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.     

Normal forms for random diffeomorphisms

Given a dynamical system (,, ,) and a random diffeomorphism (): ^{d d} with fixed point at x=0. The normal form problem is to construct a smooth near-identity nonlinear random coordinate transformation h() to make the random diffeomorphism ()=h()^–1() h() as simple as possible, preferably linear. The linearization D(, 0)=:A() generates a matrix cocycle for which the multiplicative ergodic theorem holds, providing us with stochastic analogues of eigenvalues (Lyapunov exponents) and eigenspaces. Now the development runs pretty much parallel to the deterministic one, the difference being that the appearance of turns all problems into infinite-dimensional ones. In particular, the range of the homological operator is in general not closed, making the conceptof-normal form necessary. The stochastic versions of resonance and averaging are developed. The case of simple Lyapunov spectrum is treated in detail.     

On the evolutions of the discontinuities of orders ⩾ m in solutions for quasi-linear systems of PDEs of order m,or for the linearizations of these systems

Summary One considers a system L[u]=0 of PDEs, quasi-linear (according to [1]) and of order m, which possesses a bicharacteristic line , as it happens in the hyperbolic case. For v=0, , –m (>0) let u(^v) be a discontinuity wave of order m+v that solves the system above and whose discontinuity hypersurface includes . The corresponding transport equations along are considered. Furthermore some interesting cases are pointed out, in which these equations turn out to be mutually equivalent in a suitable sense. Some theorems are stated to compare the transport equations for the discontinuities of the above kinds, that are connected with the systems d^hL[u]/dt^h=0 (h=0, , –m) and/or the linearization of the system L[u]=0 around any regular solution of it.

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Sommario Si considera un sistema L[u]=0 di equazioni alle derivate parziali, quasi lineare (secondo [1]) e di ordine m, il quale sia dotato di qualche bicaratteristica , come accade nel caso iperbolico. Per v=0, , –m(>0) sia u(^v) un'onda di discontinuità di ordine m+v risolvente il detto sistema e avente ipersuperficie di discontinuità contenente Si considerano le relative equazioni di trasporto lungo e si determinano casi interessanti in cui queste equazioni sono mutuamente equivalenti in senso opportuno. Si stabiliscono teoremi di confronto per il trasporto delle discontinuità del tipo suddetto, relative ai sistemi d^hL[u]/dt^h=0 (h=0, , –m) e/o alla linearizazione del sistema L[u]=0 attorno a qualche sua soluzione regolare.

     

Thermal and diffusive relaxation of an evaporating drop with internal heat liberation

The problem of nonsteady-state evaporation or growth of a radiating drop with uniformly distributed internal heat sources is considered. The Reynolds R=ua/v 1 and Peclet P_D= ua/D 1 numbers are assumed to be small (a is the radius of the drop, u the velocity of its relative motion, andv, D, the coefficients of viscosity, diffusion and thermal diffusivity of the vapor-gas medium). This enables the convective transfer of vapor and heat to be neglected, and the concentration and temperature fields to be regarded as spherically symmetric [1]. In view of the fact that the density of saturated vapor is less than the density of liquid the convective flow caused by the change in radius of the drop is not taken into account [2]. It has already been shown [3,4], that for _r (, _r are the coefficients of molecular and radiative thermal conductivity) there exists a bounded region ryo (1/) /_r ( is the absorption coefficient for radiation in the gas), in which the effect of radiation on the temperature relaxation of the vapor-gas medium is negligible. If the conditiona (1/) /_r is satisfied, then the temperature at the outer boundary of this region will be practically the same as the temperature at infinity T=T. This means that terms in the energy equation connected with energy transferred by radiation can be neglected. It is assumed that the free path of molecules in the gas is less than the radius of the drop, and so concentration and temperature discontinuities close to the surface of the drop can be neglected [2].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 1, pp. 78–87, January–February, 1970.The authors are grateful to V. G. Levich for discussing the results of the paper.     

