Cavity nucleation, growth and stress field in elastic-plastic medium

A finite element method is used to study the cavity nucleation and growth in an elastic-plastic medium. The critical cohesion strength, _c at IPM (interface between second phase particle and matrix) is employed as the criterion of cavity formation. Three different values of _c are taken to examine their influence on the overall mechanical behaviour and process of cavity formation.     

Flow and fracture of rocks under general triaxial compression

Recent laboratory studies of the flow and fracture of rocks under general triaxial compression are reviewed. New developments in laboratory techniques have made it possible to measure three principal stresses and strains under general triaxial stress states, in which all three principal stresses are different.Strength and ductility of isotropic rocks are markedly affected not only by the least compression ₃, but also by the intermediate compression ₂, although these two effects are rather additional in strength, but opposite in ductility. The experimental results show that dilatancy is highly anisotropic under the general triaxial stress states.Deformational properties of anisotropic rocks have been also measured under the general triaxial compression. In this case, the effect of the intermediate compression markedly depends on the orientations of the weak planes.     

Flow in porous media III: Deformable media

Stokes flow in a deformable medium is considered in terms of an isotropic, linearly elastic solid matrix. The analysis is restricted to steady forms of the momentum equations and small deformation of the solid phase. Darcy's law can be used to determine the motion of the fluid phase; however, the determination of the Darcy's law permeability tensor represents part of the closure problem in which the position of the fluid-solid interface must be determined.Roman Letters A interfacial area of the- interface contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A _e area of entrances and exits for the-phase contained within the macroscopic system, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - E Young's modulus for the-phase, N/m² - e i unit base vectors (i = 1, 2, 3) - g gravity vector, m²/s - H height of elastic, porous bed, m - k unit base vector (=e ₃) - characteristic length scale for the-phase, m - L characteristic length scale for volume-averaged quantities, m - n unit normal vector pointing from the-phase toward the-phase (n = -n ) - p pressure in the-phase, N/m² - P p – g·r, N/m² - r ₀ radius of the averaging volume, m - r position vector, m - t time, s - T total stress tensor in the-phase, N/m² - T ⁰ hydrostatic stress tensor for the-phase, N/m² - u displacement vector for the-phase, m - V averaging volume, m₃ - V volume of the-phase contained within the averaging volume, m³ - v velocity vector for the-phase, m/s Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - shear coefficient of viscosity for the-phase, Nt/m² - first Lamé coefficient for the-phase, N/m² - second Lamé coefficient for the-phase, N/m² - bulk coefficient of viscosity for the-phase, Nt/m² - T –T ⁰ , a deviatoric stress tensor for the-phase, N/m²     

Propagation of a spherical wave in nonlinearly compressible and elastoplastic materials

The problem of spherical wave propagation in soil under the action of an intense uniformly decreasing load ₀(t) applied to the boundary of a cavity with radius r₀ is considered. Soil with a high stress level is modeled either by ideally nonlinearly compressible or elastoplastic material, taking account of linear irreversible unloading for the material. In contrast to [1–7], in order to describe material movement use is made of strain theory [8] with determining functions = (), _i=_i(_i), where , _i, , _i are the first and second invariants of strain and stress tensors. During material loading these functions are presented in the form of polynomials ()=(_i+₂¦¦), _ii)=(_i-_2i)_i, in which constant coefficients _i, _i=1, 2) are determined by experiment, taking account of the triaxial stressed state of soil. Solution of the problem is constructed by an analytically reversible method, with prescribed shape for the shock-wave (SW) surface in the form of a second-degree polynomial relating to time t and a numerical method of characteristics for a prescribed arbitrarily decreasing load _i(t). On the basis of the analytical equations obtained, calculations are carried out for material parameters (including loading profile) in a computer and stresses and mass velocity of plastic and elastoplastic materials are compared.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 95–100, July–August, 1986.The authors express their sincere thanks to Kh. A. Rakhmatulin for discussing the results of this work.     

