An application of non-singularity BEM to plate bending, buckling and natural vibration problems

In this paper the fundamental solution of the singular governing equation of plate static bending is taken as the Green's function, which can satisfy the governing equation precisely in the plate region. Based on the principle of superposition, let the function values on the plate boundary, induced by a set of the Green's function sources (including the known sources in the plate region and the unknown sources in the fictitious region), satisfy the prescribed conditions on specially chosen boundary matching points, and the corresponding semi-analytical and semi-numerical solution can be obtained, which is free from the restraint of boundary forms and boundary conditions. The more matching points there are on the boundary, the better the accuracy of results is. Finally, in static bending problems a set of linear algebraic equations has to be computed; in buckling problems the minimum value of buckling eigenvalue equation has to be found; in natural vibration problems the eigenvalues of the frequency equation have to be calculated. Numerical examples are given and the results are compared with those by the analytical method and other methods. It can be seen that they are very close to each other.     

Effect of Porous Coatings on Stability of Hypersonic Boundary Layers

The influence of ultrasoundabsorbing coatings on stability of hypersonic boundary layers is considered. Two types of coatings were used in experiments: feltmetal with a random porous microstructure and a sheet perforated by blind cylindrical microchannels. The experiments were performed in a wind tunnel at a Mach number M = 5.95 on sharp cones with a 7° apex halfangle. Evolution of natural disturbances and artificially induced wave packets in the boundary layer was studied with the help of hotwire anemometry. Spatial characteristics of artificial disturbances were obtained. It is demonstrated that such coatings exert a stabilizing effect on secondmode disturbances.     

Chaos and integrability in a nonlinear wave equation

We consider the parametrized family of equations _tt ,u- _xx u-au+u ₂ ² u=O,x(0,L), with Dirichlet boundary conditions. This equation has finite-dimensional invariant manifolds of solutions. Studying the reduced equation to a four-dimensional manifold, we prove the existence of transversal homoclinic orbits to periodic solutions and of invariant sets with chaotic dynamics, provided that =2, 3, 4,.... For =1 we prove the existence of infinitely many first integrals pairwise in involution.     

Hypersonic flow past blunt-edged delta wings

A method is proposed for calculating hypersonic ideal-gas flow past blunt-edged delta wings with aspect ratios = 100–200. Systematic wing flow calculations are carried out on the intervals 6 M 20, 0 20, 60 80; the results are analyzed in terms of hypersonic similarity parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 175–179, September–October, 1990.     

Semi-similar solutions to the three-dimensional laminar boundary layer

The theory of semi-similar solutions is developed for and applied to the problem of three-dimensional laminar boundary layer flow. A number of specific examples are calculated. Particular attention is given to certain flows in which separation is approached and the nature of three-dimensional laminar boundary layer separation is inferred from the behavior of these solutions close to separation. Two types of separation are observed: singular separation characterized by the vanishing of the total shear along the line of separation and ordinary separation characterized by limiting streamlines which become parallel to the line of separation.     

Two-dimensional transonic vortex flows of an ideal gas

Equations are obtained for two-dimensional transonic adiabatic (nonisoenergetic and nonisoentropic) vortex flows of an ideal gas, using the natural coordinates (=const is the family of streamlines, and =const is the family of lines orthogonal to them). It is not required that the transonic gas flow be close to a uniform sonic flow (the derivation is given without estimates). Solutions are found for equations describing vortex flows inside a Laval nozzle and near the sonic boundary of a free stream.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 105–109, September–October, 1973.     

Wall pressure fluctuations of a turbulent separated and reattaching flow affected by an unsteady wake