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.     

Whose Thoughts Are They, Anyway? Dimensionally Exploding Bion's “Double-Headed Arrow” into Coadapting, Transitional Space

The physics and biology that found psychoanalysis account for discontinuous experience only in the presence of nonmeasurable, metaphysical operators; these include the ego and its subsystems as well as biological experience inherited through Lamarckian principles. Complex, self-organizing systems, however, can link biology to experience without metaphysics. They can also account for psychoanalytically relevant behaviors without appealing to stable internal representations. These behaviors include what W. R. Bion called transformation in O and its corollary, the appearance of the selected fact. By dimensionally exploding the double-headed arrow that he used to link the states Ps and D in his model for thinking (Ps D), we can generate a space that is, at once, psychoanalytically imaginal and dynamically coadapting. Isomorphic to D. W. Winnicott's transitional space, it is self-organizing. It is describable according to dynamics formulated by W J. Freeman, S. Kauffman and C. Langton and it can generate instantaneous conscious contents by way of a selective process analogous to spatio-temporal binding. As a whole, this model supports a clinical stance advanced by D. W. Winnicott as play, within transitional space.     

On the existence of limit cycles of Liénard equation

In this paper, we have proved several theorems which guarantee that the Liénard equation has at least one or n limit cycles without using the traditional assmuption G(±) =+. Thus some results in [3–5] are extended. The limit cycles can he located by our theorems. Theorems 3 and 4 give sufficient conditions for the existence of n limit cycles having no need of the conditions that the function F(x) is odd or nth order compatible with each other or nth order contained in each other.     

Application of the method of integral diffusion to investigation of turbulent transfer

A number of authors have critically examined semiempirical mixing length theories [1]. A defect of these theories is connected with the fact that the magnitude of the mixing length, which is assumed to be small in constructing the theory, turns out in experiments to be comparable with the characteristic dimensions of the flow region. Thus, the concept of volume convection [2–4] or integral diffusion [5], which is understood to be a transfer mechanism in which the friction stress is not expressed in terms of the velocity gradient, is introduced along with the concept of gradient diffusion. In addition, there are a number of experimental papers [6] in which it is shown that the turbulent friction stress cannot be equal to zero at the place in the flow where the derivative of the velocity is equal to zero. Mixing length theory does not describe this effect.It is possible to generalize mixing length theory [7–9] in a way which eliminates these defects. Flow of an incompressible fluid is considered.     

Electrical parameters of a channel with finite electrodes with allowance for the electrode potential drop

Spatial problems involving the electric field in an MHD channel were formulated in [1] with allowance for the electrode potential drop. It was assumed that the electrode layer had a small thickness, so that relationships on the boundary of the layer could be applied to the surface of the electrode. It was assumed that the electrode potential drop ° could be represented as a function of the current density jn at the electrode in the form of a known function ° =f (j_n) determined experimentally or deduced from the appropriate electrode-layer theory. An approximate method was then put forward for solving such problems by reducing them to the determination of the electric field from a known distribution of the magnetic field and the gas-dynamic parameters. It was shown that when =°/ E is small (E is the characteristic induced or applied potential difference), the solution can be sought in the form of series in powers of . In the zero-order approximation, the electric field is determined without taking into account the electrode processes. The first approximation gives a correction of the order of . The quantity °, which is present in the boundary conditions on the electrode in the first-order approximation, is determined from the current density calculated in the zero-order approximation.One of the problems discussed in [1] was concerned with the electric current in a channel with one pair of symmetric electrodes. Its solution was found in the first approximation in the form of the integral Keldysh-Sedov formula. In this paper we report an analysis of the solution for ° taken in the form of a step function.     