Theoretical derivation of tangential velocity profiles in a flat vortex chamber-influence of turbulence and wall friction

Summary As part of a study on the hydrodynamics of a cyclone separator, a theoretical investigation of the flow pattern in a flat box cyclone (vortex chamber) has been carried out. Expressions have been derived for the tangential velocity profile as influenced by internal friction (eddy viscosity) and wall friction. The most important parameter controlling the tangential velocity profile is = –u ₀ R/(v+ ), where u ₀ is the radial velocity at the outer radius R of the cyclone, the kinematic liquid viscosity and is the kinematic eddy viscosity. For values of greater than about 10 the tangential velocity profile is nearly hyperbolic, for smaller than 1 the tangential velocity even decreases towards the centre. It is shown how and also the wall friction coefficient may be obtained from experimental velocity profiles with the aid of suitable graphs. Because of the close relation between eddy viscosity and eddy diffusion, measurements of velocity profiles in flat box cyclones will also provide information on the eddy motion of particles in a cyclone, a motion reducing its separation efficiency.List of symbols A cross-sectional area of cyclone inlet - h height of cyclone - p static pressure in cyclone - p static pressure difference in cyclone between two points on different radius - r radius in cyclone - r ₁ radius of cyclone outlet - R radius of cyclone circumference - u radial velocity in cyclone - u ₀ radial velocity at circumference of flat box cyclone - v tangential velocity - v ₀ tangential velocity at circumference of flat box cyclone - w axial velocity - z axial co-ordinate in cyclone - friction coefficient in flat box cyclone (for definition see § 5) - ₁ value of friction coefficient for ₁<< ₂ - ₂ value of friction coefficient for ₂<<1 - = - ₁ value of for ₁<< ₂ - ₂ value of for ₂<<1 - thickness of laminar boundary layer - =/h - turbulent kinematic viscosity - ratio of z to h - k ratio of height of cyclone to radius R of cyclone - parameter describing velocity profile in cyclone =–u ₀ R/(+) - kinematic viscosity of fluid - density of fluid - ratio of r to R - ₁ value of at outlet of cyclone - ₂ value of at inner radius of cyclone inlet - _w shear stress at cyclone wall - angular momentum in cyclone/angular momentum in cyclone inlet - ₁ value of at = ₁ - ₂ value of at = ₂     

Evaluation of the dynamic-mechanical properties of solids from a cooperative model for stress relaxation

For many solid materials the stress relaxation process obeys the universal relationF = – (d/d lnt)_max = (0.1 ± 0.01) ( ₀ — _i ), regardless of the structure of the material. Here denotes the stress,t the time, ₀ the initial stress of the experiment and _i the internal stress. A cooperative model accounting for the similarity in relaxation behaviour between different materials was developed earlier. Since this model has a spectral character, the concepts of linear viscoelasticity are used here to evaluate the corresponding prediction of the dynamic mechanical properties, i.e. the frequency dependence of the storageE () and lossE () moduli. Useful numerical approximations ofE () andE () are also evaluated. It is noted that the universal relation in stress relaxation had a counterpart in the frequency dependence ofE (). The theoretical prediction of the loss factor for high-density polyethylene is compared with experimental results. The agreement is good.     

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°.     

Gas-dynamic heating in cavities under the influence of pressure pulsations at the entrance

An investigation has been made of the gas-dynamic heating of gas in nearly closed cavities (tubes, channels, etc.) under the influence of given pressure pulsations (with and without a discrete component) at the entrance. The results are given of measurements of the wall temperature of the cavities and also the power of the gas-dynamic heating as a function of the relative cavity length 10< ln/d_n < 300, the relative level of the pressure pulsations ₀/p < 0.5 at the entrance, and the magnitude of the static pressure in the range p = 2–10 kg/cm². It was established that with increasing p and especially ₀/p the power of the gas-dynamic heating increases strongly.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 177–179, November–December, 1983.     

Transport in ordered and disordered porous media V: Geometrical results for two-dimensional systems

In this paper we continue the geometrical studies of computer generated two-phase systems that were presented in Part IV. In order to reduce the computational time associated with the previous three-dimensional studies, the calculations presented in this work are restricted to two dimensions. This allows us to explore more thoroughly the influence of the size of the averaging volume and to learn something about the use of anon-representative region in the determination of averaged quantities.

