Numerical investigation of gasdynamic features of control jets

Numerical methods, based on first order difference schemes are used to investigate features of three-dimensional subsonic and supersonic flows of an inviscid non-heat-conducting gas in control jets. Elements of the nozzle channels considered are axisymmetric, and flow symmetry arises from the nonaxial feature of the prenozzle volume and the subsonic part of the nozzle, or because of nonaxiality of elements of the supersonic part. In the first case the nozzle includes an asymmetric subsonic region in which reverse-circulatory flow is observed, and in the second case it includes a region of sudden expansion of the supersonic flow from the asymmetric stagnation zone.Translated from Izvestiya Akademii NaukSSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 126–133, November–December, 1978.The authors thank A. N. Kraiko for useful comments and M. Ya. Ivanov for his interest in this work.     

Use of direct variational methods to optimize axisymmetric Laval nozzles in the case of equilibrium and nonequilibrium two-phase flows

In the construction of the optimal profile of a Laval nozzle when there are subsonic regions in the flow, the use of effective methods such as the general method of Lagrangian multipliers [1] becomes very difficult. In the present paper, direct variational methods are therefore used. For nozzles, these methods were used for the first time to profile the supersonic parts of nozzles in the case of nonequilibrium two-phase flows by Dritov and Tishin [2]. For equilibrium flows, they have been used to optimize supersonic nozzles [3, 4] and in the construction of a profile of the subsonic part of a nozzle ensuring parallel sonic flow in the minimal section of the nozzle [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 181–183, January–February, 1982.I thank A. N. Kraiko for a number of helpful comments in a discussion of the formulation of the problem.     

Development of pulsation regimes in one-dimensional unsteady MHD flows with switching off of the electrical conductivity

This paper investigates the gas flow in an electromagnetic field when the conductivity, being a function of the thermodynamic gas parameters, vanishes during the flow (switching off of the conductivity). In the case of steady supersonic flows in an expanding nozzle it was first shown analytically [1] and then confirmed by numerical experiment [2] that stable steady flow is not possible for all the problem parameters (for example, the values of the magnetic field at the exit). Instead of a steady flow a periodic regime is realized when narrow regions of conducting gas with currents flowing through them detach from the conducting region and propagate down the channel. In these papers the conductivity was assumed to be a function of only the temperature, such that for T T* (T) = 0. In [3, 4] the flows of conducting gas in the channels were calculated both with the given dependence of the gas conductivity on the temperature and on the basis of a three-component model by means of the Saha equation. At the same time, the development of periodic regimes in the flow in the nozzle was observed in both cases, but the mechanism of the origin of the current layers was not explained. The self-similar problem of the withdrawal of a nonconducting piston from a half-space occupied by a conducting gas with a magnetic field was investigated in [5] in a linear formulation. At the same time, regions of the problem parameters (the velocity of the piston and the magnetic field on it) were found when, in spite of the self-similar formulation of the problem, there is no self-similar solution. At the same time, regions exist where several solutions are possible. The possibility of the formation of isothermal rarefaction zones with low electrical conductivity when the Joule heating is balanced by the cooling of the gas on expansion (Butler waves) [6] was not taken into account in this paper, since they are unstable with respect to superheating. However, in the case of flow in a nozzle it was shown [2] that precisely the development of instabilities in these zones leads to the formation of the periodic regime. In the present paper the solution of the self-similar problem is constructed in a nonlinear formulation. The reason for the occurrence of regions in which the solution is multiply valued, which is associated with the process of arrival at self-similar boundary conditions, is explained. It is shown that a quasiperiodic regime can arise in the solution, occurring, in particular, in the regions of the problem parameters where there is no self-similar solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 115–122, July–August, 1986.     

