A visualization study of the interaction of a free vortex with the wake behind an airfoil

The two-dimensional interaction of a single vortex with a thin symmetrical airfoil and its vortex wake has been investigated in a low turbulence wind tunnel having velocity of about 2 m/s in the measuring section. The flow Reynolds number based on the airfoil chord length was 4.5 × 10³. The investigation was carried out using a smoke-wire visualization technique with some support of standard hot-wire measurements. The experiment has proved that under certain conditions the vortex-airfoil-wake interaction leads to the formation of new vortices from the part of the wake positioned closely to the vortex. After the formation, the vortices rotate in the direction opposite to that of the incident vortex.List of symbols c test airfoil chord - C vortex generator airfoil chord - TA test airfoil - TE test airfoil trailing edge - TE _G vortex generator airfoil trailing edge - t dimensionless time-interval measured from the vortex passage by the test airfoil trailing edge: gDt=(T-T- _TE)·U/c - T time-interval measured from the start of VGA rotation - U free stream velocity - U vortex induced velocity fluctuation - VGA vortex generator airfoil - y distance in which the vortex passes the test airfoil - Z vortex circulation coefficient: Z=/(U · c/2) - vortex generator airfoil inclination angle - vortex circulation - vortex strength: =/2     

Generation of vortices in a partially filled,rapidly rotating cylinder

We describe a system in which vortices are shed from a cylindrical free surface approximately centered in a rotating flow. Shedding is controlled by the parameter =2 g/ ² d, where g, , d denote gravity, rotation rate and the diameter of the free surface. We find vortex shedding for >0.162 and no vortex shedding for < 0.0847. The range depends on the aspect ratio L/d, where L is the column length, in a nonmonotonic fashion. These results are independent of viscosity and surface tension for small values of these parameters.Now at Martin Marietta, Orlando Aerospace, PO Box 5837, Mail Point 150, Orlando, FL 32855, USA     

On surface waves in generalized thermoelasticity

Knowles' representation theorem for harmonically time-dependent free surface waves on a homogeneous, isotropic elastic half-space is extended to include harmonically time-dependent free processes for thermoelastic surface waves in generalized thermoelasticity of Lord and Shulman and of Green and Lindsay.r , , r , , .This work was done when author was unemployed.     

Non-uniform recovery of vortex breakdown over delta wing in response to blowing along vortex core

The transient characteristics of vortical structure over delta wing are studied experimentally when subject to single along-core blowing perturbation. Two half delta wing models with different sweep angle = 60° and = 75° are investigated in this study. For = 75°, the transient location of the onset of vortex breakdown moves upstream monotonously toward the unperturbed location. However, for =60°, there exists a chordwise region where the upstream propagation of the onset location of vortex breakdown is temporarily delayed. This delay causes the recovery process (upstream propagation) of the onset location of vortex breakdown to be non-uniform. In fact, this non-uniform recovery may take on different appearance such as a plateau or overshoots and will last for several times of the convective time scaleC/U . Furthermore, the location of chordwise region corresponding to this delay depends strongly upon the angle of attack (AOA). In additions, the non-uniform recovering characteristic of the onset of vortex breakdown may not be observed at high AOA if the blowing rate is too low. The mechanism governing the non-uniform recovering characteristic is clearly verified through the LDA measurement and the phase-locked flow visualization. Evidently the mutual interaction between the primary vortex structure and the secondary vortex is the key mechanism that leads to non-uniform recovering character over delta wing with sweep angle of 60°The authors are grateful for the support of the this investigation from National Science Foundation of the Republic of China under the grant no. NSC-83-0424-E-005-006.     