The spatio-temporal characteristics of the wall-pressure fluctuations in separated and reattaching flows over a backward-facing step were investigated through pressure-velocity joint measurements carried out using multiple-arrayed microphones and split-film probes. A spoke-wheel-type wake generator was installed upstream of the backward-facing step. The flow structure at the effective forcing frequency (St _f=0.2) was found to be well organized in terms of wall pressure spectrum, cross-correlation, wavenumber-frequency spectrum, and wavelet auto-correlation. Introduction of the unsteady wake (St _f=0.2) reduced the reattachment length by 10%. In addition, the unsteady wake enhanced the turbulence intensity near the separation edge and, as a consequence, enhanced the quadrupole sound sources; however, the turbulence intensity near the reattachment region was weakened and the overall flow noise was attenuated. The greater organization of the flow structure induced by the unsteady wake led to a weakening of the dipole sound sources, which are the dominant sound sources in this system. The dipole sound sources generated by wall pressure fluctuations were calculated using Curles integral formula.Abbreviations AR Aspect ratio - SBF Spatial box filtering Roman symbols C _p Wall pressure fluctuation coefficient, p/0.5U ² - H Step height of backward-facing step (mm) - H _s Shape factor (H _s = ^*/) - R _s Distance from acoustic source point to observation point (m) - Re _H Reynolds number, U H/ - St The reduced frequency, fH/U - St _f Normalized forcing frequency by unsteady wake, f _p H/U - T Vortex shedding period (s) - U Free-stream velocity (m/s) - a Speed of sound (m/s) - f Frequency (Hz) - f _p Wake passing frequency (Hz) - k Turbulent kinetic energy (m²/s²) - k _x Streamwise wave number (1/m) - k _z Spanwise wave number (1/m) - l _j Cosine of angle - p Instantaneous wall pressure (Pa) - p _rms Root-mean-square of wall pressure (Pa) - p _SBF Spatial box filtered wall pressure (Pa) - p _d Dipole sound source (Pa) - p _w Conditionally-averaged wall pressure (Pa) - q Dynamic pressure, 0.5U ² (Pa) - r Distance from origin to observation point (mm) - u _c Convection velocity (m/s) - u_max Root-mean-square of streamwise velocity (m/s) - x _R Time-mean reattachment length (mm) Greek symbols _p Forward-flow time fraction - Auto-correlation of pressure at x ₀ - Two-dimensional cross-correlation of pressure with streamwise separation interval , spanwise separation interval , and time delay , at (x ₀, z ₀) - Boundary layer thickness (mm, _99%) - ^* Displacement thickness (mm, ) - _ij Kroneckers delta function - Phase angle (°) - Wavelength (mm) - Momentum thickness (mm, ) - Angle between vertical axis and observation point (°) - Density (kg/m³) - Time delay (s) - Streamwise separation interval (m) - Spanwise separation interval (m) - _p (f; x ₀) Autospectrum of pressure measured at x ₀ (Pa² s) - _pp (, ; x ₀) Streamwise cross spectrum of pressure at x ₀ (Pa² s) - _pp (, , ; x ₀, z ₀) Streamwise and spanwise cross spectrum of pressure at (x ₀, z ₀) (Pa² s) - _pp (kx, ; x ₀) Streamwise wavenumber-frequency spectrum of pressure at x ₀ (Pa² s) - _pp (kx, kz, ; x ₀, z ₀) Two-dimensional wavenumber-frequency spectrum of pressure at (x ₀, z ₀) (Pa² s)     

On the Ellipticity of the Middle Surface of a Shell and its Application to the Asymptotic Analysis of ‘Membrane’ Shells

We consider a surface S = (), where ² is a bounded, connected, open set with a smooth boundary and : ³ is a smooth map; let () denote the components of the two-dimensional linearized strain tensor of S and let 0 with length 0 > 0. We assume the the norm ,|| ()||0, in the space V0() = { H¹() × H¹() × L²(); = 0 on 0 } is equivalent to the usual product norm on this space. We then establish that this assumption implies that the surface S is uniformly elliptic and that we necessarily have 0 = .     

Calculation of the pressure on the side surface of bodies consisting of a spherical segment joined to an inverted cone in a supersonic stream

The results of investigations of inviscid flow over inverted cones with nose consisting of a spherical segment were published for the first time in Soviet literature in [1–4]. In the present paper, a numerical solution to this problem is obtained using the improved algorithms of [5, 6], which have proved themselves well in problems of exterior flow over surfaces with positive angles of inclination to the oncoming flow. It is shown that the Mach number 2 M , equilibrium and nonequilibrium physicochemical transformations in air (H = 60 km, V = 7.4 km/sec, R₀ = 1 m), and the angle of attack 0 40° influence the investigated pressure distributions. A comparison of the results of the calculations with drainage experiments for M = 6, = 0-25° confirms the extended region of applicability of the developed numerical methods. Also proposed is a simple correlation of the dependence on the Mach number in the range 1.5 M of the shape of the shock wave near a sphere in a stream of ideal gas with adiabatic exponent = 1.4.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 178–183, January–February, 1981.     

Finite difference analysis of plane Poiseuille and Couette flow developments

Summary The problem of flow development from an initially flat velocity profile in the plane Poiseuille and Couette flow geometry is investigated for a viscous fluid. The basic governing momentum and continuity equations are expressed in finite difference form and solved numerically on a high speed digital computer for a mesh network superimposed on the flow field. Results are obtained for the variations of velocity, pressure and resistance coefficient throughout the development region. A characteristic development length is defined and evaluated for both types of flow.Nomenclature h width of channel - L ratio of development length to channel width - p fluid pressure - p ₀ pressure at channel mouth - P dimensionless pressure, p/ ² - P ₀ dimensionless pressure at channel mouth - P pressure defect, P ₀–P - (P)₀ pressure defect neglecting inertia - Re Reynolds number, uh/ - u fluid velocity in x-direction - mean u velocity across channel - u ₀ wall velocity - U dimensionles u velocity u/ - U _c dimensionless centreline velocity - U ₀ dimensionless wall velocity - v fluid velocity in y-direction - V dimensionless v velocity, hv/ - x coordinate along channel - X dimensionless x-coordinate, x/h ² - y coordinate across channel - Y dimensionless y-coordinate, y/h - resistance coefficient, - ₀ resistance coefficient neglecting inertia - fluid density - fluid viscosity     

