Self-similar solutions describing the propagation of solitary waves above underwater ridges and troughs

The nonlinear ray method [1] is used to investigate the propagation of solitary waves over an uneven bottom. In the process of nonlinear evolution of the wave front, singular points develop in it; these are treated in the given model as discontinuities [2, 3]. In contrast to earlier studies, it is not assumed here that the intensity of the discontinuity is weak. Boundary conditions at the discontinuities are introduced on the basis of the results of Miles and Bakholdin [4–6], and this makes it possible to take into account the energy loss at a discontinuity and the effects of wave reflection and construct a number of new self-similar solutions for the propagation of a wave above a ridge and trough. The main attention is devoted to considering how the type of solution depends on the parameters of the wave and the relief. For certain values of the parameters, the self-similar solution of the encounter of a homogeneous wave with a ridge is not unique. The reason for this is the singularity of the relief at the end of the ridge. A numerical investigation has therefore also been made of the encounter of a wave with a ridge having a smooth relief at its end. For an under-water trough and a ridge—trough system, self-similar solutions with complete or partial reflection or transmission of the wave energy into the trough are found. A reflected wave can also arise from an encounter with a ridge.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 137–144, September–October, 1985.I thank A. G. Kulikovskii and A. A. Barmin for their interest in the work and for valuable comments made as the paper was being prepared for press.     

Effect of Localization of Pulsed Energy Addition on the Wave Drag of an Airfoil in a Transonic Flow

The possibility of controlling the aerodynamic characteristics of airfoils with the help of local pulsed-periodic energy addition into the flow near the airfoil contour at transonic flight regimes is considered. By means of the numerical solution of two-dimensional unsteady equations of gas dynamics, changes in the flow structure and wave drag of a symmetric airfoil due to changes in localization and shape of energy-addition zones are examined. It is shown that the considered method of controlling airfoil characteristics in transonic flow regimes is rather promising. For a zero angle of attack, the greatest decrease in wave drag is obtained with energy addition at the trailing edge of the airfoil.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 60–67, September–October, 2005.     

Waves in a Bed under Periodic Tangential Loading

The far-zone structure of the wave field in an elastic bed on a rigid foundation is considered. The wave field is generated by a tangential periodic force applied to the bed surface. The amplitude- frequency characteristics of surface vibrations for propagating modes are found. The partition of energy between different modes is considered.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 109–115, July– August, 2005.     

Nonreacting chemical transport in two-phase reservoirs — Factoring diffusive and wave properties

The vertical transport of mass, energy andn unreacting chemical species in a two-phase reservoir is analysed when capillarity can be ignored. This results in a singular system of equations, comprising mixed parabolic and hyperbolic equations. We derive a natural factorisation of these equations into diffusive and wave equations. If diffusive or conductive effects are important for onlyN–1 of the chemical species, then there areN parabolic equations, andn+2–N wave equations. Each wave equation allows for the corresponding variable to be discontinous, or equivalently, for shock propagation to occur. Steady flows were shown to allow for more than two (vapour and liquid dominated) saturations for a given mass, energy and chemical flux, but when thermal conduction and chemical diffusion are unimportant, only the vapour and liquid dominated cases appear likely to occur. For infinitesimal shocks there is a continuous flux vector for each diffusive variable. The existence of these continuous flux vectors results in significant simplifications of the corresponding wave equations. It remains an open question if continuous flux vectors exist for finite shocks. General expressions are obtained for the diffusion and wave matrices, which require no knowledge of continuous flux vectors.     

Convective heat transfer in laser-irradiated graphite

The gas dynamic and thermal processes developing near the surface of graphite after exposure to a 20-nsec laser pulse with an energy E- 0.1–1 J and a wavelength of 0.6943 m are investigated experimentally and by mathematical modeling. The times required for the shock wave to degenerate into an acoustic wave are also considered. Typical density profiles over the axial section of the inhomogeneity are presented for various moments of time. It is noted that the rate of ascent of the thermal inhomogeneity is much higher than the free convection velocity. The convective heat-transfer processes are studied in detail through numerical solution of the system of two-dimensional Navier-Stokes equations. The results of the calculations are in satisfactory agreement with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 180–182, May–June, 1989.     

Blast wave propagation in air from a gaseous charge

In the point explosion problem it is assumed that an instantaneous release of finite energy causing shock wave propagation in the ambient gas occurs at a space point. The results of the solution of the problem of such blasts are contained in [1–4]. This point model is applied for the determination of shock wave parameters when the initial pressure in a sphere of finite radius exceeds the ambient air pressure by 2–3 orders of magnitude. The possibility of such a flow simulation at a certain distance from the charge is shown in papers [4, 5] as applied to the blast of a charge of condensed explosive and in [6, 8] as applied to the expansion of a finite volume of strongly compressed hot gas. In certain practical problems the initial pressure in a volume of finite dimensions exceeds atmospheric pressure by a factor 10–15 only. Such cases arise, for example, in the detonation of gaseous fuel-air mixtures. The present paper considers the problem of shock wave propagation in air, caused by explosion of gaseous charge of spherical or cylindrical shape. A numerical solution is obtained in a range of values of the specific energy of the charge characteristic for fuel-air detonation mixtures by means of the method of characteristics without secondary shock wave separation. The influence of the initial conditions of the gas charge explosion (specific energy, nature of initiation, and others) is investigated and compared with the point case with respect to the pressure difference across the shock wave and the positive overpressure pulse.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 110–118, May–June, 1986.     