Vorticity meter calibration using a cross-wires survey

A vorticity meter of the vane-type has been used in this study for direct and quantitative measurements of vertical flows. The vane calibration procedure is carried out by using measurements deduced from a preliminary cross-wires survey of the flow-field generated in the wake of a half wing set at incidence in the wind tunnel. Emphasis is placed on the resulting calibration law which relates the vane rotational speed to the fluid angular velocity. Specially it is shown that this calibration law can be obtained as a linear expression by taking into account the minimum threshold rotational speed necessary for vane rotation. When comparing vorticity distributions obtained through the vortical wing wake by the indirect cross-wires technique, good agreements are found with the corresponding vorticity distributions as deduced from the so calibrated vorticity meter.List of symbols c Airfoil chord, m - K, K dimensionless constants in calibration relations [Eqs. (1, 2)] - N rotational speed of the vorticity meter, rps - N _f , N_f rotational speed of the fluid, rps - N f ₀, N_{f 0} threshold of fluid angular velocity necessary for vane rotation, rps [Eqs. (1, 2)] - r radial distance measured from the vortex center, mm - r ₀ radius of the vorticity meter, mm - R _c Reynolds number refered to wing chord - U, V, W velocity components along X, Y, Z axis respectively, m/s - V Freestream velocity, m/s - V _t Tangential velocity, V _t ² = V ² + W ², m/s - X, Y, Z axial, lateral, and vertical distances (Fig. 1) - geometric airfoil incidence, deg - relative error estimation [Eq. (3)] - streamwise vorticity component, rad/s [Eq. (4)] - efficiency factor, = N /N _f - , pitch and yaw angles, deg, (Fig. 1) - angular velocity, rad/s     

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.     

Strong vortex/boundary layer interactions

Detailed measurements with hot-wires and pressure probes are presented for the interaction between a turbulent longitudinal vortex pair with common flow down, and a turbulent boundary layer. The interaction has a larger value of the vortex circulation parameter, and therefore better represents many aircraft/vortex interactions, than those studied previously. The vortices move down towards the boundary layer, but only the outer parts of the vortices actually enter the it. Beneath the vortices the boundary layer is thinned by lateral divergence to the extent that it almost ceases to grow. Outboard of the vortices the boundary layer is thickened by lateral convergence. The changes in turbulence structure parameters in the boundary layer appear to be due to the effects of extra-rate-of-strain produced by lateral divergence (or convergence) and by free-stream turbulence. The effect of the interaction on the vortices (other than the inviscid effect of the image vortices below the surface) is small. The flow constitutes a searching test case for prediction methods for three-dimensional turbulent flows.     

Construction of exact numerical solutions of the stationary traveling wave type for viscous thin films

An effective numerical procedure, based on the Galerkin method, for finding solutions of the stationary traveling wave type in the complete formulation is proposed for the case of viscous liquid films. Examples of a viscous film flowing freely down a vertical surface have been calculated. The calculations have been made for various values of the dimensionless surface tension , including =0. The method makes it possible to predict a number of bifurcations that occur as decreases. The existence of numerous families of stationary traveling waves when 1 was demonstrated in [6]. The present study shows that as 1 all but one of these families of wave solutions disappear. The shape of the periodic and solitary waves and the pressure distribution in the film are found for various . When =0 and the wave number is fairly small, the periodic solution has a singularity, as predicted in [14]: at the crest of the wave a corner point appears; the angle between the tangents at this point =140–150. The method proposed can be used to calculate other wavy film flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 94–100, May–June, 1990.     

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.     

Interaction of Large‐Scale and Small‐Scale Oscillations During Separation of a Laminar Boundary Layer

The mutual influence of shortwave oscillations (instability waves of the separated boundary layer) and longwave disturbances at the frequency of shedding of periodic largescale vortices is experimentally studied in flow separation behind a step. The possibility of controlling the process of vortex formation by exciting amplifying disturbances in the shear layer is demonstrated.     

Boundary layer stability on a rotating disk with corotation of the surrounding fluid

The stability of the steady self-similar flow in the boundary layer on a rotating disk of infinite radius with corotation of the surrounding fluid is analyzed by the normal mode method. The spectral problem for infinitesimal three-dimensional disturbances is solved by a collocation method with expansion of the amplitude functions in Chebyshev polynomials. It is established that for all values of the parameter 0, equal to the ratio of the angular velocities of the fluid and the disk, the lower critical Reynolds number is determined byA-type, waves, whose development is governed by the parallel instability mechanism typical of an Ekman layer. TheB-type instability, associated with the presence of an inflection point on the velocity profile, disappears when 4. The neutral surfaces are calculated for Karman flow (=0) and Bödewadt flow (). It is found that in Karman flowA-type waves may grow at values of the Reynolds number several times smaller than the critical Reynolds number for spiral vortices. The results of the analysis are compared with the available experimental data.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 69–77, September–October, 1992.     

