Three-dimensional diffraction of a shock wave

The diffraction of a shock wave (M₀=4.7) at an angle close to 180 has been experimentally investigated for the three-dimensional case. Interferograms of the flow and the pressure distribution on the back wall in the course of its interaction with the diffracted wave were obtained. Rotation of the flow structure behind the shock wave relative to the axis of symmetry was observed as the flow pattern develops in time and space.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 200–201 March–April, 1993.     

Radiative cooling of plasma behind a reflected shock front

Calculation of gas flow in a shock tube on the basis of ideal theory [1] leads to results that differ from the real picture. In particular, the calculated velocity of the reflected shock wave exceeds the experimentally measured velocity [2] by about 20%. The calculated parameters of shock-heated gas agree well with the experimental results only directly behind the shock front [3]. The present paper reports a theoretical and experimental investigation of the variation of the plasma parameters behind the front of a reflected shock wave in argon. A picture of the gas-dynamic processes taking place after reflection of the incident shock wave by the end of the shock tube is determined. A method is developed for approximate analytic calculation, this making it possible to determine not only the parameters of the gas directly behind the front of the reflected shock wave for different positions of the wave relative to the end of the shock tube but also the variation of these parameters in other regions behind the reflected shock wave. The calculation takes into account the influence of the boundary layer and radiative cooling in the approximation of a low degree of ionization of the plasma and persistence of equilibrium conditions in the entire region behind the reflected shock wave. The experimental and theoretical profiles of the radiation behind the reflected shock wave are compared.     

Measurement of parameters of a gas suspension behind a shock wave in foam

A study has been made of the propagation of a shock wave in dry polyhedral foam with cell diameter 1 cm. The experiments were made in a shock tube in the range of Mach numbers M < 1.4 of the shock wave. The interaction of the shock wave with the foam was photographed. This established that the destruction of the foam by the shock wave leads to the formation of a gas-droplet flow behind the shock front. To determine the parameters of the suspension, the flow was probed by He-Ne lasers with different radiation wavelengths. The spectral-transparency method was used to find the modal diameter of the droplets of the gas suspension and the volume concentration of the droplets in the flow. The modal diameter of the droplets was 2m, and the volume concentration of the droplets decreased downstream.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 134–141, May–June, 1993.     

Initial stage of reflection of a plane shock wave from a cylinder,sphere, and ellipsoid of revolution

The stage of regular reflection of a plane shock wave from a blunt body (cylinder, sphere, and ellipsoid of revolution) is considered. At the point of intersection of the reflected shock wave and the surface of the body, analytic expressions are found for the derivative of the Mach number of the wave with respect to the time, the curvature of the wave, the normal derivatives of the density and the pressure, and the derivative of the Mach number along the wave front. It is shown that the flow has a singularity at = _* < _** (s** is the limiting angle [1] of regular reflection of a shock wave from a rigid surface). The distribution of the parameters in the region between the reflected shock wave and the surface of the body is found up to terms of third order in the time. The density distribution behind the reflected shock wave was measured experimentally, and also the shape of the reflected wave at different instants of time.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 94–100, March–April, 1980.     

