Asymptotic Research of Nonlinear Wave Processes in Saturated Porous Media

The problem of nonlinear wave dynamics of a fluid-saturated porous medium is investigated. The mathematical model proposed is based on the classical Frenkel--Biot--Nikolaevskiy theory concerning elastic wave propagation and includes mass, momentum, energy conservation laws, as well as rheological and thermodynamic relations. The model describes nonlinear, dispersive, and dissipative medium. To solve the system of differential equations, an asymptotic modified two-scales method is developed and a Cauchy problem for initial equations system is transformed to a Cauchy problem for nonlinear generalized Korteweg--de Vries--Burgers equation for modulated quick wave amplitudes and an inhomogeneous set of equations for slow background motion. Stationary solutions of the derived evolutionary equation that have been constructed numerically reflect different regimes of elastic wave attenuation: diffusive, oscillating, and soliton-like.     

Nonlinear surface waves in media with complex rheology

Weak nonlinear waves in a generalized viscoelastic medium with internal oscillators are considered. The rheological relations contain higher time derivatives of the stresses and strains as well as their tensor products. The method of expansion in a small parameter with the introduction of slow time and a running space coordinate is employed. The first approximation gives wave velocities and relations between the parameters equivalent to the results of an acoustic analysis at elastic wave fronts [1]. The second approximation leads to an evolution equation for the displacement velocity. For this a Fourier-Laplace double integral transformation is used. Reversion to the inverse transforms of the unknown functions leads to an integrodifferential evolution equation, which contains a Hubert transform and is a generalization of the Benjamin-Ono equation of deep water theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 95–103, September–October, 1990.     

Low-frequency dilatational wave propagation through unsaturated porous media containing two immiscible fluids

An analytical theory is presented for the low-frequency behavior of dilatational waves propagating through a homogeneous elastic porous medium containing two immiscible fluids. The theory is based on the Berryman–Thigpen–Chin (BTC) model, in which capillary pressure effects are neglected. We show that the BTC model equations in the frequency domain can be transformed, at sufficiently low frequencies, into a dissipative wave equation (telegraph equation) and a propagating wave equation in the time domain. These partial differential equations describe two independent modes of dilatational wave motion that are analogous to the Biot fast and slow compressional waves in a single-fluid system. The equations can be solved analytically under a variety of initial and boundary conditions. The stipulation of “low frequency” underlying the derivation of our equations in the time domain is shown to require that the excitation frequency of wave motions be much smaller than a critical frequency. This frequency is shown to be the inverse of an intrinsic time scale that depends on an effective kinematic shear viscosity of the interstitial fluids and the intrinsic permeability of the porous medium. Numerical calculations indicate that the critical frequency in both unconsolidated and consolidated materials containing water and a nonaqueous phase liquid ranges typically from kHz to MHz. Thus engineering problems involving the dynamic response of an unsaturated porous medium to low excitation frequencies (e.g., seismic wave stimulation) should be accurately modeled by our equations after suitable initial and boundary conditions are imposed.     

Rock vibration and finite oil recovery

Basic information concerning the possibility of mechanical stimulation of an oil reservoir is presented. The positive effect of vibration on the oil fraction in the output of flooded wells is demonstrated. The effect is attributed to the restoration of permeability for dispersed oil as a result of drop clusterization or breakdown. A mathematical model illustrating the special role of dominant vibration frequencies is proposed. This model is based on the nonlinear effects associated with internal viscoelastic resonance. The corresponding evolution equation of the seismic waves emitted by the vibrator is a generalization of the Burgers-Korteweg-de Vries equation. For this equation the existence of an asymptotic regular wave structure is proved. Taking the microparticle rotation effect into account leads to bimodal wave vibrations, and under conditions of long-short-wave resonance the nonlinear generation of high ultrasonic frequencies by seismic waves is possible. The ultrasonic vibrations created enable the oil drops to recover their mobility.Based on paper presented to the fluid mechanics section of the Seventh Congress on Theoretical and Applied Mechanics, Moscow, August 1991.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 110–119, September–October, 1992.     

Non-linear Evolution of P-waves in Viscous–Elastic Granular Saturated Media

The Frenkel–Biot P-wave of the first type is a seismic longitudinal wave observed in rocks fully saturated with oil, water or high-pressure gas. The P-wave of the second type is observed in unsaturated soils and other porous media saturated with gas of low pressure. Their models include properties of the skeleton, that is, its elastic modules and its own viscosity. If the non-linear terms are accounted for, the asymptotic analysis, usual for weak non-linear waves, might be applied to get the wave spectrum evolution. The wetness of grains contacts in soils and such components of oil as tars or bitumen, which attached to the skeleton, can be described by generalized viscous–elastic stress–strain connections. The latter are nominated in such a way that creates the narrow frequency interval of wave of negative dissipation where the non-linear terms begin to play the main role besides the neutral stability for waves of zero wave number. The corresponding case, relevant to single continuum model, was analyzed in the literature. Here it is shown that the interpenetrating continua with interaction of the Darcy type provide the dissipation sink in the wave evolution equation. This generalization, (Tribelsky, M.I.: Phys. Rev. Lett. (2007, submitted)), can stabilize the asymptotic solution of the evolution equation, where the dispersion terms are omitted. The asymptotic solution of the equation is invariant to initial conditions and it means a transformation of initial wave spectra to unique one while wave is spreading in the viscous–elastic medium under consideration. This explains the phenomenon, observed in wave tests at marine beach, when any dynamics action (impact, explosion, and ultrasound action) created at some distance a wave of a single frequency (~25 Hz).     

