Development of two-dimensional perturbations on the surface of a tangential discontinuity

A study is made of the asymptotic behavior at long times of initially localized small two-dimensional perturbations of the interface of two fluids in the presence of a tangential discontinuity of the velocity; surface tension is taken into account. The development of one-dimensional perturbations was considered earlier in [1]. The asymptotic behavior of the perturbed region is found, i.e., in the xyt space there is found a cone with apex at the origin such that perturbations tend to infinity with increasing t along rays within the cone, while perturbations tend to zero along the remaining rays. Conditions are found under which the instability of the tangential discontinuity is not absolute, i.e., when these conditions are satisfied, flows with tangential discontinuity of the velocity can take place. These conditions, like the shape of the cone, do not depend on the magnitude of the surface tension.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 12–16, May–June, 1979.     

Absolute instability of an interface between two magnetic fluids in the presence of a tangential velocity discontinuity and an external magnetic field

The nature of the instability of the surface of a tangential velocity discontinuity between two incompressible, inviscid, capillary, nonconducting, linearly magnetizable fluids in an external magnetic field is considered. An absolute instability criterion is obtained in analytic form. When the gravity force is negligible, this criterion does not depend on the value of the surface tension. When the surface tension is negligible, the absolute instability criterion is obtained for the region of the flow parameters in which the causality condition for the system in question is satisfied. If the magnetic field is tangential to the interface, all the criteria obtained are also applicable to the case of nonmagnetic, perfectly conducting fluids, i.e., to the case of ideal magnetohydrodynamics.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 21–26, September–October, 1995.     

On the asymptotic behavior of localized perturbations in the presence of Kelvin-Helmholtz instability

The asymptotic behavior of localized two-dimensional perturbations of the surface of a shear discontinuity separating two homogeneous steady flows of ideal incompressible fluid is studied in the linear approximation. The effect of surface tension and gravity forces is taken into account. Mathematically the problem reduces to the investigation by the method of steepest descent of the asymptotic behavior of a double integral for various values of parameters which are the components of the group velocity vector. In this problem the principal difficulty is to find the two-dimensional steepest descent contour in the space of two complex variables that determines which of the various saddle points gives the asymptotic form. First, for the Fourier component with respect to one of the variables with allowance for all the saddle points we find an asymptotic form which parametrically depends on the second variable. The choice of the second variable makes it possible to prove analytically that in the absence of gravity the asymptotic behavior of the growing perturbations is determined by a single saddle point in the plane of that variable. In this way it is possible to justify the authors' previous conclusions [1] concerning the shape of the boundary L of the region D in the group velocity plane occupied by growing perturbations. In the presence of gravity the growth rates of perturbations corresponding to different group velocities are found numerically and the region D occupied by the growing perturbations is indicated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 23–30, March–April, 1985.     

Instability of a moving plane-parallel layer

A study is made of infinitely small perturbations of a moving plane-parallel layer. It is shown that, in distinction from an isolated tangential discontinuity, a layer is unstable with any given values of the projection of the velocity of the layer on the wave vector of the perturbation. The instability of an isolated tangential discontinuity has been repeatedly investigated in detail (see, for example, [1–4]). The instability of a moving layer has remained almost unanalyzed. It is of importance to make such an analysis, the more so since the results for a layer differ qualitatively from the results for an isolated tangential discontinuity.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 11–14, May–June, 1972.     

Asymptotic behavior of localized perturbations in free shear layers

A study is made in the linear approximation, within the scope of the ideal fluid, of the asymptotic behavior of three-dimensional localized perturbations of the parameters of a shear flow which over considerable periods of time turn into growing and propagating wave packets. The behavior of the packets is studied in every possible system of coordinates moving with constant velocity parallel to the plane of the velocity shear. Mathematically, the problem reduces to using the method of steepest descent to study the asymptotic behavior of double Fourier integrals which depend parametrically on these velocities. The saddle points which determine this asymptotic behavior are found numerically. A region is indicated in a plane of flow parallel to the velocity shear which is moving and expanding linearly with time, and in which growing perturbations are found over long periods of time. The results obtained enabled us to write down the criteria for absolute and convective instability. This problem has been considered previously for flows of an ideal fluid with a shear discontinuity in the velocity [1, 2] and for flows in a wake [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, 8–14, March–April, 1987.The author wishes to express his sincere gratitude to A. G. Kulikovskii for formulating the problem and for advice on numerous occasions.     

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.     