Study of unsteady flow past bodies behind shock waves

Several theoretical and experimental studies have been devoted to the problem of the nonstationary action of the stream behind a shock wave on bodies of varied shape. In particular, in [1], the pressure and density are calculated for flow about bodies of the more typical shapes in the initial stage of the process. The basic relations which accompany the interaction of shock waves are considered in [2, 3]. The analysis of the phenomena of diffraction of shock waves on the sphere, cylinder, and cone is presented in [4]. Problems of unsteady flow about a wing are examined in [5, 6]. A detailed review of the foreign studies on unsteady flow is given in [7]. Of great practical interest is the question of the time for flow formation and the magnitudes of the unsteady loads during this period. Experimental investigations have been made recently [8, 9] in which some criteria are presented for estimating the bow shock formation time for supersonic flow about the sphere and cylinder with flat blunting. However the question of the formation time of the stationary pressure on the body surface is not referred to in these studies and no relationship is shown between the transient position of the reflected wave and the corresponding unsteady pressure on the surface. Moreover, in [8] the dimensionless time criterion is determined very approximately, independently of the Mach number of the shock wave. The present study was undertaken with the object of determining the basic criteria which characterize unsteady flow about bodies behind a plane shock wave which has time-independent parameters, and clarification of the shock wave reflected from the body and the pressure on the surface of the body during the transient period. The most typical body shapes were studied: 1) a cylinder with flat face aligned with the stream; 2) a spherically-blunted cylinder; and 3) a cylinder transverse to the stream. The experiments were conducted in a conventional shock tube using the single-diaphragm scheme. The measurements of the pressure on the models and the velocity of the incident shock wave were made using the technique analogous to that of [10, 11]. A highspeed movie camera was used to record the pattern of the wave diffraction on the body. The Mach number of the incident shock wave varied in the range from M=1.5 to M≈6.0, which corresponded to a range of Mach numbers M_∞ of the stream behind the shock wave from 0.6 to 2.1. The calculations of the required gas dynamic parameters for high temperatures were made with account for equilibrium dissociation of the air on the basis of the data of [10, 12, 13]. The magnitude of the relative maximal shock wave standoff Δ at the stagnation point obtained in the present experiments was compared with the values of Δ from other studies. In the case of the flat-blunted cylinder it was in good agreement with the results of [8–14], and in the case of the spherically-blunted cylinder and the transverse cylinder it was in agreement with the results of [15].     

Hypersonic flow past conical bodies

In [1, 2] Gonor considered the problem of imperfect inviscid gas flow about conical bodies at high supersonic speeds. The method of expansion in terms of a small parameter was used to obtain the solution. The small parameter used was the ratio of the densities in the free stream and behind the shock wave. However, this solution does not enable one to determine the velocity field in the vicinity of the cone surface.In the present paper, this problem is solved by the method described in [3], based on the use of the Poincaré-Lighthill-Ho method. The zero approximation is obtained, which is suitable throughout the entire flow region including the vortical layer. Outside of this layer, the solution transforms to the Gonor solution [1, 2].The author wishes to thank B. M. Bulakha for discussing the paper.     

Spherical shock wave reflection from a surface with a heated gas layer

The two-dimensional axisymmetric problem of the interaction between smallscale spherical shock waves initiated by a laser explosion and an absolutely rigid surface in the presence of a layer of hot gas is numerically investigated. A number of effects previously observed in physical and numerical experiments [5–8] are confirmed, in particular: the distortion of the reflected shock front and its acceleration on passage through the hot central zone of the laser explosion (lens effect), the strong deformation of this zone, and the formation of a precursor on the surface ahead of the shock wave interacting with the thermal layer. In addition, certain new anomalous effects are revealed: the formation of a pair of suspended shocks — one on the periphery of the hot central zone upon interaction with the reflected shock wave and the other behind the Mach stem in the triple point zone, etc.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 141–147, July–August, 1989.     

A study of shock wave interaction with a rotating cylinder

Shock wave reflection over a rotating circular cylinder is numerically and experimentally investigated. It is shown that the transition from the regular reflection to the Mach reflection is promoted on the cylinder surface which rotates in the same direction of the incident shock motion, whereas it is retarded on the surface that rotates to the reverse direction. Numerical calculations solving the Navier-Stokes equations using extremely fine grids also reveal that the reflected shock transition from RR MR is either advanced or retarded depending on whether or not the surface motion favors the incident shock wave. The interpretation of viscous effects on the reflected shock transition is given by the dimensional analysis and from the viewpoint of signal propagation.Received: 24 April 2002, Accepted: 16 August 2002, Published online: 25 March 2003     

Theory of nonlinear waves in a plasma

Stationary nonlinear waves propagating in a cold rarefied plasma composed of electrons and two types of ions are considered. The structure of isolated waves and shock waves is found. In recent years an intensive study has been made of finite-amplitude waves and collisionless shock waves in a rarefied plasma, in connection with laboratory experiments [1] and astrophysical applications (the problem of the interaction of the solar wind with the Earth's magnetosphere [2]). When allowance is made for dispersion effects associated with the departure of the dispersion law =(k) from the linear, and for the compensating nonlinear twisting of the wave profile, we are able to obtain the profile of stationary nonlinear waves of finite amplitude, and when allowance is made for damping we can also obtain the structure of a collisionless shock wave [3]. Such waves have been studied fairly fully for the case of a two-component plasma. The present paper examines stationary nonlinear waves propagating across a magnetic field in a cold rarefied quasi-neutral plasma composed of electrons and two types of ions.     