Weakly nonlinear magnetosonic waves and structure of low-intensity discontinuities in magnetohydrodynamics

Mathematical techniques are proposed which make it possible to reduce the system of magnetohydrodynamic equations for a viscous heat-conducting gas with finite electric conductivity and a general equation of state to the model Burgers equation. On the basis of this equation the structure of weakly nonlinear magnetohydrodynamic shock waves is studied. In particular, the width of the shock wave is estimated.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 43–48, May–June, 1993.     

Nonlinear dispersion of long internal waves

The propagation of long weakly nonlinear waves in an atmospheric waveguide is considered. A model system of Kadomtsev-Petviashvili equations [1], which describes the propagation of such waves, is derived. In the case of one excited wave mode the system of model equations goes over into the Kadomtsev-Petviashvili equation, in which, however, the variables x and t are interchanged. The reasons for this are clarified. In the two-dimensional case an approximate solution of the model equations is constructed, and steady nonlinear waves and their interaction in a collision are considered. The results of a numerical verification of the stability of the approximate steady solutions and of the solution to the problem of decay of the wave into quasisolitons are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 151–157, May–June, 1988.     

The role of boundary conditions in the instability of one-dimensional systems

We investigate the instability properties of one-dimensional systems of finite length that can be described by a local wave equation and a set of boundary conditions. A method to quantify the respective contributions of the local instability and of wave reflections in the global instability is proposed. This allows to differentiate instabilities that emanate from wave propagation from instabilities due to wave reflections. This is illustrated on three different systems, that exhibit three different behaviors. The first one is a model system in fluid mechanics (Ginzburg–Landau equation), the second one is the fluid-conveying pipe (Bourrières equation), the third one is the fluid-conveying pipe resting on an elastic foundation (Roth equation).     

Solitary waves in dissipative-dispersive systems with instability

The wave processes in a system described by a fourth-order partial differential equation with Burgers-Korteweg-de Vries nonlinearity are considered. The initial equation is reduced to a dynamical system of three equations, which is analyzed by means of a numerical method. It is shown that the equation for the waves in dissipative-dispersive systems with instability has solutions in the form of solitary waves and wave fronts.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 99–104, March–April, 1989.     

Nonlinear waves in a viscous compressible fluid with relaxation

The propagation and stability of nonlinear waves in a viscous compressible fluid with relaxation that satisfies a Theological equation of state of Oldroyd type are investigated. An equation that describes the structure of the wave perturbations and its evolution is derived subject to the condition of balance of the nonlinear dissipative and relaxation effects, and its solutions of the solitary wave type are analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 31–35, May–June, 1993.     

双相各向异性介质弹性波场有限差分正演模拟

从双相各向异性介质模型出发,以Boit理论为基础,推导了斜方晶系各向异性介质-阶弹性波动方程,引入固、流体密度比和孔隙几何参数,将Biot方程系数简化为测量简单、物理意义明确的物理量,采用交错网格技术建立了各向异性孔隙介质波动方程的高精度差分格式,并首次对这类差分格式的频散特性和稳定性作了详细分析讨论,解决了计算稳定性和边界反射问题,与解析解的对比以及理论模型的数值模拟都表明,该方法不仅大大降低了计算量,提高了正演速度,并且具有良好的稳定性和精确性。     

Projective Submodel of the Ovsyannikov Vortex

A submodel of the Ovsyannikov vortex with projective symmetry is studied. Integration of the factor system of the submodel reduces to solving a first-order differential equation which is not solved with respect to the derivative. The properties of the solutions of this equation are studied. It is shown that the submodel describes gas ow with a nonstationary source and a nonstationary sink. The problem of the motion of a gas volume between pistons of cylindrical shapes is studied, and its solution with an invariant shock wave is obtained.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 3–16, July–August, 2005.     

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.     

Flow Regimes of Interaction of a Turbulent Plane Jet into a Rectangular Cavity: Experimental Approach and Numerical Modelling

The present work is devoted to the experimental and numerical study ofthe interaction of a turbulent plane jet with a rectangular cavity.Several flow regimes have been found to occur: the non-oscillationregime, the stable oscillation regime and an unstable oscillationregime. The first two regimes have been particularly considered. Theexperimental study has been carried out using hot wire anemometry andsome visualisations. The numerical predictions based on statisticalmodelling have been made using on the one hand the standard k– model and on the other hand a two-scales split spectrum model. The structuralproperties of the flow have been described for the different situations.For the oscillatory regime, a parametrical study allowed to determinethe influence of the jet exit location and the Reynolds number on thefrequency of the jet flapping. The one point closures have been able topredict the oscillatory regime, and in particular the two-scales modelled to improved results because better account is taken of lag effectsin unsteady non-equilibrium situations.     