Hopf bifurcation in a three-dimensional system

In this paper,Liapunor-Schmidl reduction and singularity theory are employed to discuss Hopf and degenerate Hopf bifureations in global parametric region in a three-dimensional system x=-βx+y, y=-x-βy(1-kz), z=β[α(1-z)-ky²], The conditions on existence and stability are given.     

Fractal-time stochastic processes and dynamic functions of polymeric systems

Dynamic material functions of polymeric systems are calculated via a defect-diffusion model. The random motion of defects is modelled by a fractaltime stochastic process. It is shown that the dynamic functions of polymeric solutions can be approximated by the defect-diffusion process of the mixed type. The relaxation modulus of Kohlrausch type is obtained for a fractal-time defect-diffusion process, and it is shown that this modulus is capable of portraying the dynamic behavior of typical viscoelastic solutions.The Fourier transforms of the Kohlrausch function are calculated to obtain and. A three-parameter model for and is compared with the previous calculations. Experimental measurements for five polymer solutions are compared with model predictions. D rate of deformation tensor - G(t) mechanical relaxation modulus - H relaxation spectrum - I(t) flux of defects - P _n (s) probability of finding a walker ats aftern-steps - P generating function ofP _n (s) - s(t) fraction of surviving defects - , () gamma function (incomplete) - ₀ zero shear viscosity - ^* () complex viscosity - frequency - t _n n-th moment - F[] Fourier transform - f ^* (u) Laplace transform off(t) - , components of ^* - G _f, _f ^* fractional model - G ₃, ₃ ^* three parameter model - complex conjugate ofz - material time derivative ofD     

Motion of large gas bubbles ascending in a liquid

An equation is derived for the ascent velocity of large gas bubbles in a liquid. This velocity is assumed to be governed by the propagation of a wavelike perturbation caused by the bubble in the liquid.Notation w bubble (or drop) velocity - specific gravity - dynamic viscosity - kinematic viscosity - r bubble (or drop) radius - surface tension - coefficient of friction - g gravitational acceleration - D bubble (or drop) diameter - p pressure - c propagation velocity of the wavelike perturbation - wavelength     

On the measurement of lateral velocity derivatives in turbulent flows

Direct numerical simulation data for the lateral velocity derivative u/y at the centreline of a fully developed turbulent channel flow provide reasonable support for Wyngaard's analysis of the error involved in measuring this quantity using parallel hot wires. Numerical data in the wall region of the channel flow also provide a useful indication of how to select the separation between the wires. Justification for this choice is obtained by comparing several measured statistics of u/y with the corresponding numerical data.     

On the sensitive dynamical system and the transition from the apparently deterministic process to the completely random process

The detailed analysis of the dynamical process of coin tossing is made. Through calculations, it is illustrated how and why the result is extremely sensitive to the initial conditions. It is also shown that, as the initial height of the mass center of the coin increases, the final configuration, i.e. head or tail, becomes more and more sensitive to the initial parameters (the initial velocity angular velocity, and the initial orientation), the coefficient of the air drag, and the energy absorption factor of the surface on which the coin bounces. If we keep the head upward initially but allow a small range for the change of some other initial parameters, the frequency that the final configuration is head, would be 1 if the initial height h of the mass center is sufficiently small, and would be clo to 1/2 if h is sufficiently large. An interesting question is how this frequency changes continuously from 1 to 1/2 as h increases. Detailed calculations show that such a transition is very similar to the transition from laminar to turbulent flows. A basic difference between the transition stage and the completely random stage is indicated: In the completely random stage, the deterministic process of the individual case is extremely sensitive to the initial conditions and the dynamical parameters, out the statistical properties of the ensemble are insensitive to the small changes of the initial conditions and the dynamical parameters. On the contrary, in the transition stage, both the deterministic process of the individual case and the statistical properties of the ensemble are sensitive to the initial conditions and the dynamical parameters. The mechanism for this feature of the transition stage is the existence of the long-train structure in the parameter space. The illuminations of this analysis on some other random phenomena are discussed.     

