Calculation of the deceleration of a viscous supersonic gas flow in a channel

The deceleration of nonuniform viscous supersonic gas flows in planar and axisymmetric channels is investigated. A modification of Prandtl's formula for the turbulent viscosity is proposed in order to take into account the dependence of the mixing length on the value of the axial Mach number. The results of the calculations are compared with known experimental data on the deceleration of a supersonic flow in a subsonic pseudoshock.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 162–166, March–April, 1982.We thank A. N. Sekundov for discussing the work.     

Calculation of a supersonic gas jet exhausting into vacuum from a nozzle with an inclined exit

A study is made of the problem of supersonic exhausting of ideal gas into vacuum from a conical nozzle with inclined exit. The solution is sought in a region including part of the flow where the projection of the velocity vector of the gas onto the nozzle axis can be less than the local velocity of sound and take negative values.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 185–186, May–June, 1982.     

Calculation of the swirling flow of an ideal gas in a laval nozzle

The numerical solution of the problem of the motion of a swirling flow of an ideal gas in a Laval nozzle in axisymmetric formulation is obtained by the method of stabilization. As a result, a number of effects appear that are essentially not one-dimensional, in particular, the drawing-in of the sonic line into the nozzle, an effect that leads to a decrease in the nozzle's expansion coefficient. The dependence of this coefficient on the intensity of the swirling is obtained. A number of problems connected with the control of the expansion of a gas through a Laval nozzle and with variation of the thrust of a nozzle can be solved successfully in cases where a rotary motion is imparted to the flow of gas exhausted from the nozzle. Investigation of such a swirling flow in [1, 2] and a number of other papers are based on a one-dimensional model of gas flow, which makes it possible in principle to obtain integrated characteristics of the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 72–76, September–October, 1971.     

Calculation of the chemically nonequilibrium flow of a multicomponent gas through a nozzle

We shall describe a method for calculating the flow of a complex gas mixture not in chemical equilibrium, based on selection of the time scale for the problem solution and sectioning of the reaction velocities in the near-equilibrium mode of their flow. The method permits derivation of calculation programs possessing wide applicability for systems with a large number of reactions. The application of the method is illustrated through calculations on a system, the components of which contain H,O,C,N, and Cl elements. Calculation of the parameters of the flow of a multicomponent gas through a nozzle under given conditions requires consideration of the kinetics of the chemical reactions which occur in the system. Mathematically, such a problem leads to the simultaneous numerical solution of the gas-dynamics equations for the flow parameters and the chemical kinetic equations for the concentration of the individual components. Several difficulties exist, the most basic of which are calculation of the region of near equilibrium flow and the transonic flow region with transition across the sonic point, and calculation of a large number of chemical reaction velocities of greatly differing magnitudes, in the case of a complex gas mixture. In order to obtain a stable solution in the near-equilibrium flow region, several methods have recently been proposed, which permit consolidation of the integration step. We note the use of a local linearization of the chemical kinetic equation system, as employed in [1]. This method in practice is useful for relatively slow change in component concentrations. In [2] at each integration step the kinetic equations are transformed into a system of L algebraic equations (where L is the number of reactions), and with an increase in the number of reactions (L20) the laboriousness of such a calculation increases sharply. The implicit differential schemes of integration presented in [3, 4] appear more acceptable, but in fact they too have been tested only for systems with a relatively small number of reactions. The difficulty of calculating the transonic flow region, as is well known, is connected with selection of the unique value of mass flow G, at which the transition to supersonic flow is realized. This may be avoided by defining over the length of the nozzle one of the gas-dynamic functions (for example, pressure distribution [4]), which are not highly sensitive to chemical nonequilibrium, the values being taken from supplementary calculations of nonequilibrium flow through the nozzle. Several investigators have limited their examination to the supersonic flow region (see, for example, [5]), but with this method the results may lack sufficient accuracy, since in some cases (for high gradients of the gas-dynamic magnitudes) the transonic region produces a comparable contribution to the general effect of nonequilibrium. We will describe below a method with which a practically universal system of calculating the nonequilibrium flow of a complex gas mixture through a nozzle can be realized. In practice, up to 60 of the most significant reactions may be considered, out of a practically unlimited number initially present. The method is based on sectioning of the reaction velocities in the near-equilibrium mode. This permits attainment of a stable solution in the near-equilibrium flow region with acceptable machine-time expenditures. The method describes the transition through the sonic point, since the systematic error introduced by its use (within the limits of calculation accuracy) improves the convergence of the iteration process used in finding G. In applying the method it is useful to select the more important of those reactions theoretically possible, and also to conduct calculations for equilibrium flow conditions of the individual chemical reactions; the latter permits evaluation of the maximum possible contribution of reactions for which the velocity constants are unknown.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 159–163, September–October, 1971.The authors are grateful to A. I. Vol'pert and L. N. Stesik for their interest in and evaluation of the study.     