Nomenclature

Roman Letters A interfacial area of the– interface associated with the local closure problem, m² - a _i i=1, 2, gaussian probability distribution used to locate the position of particles - l unit tensor - characteristic length for the-phase particles, m - ⁰ reference characteristic length for the-phase particles, m - characteristic length for the-phase, m - _i i=1,2,3 lattice vectors, m - m convolution product weighting function - m _V special convolution product weighting function associated with a unit cell - n _i i=1, 2 integers used to locate the position of particles - n unit normal vector pointing from the-phase toward the-phase - r _p position vector locating the centroid of a particle, m - r gaussian probability distribution used to determine the size of a particle, m - r ₀ characteristic length of an averaging region, m - V averaging volume, m³ - V volume of the-phase contained in the averaging volume,V, m³ - x position of the centroid of an averaging area, m - x ₀ reference position of the centroid of an averaging area, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters V /V, volume average porosity - _{a i} standard deviation ofa _i - _r standard deviation ofr - intrinsic phase average of      

On the Shape and Dimensions of Three-Dimensional Cavities in Supercavitating Flows

The shape of long, trailing cavities behind three-dimensional headforms is discussed. The case of a flat elliptic wing is specifically treated. Three distinct shape regimes are found: quasi-planar, long-flat, spheroidal. These appear in successively higher speed ranges (lower cavitation numbers, ). It is argued that the cavities may be replaced by surrogates in the form of slender ellipsoids. The pressures on these are almost constant and correspond to a cavitation number equal to twice their longitudinal added mass coefficient, k₁. A heuristic theory based on kinetic energy fields is given, relating k₁ to the product of headform drag and cavity length. This theory correlates with an exact theory in the same form given by Garabedian for axi-symmetric cones and also with its extension to planar flows. Results are given here for the shape of the cavity behind an elliptic wing of any aspect ratio, given drag, and cavitation number. Specific formulae are given in the form, = f (C_D/AR), for the transition between the quasi-planar and long-flat regime, and the long-flat and spheroidal regime.     

Analytical evaluation of stress-strain test data

An analytical model for deducing the actual stress-strain properties from laboratory test results is discussed. As an illustration, an elastic bilinear material is used for unconfined cylindrical compression test conditions, as simulated with a finite element analysis. The results obtained are applicable for assisting in evaluating measured strength and stiffness properties of some clay soils, concrete test cylinders, concrete cores, and rock cores.The quantitative results of this study can be used for interpreting measured stress-strain data for unconfined compression test conditions. The error in measured results is shown to be influenced by Poisson's ratio, length-to-diameter ratio of the specimen, end condition, and ratio of inelastic modulus to initial elastic modulus. Curves for adjusting the measured results to the theoretical results are presented.Nomenclature D specimen diameter - E _i initial elastic stiffness modulus - E _y elastic stiffness modulus beyond the yield stress, plastic or inelastic modulus - L specimen length - axial strain - _av average strain - _g gage length strain - _y yield strain - Poisson's ratio - compressive stress - _av average stress - _t theoretical compressive stress - _y yield stress - _ym measured stress at the yield strain     

The determination of a general time creep compliance relation of linear viscoelastic materials under constant load and its extension to nonlinear viscoelastic behavior for the Burger model

Linear viscoelastic materials yield a creep function which only depends on time if creep experiments are performed under constant stress ₀. In practice, this condition is very difficult to realize, and as a consequence, the experiments are performed under constant force. For small strains the difference between the conditions of constant stress and constant force is negligible. Otherwise, the decrease in cross-section has to be taken into account and leads to increasing stress in the course of time for creep experiments under constant load. The Boltzmann superposition principle is solved under the condition of constant load and for strains . The creep complicance C(t; ₀) defined by the ratio becomes, in principle, dependent on the initial stress ₀. As a consequence, a set of creep compliance curves cannot be approximated with a simple parameter fit. Already the application of the solution on the Burger model yields a creep compliance curve with all three creep ranges. Furthermore, the mathematical structure of the time creep compliance relation of the Burger model allows nonlinear viscoelastic extension via the introduction of the yield strength _max and a nonlinearity parameter n _l . The creep behavior of PBT and PC can be described in the range of long times up to initial stresses ₀, being 75% for PBT and 60% for PC of the yield stress _max with only two or one free fit parameter, respectively.     