Surface wave generation in the process of finite deformation of the bottom of a basin

The process of generation of three-dimensional irrotational fluid motions induced by small local finite-duration displacements of part of the bottom of a basin is considered within the framework of wave linear theory for a basin of constant depth. The solution of the problem and an expression for the total wave field energy are obtained using integral transforms. The general properties of the process of unsteady wave generation induced by short-term and slow deformations of the bottom are analyzed. Within the framework of the piston generation model the energy characteristics of axisymmetric waves are compared for two time laws of bottom deformation of identical duration. In general, it is shown that under certain conditions the nature and intensity of the wave process depend on both the time law and the duration of the deformation process.Sevastopol. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 151–156, March–April, 1996.     

Calculation of reflection of a blast wave by a plane

Some results are given of calculation of the reflection of a blast wave by a rigid flat surface. A model of the explosion with a simple energy dissipation mechanism is considered, radiation being taken into account in the approximation of radiative heat conduction. The pressure distribution on the surface and the flow pattern in the region of propagation of the incident and reflected shock waves are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 179–182, September–October, 1982.We thank L. A. Chudov for helpful discussions of the work.     

Shock Compression of a Plate on a Wedged–Shaped Target

Problems of compression of a plate on a wedge–shaped target by a strong shock wave and plate acceleration are studied using the equations of dissipationless hydrodynamics of compressible media. The state of an aluminum plate accelerated or compressed by an aluminum impactor with a velocity of 5—15 km/sec is studied numerically. For a compression regime in which a shaped–charge jet forms, critical values of the wedge angle are obtained beginning with which the shaped–charge jet is in the liquid or solid state and does not contain the boiling liquid. For the jetless regime of shock–wave compression, an approximate solution with an attached shock wave is constructed that takes into account the phase composition of the plate material in the rarefaction wave. The constructed solution is compared with the solution of the original problem. The temperature behind the front of the attached shock wave was found to be considerably (severalfold) higher than the temperature behind the front of the compression wave. The fundamental possibility of initiating a thermonuclear reaction is shown for jetless compression of a plate of deuterium ice by a strong shock wave.     

Discontinuities of the variables that characterize the propagation of solitary waves in a fluid layer

A nonlinear system of equations of hyperbolic type describing the propagation of solitary waves is considered [1]. A solitary wave is characterized in this approximation by two variables — the energy density per unit length measured along its crest, and the direction of the normal to the wave crest. The evolution of a wave described by the system may lead to the appearance of discontinuities, at which there are jumps in the energy density and the direction of the wave crest [2]. To establish the conditions at the discontinuities, a solution describing the interaction of nonparallel solitons [3, 4] is used. The obtained conditions are used to solve the problem of the decay of an arbitrary discontinuity in terms of soliton variables.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 87–93, May–June, 1984.I thank A. G. Kulikovskii and A. A. Barmin for helpful discussions and valuable comments in the preparation of the paper.     

Gas dynamics with local energy release in subsonic and supersonic flow

The flows developing in the interaction of a supersonic gas stream with a continuously operating axisymmetric energy release source or as a result of the action of pulsed periodic energy injection on a subsonic gas stream are investigated numerically. For a continuously operating energy source two types of flow can be distinguished: with a shock wave detached from the source and with a shock attached to it. Approximate formulas for the gas density in the center of the energy release zone are obtained for the cases of constantly operating and periodic energy sources.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 142–148, March–April, 1995.     

Reflection of an Oblique Shock Wave in a Reacting Gas with a Finite Relaxation–Zone Length

Reflection of an oblique shock wave in a reacting gas with a finite length of the chemical–reaction zone is studied. Shock polars for an arbitrary heat release behind the oblique shock wave are constructed. Transition criteria from regular to Mach reflection and back are obtained. It is shown that transition criteria are significantly changed if the reaction–zone length is taken into account.     

Existence of Solutions for Steady Detonation of Gas Suspensions

The existence of solutions of the traveling–wave type is studied for a system of equations that describes a one–dimensional motion of a suspension of evaporating particles in a viscous and heat–conducting chemically reacting gas. Using topological methods, it is shown that solutions corresponding to weak, strong, and Chapman—Jouguet detonation exist under certain restrictions on energy release and mass transfer.     

Physical mechanisms of the rogue wave phenomenon

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrödinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.     