Flow in vicinity of blunt body stagnation point in diverging hypersonic stream

In the hypersonic thin shock layer approximation for a small ratio k of the densities before and after the normal shock wave the solution of [1] for the vicinity of the stagnation point of a smooth blunt body is extended to the case of nonuniform outer flow. It is shown that the effect of this nonuniformity can be taken into account with the aid of the effective shock wave radius of curvature R_*, whose introduction makes it possible to reduce to universal relations the data for different nonuniform outer flows with practically the same similarity criterion k. The results of the study are compared with numerical calculations of highly underexpanded jet flow past a sphere.Notations x, y a curvilinear coordinate system with axes directed respectively along and normal to the body surface with origin at the forward stagnation point - R radius of curvature of the meridional plane of the body surface - uV, vV., , p V ² respectively the velocity projections on the x, y axes, density, and pressure - and V freestream density and velocity The indices =0 and=1 apply to plane and axisymmetric flows Izv. AN SSSR, Mekhanika Zhidkosti i Gaza, Vol. 5, No. 3, pp. 102–105, 1970.     

Supersonic flow of combustible gas mixture past sphere with account for ignition delay time

Assume an axisymmetric blunt body or a symmetric profile is located in a uniform supersonic combustible gas mixture stream with the parameters M₁, p₁, and T₁. A detached shock is formed ahead of the body and the mixture passing through the, shock is subjected to compression and heating. Various flow regimes behind the shock wave may be realized, depending on the freestream conditions. For low velocities, temperatures, or pressures in the free stream, the mixture heating may not be sufficient for its ignition, and the usual adiabatic flow about the body will take place. In the other limiting case the temperature behind the adiabatic shock and the degree of gas compression in the shock are so great that the mixture ignites instantaneously and burns directly behind the shock wave in an infinitesimally thin zone, i. e., a detonation wave is formed. The intermediate case corresponds to the regime in which the width of the reaction zone is comparable with the characteristic linear dimension of the problem, for example, the radius of curvature of the body at the stagnation point.The problem of supersonic flow of a combustible mixture past a body with the formation of a detonation front has been solved in [1, 2]. The initial mixture and the combustion products were considered perfect gases with various values of the adiabatic exponent .These studies investigated the effect of the magnitude of the reaction thermal effect and flow velocity on the flow pattern and the distribution of the gasdynamic functions behind the detonation wave.In particular, the calculations showed that the strong detonation wave which is formed ahead of the sphere gradually transforms into a Chapman-Jouguet wave at a finite distance from the axis of symmetry. For planar flow in the case of flow about a circular cylinder it is shown that the Chapman-Jouguet regime is established only asymptotically, i. e., at infinity.This result corresponds to the conclusions of [3, 4], in which a theoretical analysis is given of the asymptotic behavior of unsteady flows with planar, spherical, and cylindrical detonation waves.Available experimental data show that in many cases the detonation wave does not degenerate into a Chapman-Jouguet wave as it decays, bur rather at some distance from the body it splits into an adiabatic shock wave and a slow combustion front.The position of the bifurcation point cannot be determined within the framework of the zero thickness detonation front theory [1], and for the determination of the location of this point we must consider the structure of the combustion zone in the detonation wave. Such a study was made with very simple assumptions in [5].The present paper presents a numerical solution of the problem of combustible mixture flow about a sphere with a very simple model for the structure of the combustion zone, in which the entire flow behind the bow shock wave consists of two regions of adiabatic flow-an induction region and a region of equilibrium flow of products of combustion separated by the combustion front in which the mixture burns instantaneously. The solution is presented only for subsonic and transonic flow regions.     