Supersonic nonequilibrium flow past segmental bodies of a mixture simulating the atmosphere of venus

Calculations of the flow of the mixture 0.94 CO₂+0.05 N₂+0.01 Ar past the forward portion of segmentai bodies are presented. The temperature, pressure, and concentration distributions are given as a function of the pressure ahead of the shock wave and the body velocity. Analysis of the concentration distribution makes it possible to formulate a simplified model for the chemical reaction kinetics in the shock layer that reflects the primary flow characteristics. The density distributions are used to verify the validity of the binary similarity law throughout the shock layer region calculated.The flow of a CO₂+N₂+Ar gas mixture of varying composition past a spherical nose was examined in [1]. The basic flow properties in the shock layer were studied, particularly flow dependence on the free-stream CO₂ and N₂ concentration.New revised data on the properties of the Venusian atmosphere have appeared in the literature [2, 3] One is the dominant CO₂ concentration. This finding permits more rigorous formulation of the problem of blunt body motion in the Venus atmosphere, and attention can be concentrated on revising the CO₂ thermodynamic and kinetic properties that must be used in the calculation.The problem of supersonic nonequilibrium flow past a blunt body is solved within the framework of the problem formulation of [4].Notation V body velocity - shock wave standoff - universal gas constant - ratio of frozen specific heats - hRt/m enthalpy per unit mass undisturbed stream P pressure - density - T temperature - m molecular weight - c_p specific heat at constant pressure - (X) concentration of component X (number of particles in unit mass) - R body radius of curvature at the stagnation point - _j rate of j-th chemical reaction shock layer P V ² pressure - density - TT temperature - mm molecular weight Translated from Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 5, No. 2, pp. 67–72, March–April, 1970.The author thanks V. P. Stulov for guidance in this study.     

Existence of fast traveling waves for some parabolic equations: A dynamical systems approach

We study semilinear elliptic equationsu + cu _x =f(u,u) and ² u + cu _x =f(u,u, ² u) in infinite cylinders (x,y) × ⁿ⁺¹ using methods from dynamical systems theory. We construct invariant manifolds, which contain the set of bounded solutions and then study a singular limitc, where the equations change type from elliptic to parabolic. In particular we show that on the invariant manifolds, the elliptic equation generates a smooth dynamical system, which converges to the dynamical system generated by the parabolic limit equation. Our results imply the existence of fast traveling waves for equations like a viscous reactive 2d-Burgers equation or the Cahn-Hillard equation in infinite strips.     

Wave propagation and instabilities in a rotating anisotropic plasma

Summary The effects of a uniform rotation on the propagation of small perturbations through an anisotropic collisionless plasma are investigated. These effects are present in various ways on the well-known hose, mirror and gravitational instabilities. In Part I we consider plane perturbations, in Part II cylindrical. Some remarks about stability and hydromagnetic waves in an anisotropic collisionless plasma are given in Appendix.

---

Sommario Si esaminano gli effetti di una rotazione uniforme sulla propagazione di piccole perturbazioni in un plasma anisotropo privo di urti. Tali effetti si manifestano in vario modo sui noti fenomeni di instabilità hose, mirror e magnetogravitazionale del tipo di Jeans. Nella prima parte si considerano perturbazioni piane, nella seconda di tipo cilindrico. Alcune considerazioni sulla stabilità e sulle onde idromagnetiche in un plasma anisotropo privo di urti sono svolte in Appendice.

---

---

This work was carried out while the author was a member of the Gruppo di Ricerca no. 44 per la Matematica del CNR, 1967–68.     

Numerische Untersuchung des elastisch-plastischen Verhaltens eines Fadenverbundwerkstoffes mit metallischer Matrix Eine Fallstudie

Übersicht Die Fehlervorhersage eines einachsigen Fadenverbundwerkstoffes mit metallischer Matrix erfordert die Kenntnis des nichtlinearen elastisch-plastischen Verhaltens in mikroskopischer Abmessung. Als Fallstudie wurde ein spezieller Verbundwerkstoff mit einem FEM-Programm untersucht. Im plastischen Bereich wurde das klassische, v. Mises Potential' mit dem neuen Übergangsfließpotential unter Berücksichtigung der plastischen Volumendehnung verglichen. Unter transversaler Normalbelastung zeigte der Verbund deutliche Unterschiede in der Zunahme und der Ausdehnung der lokalen plastischen Zonen. Alle kritischen Verformungsstadien wurden von dem Übergangsfließpotential bei geringerer Belastung erreicht.

---

Numerical investigation of the elastic-plastic behaviour of a fibre-reinforced composite with a metallic matrix

Summary Failure prediction of unidirectional fibre-reinforced composite with a metallic matrix needs knowledge about the nonlinear elastic-plastic behaviour in a microscopic scale. A specific composite was investigated using a FEM-program as a case study. In the plastic range the classical v. Mises Potential was compared with the new Transition Flow Potential, taking into account the plastic volume dilatation. Subjected to transverse normal loading the composite showed evident differences in the increase and the spread of locally plastic regions. All critical deformation states were reached by the Transition Flow Potential at lower loading.