correlation function for fluctuations of refractive index and the distorting PSF as a result. Grid-generated turbulence is a classic example of homogeneous and isotropic turbulence. Statistical properties of this flow have been investigated experimentally. In our case of grid-generated turbulence the statistical properties are distinct from the Kolmogorovs two-thirds law. Calculations performed on the basis of the linearized three-dimensional unsteady Navier-Stokes equations gave similar results. We modelled laser beam propagation through turbulent atmosphere and obtained numerical data for the distortion of images. The distortion of PSF and the set of resolving functions were found according to the structure function. The problem of compensation of distortions caused by turbulence was solved with the aid of a new local-linear super-resolution method. This method allows to resolve turbulent distortions of PSF at low signal-to-noise ratio.

The Evolution of Acoustic Radiation by an Ensemble of Vortex Rings in Air

The evolution of acoustic radiation emitted by an ensemble of vortex rings in air is studied on the basis of nonstationary Navier–Stokes equations. We use the expansions of required functions into a power series of the initial vorticity which is a small value. The Navier–Stokes equation system reduces to a parabolic system with constant coefficients for the higher derivatives. The problem is posed as follows. The vorticity is defined inside the toroid at t = 0. The other parameters of the gas are assumed to be constant throughout the space at the initial instant of time. The solution is expressed in terms of multiple integrals, which are calculated using Korobov grids. The density oscillations were investigated. The results show that the frequency spectrum depends on time; high-frequency oscillations are observed at small times and low-frequency oscillations then occur. At the same time, the amplitude of high-frequency oscillations decreases in comparison with low-frequency oscillations. Thus, a transition of energy from the high-frequency spectrum to the lowfrequency spectrum occurs. These results can be useful for modeling decaying grid turbulence.

Propagation of the non-paraxial Gaussian beam through the inhomogeneous atmosphere

Turbulent fluctuations in the atmosphere distort a laser beam during its propagation. There exist two problems : (i) adequate description of the atmospheric turbulence and (ii) analysis of the propagation of the beam through turbulence, investigation of the beam spreading and distribution of its intensity. Unfortunately, only the scalar case of beam propagation has been considered often in the literature. Most part of authors studied only paraxial beams. Non-paraxial beams are considered in [1, 2]. Below we consider the propagation of a non-paraxial laser beam. The analysis has been made on the basis of the Maxwell equations. Two cases have been considered: (i) the permittivity and permeability are constant (the homogeneous atmosphere); (ii) the case when the permeability is equal to unity, the permittivity being dependent on coordinates. We assume that the permittivity is close to unity. Let us consider the first case. Some details of the solution were recently published in [3]. We have a linear system of ordinary differential equations with constant coefficients due to the Fourier transform. The unknown functions are determined from the condition at the plain x3 = 0 ( x3 being the coordinate along the axis of the beam). In the second case (the permittivity is the function of coordinates being close to unity) we have a system of linear ordinary differential equations after the Fourier transform, too. The right-hand terms depend on the previous solution which was obtained for the homogeneous atmosphere. The solution is the sum of that one for the homogeneous atmosphere and that one for the variable part of the permittivity. Thus we have the solution which describes the propagation of the non-paraxial beam through the inhomogeneous atmosphere on condition that the variation of the refractive index is small. Numerical calculations were fulfilled for the components of the electric field.

Modelling of propagation and scintillation of a laser beam through atmospheric turbulence

The investigation was fulfilled on the basis of the Navier-Stokes equations for viscous heat-conducting gas. The Helmholtz decomposition of the velocity field into a potential part and a solenoidal one was used. We considered initial vorticity to be small. So the results refer only to weak turbulence. The solution has been represented in the form of power series over the initial vorticity, the coefficients being multiple integrals. In such a manner the system of the Navier- Stokes equations was reduced to a parabolic system with constant coefficients at high derivatives. The first terms of the series are the main ones that determine the properties of acoustic radiation at small vorticity. We modelled turbulence with the aid of an ensemble of vortical structures (vortical rings). Two problems have been considered : (i) density oscillations (and therefore the oscillations of the refractive index) in the case of a single vortex ring; (ii) oscillations in the case of an ensemble of vortex rings (ten in number). We considered vortex rings with helicity, too. The calculations were fulfilled for a wide range of vortex sizes (radii from 0.1 mm to several cm). As shown, density oscillations arise. High-frequency oscillations are modulated by a low-frequency signal. The value of the high frequency remains constant during the whole process excluding its final stage. The amplitude of the low-frequency oscillations grows with time as compared to the high-frequency ones. The low frequency lies within the spectrum of atmospheric turbulent fluctuations, if the radius of the vortex ring is equal to several cm. The value of the high frequency oscillations corresponds satisfactorily to experimental data. The results of the calculations may be used for the modelling of the Gaussian beam propagation through turbulence (including beam distortion, scintillation, beam wandering). A method is set forth which describes the propagation of non-paraxial beams. The method admits generalization to the case of inhomogeneous medium.

Grid turbulence and its interaction with a shock wave

Turbulent fluctuations of density and pressure in air and argon in a shock tube have been investigated as well as their interaction with a shock wave reflected from a perforated plate at the end of a shock tube. Air and argon were used as test gases. The Mach number of the incident shock was 1.9–3.9, that one of the reflected shock was 1.4–2.4. The turbulent length scale behind the incident shock was measured as well as that one behind the reflected shock. The last value is a few times less than the former. It was established that there is overpressure in the turbulent flow behind the reflected shock. The value of the overpressure is 12% in argon and 9% in air.

Calculation of the Acoustic Spectrum of a Cylindrical Vortex in Viscous Heat-Conducting Gas Based on the Navier-Stokes Equations

An extremely interesting problem in aero-hydrodynamics is the sound radiation of a single vortical structure. Currently, this type of problem is mainly considered for an incompressible medium. In this paper a method was developed to take into account the viscosity and thermal conductivity of gas. The acoustic radiation frequency of a cylindrical vortex on a flat wall in viscous heat-conducting gas (air) has been investigated. The problem is solved on the basis of the Navier–Stokes equations using the small initial vorticity approach. The power expansion of unknown functions in a series with a small parameter (vorticity) is used. It is shown that there are high-frequency oscillations modulated by a low-frequency signal. The value of the high frequency remains constant for a long period of time. Thus the high frequency can be considered a natural frequency of the vortex radiation. The value of the natural frequency depends only on the initial radius of the cylindrical vortex, and does not depend on the intensity of the initial vorticity. As expected from physical considerations, the natural frequency decreases exponentially as the initial radius of the cylinder increases. Furthermore, the natural frequency differs from that of the oscillations inside the initial cylinder and in the outer domain. The results of the paper may be of interest for aeroacoustics and tornado modeling.