Effect of the form of the electrodes on the current distribution in a magnetogasdynamic channel

A solution is given to the plane problem of the flow of a conducting gas across a homogeneous magnetic field in a magnetogasdynamic channel taking account of the Hall effect at small magnetic Reynolds numbers. The channel is formed by two long electrodes, and the cross section of the channel varies slightly and periodically along the gas flow. It is assumed that the electromagnetic forces are small. It is shown that the current distribution in the channel is nonuniform to a consider able degree and that inverse currents can form at the electrodes, with both subsonic and supersonic flows of the conducting gas. Transverse motion of the gas, due to a change in the cross section of the channel, leads to an increase of Joule energy losses. In [1] the current distribution was obtained in a flat channel formed by infinite dielectric walls, with the flow of a steady-state stream of plasma through the channel across a homogeneous magnetic field. With interaction between the flow and the magnetic field, closed current loops develop in the channel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 26–33, November–December, 1970.     

Asymptotic analysis of transonic flows

In this paper we derive the equations of the second and third approximations for the stream function of two-dimensional and axisymmetric potential transonic flow of an inviscid gas and find their particular solutions corresponding to certain transonic flows.A similar study concerning the second approximation of subsonic and supersonic flow was made by Van Dyke [1] and Hayes [2]. The second approximation for the velocity potential of transonic flow has been examined in detail by Hayes [3]. Euvrard [4, 5] has investigated the asymptotic behavior of transonic flow far from a body, while Fal'kovich, Chernov, and Gorskii [6] have studied the flow in a nozzle throat.The transonic asymptotic analysis for the stream function is presented in this paper.     

Nozzle startup in an oncoming flow

Three variants of the startup of an axisymmetric convergent-divergent nozzle are considered with the static pressures at the entry and exit of the nozzle being the same at the beginning of the process. The subsonic startup corresponds to open nozzle acceleration in air. The supersonic startup simulates the sudden opening of a cover at the nozzle inlet under supersonic flight conditions. A successful nozzle startup with the formation of steady supersonic flow along the whole channel is realized in the third variant of supersonic startup with gas injection through a small region of the wall of the divergent nozzle section. The investigation is performed numerically, on the basis of the Euler equations for axisymmetric gas flows.     

Laval nozzle flow calculation

The inverse problem of the theory of the Laval nozzle is considered, which leads to the Cauchy problem for the gasdynamic equations; the streamlines and the flow parameters are found from the known velocity distribution on the axis of symmetry.The inverse problem of Laval nozzle theory was considered in 1908 by Meyer [1], who expanded the velocity potential into a series in powers of the Cartesian coordinates and constructed the subsonic and supersonic solutions in the vicinity of the center of the nozzle. Taylor [2] used a similar method to construct a flowfield which is subsonic but has local supersonic zones in the vicinity of the minimal section. Frankl [3] and Fal'kovich [4] studied the flow in the vicinity of the nozzle center in the hodograph plane. Their solution, just as the Meyer solution, made it possible to obtain an idea of the structure of the transonic flow in the vicinity of the center of the nozzle.A large number of studies on transonic flow in the vicinity of the center of the nozzle have been made using the method of small perturbations. The approximate equation for the transonic velocity potential in the physical plane, obtained in [3–6], has been studied in detail for the plane and axisymmetric cases. In [7] Ryzhov used this equation to study the question of the formation of shock waves in the vicinity of the center of the nozzle, and conditions were formulated for the plane and axisymmetric cases under which the flow will not contain shock waves. However, none of the solutions listed above for the inverse problem of Laval nozzle theory makes it possible to calculate the flow in the subsonic and transonic parts of the nozzles with large gradients of the gasdynamic parameters along the normal to the axis of symmetry.Among the studies devoted to the numerical calculation of the flow in the subsonic portion of the Laval nozzle we should note the study of Alikhashkin et al., and the work of Favorskii [9], in which the method of integral relations was used to solve the direct problem for the plane and axisymmetric cases.The present paper provides a numerical solution of the inverse problem of Laval nozzle theory. A stable difference scheme is presented which permits analysis with a high degree of accuracy of the subsonic, transonic, and supersonic flow regions. The result of the calculations is a series of nozzles with rectilinear and curvilinear transition surfaces in which the flow is significantly different from the one-dimensional flow. The flowfield in the subsonic and transonic portions of the nozzles is studied. Several asymptotic solutions are obtained and a comparison is made of these solutions with the numerical solution.The author wishes to thank G. D. Vladimirov for compiling the large number of programs and carrying out the calculations on the M-20 computer.     