On elastic-plastic structures with associated stress-strain relations allowing for work softening

Summary For the elementary (finite or infinitesimal) constituents of the structure are assumed generalized stress-strain relations which satisfy the condition of normality but which may exhibit work-softening, concavity of yield surfaces, variation of elastic coefficients with stress and/or plastic strain.The usual phenomena of geometric instability are excluded. Sufficient conditions are formulated for overall stability in spite of the presence of unstable elements, and for uniqueness of the incremental boundary-values problem. Conditions are discussed with a view to applications and expressed in terms of positive definiteness of appropriate quadratic forms.Finally, yield surfaces and flow laws for the structure are examined, and among other things their necessary association is shown.

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Sommario Per i costituenti elementari (finiti o infinitesimi) della struttura si assumono legami incrementali tra sforzi e deformazioni generalizzati che soddisfano alla condizione di normalità ma che per il resto sono generici, cioè tali da presentare eventualmente incrudimento negativo, concavità del campo elastico, variazione dei coefficienti elastici con gli sforzi e/o con le deformazioni plastiche.Esclusi per il sistema i fenomeni usuali di instabilità geometrica si formulano condizioni sufficienti per la stabilità del complesso nonostante la presenza di parti a funzionamento instabile, e per l'unicità del problema incrementale al contorno. Le condizioni sono discusse in vista delle applicazioni ed espresse in termini di definizione positiva di forme quadratiche opportune.Si esaminano infine le superfici di snervamento e le leggi di scorrimento per l'intera struttura e, tra l'altro, se ne dimostra la necessaria associazione.

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First published in Italian in Rendiconti dell'Istituto Lombardo Classe Scienze e Lettere, A 100, 1966.The present investigation has been promoted and financed by the Consiglio Nazionale delle Ricerche (C.N.R.) at the Istituto di Scienza delle Costruzioni of the Facoltà di Architettura, Politecnico of Milano. Part of it was carried out at Brown University (Providence R. I. - U.S.A.). The author acknowledges with tanks the NATO Fellowship assigned by the C.N.R. in 1964, the encouragement and kind hospitality he received from Professor D.C. Drucker, and the interesting discussions he had with Professor Drucker and Dr. A. C. Palmer.     

Flow over a body with development of a steady separation zone as re → ∞

This paper discusses formulation of the total problem of flow of an incompressible liquid over a body, with formation of a closed stationary separation zone as Re . The scheme used is based on the method of matched asymptotic expansions [1]. Following [1], it is postulated that the separated zone is developed (i.e., it is not infinitely fragmented and does not vanish as Re ), and the flow inside it has a definite degree of regularity with respect to Re. With these hypotheses we can use the Prandtl-Batchelor theorem [2], which states that, in the limit as Re , a region of circulating flow becomes vortex flow of an inviscid liquid with constant vorticity . Therefore, a basis for constructing matched asymptotic expansions is the vortex-potential problem (the problem of determining a stream function , satisfying the equation = 0 in the region of translational motion and the equation = in a certain region, unknowna priori, of circulating motion). In the general case the solution of the vortex-potential problem depends on two parameters: the total pressure p_o and the vorticity in the separated zone. These parameters appear in the condition for matching the solutions of the first and second boundary-layer approximations (at the boundary of the separated zone for the end Re values) with the corresponding solutions for the inviscid flow. It is shown in the present paper that the conditions for matching the cyclic boundary layer with the external translational flow are the same additional relations which allow us to close the total problem. Thus, in using the method of matched asymptotic expansions to solve the problem of flow over a body with closed stationary separation zones one must simultaneously consider no less than two approximations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 28–37, March–April, 1978.The authors thank G. Yu. Stepanov for discussion of the paper and valuable comments.     

