Wave motion of an ideal fluid in a narrow open channel

This paper considers a nonlinear integrodifferential model constructed for the motion of an ideal incompressible fluid in an open channel of variable section using the long-wave approximation. A characteristic equation for describing the perturbation propagation velocity in the fluid is derived. Necessary and sufficient conditions of generalized hyperbolicity for the equations of motion are formulated, and the characteristic form of the system is calculated. In the case of a channel of constant width, the model reduces to the Riemann integral invariants which are conserved along the characteristics. It is found that, during the evolution of the flow, the type of the equations of motion can change, which corresponds to long-wave instability for a certain velocity distribution along the channel width. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 61–71, March–April, 2009.     

Model of a strong discontinuity for the equations of spatial long waves propagating in a free-boundary shear flow

The long-wave equations describing three-dimensional shear wave motion of a free-surface ideal fluid are rearranged to a special form and used to describe discontinuous solutions. Relations at the discontinuity front are derived, and stability conditions for the discontinuity are formulated. The problem of determining the flow parameters behind the discontinuity front from known parameters before the front and specified velocity of motion of the front are investigated. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 206–213, July–August, 2008.     

Gas-dynamic analogy in the theory of stratified liquid flows with a free boundary

Novel approximate mathematical models of long-wave theory describing flows of a density-stratified liquid with a free boundary are proposed. It is shown that in certain cases the equations of the novel models coincide with either the equations of nonisentropic gas dynamics with a polytropic equation of state for γ = 2 or the equations describing the dynamics of a mixture of two perfect gases.     

Unsteady interaction of uniformly vortex flows

The problem of the decay of an arbitrary discontinuity (the Riemann problem) for the system of equations describing vortex plane-parallel flows of an ideal incompressible liquid with a free boundary is studied in a long-wave approximation. A class of particular solutions that correspond to flows with piecewise-constant vorticity is considered. Under certain restrictions on the initial data of the problem, it is proved that this class contains self-similar solutions that describe the propagation of strong and weak discontinuities and the simple waves resulting from the nonlinear interaction of the specified vortex flows. An algorithm for determining the type of resulting wave configurations from initial data is proposed. It extends the known approaches of the theory of one-dimensional gas flows to the case of substantially two-dimensional flows. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 55–66, September–October, 1998.     

Nonlinear periodic waves in a gas

A system of equations describing the one-dimensional time-dependent polytropic motion of a gas is considered. In special cases the general solutions of this system of equations are obtained and exact solutions with the initial conditions which are periodic with respect to the spatial variable are found. For an arbitrary polytropic exponent an asymptotic solution, which is uniformly suitable till the onset of the gradient catastrophe, is constructed in the form of expansions in series in a small parameter, namely, the initial wave amplitude. Asymptotic dependences of the time of onset and the location of the gradient catastrophe are obtained. The complex correspondence between the initial system of equations and the system of equations describing the motion of quasi-gas media is given. An example of using this correspondence is considered.     

Equations of the shallow water model on a rotating attracting sphere. 1. Derivation and general properties

A shallow water model on a rotating attracting sphere is proposed to describe large-scale motions of the gas in planetary atmospheres and of the liquid in the world ocean. The equations of the model coincide with the equations of gas-dynamic of a polytropic gas in the case of spherical gas motions on the surface of a rotating sphere. The range of applicability of the model is discussed, and the conservation of potential vorticity along the trajectories is proved. The equations of stationary shallow water motions are presented in the form of Bernoulli and potential vorticity integrals, which relate the liquid depth to the stream function. The simplest stationary solutions that describe the equilibrium state differing from the spherically symmetric state and the zonal flows along the parallels are found. It is demonstrated that the stationary equations of the model admit the infinitely dimensional Lie group of equivalence. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 24–36, March–April, 2009.     

Symmetries and exact solutions of the shallow water equations for a two-dimensional shear flow

This paper considers nonlinear equations describing the propagation of long waves in two-dimensional shear flow of a heavy ideal incompressible fluid with a free boundary. A nine-dimensional group of transformations admitted by the equations of motion is found by symmetry methods. Two-dimensional subgroups are used to find simpler integrodifferential submodels which define classes of exact solutions, some of which are integrated. New steady-state and unsteady rotationally symmetric solutions with a nontrivial velocity distribution along the depth are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 41–54, September–October, 2008.     

Interaction of shear flows of an ideal incompressible fluid in a channel

The problem of the decay of an arbitrary discontinuity for the equations describing plane-parallel shear flows of an ideal fluid in a narrow channel is considered. The class of particular solutions corresponding to fluid flows with piecewise constant vorticity is studied. In this class, the existence of self-similar solutions describing all possible unsteady wave configurations resulting from the nonlinear interaction of the specified shear flows is established. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 34–47, November–December, 2006.     

Equation of a shock wave front

Second-order differential equations of the hyperbolic type are derived for describing the local law of shock wave propagation. The shock waves are assumed to be two-dimensional unsteady in a stationary gas flow and three-dimensional steady in a supersonic flow. The behavior of the characteristics of these equations is investigated as a function of the governing flow parameters and their relative position with respect to the typical bicharacteristics of the characteristic cone behind the shock is analyzed. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 159–165, May–June, 2000.     

Simple waves on a shear gas flow in a channel of constant cross section

The plane-parallel unsteady-state shear gas flow in a narrow channel of constant cross section is considered. The existence theorem of solutions in the form of simple waves of a set of equations of motion is proved for a class of isentropic flows with a monotone velocity profile over the channel depth. The exact solution described by incomplete beta-functions is found for a polytropic equation of state in a class of isentropic flows. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 36–43, January–February, 1999.     