Surface shear stress measurements around multiple jets in crossflow using the fringe imaging skin friction technique

The skin friction distribution around multiple jets in crossflow determined using the fringe imaging skin friction (FISF) technique is examined experimentally for two supply channel configurations. The velocity field associated with the jets is correlated to the skin friction distribution. A counterflow supply channel configuration has a pronounced skin friction deficit downstream of the injection holes caused by the lifting of fluid away from the wall by the counterrotating vortex pair (CRVP). In the counterflow case, the CRVP remains closer to the wall than in the coflow case. A potential flow analysis of a pair of counterrotating vortices above a wall demonstrates that the distance of the CRVP from the wall has a greater effect on the wall pressure distribution, and hence the separation phenomena, than does the strength of the vortices.Nomenclature a Distance between vortices in Y direction (potential flow analysis) - b Distance between vortices in Z direction (potential flow analysis) - C _f Skin friction coefficient - D Hole diameter - L/D Hole length-to-diameter ratio - M Momentum (blowing) ratio=jet momentum/free stream momentum - n _O Oil index of refraction - p Static pressure - P/D Spanwise spacing of injection-hole-to-diameter ratio - q Dynamic head=1/2U ² - Re Reynolds number based on momentum thickness - s Fringe spacing - t Time - v, w Wall-normal and spanwise velocity components - X Streamwise distance downstream from hole centerline - Y Height from test section floor - Z Spanwise distance from hole centerline - U Free stream velocity - Wavelength of incident radiation - _O Oil dynamic viscosity - Air density - _w Wall shear stress - _r Angle of incidence of illumination for FISF - ₀ Initial boundary layer momentum thickness Acronyms CRVP Counterrotating vortex pair - FISF Fringe imaging skin friction - SFD Skin friction deficit     

The effect of background particulates on the delayed transition of a heated 9:1 ellipsoid

An experimental study was done to quantify the effects of a variety of background particulates on the delayed laminar-turbulent transition of a thermally stabilized boundary layer in water. A Laser-Doppler Velocimeter system was used to measure the location of boundary layer transition on a 50 mm diameter, 9:1 fineness ratio ellipsoid. The ellipsoid had a 0.15 m RMS surface finish. Boundary layer transition locations were determined for length Reynolds numbers ranging from 3.0 × 10⁶ to 7.5 × 106. The ellipsoid was tested in three different heating conditions in water seeded with particles of four distinct size ranges. For each level of boundary layer heating, measurements of transition were made for clean water and subsequently, water seeded with 12.5 m, 38.9 m, 85.5 m and 123.2 m particles, alternately. The three surface heating conditions tested were no heating, T = 10°C and T = 15°C where T is the difference between the inlet model heating water temperature, T _i, and free stream water temperature, T . The effects of particle concentration were studied for 85.5 m and 123.2 m particulates.The results of the study can be summarized as follows. The 12.5 m and 38.9 m particles has no measurable effect on transition for any of the test conditions. However, transition was significantly affected by the 85.5 m and 123.2 m particles. Above a length Reynolds number of 4 × 10⁶ the boundary layer transition location moved forward on the body due to the effect of the 85.5 m particles for all heating conditions. The largest percentage changes in transition location from clean water, were observed for 85.5 m particles seeded water.Transition measurements made with varied concentrations of background particulates indicated that the effect of the 85.5 m particles on the transition of the model reached a plateau between 2.65 particulates/ml concentration and 4.2 particles/ml. Measurements made with 123.3 m particles at concentrations up to 0.3 part/ml indicated no similar plateau.     

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.     