Theory of short waves in gas dynamics

An examination is made of the two-dimensional, almost stationary flow of an ideal gas with small but clear variations in its parameters. Such gas motion is described by a system of two quasilinear equations of mixed type for the radial and tangential velocity components [1, 2]. Partial solutions [3, 4], characterizing the variation in the gas parameters in the vicinity of the shock wave front (in the short-wave region), are known for this system of equations. The motion of the initial discontinuity of the short waves derived from the velocity components with respect to polar angle and their damping are studied in the report. A solution of the equations characterizing the arrangement of the initial discontinuity derived from the velocities is presented for one particular case of the class of exact solutions of the two parameter type [4]. Functions are obtained which express the nature of the variation in velocity of the front of the damped wave and its curvature.Translation from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 55–58, May–June, 1973.     

Laminar boundary layer on a partly moving surface in the presence of blowing or suction

Planar and axisymmetric flows of a multicomponent compressible gas in a laminar boundary layer with nonzero tangential component of the velocity on a permeable surface are considered. The asymptotic solutions of the boundary-layer equations obtained earlier [1–4] for large values of the blowing and suction parameters are generalized to the case when the velocity vector of the blown or extracted gas makes an acute angle with the surface of the body, this angle depending on the longitudinal coordinate. The region of applicability of the asymptotic formulas is estimated on the basis of the results of numerical solution of the boundary-layer equations. The results are given of some calculations of the boundary layer on a partly moving surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 28–36, September–October, 1979.We thank G. A. Tirskii and G. G. Chernyi for a helpful discussion of the results.     

Asymptotic behavior of solutions of the Orr-Sommerfeld equations in regions adjacent to two branches of the neutral curve

The asymptotic behavior of the upper and lower branches of the neutral stability curve of a boundary layer found by Lin [1] was determined more accurately by various authors [2–4], who, on the basis of the linearized Navien-Stokes equations, analyzed the higher approximations in the Reynolds number R. In the limit R , neutral perturbations have wavelengths that exceed in order of magnitude the boundary layer thickness. The long-wavelength asymptotic behavior of the Orr-Sommerfeld equation is, in particular, of interest because the characteristic solutions of the linearized equations of free interaction (triple-deck theory) [5–7] are a limiting form of Tollmierr-Schlichting waves in an incompressible fluid with critical layers next to the wall [8–9]. At the same time, the dispersion relation, which is identical to the secular equation of the Orr-Sommerfeld problem, contains an entire spectrum of solutions not considered in the earlier studies [2–4]. The first oscillation mode in the spectrum may be either stable or unstable. In the present paper, solutions are constructed for each of the subregions (including the critical layer) into which the perturbed velocity field in the linear stability problem is divided at large Reynolds numbers. Dispersion relations describing the neighborhood of the upper and lower branches of the neutral curve for the boundary layer are derived. These relations, which contain neutral solutions as a special case, go over asymptotically into each other in the unstable region between the two branches.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–11, July–August, 1984.     

Nonlinear analysis of the flow initiated by the sudden motion of a wedge

Certain self-similar problems involving the sudden motion of a wedge which were treated in the linear approximation in [1–3] are studied by the method of matched asymptotic expansions. The nature of the wave boundary of the perturbed region is determined. Second-approximation solutions are constructed which describe flows behind weak shock fronts propagating in a stationary gas and behind fronts of weak discontinuity lines propagating by known uniform flows. A boundary-value problem is formulated whose solution describes, in first approximation, flows in the neighborhoods of points of interaction of the fronts. The existence of similarity rules of flows in these nieghborhoods is estimated. An approximate solution of the problems is given.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 37–47, May–June, 1976.     

Instability of an Unbounded Elastic Plate in a Gas Flow

The stability of an unbounded plane elastic plate in gas moving on one side of the plate and at rest on the other is analyzed. The gases are inviscid and in general different. The plate is under tension and has flexural stiffness. It is shown that the system is always unstable to plane sinusoidal perturbations with wave vector parallel to the velocity. As limiting cases, a tangential discontinuity between the two gases and unilateral flow past a plate with constant pressure on the opposite side are considered. In these cases, the conditions of stability to plane perturbations are non-trivial and are investigated below.     

Surface waves and stability of tangential velocity discontinuity on a solid-fluid boundary

Solutions of the Rayleigh-wave type on the boundary of an elastic half-space and a moving layer of ideal fluid are obtained. The limiting cases of zero flow velocity and a tangential velocity discontinuity in the fluid were investigated in [1–3]. In [4] the order of magnitude of the critical flow velocity was estimated. An increase in the velocity scales used in engineering and experimental practice (see [5], for instance) has aroused interest in a more thorough analysis of the effect.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 43–46, May–June, 1981.     

Stability of one-dimensional gas flows in expanding regions

The stability of gas flows produced by the motion of a flat piston or the decay of an arbitrary discontinuity is considered. The boundaries of the region (or regions) in which the development of perturbations is considered are planes (shock wave, contact discontinuity, piston, etc.) which move away from each other.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 112–119, March–April, 1981.     