Experimental and theoretical/numerical investigations of thin films bonding strength

A laser spallation facility has been developed to measure the strength of planar interfaces between a substrate and a thin coating. This quantity is a central requirement in contemporary thin film and protective coatings technology and its successful measurement should improve the scientific/technological potential for the design of advanced composites, protective coatings of composites that operate in hostile environments, and in joining of dissimilar materials. The technique involves impinging a laser pulse of ultra short duration on the rear surface of the substrate, which is coated by a thin layer of energy absorbing metal such as Sn and Pb. The explosive evaporation of the metallic layer, confined between a fused quartz crystal and the substrate, induces a compressive shock wave, which propagates through the substrate toward the material interface. Upon reflection from the free surface of the coating, the pressure pulse is converted into a tensile wave which, under certain conditions, can lead to spallation at the interface. It is shown by mathematical simulation that atomic bond rupture is the mechanism of separation in this experiment. Since the interaction of laser energy with matter is a complicated, highly non-linear process, our investigations, at first, were based on measurement of the pressure pulse generated by the threshold flux level that leads to spallation, by using a micro-electronics device with a piezo-electric crystal, and on computation of the tensile stress experienced at the material interface, by numerical simulation of the induced stress wave propagation. Several substrate/coating (ceramic/ceramic and ceramic/metal) systems have been investigated such as, 1–15 μm SiC by CVD, 1–4 μm TiC and TiN by PVD coatings on sapphire substrates, as well as 1–2 μm Au, Sn and Ag coatings by sputtering on sapphire, fused quartz and glass substrates. For identically prepared specimens, the measured threshold energy levels are reproducible, thus leading to reproducible bond strength values, while the spall size, as expected, is dependent on the laser pulse energy level. Finally, the bond strength values obtained are in very good agreement with similar data derived by direct experimental techniques based on Laser-Doppler-Interferometry.     

Influence of an underwater ridge on a nonlinear wave expanding from a point on the surface of a liquid

A study is made of the influence of an underwater ridge on a solitary wave that prior to the interaction with the ridge has the form of a circle situated outside the ridge. It is shown that the nonlinear effects lead to a concentration of the wave energy above the ridge. As they move away from the source, the waves propagating above the ridge are not damped in the considered approximation but are damped everywhere away from the ridge. An analogy is pointed out between the propagation of the wave and two-dimensional steady flows of a fluid, and this makes it possible to use hydrodynamic intuition for qualitative predictions about the nature of the wave propagation in various cases. All the results of the paper can be extended to the case of waves that are periodic in time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 100–105, July–August, 1982.     

Motion of gas due to a point explosion in an inhomogeneous atmosphere

The problem of a strong point explosion in an atmosphere with exponential dependence of the density on the altitude was considered in detail in the monograph [1]. As the blast wave becomes weaker, it is necessary to take into account the counterpressure and the gravitational force. Then in the central hot region of the explosion the gas-dynamic processes are intensified, and the method of calculating the flow parameters in the neighborhood of the point of energy release becomes important. The singularity at this point can be eliminated by including dissipative factors in the treatment; it is then possible to use standard difference methods in the hot central region. In the present paper, the qualitative and quantitative effects of the counterpressure and gravitation are analyzed. A model of the explosion with a simple mechanism of energy dissipation — heat transfer by radiation — is considered. The calculations of the explosion are continued until the formation of the mushroom cloud. The results are compared with the data of other authors.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zfaidkosti i Gaza, No. 6, pp. 144–151, October–December, 1981.     

A fast method to generate collisional excitation cross-sections of highly charged ions in a hot dense matter

A method to compute collisional excitation cross-sections in jj-averaged configuration sets is presented in the framework of plane-wave Born approximation using Dirac–Hartree–Slater wave functions with appropriate low-energy corrections. When averaged into the ls configuration or hydrogenic superconfiguration sets, the results are found to compare with distorted wave calculations well within 30% on average. The cross-sections are averaged into hydrogenic cross-sections and fitted using the Gaunt factor formalism. We present analytic fit coefficients of Gaunt factors for 12 atoms of Z between 5 and 79 for hydrogenic transitions.     

DYNAMICS AND STABILITY OF PINNED–CLAMPED AND CLAMPED–PINNED CYLINDRICAL SHELLS CONVEYING FLUID

The paper examines the dynamics and stability of fluid-conveying cylindrical shells having pinned–clamped or clamped–pinned boundary conditions, where “pinned” is an abbreviation for “simply supported”. Flügge's equations are used to describe the shell motion, while the fluid-dynamic perturbation pressure is obtained utilizing the linearized potential flow theory. The solution is obtained using two methods — the travelling wave method and the Fourier-transform approach. The results obtained by both methods suggest that the negative damping of the clamped–pinned systems and positive damping of the pinned–clamped systems, observed by previous investigators for any arbitrarily small flow velocity, are simply numerical artefacts; this is reinforced by energy considerations, in which the work done by the fluid on the shell is shown to be zero. Hence, it is concluded that both systems are conservative.     

Shock strengthening in a wedge-shaped cavity

The unsteady problem of the entry of a shock wave of arbitrary intensity into a wedge-shaped cavity is examined. An exact solution of the non-linear problem of reflection of a plane wave from a nonplanar wall is found for certain cavity angles. Numerical wave focusing calculations are carried out for arbitrary cavity angles. A single scaling law is obtained for gas flows with waves of moderate and high intensity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 123–129, September–October, 1987.