Unsteady flow past a cascade of staggered thin profiles

Many studies have been made of plane flow of an incompressible inviscid fluid past a cascade of profiles with arbitrary stagger angle 0. For example, in the particular case of the motion of a cascade with the stagger angle at zero-oscillation phase-shift angle =0 Khaskind [1] determined the unsteady lift force theoretically by isolating the singularities with the Sedov method [2], applying a conformal mapping to the cascade of unstaggered flat plates. Belotserkovskii et al. [3] calculated the over-all unsteady aerodynamic characteristics of a cascade in the particular case =0 and for any on a computer by the method of discrete vortices, and for the more general case (0) Whitehead [4] has done the same using a vortex method. Gorelov and Dominas [5] calculated the over-all unsteady force and moment coefficients of a profile in a cascade with stagger angle 0 and phase shift 0.The calculation method was based on unsteady theory for a slender isolated profile whose flow pattern is known, with subsequent account for the interference of the profiles and the vortex wakes behind them.In the present study the singularity isolation method [2] is extended to slender profiles with arbitrary stagger angle 0 and arbitrary phase shift 0 of the oscillations between neighboring profiles. It is shown that the solution reduces to the solution of a Fredholm integral equation of the first kind in terms of the sum of the tangential velocity components along the profile. It is found that the relative effect of the unsteady flow due to the system of vortex trails behind the cascade with stagger may be determined without solving this integral equation. However, this solution must be found to calculate the added masses of the cascade and the total magnitudes of the unsteady forces. It is found that regularization transforms the integral equation of the first kind to an integral equation of the second kind, for which solution methods are known.Thus the expressions for the unsteady forces are determined in the form of separate terms, each of which has a physical significance: as a result we obtain finite formulas (improper integrals) for calculating the variable forces; from these formulas are derived the asymptotic expressions for the forces in the limiting cases of high and low solidities and Strouhal numbers, which as a rule are lost in numerical calculations. The proposed method may be considered as one of the techniques for improving the convergence of the numerical methods (elimination of singularities). Moreover, this method may be used to solve the problems of unsteady flow past cascades of arbitrary systems of slender profiles for various profile incidence angles relative to the x-axis and in the presence of a finite cavitation zone on the profiles.The limited practical application of this method is explained by the extreme theoretical difficulties in its applications to cascades with stagger angle. In the present studies these difficulties are examined using the example of a cascade with stagger angle.     

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.     

Factors influencing non-circular ring vortex motion

Non-circular ring vortices are innately unstable, giving rise to a range of new phenomena. Here we report on our and Heertsch's [1] experiments in which vortices were generated at rectangular holes and nozzles with aspect ratios 2<<20. Different piston histories were also used. For forestrokes alone we were able to confirm the typical non-splitting motion of the primary vortex. On introducing a backstroke following the forestroke even for values of as low as 2 — values which should not give rise to splitting vortices — vortices could be made to split into 2, 3 or 4 secondary vortices. For cases where they rejoined the process was significantly different to that predicted by theory [2]. For 3 for a nozzle geometry the splitting angle is extremely sensitive to the stroke (length) so long as splitting takes place, whereas for 9>>5 the splitting angle tends to become independent of the stroke. This sensitivity on the stroke is reduced for vortices generated at a hole geometry. For all cases investigated here the splitting angle seems to be relatively insensitive to the Reynolds number. Vortices generated at hole geometries also tend to be less stable than those generated at tube geometries. Finally, the dependence of the splitting angle on the stroke length only scales with the nozzle breadth for 7>>5.

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Sommario Vortici ad anello non circolari sono intrinsecamente instabili e danno luogo ad una gamma di nuovi fenomeni. In questo articolo vengono riportati gli esperimenti degli Autori e di Heertsch in cui sono generati vortici in ugelli e fori rettangolari con rapporti geometrici =2÷20. Sono state anche usate differenti storie del moto del pistone. Nel caso in cui si usi solo la corsa in avanti si è stati capaci di confermare il moto tipico del vortice primario senza divisione del vortice stesso. Introducendo una corsa inversa, subito dopo la corsa in avanti, persino per pari circa a 2 — valore in cui il vortice non si dovrebbe dividere — i vortici si potevano dividere in due, tre o quattro vortici secondari. Nei casi in cui si verificava la riconnessione, l'evoluzione del processo era molto differente rispetto alla teoria. Per <3, per una data geometria dell'ugello, l'angolo di separazione è estremamente sensibile alla lunghezza della corsa, mentre per =5÷9 l'angolo di separazione tende a diventare indipendente dalla corsa. Questa sensibilità è ridotta per vortici generati in fori. In tutti i casi l'angolo di separazione sembra abbastanza indipendente dal numero di Reynolds. Vortici generati in corrispondenza di fori tendono ad essere meno stabili di quelli generati in ugelli. Infine, la dipendenza dall'angolo di separazione sulla lunghezza della corsa scala con l'ampiezza dell'ugello solamente per =5÷7.

     

On flow and stability of a Newtonian fluid past a rotating plane

An exact solution is given for the steady flow of a Newtonian fluid occupying the halfspace past the plane z=0 uniformly rotating about a fixed normal axis (Oz). This solution is obtained in a velocity field of the form considered by Berker [2] and can be deduced as a limiting case, as h+, of the solution to the problem relative to the strip 0zh imposing at z=h either the adherence boundary conditions or the free surface conditions. Furthermore, the stability of this flow, subject to periodic disturbances of finite amplitude, is studied using the energy method and the result is compared with those corresponding to stability of flows in the strip 0zh.