General expressions for unsteady forces in a lattice of profiles

A large number of studies have been devoted to the unsteady flow of a viscid incompressible fluid past a lattice of thin profiles and the determination of the resulting aerodynamic forces and moments. For example, in the particular case of the motion of a lattice with stagger with zero phase shift of the oscillations between neighboring profiles, Haskind [1] determined the unsteady lift force and moment. Popescu [2] suggested expressions for the force and moment in the case when =0 and =0, using the method of conformal mapping. Samoilovich [3] obtained equations for the unsteady lift force and moment by the method of the acceleration potential for phase shift =0 and = of the oscillations between neighboring profiles. Musatov [4] used an electronic digital computer to calculate the overall unsteady aerodynamic characteristics of a grid by the vortex method, taking into account the amplitude of the oscillations and the initial circulation for =m (m1). Gorelov [5] determined the coefficients of the over-all unsteady aerodynamic force and moment of a profile in a lattice with the stagger and any value of =m. He used a method based on the unsteady flow past an isolated profile with subsequent account for the interference of the profiles in the lattice.In the following we find general expressions for the unsteady lift force and moment acting on a lattice moving in an incompressible fluid with the constant velocity U. These formulas generalize the known formulas for the isolated profile [6]. The profiles of a staggered grid (Section 1) are considered to be thin and slightly curved, and perform oscillations with a phase shift of the oscillations between neighboring profiles. The method of separation of singularities is used to obtain the solution in closed form. The coefficients of the expansion of the complex velocity in a series in the derivatives of a function are calculated. An integral equation relative to the unknown tangential velocity component in the wake is derived (Section 2), and its analytic solution is given (Section 3). For =0 the solution coincides with the solution obtained earlier in [7]. Expressions are obtained for the forces and moments (Section 4) in the form of four terms. The first two terms determine the force and moment for motion with constant circulation, and the last two determine these characteristics for motion with variable circulation. The suction force acting at the leading edges of the profiles is found in a general form. Particular cases of closely and widely spaced lattices are considered. Computational results are presented.     

Supersonic air flow past a sphere with account for vibrational relaxation

A numerical solution is obtained for the problem of air flow past a sphere under conditions when nonequilibrium excitation of the vibrational degrees of freedom of the molecular components takes place in the shock layer. The problem is solved using the method of [1]. In calculating the relaxation rates account was taken of two processes: 1) transition of the molecular translational energy into vibrational energy during collision; 2) exchange of vibrational energy between the air components. Expressions for the relaxation rates were computed in [2]. The solution indicates that in the state far from equilibrium a relaxation layer is formed near the sphere surface. A comparison is made of the calculated values of the shock standoff with the experimental data of [3].Notation uV_max, vV_max velocity components normal and tangential to the sphere surface - V_max maximal velocity - P V _max ² pressure - density - TT temperature - e_viRT vibrational energy of the i-th component per mole (i=–O₂, N₂) - =rb–1 shock wave shape - a _f the frozen speed of sound - HRT/m gas total enthalpy     

Heat Transfer on the Surface on a Spherically Blunted Cone Exposed to a Supersonic Spatial Flow with Injection of a Coolant Gas

The heattransfer processes in a supersonic spatial flow around a spherically blunted cone with allowance for heat overflow along the longitudinal and circumferential coordinates and injection of a coolant gas are studied numerically. The prospects of using highly heatconducting materials and injection of a coolant gas for reduction of the maximum temperatures at the body surface are demonstrated. The solutions of the direct and inverse problems in one, two, and threedimensional formulations for different shell materials are compared. The error of the thinwall method in determining the heat flux on the heatloaded boundary of the body is estimated.     

Direct derivation of the two-shaft system balance conditions for the second order reciprocal inertia forces on the V-type eight cylinders internal combustion engines with a plane crankshaft

By employing the four shafts balance concept paper [1] has reported a balance regime for the second order reciprocal inertia forces on the V-type eight cylinder internal combustion engines with a plane crankshaft. Thereafter, paper [2] has acquired a two-shafts balance regime, but through a rather tedious roudabout degenerating manipulation. The present article has, but starting out directly from the two-shafts balance concept, successfully acquired the same results as those in paper [2]. In addition, we propose, herein, a third balance system which might be, in general, called the slipper balance regime.     

