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.     

Plug flow of a fibrous suspension

A model of a fibrous suspension with plug flow is constructed. A solution to the problem of the flow of a suspension in a straight round tube is obtained for two partial cases and is compared with experiment. With the flow of a fibrous suspension in a round tube, several sets of flow conditions can be distinguished [1–3]. If the flow rate is relatively small, the so-called plug flow is established. It is characterized by the fact that two flow regions are formed in the tube: the core of the flow, or the plug [1–3], in which the mass of the fibers is concentrated, and a layer near the wall in which only the liquid phase of the suspension flows. When the suspension has attained a determined velocity, the plug starts to break down, and the flow ceases to be of the plug type.Petrozavodsk. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 65–71, July–August, 1972.     

Investigation of isothermal flow of a mixture of gas and particles in a channel of variable section

Isothermal flow of a gas with particles is investigated analytically, which makes it possible to analyze all possible flow regimes in channels of different shapes. It is shown that in a channel of constant section there are two possibilities: either an equilibrium regime is established with constant flow parameters, or the gas reaches the velocity of sound, and then further flow in the channel is impossible (blocking of the channel). In a contracting nozzle, blocking also occurs if the channel is sufficiently long. In an expanding nozzle when there are particles in the gas with a velocity lower than the gas velocity, it is possible to have flow regimes with transition through the velocity of sound: a subsonic flow goes over into a supersonic flow and, conversely, it is also possible to have a flow in which there is blocking of the channel, which is quite different from the flow of a pure gas in an expanding nozzle and is due to the influence of interphase friction on the flow. The variation of the pressure along the flow can be nonmonotonic with points of local maximum or minimum which do not coincide with the singular point at which the gas velocity reaches the velocity of sound. In the case of nonequilibrium gas flows with particles in a Laval nozzle, the velocity of the gas may become equal to the isothermal velocity of sound not only in the exit section of the nozzle or in its expanding part, as noted in [4–6], but also at the minimal section, since it is possible to have flows for which the velocities of the phases are equalized at this section.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 61–68, October–December, 1981.     

On the interference of a weak and a strong shock wave in a dissociating nitrogen flow

The influence of the nitrogen dissociation on the interactions due to the interference of two planar shock waves in a hypersonic high enthalpy flow is theoretically investigated for infinite reaction rates. The two limiting cases of infinitely slow and infinitely fast reactions are modelled as a perfect gas and an ideal dissociating gas in chemical equilibrium.To investigate the influence of finite reaction rates on the interactions of shock waves, experiments are performed in the high enthalpy shock tunnel Göttingen (HEG) with a wind tunnel model consisting of a wedge type shock generator and a transversally mounted cylinder. The pressure and heat transfer loads resulting from the shock wave interferences are measured and the flow field is visualized by means of interferograms. The experimental results are compared with the results of a numerical simulation for a dissociating nitrogen flow and with the experimental results for a perfect gas flow.     

Effect of precursor radiation on the flow structure and ionization behind the shock front in inert gases

The problem of the propagation of strong, intensely radiating shock waves in inert gases is considered. It is shown that the heating of the shock tube walls by the precursor radiation, accompanied by an increase in the temperature of the adjacent gas, leads to the transverse stratification of the medium and to the disturbance of the one-dimensionality of the flow of shock-heated gas behind the wave front. Ionization kinetics calculations which take this into account indicate an acceleration of ionization near the tube walls, which is consistent with experiment. On the basis of the gas heating values obtained it is possible to establish critical values of the gas pressures ahead of the front and the shock wave Mach numbers, transition through which is accompanied by a radical restructuring of the flow with the formation of a configuration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 124–131, May–June, 1991.     