Occurrence of periodic regimes in steady supersonic mid flows due to loss of electrical conductivity of medium

A study is made of the features of supersonic magnetohydrodynamic (MHD) flows due to the vanishing of the electrical conductivity of the gas as a result of its cooling. The study is based on the example of the exhausting from an expanding nozzle of gas into which a magnetic field (Re_m 1) perpendicular to the plane of the flow is initially frozen. It is demonstrated analytically on the basis of a qualitative model [1] and by numerical experiment that besides the steady flow there is also a periodic regime in which a layer of heated gas of electric arc type periodically separates from the conducting region in the upper part of the nozzle. A gas-dynamic flow zone with homogeneous magnetic field different from that at the exit from the nozzle forms between this layer and the conducting gas in the initial section. After the layer has left the nozzle, the process is repeated. It is established that the occurrence of such layers is due to the development of overheating instability in the regions with low electrical conductivity, in which the temperature is approximately constant due to the competition of the processes of Joule heating and cooling as a result of expansion. The periodic regimes occur for magnetic fields at the exit from the nozzle both greater and smaller than the initial field when the above-mentioned Isothermal zones exist in the steady flow. The formation of periodic regimes in steady MHD flows in a Laval nozzle when the conductivity of the gas grows from a small quantity at the entrance due to Joule heating has been observed in numerical experiments [2, 3]. It appears that the oscillations which occur here are due to the boundary condition. The occurrence of narrow highly-conductive layers of plasma due to an initial perturbation of the temperature in the nonconducting gas has previously been observed in numerical studies of one-dimensional flows in a pulsed accelerator [4–6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 138–149, July–August, 1985.     

Starting of an axisymmetric convergent-divergent nozzle in a hypersonic flow

The starting of an axisymmetric convergent-divergent nozzle, with the result that supersonic flow is formed within almost the entire channel, is modeled, as applied to the hypersonic aerodynamic setup of the Institute of Mechanics of Moscow State University. A successful starting is realized when the nozzle is thrown in a uniform supersonic air flow at a fairly high Mach number. The steady flow structure is studied. It is numerically shown that in the convergent section of the channel there arises an oblique shock wave whose interaction with the nozzle axis leads to the formation of a reflected shock and a curvilinear Mach disk with a region of unsteady subsonic flow in the vicinity of the throat. The mathematical model is based on the two-dimensional Euler equations for axisymmetric gas flows.     

“Lateral” interaction of a supersonic underexpanded ideal-gas jet with surfaces of different shape

A numerical investigation is made of the interaction of an underexpanded jet of an inviscid and nonheat-conducting gas issuing from an axisymmetric conical nozzle with plane, cylindrical, and spherical surfaces. It is assumed that the flow turning angle for flow about a barrier is smaller than the critical angle, and subsonic regions are absent in the flow field studied. The effect of the characteristic parameters (Mach number at the nozzle exit, jet underexpansion) on the flow pattern and jet forces is analyzed. The results of numerical calculations are compared to the results of approximate theories and experimental data. A theoretical solution of the problem of the effect of a supersonic jet on a surface of given shape, even in the approximation of an inviscid, nonheat-conducting gas, is quite difficult. A reason for this is that the flow region contains shock waves interacting with each other, contact discontinuities, and zones of mixed sub-and supersonic flow. As far as is known to the authors, the results obtained for three-dimensional problems for the interaction of supersonic jets with each other or with barriers are primarily experimental (for example, [1–6]). A numerical analysis of the interaction of axisymmetric ideal-gas jets was carried out in [7–10]. In [7] a three-dimensional form of the method of characteristics was used to calculate the initial interaction region for two supersonic cylindrical jets (with Mach number M=10) intersecting at an angle of 60. The interaction of several jets has been considered in [8, 9], where the solution was obtained according to the Lax—Wendroff method without elimination of the discontinuity lines of flow parameters. In [10] the lateral interaction of axisymmetric supersonic jets with each other and with a plate is investigated by means of a straight-through calculationTranslated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–8, November–December, 1974.The authors thank A. N. Kraiko for useful discussions of the results, and A. L. Isakov and É. N. Gasparyan for kindly providing the experimental data.     