A power-law fluid flow past a porous sphere

Summary A model has been developed for the flow of a non-Newtonian fluid past a porous sphere. The drag force exerted on a porous sphere moving in a power-law fluid is obtained by an approximate solution of equations of motion in the creeping flow regime. It is predicted that the effect of the pseudoplastic anomaly on the drag force is more pronounced at large porosity parameters.

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Zusammenfassung Es wird ein Modell für die Strömung einer nichtnewtonschen Flüssigkeit längs einer porösen Kugel entwickelt. Die auf die in einer Ostwald-DeWaele-Flüssigkeit bewegte Kugel ausgeübte Reibungskraft wird durch eine Näherungslösung der Bewegungsgleichungen für schleichende Strömung gewonnen. Man findet, daß der Einfluß der Abweichung vom newtonschen Verhalten um so ausgeprägter wird, je größer die Porosität ist.

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A, B, C, D a, b, c, d coefficients in eqs. [10] and [18] - F _D drag force - K consistency index in power-law model - k ₁ ,k ₂ coefficients defined by eq. [18] - m porosity parameter - n flow index in power-law model - P pressure - P ^* dimensionless pressure defined by eq. [4] - P pressure difference - R radius of porous sphere - r radial distance from the center of the sphere - U velocity of uniform stream - u _i velocity component - u _i ^* dimensionless velocity component defined by eq. [4] - Y drag force correction factor defined by eq. [27] - _ij rate of deformation tensor - _ij ^* dimensionless rate of deformation tensor defined by eq. [4] - , spherical coordinates - dimensionless radial distance defined by eq. [4] - second invariant of rate of deformation tensor - ^* dimensionless second invariant of rate of deformation tensor defined by eq. [4] - _ij stress tensor - _ij ^* dimensionless stress tensor defined by eq. [4] - stream function - ^* dimensionless stream function defined by eq. [4] - i inside the surface of the sphere - o outside the surface of the sphere With 1 figure and 1 table     

Unsteady supersonic flow around blunt bodies with detached shock wave

The numerical method of calculating the supersonic three-dimensional flow about blunt bodies with detached shock wave presented in [1–3] is applied to the case of unsteady flow. The formulation of the unsteady problem is analogous to that of [4], which assumes smallness of the unsteady disturbances.The paper presents some results of a study of the unsteady flow about blunt bodies over a wide range of variation of the Mach number M=1.50– and dimensionless oscillation frequency l/V=0–1.0. A comparison is made with the results obtained from the Newton theory.     

Particle-in-a-box study of the evolution of an ion-acoustic shock wave

In an analysis of a one-dimensional numerical model of a nonisothermal plasma it is shown that an ion-acoustic shock wave of subcritical amplitude separates a soliton from the shock front after the reversing stage. This process is accompanied by turbulent flow behind the front and by trapping of ions in potential wells. The numerical particle-in-a-box method is being used widely to study plasma phenomena. One field in which this method has been found fruitful is in the study of a nonisothermal plasma, characterized by an ion-acoustic wave branch.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 3–5, May–June, 1971.The authors thank R. Z. Sagdeev for support and interest in this study.     

Propagation of cylindrical shock waves in a gravitating medium

We study gas motion behind the front of a cylindrical shock wave created by the motion of a piston in a gravitating medium. The problem is self-similar, but the solution cannot be obtained in closed form. A numerical calculation is made for various Mach numbers. The calculation shows that the central part of the configuration is displaced a definite distance from the axis of symmetry.Cylindrical shock waves through a compressible homogeneous medium in a gravity field have been examined by Sedov [1] and Lin [2], However, these studies contain the essential assumption that the total energy (i. e., the sum of the kinetic and thermal energies) within the region bounded by the expanding shock wave is independent of time.In the following we extend the previous studies to the case of shock waves in nonhomogenous media, which propagate in the fluctuating gravity field created by the disturbed mass itself. The shock wave is created by the motion of a piston whose velocity varies as some power of the time, i. e., v. The total energy of the configuration also depends on the time.The authors wish to thank M. P. Murgai for cooperation and assistance and C. D. Ghildyal for valuable advice, as well as L. I. Sedov and G. I. Petrov for their critical comments.     