Radial diffusion in the poiseuille flow of a gas mixture

In a number of experiments (see [1], in which experimental papers are listed), diffusion has been observed in the radial direction in the process of flow of a mixture along tubes at low pressures. The heavier molecules accumulate near the tube axis. The attempt made in [1] to explain this phenomenon by the influence of the Burnett contribution to the diffusion did not lead to success, and the Burnett terms in the radial diffusion velocity indicate a motion of heavy molecules away from the tube axis. In the present paper, a complete analysis is given of this phenomenon. We consider the problem of the flow of a mixture along a cylindrical tube of finite length for given pressure difference p between its ends. On the basis of the hydrodynamic equations of the Burnett and super-Burnett approximations, a consistent asymptotic (with respect to the small parameter ) solution is given; = (p/p)R/L is the relative change in the pressure along the tube at a distance of order R (R and L are the radii and length of the tube). Radial diffusion occurs in the quadratic approximation in . It is shown that the radial diffusion velocity contains new terms not present in [1]; these are due to the inhomogeneity of the temperature and the pressure over the tube section, the expansion of the gas, and the super-Burnett correction to the diffusion velocity. The most important is the thermodiffusion term, which is determined by the hydrodynamic equations of the Navier-Stokes approximation. The remaining terms have order relative to it of Kn² (Kn = 1 /R is the Knudsen number, and 1 is the mean free path of the molecules). The expression obtained for the diffusion velocity agrees in sign with the experiment.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 91–96, September–October, 1979.I am grateful to G. E. Skvortsov, who drew my attention to this problem, and Yu. N. Grigor'ev for discussing the results.     

Reflection of a strong shock wave from a sphere and a cylinder

We examine the flow on the axis in the vicinity of the stagnation point for reflection of a strong plane shock wave (with uniform parameters behind the wave) from a sphere and a circular cylinder whose generators are parallel to the incident wave front.The small parameter method [1, 2] is used to obtain, in closed form, relations which define the time variation of the velocity profile, pressure, enthalpy, and reflected shock wave standoff.As the time t , these relations reduce to the known formulas [3, 4] which define the steady flow on the axis for the flow behind the incident shock wave about a body, if account is taken of the leading terms containing the small parameter.The solution is extended to the case in which account for equilibrium dissociation and ionization is necessary.Comparison of the results with measurement [5] of the reflected shock wave distance from a sphere, as a function of time, shows satisfactory agreement.     

On the properties of compressible gas flow in a porous media

The flow of an adiabatic gas through a porous media is treated analytically for steady one- and two-dimensional flows. The effect on a compressible Darcy flow by inertia and Forchheimer terms is studied. Finally, wave solutions are found which exhibit a cut-off frequency and a phase shift between pressure and velocity of the gas, with the velocity lagging behind the pressure.Nomenclature A area of tube for one-dimensional flow - B drag coefficient associated with Forchheimer term - c speed of sound - M Mach number - p ^* gas pressure - p dimensionless gas pressure - s coordinate along the axis of tube - t ^* time variable - t dimensionless time variable - V^* gas velocity in the porous media - V dimensionless gas velocity Greek Letters ratio of specific heat capacities - phase angle between gas pressure and velocity for linear waves - parameter indicating the importance of the inertia term - viscosity - _p natural frequency of the porous media - ^* gas density - dimensionless gas density - parameter indicating the importance of the Forchheimer term - porosity of porous media - velocity potential - stream function     

Shock velocity measurements using microwave Doppler shift

A commercially available 10.587 GHz microwave Doppler module is used for the measurement of shock velocity in a conventional shock tube. With proper electronic circuits the Doppler frequency obtained is found to be quite noisefree and consistent for shock velocities in the range of 1.8 mm/sec to 2.0 mm/sec.     

Effect of hall currents on the flow of a conducting gas at high flow velocities

In [1] the flow of a compressible fluid was examined for the case when the conductivity = with account for the Hall effect. Oates [2] solves the problem of the influence of Hall currents on the flow in an accelerator for channels having a very small ratio of height to length when the velocity component in the direction of the channel height may be assumed to be zero. The problem of the influence of Hall currents on the flow of a conducting gas of finite conductivity is solved below for the case when the gas is accelerated to high velocities ( 50–100 km/sec) with account for the presence of two velocity components.     

