Effect of leading radiation on the stability of strong shock waves in gases with an arbitrary equation of state

The effect of leading radiation on the stability of a strong shock wave in an ideal gas with an arbitrary equation of state is investigated. The ionization ahead of and behind the shock front and the radiation are assumed to be in equilibrium. The investigation is carried out in the linear approximation with respect to amplitude for disturbances with a wavelength much greater than the width of the relaxation zones ahead of and behind the shock. The conditions under which the leading radiation has a destabilizing effect on the shock wave are established. It is shown, in particular, that neutrally stable shock waves become unstable. The conditions under which the onset of instability is of the threshold type with respect to the radiation intensity are determined. It is found that the radiation also has a destabilizing effect on stable shock waves, including shock waves in a perfect gas. However, in this case instability can develop only when the disturbances have a wavelength comparable with the width of the relaxation zone. A simple physical mechanism of the onset of instability under the influence of leading radiation is proposed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 125–133, May–June, 1990.The authors are grateful to A. G. Kulikovskii and A. A. Barmin for their constant interest and useful discussions.     

Airflow behind strong shock front with account for nonequilibrium ionization and radiation

We consider the gas state behind a shock wave front in air with a velocity v10 km/sec. Nonequilibrium ionization and radiative transport are taken into account. We take into consideration the real air spectrum — the numerous lines, bands, and continuua. Account for the radiation leads to an integrodifferential system of equations for which a solution method is developed. As a result we obtain the gas parameter profiles behind the shock wave, which are affected by the relaxation processes and radiative cooling. The calculations were made for v=10–16 km/sec and a pressure p=10^–5–10^–2 atm ahead of the front.In order to obtain realistic results, we consider only the gas layer bounded by the shock and a surface parallel to it. It is assumed that the gas bounded by these planes is not irradiated from without. In this formulation still another defining parameter appears—the distancel between the planes. The calculations were made forl=1–100 cm.     

Experimental study of strong shock propagation in a channel

The widespread use of shock tubes in laboratory practice is well known. However, despite existing information [1] about shock-wave velocities of 100 km/sec, experimental data on the size of the shock-heated region behind the shock front are confined to the Mach numbers M = 10 [2]. Theoretical data do not go beyond the limit of this range except for air where calculations were performed up to M = 20 [3, 4]. Behind strong shocks, the effects resulting from viscosity, thermal conductivity, and radiation of the medium should lead to serious deviation of the actual flow from the idealized pattern for uniform motion of a piston in a channel filled with anonviscous, thermally nonconducting, and nonradiating medium. It is therefore practical to make an experimental study of the behavior of density and of the size of the shock-heated region behind a shock front propagating down the channel of a shock tube that is capable of producing velocities up to 8 km/sec.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 23–28, July–August, 1976.     

Experimental investigation of the emission in the unsteady flow region behind strong shock fronts in mixtures of co and CO2 with nitrogen and argon

Mixtures of CO (or CO₂) gases and N₂ behind strong shock fronts at temperatures 4000–10 000 ° K have been investigated with a view to elucidating the mechanism of the physicochemical processes in the unsteady region of the gas flow behind a shock front leading to the behavior of strongly radiating CN and C₂ molecules and C atoms and also determining the quantitative characteristics of the chemical reactions. A shock tube was used in the investigations, which made it possible to obtain the intensity distribution of the radiation of several components — CN, C₂, and C — behind shock fronts.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 120–129, March–April, 1981.We thank S. A. Losev and O. P. Shatalov for assisting in the work and for valuable discussions.     

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.     

Pressure of an arbitrary acoustical wave on a plane

A study is made of the perturbed flow of a gas, brought about by a weak shock wave, falling on a fixed surface at an arbitrary angle. A solution determining the field of the velocities behind the front of the wave in an initially boundary-value problem with movable boundaries for a three-dimensional wave equation is obtained in the form of a double integral, containing an arbitrarily given function determining the parameters of the gas in the incident wave. The region of integration is a region included within an ellipse, whose relative eccentricity is equal to the sine of the angle of inclination of the front of the incident wave. A formula is obtained for the distribution of the pressure at the plane.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 114–116, January–February, 1975.     