Viscometric properties of viscous theta solutions of monodisperse polystyrene

Theta solvents for polystyrene are prepared from high-viscosity blends of styrene and low-molecular-weight polystyrene, and then used to make dilute solutions with monodisperse polystyrene solutes of high-M = 2.3, 6.0, 9.0, 18.0 · 10⁵. A Weissenberg rheogoniometer is used to measure the non-Newtonian viscosity as a function of shear stress, for low values, and also the complex viscosity components and as functions of frequency. A capillary viscometer is used for high- measurements of(). Viscometric properties, at room temperature, are analyzed as functions of high-molecular-weight solute concentrationc with parameters of constant or to obtain [()], [ ()], and [ ()]. Such a collection of data has apparently not previously been available for polymers in theta solvents (in which Gaussian chain statistics prevail). Also unique is the achievement of high stress ( = 2 10⁴ Pa) at low shear rate, by virtue of high solvent viscosity which is not characteristic of other known theta solvents.     

The problems of nonlinear bending for orthotropic rectangular plate with four clamped edges

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Shear viscosity behavior of highly concentrated suspensions at low and high shear-rates

A method is proposed for determining the shear viscosity behavior of highly concentrated suspensions at low and high shear-rates through the use of a formulation that is a function of three parameters signifying the effects of particle size distribution. These parameters are the intrinsic viscosity [], a parametern that reflects the level of particle association at the initiation of motion and the maximum packing concentration _m. The formulation reduces to the modified Eilers equation withn = 2 for high shear rates. An analytical method was used for the calculation of maximum packing concentration which was subsequently correlated with the experimental values to account for the surface induced interaction of particles with the fluid. The calculated values of viscosities at low and high shear-rates were found to be in good agreement with various experimental data reported in literature. A brief discussion is also offered on the reliability of the methods of measuring the maximum packing concentration. _r = /₀ relative viscosity of the suspension - volumetric concentration of solids - k _n coefficient which characterizes a specific effect of particle interactions - _m maximum packing concentration - _r,0 relative viscosity at low shear-rates - [] intrinsic viscosity - n, n parameter that reflects the level of particle interactions at low and high shear-rates, respectively - _r, relative viscosity at high shear-rates - (_m)_s, (_m)_i, (_m)_l packing factors for small, intermediate and large diameter classes - v _s, v_i, v_l volume fractions of small, intermediate and large diameter classes, respectively - _si, _sl coefficient to be used in relating a smaller to an intermediate and larger particle group, respectively - _is, _il coefficient to be used in relating an intermediate to a smaller and larger particle group, respectively - _ls, _li coefficient to be used in relating a larger to a smaller and intermediate particle group, respectively - _m0 maximum packing concentration for binary mixtures - _m,e measured maximum packing concentration - _m,c calculated maximum packing concentration     

Temperament Development Modeled as a Nonlinear Complex Adaptive System

Using approach-withdrawal (AW) as a specific instance of temperament, a theoretical model of temperament as a complex dynamic system is proposed. Developmental contextualism (Lerner, 1998) serves as a guiding theory in determining the structural components of the system and Kauffman's (1993) Boolean models of self-organization are adapted to estimate the parameter functions. In this model P(AW) = f(, ) where P(AW) is the probability density function of an approach or a withdrawal response, ( is a standardized parameter estimate of the biological sensitivity to stimulation, and is a standardized parameter estimate of the contextual response to an approach or withdrawal response. It is theorized that the functions of ( and follow a Hill function of the forms: d /dt = (²/c² + ²) – K₁ d /dt = ( ²/c² + ²) – K₂, where K₁, K₂, and c are system constants. This results in a double sigmoid function in which at extreme values of and the system stabilizes on a steady state of either approach or withdrawal response patterns. At intermediate parameter values the probability density functions of approach and withdrawal responses are wider. Thus, AW can be modeled as representing two basins of attraction. In addition, considerations are given to the systems sensitivity to initial conditions.