Wedging of an elastic medium with the formation of separation zones

The problem of motion of a rigid body in an elastic medium is solved analytically for the case when a separation zone caused by asymmetry is formed in front of the body. A scheme of flow around wedge-shaped and ogive bodies is given for the entire range of the velocities under consideration. It is shown that there exists a limit velocity such that the separation zone disappears when the body moves at a velocity greater than the velocity of transverse waves. The forces exerted on a wedge-shaped body and on an ogive body are the same in the case of the limit velocity.     

Stability of spatial motion of a body of revolution in an elastoplastic medium

A previously constructed model that describes the spatial motion of a body of revolution in an elastoplastic medium (without flow separation and with nonsymmetric separation of the medium flow taken into account) is used to study the Lyapunov stability of rectilinear motion of a body in the case of frozen axial velocity on a half-infinite time interval. Some stability criteria are obtained and the influence of tangential stresses is analyzed.     

The supersonic motion of a rigid body in an elastic medium with friction

The plane problem of supersonic steady motion of a body in an elastic medium is solved. Two possible cases of body motion are considered depending on its velocity. In the first case, the body moves at a velocity greater than the velocity of transverse waves but smaller than the velocity of longitudinal waves. In the second case, the body moves at a velocity greater than the velocity of longitudinal waves. An analytic solution of the problem under study is obtained and analyzed. It is shown that friction substantially influences the penetration process.     

Stability of spatial motion of a body with flow separation and rotation about the symmetry axis

A model describing the spatial motion (without separation and with nonsymmetric separation of the flow in the medium) of a body rotating about its symmetry axis in a resisting medium is constructed. Several criteria for stability of the body rectilinear motion are obtained in the case of frozen axial velocity. The influence of retardation on the stability of rectilinear motion of a cone is considered.     

Stability, Paths, and Dynamic bending of a Blunt Body of Revolution Penetrating into an Elastoplastic Medium

The deep penetration of a thin body with a blunt nose and rear into a lowstrength medium is explored. The motion of the body is described by a system of autonomous integrodifferential equations using the physical model of a separated asymmetric flow over the body and the localinteraction method. An analytical calculation of the Lyapunov stability boundary for straightline motion is performed for bodies with a parabolic meridian. The dependences of the dynamic stability of the body on various parameters are studied numerically. Curved motion paths are constructed in the region of instability, and the classification of paths proposed in previous studies of the motion of pointed bodies is confirmed. It is shown that an reverse ejection is possible when a blunt impactor enters a semiinfinite target. It is established that there is a fundamental possibility of attaining a path close to a specified one and that there is a weak dependence of motion characteristics with a developed separation on the separation angle. Examples are given of calculations of the evolution of the lateral load, the transverse force and moment, and the strength margin of the body using the theory of dynamic bending of a nonuniform rod.     

Vibrational dynamics of a light body in a liquid-filled rotating cylinder

The behavior of a light free cylindrical body in a rapidly rotating horizontal cylinder containing a liquid under vibrational action (the vibration direction is perpendicular to the rotation axis) is investigated. An intense rotation of the body relative to the cavity is detected. Depending on the vibration frequency, the body rotation velocity in the laboratory reference system may be higher or lower than the cavity rotation velocity and in the resonance region they may differ by several times. The mechanism of motion generation is theoretically described. It is shown that the motion is related with the excitation of inertial oscillations of the body: the cause of the motion is an average vibrational force generated due to nonlinear effects in the Stokes boundary layer near the oscillating body. The formation of large-scale axisymmetric vortex structures periodic along the rotation axis, which appear under conditions of inertial oscillation of the body during its motion, both leading and lagging, is detected.     