Three-dimensional effects in beams: Isodyne assessment of a plane solution

The stress distribution in a homogeneous beam subjected to three-point bending is investigated using the method of optical isodynes. The three stress components _xx,_yy and _xy acting in the planes formed by the longitudinal and vertical axes of the beam are determined in three planes situated at different through the thickness locations with respect to the beam's midplane. The experimental results are subsequently correlated with the two-dimensional elasticity solution. It is illustrated that at locations sufficiently removed from the centrally applied concentrated load, good correlation between theory and experiment is obtained. In the regions where high stress gradients exist however, differences are observed in the in-plane stress distributions in the different planes. These differences are explained by the presence of the out of plane normal stress _zz using the relations of optical isodynes. Greatest differences between theory and experiment are obtained for the in-plane shear stress component _xy.     

Zur Variationsformulierung nichtlinearer Randwertprobleme

Übersicht Es werden verschiedene Bedingungen aufgestellt, die es erlauben, die durch die beiden (Systeme von) nichtlinearen DifferentialgleichungenA (u, ) = q, B (u, ) = und Randbedingungen zusammen mit den nichtlinearen algebraischen Relationenq = C(u, ), = D(u, ) beschriebene Aufgabe durch äquivalente Variationsprobleme zu ersetzen. Dabei zeigt sich ein enger Zusammenhang mit den in der Festkörpermechanik wohlbekannten Prinzipien der virtuellen Verschiebungen und der virtuellen Kräfte. Die auf systematischem Weg konstruierten Variationsfunktionale enthalten viele in der Physik bekannte Funktionale als Sonderfälle, insbesondere jene, die in der Elastomechanik nach Green, Castigliano, Hellinger, Reißner, Hu und Washizu benannt werden.

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Summary In this paper there are established various conditions which allow a variational formulation of the problem described by the two (systems of) nonlinear differential equationsA(u, ) = q, B(u, ) = and boundary conditions together with the nonlinear algebraic relationsq = C(u, ), = D(u, ). Besides a close relationship is revealed to the principles of virtual displacements and virtual forces which are wellknown in solid mechanics. The systematically constructed variational functional contain many functionals in physics as special cases, mainly those of Green, Castigliano, Hellinger, Reißner, Hu and Washizu in elastomechanics.

     

On Perturbative Expansions to the Stochastic Flow Problem

When analyzing stochastic steady flow, the hydraulic conductivity naturally appears logarithmically. Often the log conductivity is represented as the sum of an average plus a stochastic fluctuation. To make the problem tractable, the log conductivity fluctuation, f, about the mean log conductivity, lnK _G, is assumed to have finite variance, _f ². Historically, perturbation schemes have involved the assumption that _f ²<1. Here it is shown that _f may not be the most judicious choice of perturbation parameters for steady flow. Instead, we posit that the variance of the gradient of the conductivity fluctuation, _f ², is more appropriate hoice. By solving the problem withthis parameter and studying the solution, this conjecture can be refined and an even more appropriate perturbation parameter, , defined. Since the processes f and f can often be considered independent, further assumptions on f are necessary. In particular, when the two point correlation function for the conductivity is assumed to be exponential or Gaussian, it is possible to estimate the magnitude of f in terms of _f and various length scales. The ratio of the integral scale in the main direction of flow ( _x ) to the total domain length (L^*), _x ²=_x/L^*, plays an important role in the convergence of the perturbation scheme. For _x smaller than a critical value _c, _x < _c, the scheme's perturbation parameter is =_f/_x for one- dimensional flow, and =_f/_x ² for two-dimensional flow with mean flow in the x direction. For _x > _c, the parameter =_f/_x ³ may be thought as the perturbation parameter for two-dimensional flow. The shape of the log conductivity fluctuation two point correlation function, and boundary conditions influence the convergence of the perturbation scheme.     