Direct numerical simulation of a liquid sheet in a compressible gas stream in axisymmetric and planar configurations

A thin liquid sheet present in the shear layer of a compressible gas jet is investigated using an Eulerian approach with mixed-fluid treatment for the governing equations describing the gas–liquid two-phase flow system, where the gas is treated as fully compressible and the liquid as incompressible. The effects of different topological configurations, surface tension, gas pressure and liquid sheet thickness on the flow development of the gas–liquid two-phase flow system have been examined by direct solution of the compressible Navier–Stokes equations using highly accurate numerical schemes. The interface dynamics are captured using volume of fluid and continuum surface force models. The simulations show that the dispersion of the liquid sheet is dominated by vortical structures formed at the jet shear layer due to the Kelvin–Helmholtz instability. The axisymmetric case is less vortical than its planar counterpart that exhibits formation of larger vortical structures and larger liquid dispersion. It has been identified that the vorticity development and the liquid dispersion in a planar configuration are increased at the absence of surface tension, which when present, tends to oppose the development of the Kelvin–Helmholtz instability. An opposite trend was observed for an axisymmetric configuration where surface tension tends to promote the development of vorticity. An increase in vorticity development and liquid dispersion was observed for increased liquid sheet thickness, while a decreasing trend was observed for higher gas pressure. Therefore surface tension, liquid sheet thickness and gas pressure factors all affect the flow vorticity which consequently affects the dispersion of the liquid.      

Reptation Motion of Large Animals in a Fluid

An asymptotic analysis of the plane problem of reptation motion of animals in a fluid is performed in a long-wave approximation. Turbulent motion is considered. Asymptotic estimates are obtained for the axial and shear forces, expended energy, and motion trajectory. Results of numerical analysis are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 60–69, March–April, 2006.     

Shallow waves in a two-layer vortex fluid under a lid

A mathematical model of the vortex motion of an ideal two-layer fluid in a narrow straight channel is considered. The fluid motion in the Eulerian-Lagrangian coordinate system is described by quasilinear integrodifferential equations. Transformations of a set of the equations of motion which make it possible to apply the general method of studying integrodifferential equations of shallow-water theory, which is based on the generalization of the concepts of characteristics and the hyperbolicity for systems with operator functionals, are found. A characteristic equation is derived and analyzed. The necessary hyperbolicity conditions for a set of equations of motion of flows with a monotone-in-depth velocity profile are formulated. It is shown that the problem of sufficient hyperbolicity conditions is equivalent to the solution of a certain singular integral equation. In addition, the case of a strong jump in density (a heavy fluid in the lower layer and a quite lightweight fluid in the upper layer) is considered. A modeling that results in simplification of the system of equations of motion with its physical meaning preserved is carried out. For this system, the necessary and sufficient hyperbolicity conditions are given. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 68–80, May–June, 1999.     

Propagation of nonlinear waves in an inhomogeneous gas-liquid medium. Derivation of wave equations in the Korteweg-de Vries approximation

Equations describing the propagation of waves of small but finite amplitude in a liquid with gas bubbles are derived. The bubble distribution density is a continuous function of bubble size and spatial coordinates. It is found that, for a uniform bubble distribution, the obtained equations become the Korteweg-de Vries, Kadomtsev-Petviashvili and Khokhlov-Zabolotskaya equations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 188–197, March–April, 2009.     

Two-dimensional gas vortices and twisted gas jets

This paper studies an invariant solution of rank one of the equations of motion of a polytropic gas that describes two-dimensional gas vortices and twisted gas jets. Flow types are classified according to the governing parameter: vortices in the form of sources and sinks, unlimited expansion, and collapse. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 71–81, May–June, 2009.     

The Motion of a Fluid–Rigid Disc System at the Zero Limit of the Rigid Disc Radius

We consider the two-dimensional motion of the coupled system of a viscous incompressible fluid and a rigid disc moving with the fluid, in the whole plane. The fluid motion is described by the Navier–Stokes equations and the motion of the rigid body by conservation laws of linear and angular momentum. We show that, assuming that the rigid disc is not allowed to rotate, as the radius of the disc goes to zero, the solution of this system converges, in an appropriate sense, to the solution of the Navier–Stokes equations describing the motion of only fluid in the whole plane. We also prove that the trajectory of the centre of the disc, at the zero limit of its radius, coincides with a fluid particle trajectory.     

Exact solutions of the one-dimensional Russo-Smereka kinetic equation

We obtain new classes of invariant solutions of the integrodifferential equations describing the propagation of nonlinear concentration waves in a rarefied bubbly fluid. For all the solutions obtained, trajectories of particle motion in phase space are calculated. The stability of some flows is studied in a linear approximation. In several cases, the construction of solutions reduces to an integrodifferential equation of the second kind, which can be solved by the iteration method. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 21–32, July–August, 2000.     

Attenuated Wave Field in Fluid-Saturated Porous Medium with Excitations of Multiple Sources

This research addresses the investigation of an elastic wave field in a homogeneous and isotropic porous medium which is fully saturated by a Newtonian viscous fluid. A new methodology is developed for describing the wave field in the medium excited by multiple energy sources. To quantify the relative displacements between the fluid and solid of the medium, the governing equations of the elastic wave propagation are derived in the form of displacements specially. The velocities and attenuation of the waves are considered as functions of viscosity and frequency. Making use of the Hankel function and the moving-coordinate method, a model of the wave motion with multiple cylindrical wave sources is built. Making use of the model established in this research, the relative displacement between the fluid and the solid can be quantified, and the wave field in the porous media can then be determined with the given energy sources. Numerical simulations of cylindrical waves from multiple energy sources propagating in the porous medium saturated by viscous fluid are performed for demonstrating the practicability of the model developed.