The structure of particle-laden,underexpanded free jets

Underexpanded, supersonic gas-particle jets were experimentally studied using the shadowgraph technique in order to examine the influence of the dispersed particles on the shape of the free jet and the structure of the imbedded shock waves. The particle mass loading at the nozzle exit was varied between zero and one, and two sizes of particles (i.e. spherical glass beads) with mean number diameters of 26 and 45 m were used. It was found that the Mach-disc moves upstream towards the orifice with increasing particle loading. The laser light sheet technique was also used to visualize the particle concentration distribution within the particle jet and the spreading rate of the particle jet. Furthermore, the particle velocity along the jet centerline was measured with a modified laser-Doppler anemometer. These measurements revealed that the particles move considerably slower than the gas flow at the nozzle exit. This is mainly the result of the particle inertia, whereby the particles are not accelerated to sonic speed in the converging part of the nozzle.In order to further explore the particle behavior in the free jet, numerical studies were performed by a combined Eulerian/Lagrangian approach for the gas and particle phases, including full coupling between the two phases. The numerical results showed that the application of different particle velocities at the nozzle exit as the inlet conditions, which were below the sonic speed of the gas phase has a significant influence on the free jet shape and the configuration of the shock waves. These results demonstrate that the assumption of equilibrium flow (i.e. zero slip between the phases) at the nozzle exit which has been applied in most of the previous numerical studies is not justified in most cases. Furthermore, the numerical calculations of the free jet shape and the particle velocity along the jet axis were compared with the measurements. Although correlations for rarefaction and compressibility effects in the drag coefficient were taken into account, the particle velocity along the center line was considerably overpredicted.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Investigation of flow beyond a supersonic nozzle by an electron-beam technique

The results of an experimental study of a flow of rarefied gas of density 10^–5 g/cm³ beyond the cutoff of a hypersonic nozzle (M11) by means of an electron beam with energy up to 43 keV are presented. The density and velocity fields at different distances from the nozzle and various receiver pressures were measured using this method and the static and total pressure fields were also measured with the help of a Pitot tube. The flow parameters beyond the nozzle were calculated for two limiting cases: with equilibrium condensation and without condensation. This calculation is compared with the experimental results.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 111–117, January–February, 1976.The authors thank S. N. Romanenko for help with the electron-beam experiments.     

Optimal Profiling of the Supersonic Part of a Plug Nozzle Contour

The problem of the optimal profiling of the supersonic part of a plug nozzle contour is solved within the framework of the ideal (inviscid and non-heat-conducting) gas model. The contours obtained provide a thrust maximum for given uniform sonic flow in the radial critical section of the nozzle, given constraints on the nozzle dimensions, and a given outer pressure (counterpressure). The initial sonic regions of the optimal contours are profiled on the basis of the condition that there the flow Mach number is unity. Varying the initial sonic region length makes it possible to construct nozzles of different sizes. The possibilities of the computational programs developed are demonstrated with reference to the example of plug nozzles, optimal when operated in a vacuum. It is shown that low thrust losses are obtained even for moderate nozzle dimensions. In the examples calculated, the optimal plug nozzles provide a greater thrust than the optimal axisymmetric and two-dimensional nozzles with an axial sonic flow for the same lengths and gas flow rates.     

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.     

Self-similar solutions for unsteady flow behind an exponential shock in an axisymmetric rotating dusty gas

Similarity solutions are obtained for one-dimensional unsteady isothermal and adiabatic flows behind a strong exponential cylindrical shock wave propagating in a rotational axisymmetric dusty gas, which has variable azimuthal and axial fluid velocities. The shock wave is driven by a piston moving with time according to an exponential law. Similarity solutions exist only when the surrounding medium is of constant density. The azimuthal and axial components of the fluid velocity in the ambient medium are assumed to obey exponential laws. The dusty gas is assumed to be a mixture of small solid particles and a perfect gas. To obtain some essential features of the shock propagation, small solid particles are considered as a pseudo-fluid; it is assumed that the equilibrium flow conditions are maintained in the flow field, and that the viscous stresses and heat conduction in the mixture are negligible. Solutions are obtained for the cases when the flow between the shock and the piston is either isothermal or adiabatic, by taking into account the components of the vorticity vector. It is found that the assumption of zero temperature gradient results in a profound change in the density distribution as compared to that for the adiabatic case. The effects of the variation of the mass concentration of solid particles in the mixture \(K_p\) , and the ratio of the density of solid particles to the initial density of the gas \(G_a\) are investigated. A comparison between the solutions for the isothermal and adiabatic cases is also made.     