UNSTEADY LIFT AND SOUND PRODUCED BY AN AIRFOIL IN A TURBULENT BOUNDARY LAYER

An analysis is made of the unsteady lift exerted on a stationary rigid body immersed in an incompressible, plane-wall turbulent boundary layer. The lift is expressed as a surface integral over the body involving theupwash velocity induced by the “free” vorticity Ω (found by taking explicit account of the interaction of the body with the flow and excluding the bound vorticity) and a harmonic function X₂that depends only on the shape of the body. The upwash velocity is the free-field velocity given in terms of Ω by the Biot–Savart formula, augmented by the velocity field of a conventional distribution of image vortices in the wall. The function X₂can be interpreted as the velocity potential of flow past the body, produced by motion of the wall at unit speed towards the body. Detailed predictions are made of the lift on a slender airfoil placed in the outer region of the boundary-layer. When the airfoil chord is large compared to the boundary-layer thickness, vortex shedding into the wake causes the magnitude of the net upwash velocity near the trailing edge to be small. The main contributions to the surface integral are then from the nose region, where the upwash velocity may be estimated independently of the fluctuations near the trailing edge. Analytical results for a thin plate airfoil of chord 2a at distance h from the wall show that the lift increases as a/h increases; it is ultimately independent of a and scales with the ratio of h to the hydrodynamic wavelength. Application is made to determine the sound generated by the airfoil in a weakly compressible boundary layer flow over a finite elastic plate.     

In-plane waves in an elastic solid containing a cracked slab region

Propagation of longitudinal and transverse waves in an elastic solid that contains a cracked slab region is investigated. The cracks have a uniform probability density in the slab region, are parallel to the boundaries of the slab, and the solid is uncracked on either side of the slab. The waves are normally incident on the cracks. It is shown that the resulting average total motion in the solid is governed by a pair of coupled integral equations. These equations are solved under the special assumption that the average exciting motion near a fixed crack is equal to the average total motion. In this case, one finds that in the cracked region, where multiple scattering occurs, there is a forward motion and a backward motion. The two motions have identical frequency-dependent velocity and attenuation, for which simple closed-form formulae are obtained. Simple formulae are also obtained for the wave amplitudes outside the slab. Numerical results corresponding to the velocity, attenuation, reflection amplitude, and transmission amplitude are presented for several values of crack density and slab thickness.     

Numerical calculation of the motion of a gas from a surface explosion

The motion of a gas by the normal impact of a high-speed body at the interface between a dense half-space and a vacuum is investigated numerically. The motion of the shock wave and the shape and distribution of the parameters of the gas dispersing in the vacuum are obtained. The motion is studied during the formation of a region with high pressure at the boundary with the vacuum of a gas occupying the half-space z > 0. The assumption of cylindrical symmetry relative to the z axis enables this three-dimensional nonsteady-state problem in the general case to be solved as a two-dimensional problem. For the corresponding one-dimensional problem, the numerical solution and, for certain gases also, the analytic solutions are well known and are considered in detail in [1]. As a result of solving the two-dimensional problem, profiles of the gasdynamic quantities are obtained which are similar to the solutions in the one-dimensional case and the result of the solution by a self-similar method. The cup-shaped surface of the shock wave front with a pressure gradient on it “focusses” the dispersing gas so that its velocity component normal to the surface z = 0 is greater by an order of magnitude than the component parallel to the surface of separation of the medium, and only at individual points is their ratio close to 0.4. Therefore, the dispersing gas is formed into the shape of a “jet”, the pressure and density profiles on the axis of which have a shape similar to the one-dimensional problem of a brief shock, but in the plane z = 0 the pressure and density distributions are similar to the distributions of these quantities in the case of a powerful point explosion in an unbounded medium. The initial disturbance in the symmetrical problem being considered may be the result of either the normal impact of the body with a high velocity at the surface of the dense medium, or the consequence of the effect of a giant laser pulse, or some other process when a certain volume is formed with a high pressure at the interface between the dense medium and a vacuum, or with another low-density medium.     