Heat transfer of gas-particle flow in a supersonic convergent-divergent nozzle

Summary Heat flux, wall heat transfer coefficients, and wall pressures are determined for high velocity flow of gas-solid mixtures in a converging-diverging nozzle. Flow separation accompanied with oblique shock formation occurs in the diverging section of the nozzle. The shock strength is reduced upon the addition of solid particles. The wall pressure in the convergent section of the nozzle appears unaffected by the presence of solid particles. In the divergent section, however, the wall pressure is slightly lowered. At the maximum ratio of solid to air flow used in the experiments (3.7) increases in the heat transfer rate of up to 20 and 50 percent are obtained in the convergent and separated (divergent) regions of the nozzle, respectively. Slightly larger increases in the wall heat transfer coefficients are also obtained. It is concluded that the wall heat flux and heat transfer coefficients are influenced strongly by the presence of disturbances upstream of the nozzle inlet.Nomenclature W _a air flow rate - W _s solids flow rate - x axial distance from nozzle entrance - L axial length of nozzle - specific heat ratio of fluid - A _e exit cross section of flow - A _* throat cross section of flow - P ₀ inlet pressure - P _s wall separation pressure - P _a ambient exhaust pressure - shock wave angle - shock wave deflection angle - M ₁ Mach number upstream of shock wave - Mach number normal to shock wave - q heat flux - k _f thermal conductivity of fluid - T _wi inside wall temperature - T _wo outside wall temperature - T _ad adiabatic wall temperature - h wall heat transfer coefficient - C nozzle constant - A local cross section of flow - c _p specific heat of fluid - Pr Prandtl number - viscosity of fluid - r _c throat radius of curvature - factor accounting for variation of and Units absolute temperature °R(ankine) °F+459.7 - conductivity 1 BTU (hr ft °F)^–1 4.137×10^–3 cal (s cm °C)^–1 - specific heat 1 BTU (1b °F)^–1 1 cal (g °C)^–1 - absolute pressure 1 psia 0.0680 atm Supported in part by aid provided by the UCLA Space Science Center (Grant NsG 236-62 Libby).Listed for readers not familiar with the units adopted in this paper (editor).     

Calculation of gas flow in a laval nozzle in the presence of nonequilibrium physicochemical processes

An explicit divergence difference scheme of third order of approximation with respect to the spatial variables is used to calculate two-dimensional steady flows of an inviscid gas that does not conduct heat in contracting-expanding nozzles in the presence of nonequilibrium physicochemical processes. The flow structure is demonstrated for a three-component vibrationally nonequilibrium gas mixture in planar Laval nozzles with different radii of curvature of the nozzle profile in the neighborhood of the critical section.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 173–177, July–August, 1982.We thank A. N. Kraiko and Yu. V. Kurochkin for helpful advice and interest in the work.     

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.     

Fluctuations in cavities in a supersonic gas flow

Numerical calculations are carried out within the framework of the ideal compressible gas model for the unsteady flows in rectangular cavities exposed to a supersonic external stream. The Euler equations are integrated by means of Godunov's finite-difference method [6]. On the basis of an analysis of the calculation results an expression is proposed for determining the possible frequencies of the flow rate oscillations in the cavity as a function of the free-stream Mach number M and the geometry of the cavity. The results obtained are compared with the experimental data and calculations of other authors.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 121–127, March–April, 1990.     

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.     

Flutter of a viscoelastic plate in a supersonic gas flow

The flutter of a viscoelastic plate in a supersonic gas flow is studied. A technique and algorithm for numerical solution of nonlinear integro-differential equations with weakly singular kernels are elaborated. The critical flutter speed of viscoelastic plates is determined