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Sommario In questa nota si mostra che-oltre alla calssica soluzione di von Karman [1] — esiste, per opportuni valori del gradiente di pressione all'infinito, una soluzione esatta per il moto stazionario di un fluido Newtoniano posto nel semispazio limitato dal piano z=0 uniformemente rotante attorno ad un asse ad esso perpendicolare (Oz). Tale soluzione, ottenuta sulla scia del lavoro di Berker [2], si può dedurre anche come limite, per h+, della soluzione del problema relativo alla striscia 0zh quando sul piano z=h si assegnano o le condizioni di aderenza o le condizioni di frontiera libera. Si studia poi la stabilità di tale moto rispetto a perturbazioni spazialmente periodiche di ampiezza finita col metodo dell'energia e si confronta il risultato ottenuto con quelli relativi alla stabilità dei moti nella striscia 0zh.

     

The extension of Beltrami's ellipsoid to anisotropic bodies

Summary The spectral decomposition of the compliance, stiffness, and failure tensors for transversely isotropic materials was studied and their characteristic values were calculated using the components of these fourth-rank tensors in a Cartesian frame defining the principal material directions. The spectrally decomposed compliance and stiffness or failure tensors for a transversely isotropic body (fiber-reinforced composite), and the eigenvalues derived from them define in a simple and efficient way the respective elastic eigenstates of the loading of the material. It has been shown that, for the general orthotropic or transversely isotropic body, these eigenstates consist of two double components, ₁ and ₂ which are shears (₂ being a simple shear and ₁, a superposition of simple and pure shears), and that they are associated with distortional components of energy. The remaining two eigenstates, with stress components ₃, and ₄, are the orthogonal supplements to the shear subspace of ₁ and ₂ and consist of an equilateral stress in the plane of isotropy, on which is superimposed a prescribed tension or compression along the symmetry axis of the material. The relationship between these superimposed loading modes is governed by another eigenquantity, the eigenangle .The spectral type of decomposition of the elastic stiffness or compliance tensors in elementary fourth-rank tensors thus serves as a means for the energy-orthogonal decomposition of the energy function. The advantage of this type of decomposition is that the elementary idempotent tensors to which the fourth-rank tensors are decomposed have the interesting property of defining energy-orthogonal stress states. That is, the stress-idempotent tensors are mutually orthogonal and at the same time collinear with their respective strain tensors, and therefore correspond to energy-orthogonal stress states, which are therefore independent of each other. Since the failure tensor is the limiting case for the respective _x, which are eigenstates of the compliance tensor S, this tensor also possesses the same remarkable property.An interesting geometric interpretation arises for the energy-orthogonal stress states if we consider the projections of _x in the principal₃D stress space. Then, the characteristic state ₂ vanishes, whereas stress states ₁, ₃ and ₄ are represented by three mutually orthogonal vectors, oriented as follows: The ₃ and ₄ lie on the principal diagonal plane (₃₁₂) with subtending angles equaling (–/2) and (-), respectively. On the positive principal ₃-axis, is the eigenangle of the orthotropic material, whereas the ₁-vector is normal to the (₃₁₂)-plane and lies on the deviatoric -plane. Vector ₂ is equal to zero.It was additionally conclusively proved that the four eigenvalues of the compliance, stiffness, and failure tensors for a transversely isotropic body, together with value of the eigenangle , constitute the five necessary and simplest parameters with which invariantly to describe either the elastic or the failure behavior of the body. The expressions for the _x-vector thus established represent an ellipsoid centered at the origin of the Cartesian frame, whose principal axes are the directions of the ₁-, ₃- and ₄-vectors. This ellipsoid is a generalization of the Beltrami ellipsoid for isotropic materials.Furthermore, in combination with extensive experimental evidence, this theory indicates that the eigenangle alone monoparametrically characterizes the degree of anisotropy for each transversely isotropic material. Thus, while the angle for isotropic materials is always equal to _i = 125.26° and constitutes a minimum, the angle || progressively increases within the interval 90–180° as the anisotropy of the material is increased. The anisotropy of the various materials, exemplified by their ratiosE _L/2G_L of the longitudinal elastic modulus to the double of the longitudinal shear modulus, increases rapidly tending asymptotically to very high values as the angle approaches its limits of 90 or 180°.     

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.