On the expansion into a vacuum of a rarefied plasma with two sorts of ions

The one-dimensional expansion of a plasma with different temperatures and two sorts of ions into a vacuum is examined. When the ion velocity distribution in the plasma is Maxwellian, propagation of a rarefaction wave is observed, the boundary of which is a weak discontinuity moving with the velocity of ionic sound in the plasma. The value of this velocity is found for the plasma in question. Attention is mainly focused on finding the first two moments of the distribution functions, i.e., the mean velocities and the densities of the heavy particles. An approximate asymptotic solution is obtained for the system of transport equations in the case when the two kinds of ions have similar masses, and the system is solved numerically by computer. Some features of the solutions, typifying a plasma in which the different sorts of ions have different masses, are analyzed in detail.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 1, pp. 32–38, January–February, 1970.In conclusion the author thanks N. T. Pashchenko for suggesting the problem and showing unfailing interest.     

Taylor instability in the discontinuity between two fluids in an electromagnetic field

The gravitational instability of the discontinuity between two compressible (or incompressible) fluids is investigated. The fluids are exposed to an electromagnetic field, and one of them is nonconducting, while the other has a finite conductivity. The magnetic Reynolds number is assumed to be small. It is shown that in contrast to the cases investigated in [1, 2], where compressible, infinitely conducting fluids were considered on both sides of the discontinuity, in the present case the electromagnetic field is not able to stabilize the discontinuity and the perturbations can propagate in fixed directions. The presence of walls inhibits the perturbation growth [2, 3], while their conductivity does not affect the instability of the discontinuity. The greatest perturbation growth is found to occur in a wave propagating along the magnetic field, when the electromagnetic field does not influence these perturbations in the case of incompressible fluids, but does influence them in the compressible case.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 24–28, September–October, 1976.The author wishes to express his appreciation to A. A. Barmin and A. G. Kulikovskii for suggesting the problem and for their continued interest in the work.     

Theory of slightly perturbed three-dimensional flows in nozzles

The theory of slightly perturbed flows in conical nozzles is used to determine the transverse force and moment generated in the presence of asymmetric perturbations. A system of ordinary differential equations is derived for finding the transverse force and moment. An approximate analytical solution of this system is constructed and its qualitative features are studied. A comparison is made with a numerical solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 146–154, January–February, 1977.     

Rayleigh-Taylor instability at large times

The potential flow of an inviscid incompressible heavy fluid lying above a light one is investigated. The asymptotic stage is described by an unsteady discontinuity, which approximates the flow in the neighborhood of the tongue, and by a steady flow outside this narrow region. Consequences of the conservation laws which make it possible to check the accuracy of the solution of the steady-state problem are obtained. A steady-state solution is constructed for Froude numbers 0相似文献   

Asymptotic Solutions of the Equations for a Viscous Heat-Conducting Compressible Medium

Small nonstationary perturbations in a viscous heat-conducting compressible medium are analyzed on the basis of the linearization of the complete system of hydrodynamic equations for small Knudsen numbers (Kn ≪ 1). It is shown that the density and temperature perturbations (elastic perturbations) satisfy the same wave equation which is an asymptotic limit of the hydrodynamic equations far from the inhomogeneity regions of the medium (rigid, elastic or fluid boundaries) as M _a = v/a → 0, where v is the perturbed velocity and a is the adiabatic speed of sound. The solutions of the new equation satisfy the first and second laws of thermodynamics and are valid up to the frequencies determined by the applicability limits of continuum models. Fundamental solutions of the equation are obtained and analyzed. The boundary conditions are formulated and the problem of the interaction of a spherical elastic harmonic wave with an infinite flat surface is solved. Important physical effects which cannot be described within the framework of the ideal fluid model are discussed.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 76–87.Original Russian Text Copyright © 2005 by Stolyarov.     

The field near the wave front in a Cauchy-Poisson problem

A study is made of the asymptotic behavior of the Green's function of the Cauchy—Poisson problem in the far zone near the wave front, i.e., for r c₀t, where is the maximal group velocity of a surface wave. It is shown that the solution to this problem given in the book by LeBlond and Mysak [1] is incorrect, and the correct asymptotic behavior, expressed in terms of the square of an Airy function, is given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 173–174, March–April, 1984.We thank Yu. L. Gazaryan for interest in the present work.     

Stability of flow of a viscous liquid down an inclined plane

Using the Navier-Stokes equation the stability of a layer of viscous liquid flowing down a solid surface under gravity is studied in the linear formulation. The effect of surface tension and the inclination of the solid surface on the limits of stability are examined also. Curves are calculated for the neutral stability with respect to two types of perturbations — surface waves and shear waves.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskol Fiziki, No. 2, pp. 172–176, March–April, 1975.