Propagation of nonlinear waves in a fluidized bed in the presence of interaction between the particles of the dispersed phase

A model of a fluidized bed as a medium consisting of two interacting interpenetrating ideal fluids is used to investigate the propagation of one-dimensional linear and nonlinear perturbations of the particle concentration in a gas-fluidized bed. The interaction of the particles with each other is taken into account by introducing into the momentum conservation equation for the dispersed phase an effective pressure that depends on the local porosity of the bed and the relative velocity of the dispersed and dispersion phases. The conditions of hyperbolicity of the system of equations describing wave propagation are determined. The stability of the uniform state is investigated. Dispersion effects in the fluidized bed are considered. The propagation of a steady dispersed-phase concentration wave is investigated. The conditions of formation of concentration discontinuities at the steady wave front are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 81–89, March–April, 1987.     

Unsteady waves in a liquid containing vapor bubbles

Unsteady wave processes in vapor-liquid media containing bubbles are investigated taking into account the unsteady interphase heat and mass transfer. A single velocity model of the medium with two pressures is used for this, which takes into account the radial inertia of the liquid with a change in volume of the medium and the temperature distribution in it [1]. The system of original differential equations of the model is converted into a form suitable for carrying out numerical integration. The basic principles governing the evolution of unsteady waves are studied. The determining influence of the interphase heat and mass transfer on the wave behavior is demonstrated. It is found that the time and distance at which the waves reach a steady configuration in a vapor-liquid bubble medium are considerably less than the correponding characteristics in a gas-liquid medium. The results of the calculation are compared with experimental data. The propagation of acoustic disturbances in a liquid with vapor bubbles was studied theoretically in [2]. The evolution of waves of small but finite amplitude propagating in one direction in a bubbling vapor-liquid medium is investigated in [3, 4] on the basis of the generalization of the Burgers-Korteweg-de Vries equation obtained by the authors. An experimental investigation of shock waves in such a medium is reported in [5, 6], and the structure of steady shock waves is discussed [7].Translated from Izvestiya Akademii Nauk SSSR, Hekhanika Zhidkosti i Gaza, No. 5, pp. 117–125, September–October, 1984.     

Types of wave structure in the interaction of a convergent jet with an infinite two-dimensional obstruction

Possible types of wave structure formed in the first roll of a convergent supersonic jet as it interacts with an infinite two-dimensional object are indicated in this work based on a generalization of results from theoretical and experimental studies. The influence of the Mach number, over pressure parameter n, isentropic exponent k, and the location h of the obstacle on the wave structure is considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 45–50, January–February, 1976.     

Molecular gas dynamics applied to phase change processes at a vapor–liquid interface: shock-tube experiment and MGD computation for methanol

This paper deals with a molecular gas-dynamics method applied to the accurate determination of the condensation coefficient of methanol vapor. The method consisted of an experiment using a shock tube and computations using a molecular gas-dynamics equation. The experiments were performed in such situations where the shift from a vapor–liquid equilibrium state to a nonequilibrium one is realized by a shock wave in a scale of molecular mean free time of vapor molecules. The temporal evolution in thickness of a liquid film formed on the shock-tube endwall behind a reflected shock wave is measured by an optical interferometer. By comparing the measured liquid-film thickness with numerical solutions for a polyatomic version of the Gaussian–BGK model of the Boltzmann equation, the condensation coefficient of methanol vapor is accurately determined in vapor–liquid nonequilibrium states. As a result, it is clear that the condensation coefficient is just unity very near to an equilibrium state, but is smaller far from the equilibrium state.     

Characteristics of the solitary waves and lump waves with interaction phenomena in a (2 + 1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation

In this paper, we consider a (\(2+1\))-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada (gCDGKS) equation, which is a higher-order generalization of the celebrated Kadomtsev–Petviashvili (KP) equation. By considering the Hirota bilinear form of the CDGKS equation, we study a type of exact interaction waves by the way of vector notations. The interaction solutions, which possess extensive applications in the nonlinear system, are composed by lump wave parts and soliton wave parts, respectively. Under certain conditions, this kind of solutions can be transformed into the pure lump waves or the stripe solitons. Moreover, we provide the graphical analysis of such solutions in order to better understand their dynamical behavior.     

Effect of nonlinearity on the development of oscillatory magnetohydrodynamic Kelvin-Helmholtz instability in the weakly supercritical regime

The nonlinear stage of development of perturbations at a tangential magnetohydrodynamic discontinuity is investigated in the weakly subcritical and supercritical regimes. It is assumed that the fluid is incompressible and that the density and magnetic field, as well as the velocity, suffer a discontinuity. An equation describing the evolution of low-amplitude nonlinear perturbations is obtained. For periodic perturbations this equation reduces to an infinite system of ordinary differential equations for the amplitudes of the Fourier harmonics. The system is reduced to finite form by truncation and then integrated numerically. Calculations show that the evolution of an initially sinusoidal perturbation always ends with the appearance in the wave profile of an infinite derivative. This can take the form of either an infinitely sharp peak (knife-edge) or wave breaking.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 30–39, May–June, 1988.