An “ideal equivalent gas method” for the study of shock waves in supersonic real gas flows

Summary A method developed by the author for the systematic study of the thermodynamic and dynamic properties of the gas behind a shock wave is reported.The method is applicable to supersonic flow regimes for which the excitation, dissociation and ionization effects invalidate the usually adopted hypothesis of ideal gas.An Ideal Equivalent Gas, having the ratio of the specific heats _s dependent on Mach number and altitude of flight is postulated.On the basis of the mass, momentum and energy conservation equations, valid through the shock wave, the relations defining the thermodynamic and dynamic state of the gas behind the shock wave are derived. These relations establish an extension of the classic relations valid for the ideal gas and reduce to them identically for _s=.The dependence of the ratio of specific heats _s of the Ideal Equivalent Gas on Mach number and altitude has been established, over a wide range, on the basis of the real gas solutions derived by Huber.

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Sommario Nella presente nota viene esposto un metodo sviluppato dall'autore per lo studio sistematico dello stato termodinamico e dinamico del gas a valle di un'onda d'urto in regime supersonico, allorchè cioè gli effetti dell'eccitazione dei gradi di libertà vibrazionali delle molecole e della loro dissociazione e successiva ionizzazione invalidano l'ipotesi di gas ideale generalmente adottata.Viene definito un gas ideale equivalente avente rapporto dei calori specifici _s funzione del numero di Mach e della quota di volo ed in base alle equazioni di conservazione della massa, della quantità di moto e dell'energia, valide attraverso all'onda d'urto, vengono derivate delle relazioni definenti lo stato termodinamico e dinamico del gas a valle dell'onda d'urto. Tali relazioni costituiscono una estensione delle classiche relazioni dell'urto valide per il gas ideale alle quali si riducono per _s=.La dipendenza del rapporto dei calori specifici _s del gas ideale equivalente, dal numero di Mach e dalla quota è stata stabilita sulla base delle soluzioni ottenute da Huber per il gas reale.

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Ionization and nonequilibrium radiation of air behind strong shock waves

It is shown that at high velocities of shock waves (V 9.5 km/sec) an important factor influencing the rate of ionization is the depletion of the number of excited states of the atoms through de-excitation. In the case of low pressures (p 1 torr) and for a bounded and optically transparent region of gas heated by the shock wave (for example, for the motion of gas in a shock tube or in a shock layer near a blunt body), the effective ionization rate k_f depends on the pressure [1], which leads to violation of the law of binary similarity which holds under these conditions without allowance for de-excitation. On leaving the relaxation zone, the gas arrives at a stationary state with constant parameters differing from those in thermodynamic equilibrium. The electron concentration and also the radiation intensity in the continuum and the lines are lower than the values for thermodynamic equilibrium. These considerations explain the results of known experiments and some new experiments on ionization and radiation of air behind a travelling shock wave.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 105–112, January–February, 1980.     

Effect of freestream nonequilibrium on flow past a wedge

We examine the effect of flow freezing in the nozzle of a hypersonic wind tunnel on the parameters of nonequilibrium flow past a wedge.Zhigulev's solution [1] for vibrational relaxation is extended to the nonequilibrium freestream case.It is shown that in this case the perturbations of the frozen-flow parameters behind the oblique shock wave can change sign if the flow in the working section deviates significantly from equilibrium.A method is proposed for converting experimental results obtained in tunnels with frozen flow to the case of equilibrium freestream flow past the body.The author wishes to thank O. Yu. Polyanskyi and V. P. Agavonov for their interest in this study.     

Special case of the density distribution behind a nonstationary shock wave

The density distribution behind a nonstationary shock wave for a definite value of the Mach number M_*, which depends on = c_p/c_v, is considered. Use is made of the previously established fact [1] that for M = M_*() there exists a connection between the first and second derivatives of the density along the normal behind the wave. An investigation is made into the density profile in dimensionless variables behind plane, cylindrical, and spherical shock waves in the neighborhood of the shock front. In the first case, if the gas in front of the wave is homogeneous, only two types of density profile are possible (up to small quantities of third order in the coordinate). In the second and third cases, the form of the density distribution also depends on a parameter, the ratio of the first derivative along the normal of the density behind the wave to the radius of curvature of the wave.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 163–167, November–December, 1979.