Solution of the direct problem of the flow of a two-phase mixture of a gas and foreign solid or liquid particles in a laval nozzle

Within the framework of a two-liquid (two-velocity and two-temperature) model of a continuous medium, the article considers the flow of a mixture of a gas and foreign particles in the subsonic, transonic, and supersonic parts of a Laval nozzle. In the case of a thin layer of pure gas near the wall, the problem is solved in two stages. First, the method of establishment is used to calculate the core of the flow, where the gas with the particles is flowing; under these circumstances, the parameters in the layer of pure gas are determined approximately; then simplified equations (of the type of the equations of the boundary layer) are used to find the distribution of the parameters in the zone of pure gas, and the flow in the core of the stream is refined. Examples of the calculation are given. Use of the method developed permitted establishing some of the special characteristics of the flow of a mixture of gas with particles in a Laval nozzle in the case of Stokes flow around the foreign particles.     

Development of magnetohydrodynamic boundary layers associated with the sudden initiation or deceleration of a supersonic flow at the boundary of a half-space

The problem of nonstationary magnetohydrodynamic flow of a viscous fluid in a half-space resulting from the motion of an infinite plate has received much attention. In [1], for example, solutions are presented for the case of isotropic conductivity, while in [2] a solution of the Rayleigh problem is offered for the case of anisotropic conductivity. In these instances the fluid was assumed incompressible and uniform, and the system of equations was found to be linear. In problems involving nonstationary flow of a gas in a transverse magnetic field resulting from the deceleration of a high-velocity gas flow at the boundary of a half-space or the motion of an infinite plate at supersonic speed relative to a stationary gas it becomes necessary to take into account the compressibility of the gas and the temperature dependence of the conductivity. It is then possible to have flows in which the gas becomes electrically conducting and begins to interact with the magnetic field solely as a result of the increase in temperature due to viscous dissipation of energy. The magnetic field, interacting with the conducting gas, exerts an effect on the drag and heat transfer to the surface of the plate. At sufficiently low gas pressures and strong magnetic fields a Hall effect may be observed. The system of equations describing the motion of a compressible gas with variable conductivity is essentially nonlinear. The present article is devoted to a study of such motions.     

Profiling the Supersonic Part of a Plug Nozzle with a Nonuniform Transonic Flow

The effect of transonic flow nonuniformity on the profiling of optimal plug nozzles is studied in the inviscid gas approximation. Sonic and supersonic regions providing maximum thrust for given nozzle dimensions and a given outer pressure are designed for given subsonic contours and calculated nonuniform transonic flows. As in the case of uniform flow on a cylindrical sonic surface, the initial regions of the designed contours satisfy the condition that in these regions the flow Mach number is unity or near-unity. In all the examples calculated, the optimal plug nozzles produce a greater thrust than the optimal axisymmetric and annular nozzles with a near-axial flow for the same lengths and the same gas flow rates through the nozzle. It is established that contouring without regard for transonic flow nonuniformity can result in considerable thrust losses. However, these losses are due only to a decrease in the flow rate, while the specific thrust may even increase slightly.     

Calculation of hydrogen-air combustion behind detached shock wave for supersonic flow past a sphere

Many of the published theoretical studies of quasi-one-dimensional flows with combustion have been devoted to combustion in a nozzle, wake, or streamtube behind a normal shock wave [1–6].Recently, considerable interest has developed in the study of two-dimensional problems, specifically, the effective combustion of fuel in a supersonic air stream.In connection with experimental studies of the motion of bodies in combustible gas mixtures using ballistic facilities [7–9], the requirement has arisen for computer calculations of two-dimensional supersonic gas flow past bodies in the presence of combustion.In preceding studies [10–12] the present author has solved the steady-state problem under very simple assumptions concerning the structure of the combustion zone in a detonation wave.In the present paper we obtain a numerical solution of the problem of supersonic hydrogen-air flow past a sphere with account for the nonequilibrium nature of eight chemical reactions. The computations encompass only the subsonic and transonic flow regions.The author thanks G. G. Chernyi for valuable comments during discussion of the article.     