An analogue study for flame flickering

An analogue experiment is proposed to simulate flame flickering comprising a free ascending column fed on its side with a light gas (helium) emerging from a vertical slot in ambient air. The convective motion of the helium jet is considered to represent the motion of burnt gases of buoyant jet flames. The helium jet is accelerated by buoyancy effects and the flow field is similar to that of burnt gases observed for real buoyant flames. The vertical velocity profile of the steady helium jet is measured at different vertical distances. The unsteady helium jet is also studied by measuring the instability frequency as a function of ambient pressure at different injection flow rates, and by analyzing the tomography images of the helium jet. The instability morphology is the same as that observed on real buoyant flames. We conclude that this type of instability can be approximately characterized by the maximum vertical velocityu _max, and the distance betweenu _max in the helium ascending column andu = o in the ambient air. For this type of instability the local vorticity is proportional to which can be influenced by gravity and ambient pressure. Theoretical prediction of the instability frequency as a function of gravity and ambient pressure has been obtained, and is in good agreement with the experimental results.List of symbols C ₁,C ₂ constants - F instability frequency - F _c critical frequency - F _m the most amplified frequency - F (K, ) function defined in (11) - g gravitational acceleration - g reduced gravity acceleration g(₀-^*)/^* - k real wave number of the disturbance - K reduced wave numberK=2k - K _c reduced wave number of the critical instability mode - K _m nondimensional wavenumber of the most amplified mode - L vertical characteristic length (in x direction) - P ambient pressure - u local vertical buoyant velocity (inx direction) - u _max local maximum vertical velocity - v local velocity component iny direction (horizontal) - V ₀ injection velocity of helium (iny direction) - x vertical distance measured from the leading edge of boundary layer - y horizontal distance measured from the exit plane of the vertical slot - Z(K, ) function defined in equation (11) Greek symbols distance betweenu _max in the helium ascending column andu = o in the ambient air - - wavelength of instability - _c critical wavelength - _m the most amplified wavelength - ^* helium density at slot exit - ₀ ambient air density - ^* helium dynamic viscosity at slot exit - v ^* helium kinematic viscosity at slot exit - complex number presented in disturbancee ^i(kx+t) - _i imaginary part of , representing the amplification rate of disturbance - _r real part of , where ( _r /k) represents the group velocity - reduced complex number of , defined      

On the linear instability of the Hall-Stewartson vortex

It is demonstrated that the Hall-Stewartson leading-edge vortex is linearly unstable to viscous perturbations of the center-mode type. Center modes are found to occur in two reigons of Reynolds-number-wave-number space, in limits in which the axial wave number is large. The appropriate center-mode equations in these neighborhoods are established, and it emerges that the two sets are identical. The single system of equations, which depends on the azimuthal wave number m and a distance parameter only, is solved numerically for various values of m and . Highly unstable modes are found for large positive , and the results are shown to be in good agreement with proposed asymptotic expansions when >1. To lowest order, unstable modes have phase surfaces that rotate with the fluid: in addition constant phase surfaces propagate upstream but the group velocity is directed downstream. The growth rate of the instability increases faster than Reynolds number to the quarter power. This, together with the finding that the length scale of the unstable modes found goes to zero as the Reynolds number tends to infinity, makes this instability an unusual one.This work was supported by the Air Force Office of Scientific Research under contract AFOSR-89-0346 monitored by Dr. L. Sakell, and by the U.S. Army Research Office at the Mathematical Sciences Institute of Cornell University.     

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.     

Pressure pulsations during interaction between a subsonic jet and the subsonic section of a supersonic jeton a flat obstacle

Pressure pulsations were measured during in-leakage of a subsonic jet and the subsonic section of a heated supersonic jet on a flat obstacle. Data have been obtained on the total and spectrum levels of the pressure pulsations at different spacings X of the obstacle from the nozzle exit. It is shown that when the obstacle is disposed at the section of the jet where the local velocity is subsonic, the pulsation levels outside the dependence on the conditions at the nozzle exit (Mach number Maxa 0 a 3.0; stagnation temperature T₀=280–1200K) vary in direct proportion to the local velocity head q. The ratio between the total level and q is (/g)=0.2–0.3. It is established that for a subsonic velocity ahead of the obstacle, all the spectra obtained for different values of M _a , T₀, d _a and X in the coordinates Sh=f(d/V) and (₁*/q)(V/d) will lie on a single generalized spectrum. Here ₁* is the pulsation level in a 1-Hz band, and d and V are, respectively, the jet diameter and velocity directly in front of the obstacle.Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 172–174, September–October, 1975.