Distribution of radiant thermal flux over the surface of a sphere for hypersonic flow of a nonviscous radiating gas past the sphere

In the present study using the Newtonian approximation [1] we obtain an analytical solution to the problem of flow of a steady, uniform, hypersonic, nonviscous, radiating gas past a sphere. The three-dimensional radiative-loss approximation is used. A distribution is found for the gasdynamic parameters in the shock layer, the withdrawal of the shock wave and the radiant thermal flux to the surface of the sphere. The Newtonian approximation was used earlier in [2, 3] to analyze a gas flow with radiation near the critical line. In [2] the radiation field was considered in the differential approximation, with the optical absorption coefficient being assumed constant. In [3] the integrodifferential energy equation with account of radiation was solved numerically for a gray gas. In [4–7] the problem of the flow of a nonviscous, nonheat-conducting gas behind a shock wave with account of radiation was solved numerically. To calculate the radiation field in [4, 7] the three-dimensional radiative-loss approximation was used; in [5, 6] the self-absorption of the gas was taken into account. A comparison of the equations obtained in the present study for radiant flow from radiating air to a sphere with the numerical calculations [4–7] shows them to have satisfactory accuracy.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 44–49, November–December, 1972.In conclusion the author thanks G. A. Tirskii and É. A. Gershbein for discussion and valuable remarks.     

Investigation of flow past blunt bodies with allowance for radiation by the large-particle method

Much recent work has been done on developing methods of solving gas-dynamic problems in which radiation plays a part (see, for example, [1–7]). This is because the temperature in the shock layer associated with flight in the atmosphere at hypersonic velocities can reach values exceeding 10⁴ °K. In such a case, heat transfer by radiation can make an important contribution to the total heat transfer. With increasing flight velocities, the importance of radiation in heat transfer increases and then becomes predominant. In the present paper, the large-particle method as developed by Belotserkovskii and Davydov [8] is developed to calculate flows with radiation around blunt bodies, including the case when there is distributed blowing from the surface of the bodies into the shock layer, which simulates ablation of a heat-shielding covering under the influence of strong heating by radiation. The results are given of systematic calculations of flow past blunt bodies of various shapes by a stream of radiating air, and the results are compared with the data of other methods.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 106–112, July–August, 1982.     

The behavior of a gas bubble in a shock wave

The phenomenon of thermal relaxation of the gas bubbles in a fluid behind a shock front is analyzed. The approach to solving the problem of heat transfer between a gas bubble and a fluid developed by the author is used to obtain a solution describing the initial stage of bubble collapse behind the shock front.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 187–189, September–October, 1993.     

Motion of a gas shock layer with a free boundary under nonadiabatic conditions

This paper is a study of the effect of heat input (removal) on the characteristics of a shock layer produced by a gas at high supersonic velocity encountering a mobile boundary, which for generality is assumed to be free. We will use the Chernyi method, which was employed previously to solve the problem of a shock layer in an adiabatic flow [1, 2]. The results obtained can be useful for analysis of the effect of radiation (absorption) and processes involving the relaxation of internal degrees of freedom of molecules, condensation, chemical reactions, etc., whose effect on the gasdynamics of the flow in a shock layer may be similar to heat input or removal [3–5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 152–154, May–June, 1976.The author thanks A. K. Rebrov for discussion of the results.     

Nonequilibrium molecular radiation behind the front of a strong shock wave in the CO<Subscript>2</Subscript>-N<Subscript>2</Subscript>-O<Subscript>2</Subscript> mixture

Models of population of some radiating electron-vibrational states of CO, CN, and C₂ molecules are developed. The characteristics of radiation in a chemically nonequilibrium flow behind the front of a strong shock wave in a mixture of gases constituting the Martian atmosphere are calculated. The numerical data are compared with experimental results.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 13–22, March–April, 2005     

Specific features of modeling of nonequilibrium radiation behind the shock wave in air in the vacuum ultraviolet spectral range

A physical-chemical model of generation of nonequilibrium molecular radiation in the vacuum ultraviolet (VUV) spectral range behind the shock wave in air for shock wave velocities from 4.5 to 9.5 km/s is developed. Experimental results obtained in a shock tube in investigations of photoionization of air ahead of the shock wave front are used for verification of the numerical model of VUV radiation in the wavelength range from 85 to 105 nm. Model calculations show that nonequilibrium VUV radiation arises in a very thin high-temperature layer behind the shock wave front and is affected by heavy particles and electrons.     

Radiative relaxation kinetics of shock-heated xenon

The kinetics of the ionization and radiation processes taking place in xenon heated in strong shock waves (Mach numbers on the interval 10–20) are numerically investigated. A fairly complete model of the medium and the processes is used. The influence of the radiation efficiency is taken into account on the assumption that the radiation layer is homogeneous and isotropic at every instant of time. The calculated distribution of plasma brightness temperature along the luminous layer is compared with that measured in shock-tube experiments. The experimental data are satisfactorily described by the modeling.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 157–163, November–December, 1992.     