External stability of resonances in the motion of an asymmetric rigid body with a strong magnet in the geomagnetic field

We consider a precession motion, close to the classical Lagrange case, of an asymmetric rigid body with a strong magnet in an orbit in the geomagnetic field. For the principal moment we take the restoring torque due to the interaction between the planet magnetic fields and the rigid body. The perturbing actions are due to small moments of the rigid body mass-inertial asymmetry and small constant moments. We show that these perturbations result in the realization of secondary resonance effects in the rotational motion of the rigid body caused by the influence of resonance denominators in higher-order approximations of the averaging method. These effects were discovered in the study of rotational motion of a satellite with a magnetic damper in the nearly Euler case. In the present paper, we analyze both the secondary resonance effects themselves and the external stability of resonances. We obtain conditions ensuring a decrease in the angular velocity of the rigid body rotation about its center of mass. We also discover several new laws of influence of resonances on the nonresonance evolution of slow variables, which is related to the appearance of stable resonances.     

Dynamic properties of a body with discontinual mass variation

In this paper the procedure for the dynamic analysis of body separation is introduced. Based on the general laws of classical dynamics, the method for obtaining the velocity and the angular velocity of the remainder body during separation is developed. Due to the discontinual mass variation, the jump-like change of the velocity and the angular velocity of the body is evident. Various types of motion of the separated body are considered. Depending on the type of motion of the separated body the dynamic properties of the remainder body are obtained. As a special case the in-plane motion of the body before and after separation is considered. The theoretical considerations are applied for the separation analysis of a rotor (a shaft-disc system). The transient motion of the body after separation is investigated. To prove the correctness of the procedure suggested in the paper, the case when the mass and the moment of inertia of the separated body are infinitesimal is analyzed. The obtained differential equations are the same as those previously obtained.     

A new expression of force on a body in viscous vortex flow and asymptotic pressure field

Unsteady vortex flow around a fixed solid body in a viscous incompressible fluid is investigated for the case where the velocity field is assumed to vanish at infinity. Consideration of the asymptotic pressure field far from the body leads to a new formula for the force acting on the body, which is given by a volume integral whose integrand is linear with respect to the vorticity and does not include the velocity. This is facilitated by using a renormalized Green's function introduced by Howe. The formula offers an interesting interpretation for the force in the case of inviscid vortex rings moving near the body: that is, the force is proportional to the rate of change of volume flux through the rings of an imaginary potential flow around the body. The relation of the present subject to the excitation of acoustic waves by vortex motion moving near a compact body is considered.     

Laminar boundary layer on the moving surface of a rankine oval

A numerical investigation has been made of the laminar boundary layer that arises on the moving surface of a cylindrical body (Rankine oval with relative elongation 4) that moves with constant velocity in an incompressible fluid. The distributions of the frictional stress on the surface of the cylinder for different velocities of the wall motion are found. Numerical integration was employed to determine the work needed to overcome the frictional drag, the pressure, and also the work expended on the motion of the moving surface of the body in the case of constant velocity. In the presence of a separation region the drag forces are calculated under the assumption that in the separation region the pressure and the frictional stress on the wall are constant and equal to the corresponding values at the singular point of the solution of the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 3, pp. 171–174, May–June, 1984.I thank G. G. Chernyi for constant interest in the work and discussing the results.     

Investigation of steady laminar flows with an initial disturbance

Results of a parametric study of steady asymmetric flows are analyzed. Three-dimensional unsteady equations of hydromechanics for a compressible medium are solved by a time-dependent method. The range of the characteristic Reynolds number Re = 60–350 is considered. It is shown that a symmetric flow becomes asymmetric at Re = 90. This value can be considered as a threshold value for air. In the examples considered, the upper separation region is always smaller than the lower separation region owing to flow asymmetry in the vicinity of the left boundary of the domain of integration. The dependence of the separation region size on the Reynolds number is found. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 11–19, May–June, 2008.     