Supersonic Gas Flows in Radial Nozzles

Results of experimental investigations and numerical simulations of supersonic gas flows in radial nozzles with different nozzle widths are presented. It is demonstrated that different types of the flow are formed in the nozzle with a fixed nozzle radius and different nozzle widths: supersonic flows with oblique shock waves inducing boundary layer separation are formed in wide nozzles, and flows with a normal pseudoshock separating the supersonic and subsonic flow domains are formed in narrow nozzles (micronozzles). The pseudoshock structure is studied, and the total pressure loss in the case of the gas flow in a micronozzle is determined.     

Flow of gas mixtures with vibrational energy relaxation in plane and axisymmetric nozzles

A method is described for the calculation of plane and axisymmetric flows of gas mixtures with vibrational energy relaxation in the subsonic, transonic, and supersonic regions of the nozzle. The method is based on numerical solution of the inverse problem of nozzle theory. Results are given for the flow of a C0₂-N₂-H₂O-He mixture with vibrational relaxation and compared with the results of one-dimensional calculations. It is found that vibrational-energy relaxation has a significant effect on the gasdynamic parameters of flow in nozzles with large, relative expansion and therefore in choosing a nozzle shape, especially in the supersonic region, it is necessary to calculate the nonequilibrium flow. It is shown that the geometry of the transonic and supersonic regions of the nozzle has a considerable effect on the distribution of the inverse population of the level and the amplification factor.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 125–131, September–October, 1977.     

Study of spatial mixed flows in nozzles with asymmetrical inlets

The results of some investigations into the behavior of the spatial mixed flows of a nonviscous and nonheat-conducting gas in nozzles deviating from axisymmetrical shape in the subsonic region are presented. The investigation is based on the numerical integration of the transient gasdynamic equations using the Godunov scheme [1, 2] generalized to the spatial case.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 167–169, March–April, 1975.The author wishes to thank M. Ya. Ivanov, A. N., Kraiko, and G. G. Chernyi for help and discussion of the results.     

Calculation of supersonic gas flow over a ducted body in regimes with a detached shock wave

Calculations were made of the supersonic flow of an inviscid gas which does not conduct heat over two-dimensional and axisymmetric ducted bodies in regimes with a detached shock wave and completely subsonic gas velocity in the cylindrical duct. The investigated bodies have a pointed leading edge. The flow rate of the gas through the duct is assumed to be given. This corresponds to the presence in the exit section of the duct of a throttle or an impermeable barrier (in which case the flow rate is zero). The numerical algorithm used in the calculations is based on stabilization in time and Godunov's difference scheme [1] with separation of the shock wave. The integrated flow characteristics are given. The values of the wave resistance coefficient obtained in the calculations are compared with the values found using Taganov's approximate approach.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 160–163, May–June, 1981.I thank A. N. Kraiko for regular consultations, Yu. B. Lifshits for a helpful comment, and V. A. Vostretsov for assisting in the work.     

Slender axisymmetric cavities in a supersonic flow

Slender axisymmetric cavities in a subsonic flow of compressible fluid were investigated in [1–4]. In [5] a finite-difference method was used to calculate the drag coefficient of a circular cone, near which the shape of the cavity was determined for subsonic, transonic, and supersonic water flows; however, in the supersonic case the entire shape of the cavity was not determined. Here, on the basis of slender body theory an integrodifferential equation is obtained for the profile of the cavity in a supersonic flow. The dependence of the cavity elongation on the cavitation number and the Mach number is determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 179–181, January–February, 1989.