Decay of a plane shock wave in a two-parameter medium with arbitrary equation of state

Special curves, called shock polars, are frequently used to determine the state of the gas behind an oblique shock wave from known parameters of the oncoming flow. For a perfect gas, these curves have been constructed and investigated in detail [1]. However, for the solution of problems associated with gas flow at high velocities and high temperatures it is necessary to use models of gases with complicated equations of state. It is therefore of interest to study the properties of oblique shocks in such media. In the present paper, a study is made of the form of the shock polars for two-parameter media with arbitrary equation of state, these satisfying the conditions of Cemplen's theorem. Some properties of oblique shocks in such media that are new compared with a perfect gas are established. On the basis of the obtained results, the existence of triple configurations in steady supersonic flows obtained by the decay of plane shock waves is considered. It is shown that D'yakov-unstable discontinuities decompose into an oblique shock and a centered rarefaction wave, while spontaneously radiating discontinuities decompose into two shocks or into a shock and a rarefaction wave.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 147–153, November–December, 1982.     

On the flow around a moving plate by a traveling shock

The incidence of a weak shock on a plate moving at supersonic speed is examined theoretically. The shock overtakes the plate. Cases with constant stream parameters behind the shock front and with a nonuniform stream are considered. Formulas are obtained for the time dependence of the plate lift.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 173–176, March–April, 1972.     

Thermal stresses and the shattering of solid particles in a shock layer

The motion of blunt bodies through two-phase media at high supersonic velocities is accompanied by strong heating of particles when they enter the shock layer. Because the ratio of the heating time of nonmetallic particles to the time of their thermal relaxation with the gas exceeds unity, large temperature gradients are developed in the particles, which are stressed and deformed and under the influence of the force and inertial loads they can then shatter, which significantly changes their force and thermal effect on the supersonic body. A special case of this problem — the shattering of ice particles in a shock layer under the influence of pressure forces — was investigated in [1]. In the present paper, the results of numerical calculations and known analytic solutions are used in the development of an approximate method for estimating the stresses that arise in spherical particles. Simple criteria are established for determining when the tensile stresses in the particles reach critical values above which the particles may shatter. As an example, the distribution of the temperature and stresses in silicon dioxide particles is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 66–73, January–February, 1981.We thank V. G. Pchelkina for assistance in calculating the temperature fields.     

Lifting bodies using the flow behind axisymmetric conical shock waves

One of the methods of designing aircraft with supersonic flight speeds involves solving an inverse problem by means of the well-known flow schemes and the substitution of rigid surfaces for the flow surfaces. Lifting bodies using the flows behind axisymmetric shock waves belong to these configurations. All lifting bodies using the flow behind a conical shock wave can be divided into two types [1]. Bodies whose leading edge passes through the apex of the conical shock wave pertain to the first type and those whose leading edge lies below the apex of the conical shock wave, to the second. For small apex angles of the basic cone at hypersonic flow velocities an approximate solution of the variation problem was obtained, which showed that the lift-drag ratio of lifting bodies of the second type is higher than that of the first [2]. The present paper gives a numerical solution of the problem for flow past lifting bodies of the second type using the flow behind axisymmetric conical shock waves with half-angles of the basic cone _S=9.5 and 18° The upper surfaces of the bodies are formed by intersecting planes parallel to the velocity vector of the oncoming flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 135–138, March–April, 1986.     

Compression and rarefaction waves in shock-compressed metals

The behavior of duralumin and copper is studied under conditions of specimen loading by two successive shock waves and during unloading after the shock compression. The amplitude of the first shock wave was 150–250 kbar. Direct measurements were performed of the difference in main stresses behind the shock front in duralumin. The results obtained do not agree with existing concepts of the behavior of solids under dynamic loading. Possible causes of this divergence are considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 146–153, March–April, 1976.In conclusion, the authors thank G. A. Savel'ev for his aid in the preparation and performance of the experiments.     

Supersonic flow past a blunt wedge

A study is made of the asymptotic solution of the problem of flow past a blunt wedge by a uniform supersonic stream of perfect gas. By separation of variables it is shown that at large distances the disturbance of the flow is damped exponentially. In the case of subsonic flow behind the shock wave the exponent of the leading correction term in the expansion of the shock front is calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–140, July–August, 1984.     

Reflection of a blast-profile shock wave from the end wall of a shock tube

The flow in the shock tube, which models the flow conditions behind a blast wave, is calculated for an inviscid non-heat-conducting perfect gas with a constant value of the specific heat ratio. The results are compared with the authors' experimental data.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 141–148, May–June, 1992.The authors are grateful to V. A. Levin for assisting with the work and useful discussions.