Impact of a body with a plane bottom on a thin liquid layer at a small angle

The problem of the impact of a body with a plane bottom (of the type of a box) on a thin liquid layer at a small angle is solved in the two-dimensional formulation. The nonlinear shallow water equations are used, together with the method of matched asymptotic expansions. It is found that at certain values of the input parameters of the problem the liquid pressure diminishes near the lower end of the body and becomes smaller than the atmospheric pressure, which results in liquid separation from the box bottom. The numerical results show that all input parameters of the problem have a considerable effect on the nature of body motion. The liquid separation effect on body motion is analyzed.     

A method for calculating intense injection from surfaces of blunt bodies entering planetary atmospheres at hypersonic velocities

Flow past blunt bodies entering planetary atmospheres at hypersonic velocities is studied. A method for calculating the flowfield near the body nose is developed which allows for radiative heat transfer in the P ₁ approximation of the spherical harmonics method but does not take gas viscosity and heat conduction into account. The solution is constructed on the basis of a two-layer flow model, with account for intense injection of ablation products from the body nose due to radiative heat fluxes from the shock layer. The advantages of the method are that the multi-dimensional character of the radiation field is taken into account and the general problem of radiation gasdynamics is solved on the basis of a unified algorithm. The flow past a spherical segment and a spherically-blunted cone re-entering the Earth’s atmosphere at a velocity of 20 km/s and an entry angle of ?10° is calculated.     

Motion of a slender body near the free surface of a compressible fluid

An analytic solution to the problem of motion of a slender rigid body in a semi-infinite domain of a compressible fluid is obtained for the case when the body moves in parallel to the free surface at a constant velocity. This problem is similar to the problem of motion of a hydrofoil ship whose wing-like device allows it to lift its hull above the water surface and to decrease the friction and drag forces limiting the speed of usual ships. During its motion in water, a hydrofoil produces a lift force. The obtained analytic solution allows one to derive explicit expressions for the drag force and for the lift force in the limiting cases of relatively small and large depths. When depth is small, the drag force is greater than that in an infinite medium, since the wave drag is additionally evolved. When the velocity increases and approaches the sound velocity, the forces exerted on the body increase without limit, which is typical for a linear formulation of the problem.     

Von Kàrmàn vortices behind a bluff body in a wall-bounded turbulized flow with a turbulized boundary layer

The results of the experimental investigation of vortex formation behind a body set transversely in a highly-turbulized channel flow in the presence of a fixed separation line and a turbulized boundary layer on the body nose are presented. It is shown that the action of the turbulizing factors can lead both to an increase and a decrease in von Kàrmàn vortex formation frequency, depending on the cross-sectional shape of the body. A simplified model explaining the effects established and based on the relation between the transverse dimensions of the wake downstream the body and the velocity at its outer boundary is presented.     

Asymmetry in the turbulent flow of a viscoelastic liquid through an axisymmetric sudden expansion

This paper concerns the asymmetry in mean axial velocity distributions for the flow through an axisymmetric sudden expansion of a viscoelastic, shear-thinning aqueous solution of a polyacrylamide (PAA). The asymmetry manifests itself as an azimuthal variation in the length of the recirculation region of the separated flow downstream of the expansion inlet. For water, the flow is found to be axisymmetric. The asymmetry for the PAA flow, which remained unchanged despite alterations to the flow facility, is attributed to the high viscoelasticity of the polymer solution. The conclusion is drawn that the asymmetry is a purely physical feature of such a flow, and not the product of upstream or downstream flow conditions deriving from the flow facility, or the result of geometrical imperfections in the axisymmetric sudden expansion set-up.     

Theory of nonstationary vortices in an ideal fluid

Equations that describe the evolution of a region with nonzero vorticity are formulated. These equations are solved on a bounded time interval for regions having the shape of a sphere or a circular cylinder at the initial time. It is shown that a spherical vortex formed in a medium at rest begins to move, and is stretched in the direction of the motion; a cylindrical vortex, under the influence of the nonuniform intensity of the vorticity on its boundary, changes both the magnitude and direction of its velocity, and describes a curvilinear trajectory. Expressions are obtained which describe the initial evolution of a fluid sphere of one density in a fluid medium of another density.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–11, November–December, 1980.