Numerical modeling of flow with chemical reactions in a strong shock wave with approximate allowance for translational nonequilibrium

The effect of translational nonequilibrium on the course of chemical reactions in a shock wave is studied using the beam–gas model extended to the case of a multicomponent gas. For Arrhenius reactions of general form with collisions between beam and gas molecules, a modified expression of reaction rate is obtained that takes into account the relative motion of the two media. A procedure for numerical solution of the problem is considered, and calculation results for a shock wave in a dissociating air at an oncoming flow velocity of 6000 m/sec are given.     

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.     

The influence of boundary layer growth on shock tube test times

Summary A theoretical and experimental investigation of the limitation on shock tube test times which is caused by the development of laminar and turbulent boundary layers behind the incident shock is presented. Two theoretical methods of predicting the test time have been developed. In the first a linearised solution of the unsteady one-dimensional conservation equations is obtained which describes the variations in the average flow properties external to the boundary layer. The boundary layer growth behind the shock is related to the actual extent of the hot flow and not, as in previous unsteady analyses, to its ideal extent. This new unsteady analysis is consequently not restricted to regions close to the diaphragm. Shock tube test times are determined from calculations of the perturbed shock and interface trajectories. In the second method a constant velocity shock is assumed and test times are determined by approximately satisfying only the condition of mass continuity between the shock and the interface. A critical comparison is made between this and previous theories which assume a constant velocity shock. Test times predicted by the constant shock speed theory are generally in agreement with those predicted by the unsteady theory, although the latter predicts a transient maximum test time in excess of the final asymptotic value. Shock tube test times have also been measured over a wide range of operating conditions and these measurements, supplemented by those reported elsewhere, are compared with the predictions of the theories; good agreement is generally obtained. Finally, a simple method of estimating shock tube test times is outlined, based on self similar solutions of the constant shock speed analysis.Nomenclature a speed of sound - A, B, C constants defined in section 5.3 - D shock tube diameter - K =/q ^m, boundary layer growth constant, see Appendices A and B - l hot flow length - m constant, =1/2 or 4/5 for laminar or turbulent boundary layers, respectively - M ₀ initial shock Mach number at the diaphragm - M _s shock Mach number at station x _s - M ₂ =(U ₀–u ₂)/a ₂, hot flow Mach number relative to the shock front - N = ₂ a ₂/ ₃ a ₃, the ratio of acoustic impedances across the interface - P pressure - P* =P _e–P ₂, perturbation pressure - q boundary layer growth coordinate defined in § 2 - Q =(W–1+S) K - r radial distance from shock tube axis - S boundary layer integral defined by equation A6 - t time - t =/ , dimensionless ratio of test times - T =l/l , Roshko's dimensionless ratio of hot flow lengths - u axial flow velocity in laboratory coordinate system, see figure 1a - u* =u _e–u₂, perturbation axial flow velocity - U shock velocity - U ₀ initial shock velocity at the diaphragm - U* =U–U ₀, perturbation shock velocity - v axial flow velocity in shock-fixed coordinate system, see figure 1b - w radial flow velocity - W =U ₀/(U ₀–u ₂), density ratio across the shock - x axial distance from shock tube diaphragm - x _s, x _s axial distance between shock wave and diaphragm - t = _I/ , dimensionless ratio of test times - X =l _I/l , Roshko's dimensionless ratio of hot flow lengths - y =(D/2)–r, radial distance from the shock tube wall - ratio of specific heats - boundary layer thickness; undefined - boundary layer displacement thickness - boundary layer thickness defined by equation A2 - characteristic direction defined by dx/dt = (u ₂–a ₂) - =(M ₀ ² +1)/(M ₀ ² –1) - viscosity - characteristic direction defined by dx/dt=(u ₂+a ₂) - density - * = _te–₂, perturbation density - Prandtl number - shock tube test time - =M ₀ ² /(M ₀ ² –1) Suffices 1 conditions in the undisturbed flow ahead of the shock - 2 conditions immediately behind an unattenuated shock - 3 conditions in the expanded driver gas - 4 conditions in the undisturbed driver gas - e conditions between the shock and the interface, averaged across the inviscid core flow - i conditions at the interface - I denotes the predictions of ideal shock tube theory - asymptotic conditions given when x _s and t - s conditions at or immediately behind the shock - w conditions at the shock tube wall - a, b, b ¹, c, d, d ¹, f, f ¹, g, g ¹, j, k, k ¹ conditions at the points indicated in figure 2