Theory of slightly perturbed three-dimensional flows in nozzles

The theory of slightly perturbed flows in conical nozzles is used to determine the transverse force and moment generated in the presence of asymmetric perturbations. A system of ordinary differential equations is derived for finding the transverse force and moment. An approximate analytical solution of this system is constructed and its qualitative features are studied. A comparison is made with a numerical solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 146–154, January–February, 1977.     

Construction of asymmetric nozzles of maximum moment given additional conditions on the geometrical and force characteristics

A solution is given to the variation problem of constructing asymmetric plane nossles which realize the maximum moment relative to some point. The contours of the nozzle are assumed to be noninteracting. The method of the undetermined control contour is used [1]. The solution of this problem contains as a special case the solution to the problem of constructing a nozzle of maximum thrust, including also the case of a given lifting force [1–3]. It is shown that the construction of a nozzle of maximum moment under additional conditions on the thrust and the lifting force, or on the moment relative to another point, reduces to the construction of a nozzle of maximun moment relative to some auxiliary point.Translated from Izvestiya Akademi Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 148–152, September–October, 1984.The author thanks A. N. Kraiko for useful discussions and for his appraisal of the study.     

Investigation of supersonic flows in conical nozzles

The article gives the results of an investigation of flows in supersonic axisymmetric conical nozzles with the presence of shock waves in the flow. The method of straight-through calculation [1] and the method of small perturbations [2] are used. An investigation is made of the effect of various geometric parameters and of the adiabatic index on the flow of a gas in conical nozzles. A comparison is made with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 101–107, May–June, 1974.In conclusion, the authors thank N. V. Drozdov for his participation in carrying out the calculations.     

A variational problem in one-dimensional unsteady gas dynamics

The problem of finding the optimal motion of a piston that confines a certain volume of gas at the initial time is solved. The motion with which the piston performs the maximum work subject to constraints on its motion and the duration of this motion is found. Cases of plane, cylindrical, and spherical symmetry are considered. Numerical examples are given. It is noted that there is some analogy between the solution obtained and the known solutions for two-dimensional supersonic optimal nozzles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 171–175, July–August, 1984.I thank A. N. Kraiko for helpful discussions of the work.     

Analysis of some supersonic nonsymmetric nozzles

Supersonic nonsymmetric plane nozzles, which are characterized mainly by centered compression and rarefaction waves, were constructed earlier in [1]. An intensive compression wave is undesirable in a number of cases, because of the possibility of boundary-layer detachment on the nozzle walls, for instance. Hence, some constraints on the pressure gradient, for example, the condition that the pressure does not grow at the walls, must be included in the formulation of the problem. The presence of this last condition distinguishes the results in [1] only by the fact that the centered compression waves in the nozzles are replaced by constant parameter sections. Considered below under identical conditions are nonsymmetric nozzles with compression waves, with constant parameter sections, and, also, known nozzles with a straight lower wall for comparison.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 154–156, May–June, 1976.The author is grateful to A. N. Kraiko for his interest and attention to the research and for useful discussions.     

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.     

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.     

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.     

Atmospheric reentry: Flow regime and optical radiation

The possibility of determining the moment of laminar-turbulent transition under real conditions of hypersonic flight in the earth's atmosphere on the basis of an analysis of the body radiation as a function of flight altitude is discussed. The simple method of detecting the moment of laminar-turbulent transition using the difference between turbulent and laminar heat flow is justified.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 202–204, September–October, 1993.The authors are grateful to A. N. Krasnokutskaya for carrying out the computer calculations and to Yu. A. Epishkin, V. M. Shmanenkov, and G. E. Starchenko for the discussion of the transition problem which prompted them to write this paper.     

Influence of nozzle profile on the characteristics of a gas-dynamic laser

The results are given of numerical profiling and analysis of the influence of nozzle shape and the gas-dynamic parameters on the characteristics of gas-dynamic lasers. Investigation of the two-dimensional nonequilbrium flow in a family of similar nozzles and nozzles with different angles of inclination of the contracting part show that it is expedient to choose a shape of the subsonic part that ensures a straight sonic line. Relationships between the geometrical parameters of the subsonic and transonic part of the nozzle are recommended which ensure separationless flow and a shape of the sonic surface that is nearly flat. A parametric investigation was made of the supersonic section of two classes of planar gas-dynamic laser nozzles constructed on the basis of uniform and symmetric characteristics at the exit. The parametric investigations of the influence of the degree of expansion, the total pressure and the temperature, and also the gas composition show that the smallest losses of useful vibrational energy in the cavity are achieved for nozzles constructed on the basis of uniform characteristics.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 163–167, November–December, 1982.     

Mixed boundary-value problems of profiling of supersonic nozzles and channels

In the framework of the inverse method of numerical profiling of the supersonic part of nozzles and channels, two new mixed boundary-value problems of gas dynamics are formulated and solved with boundary conditions specified both along and across the flow. A modification of the grid-characteristic method with respect to layers formed by streamlines is constructed. The solution of the posed mixed problems makes it possible to profile a class of flat and axisymmetric nozzles and channels larger than the known class. Transition channels, which transform one expanding flow into another for which the flow at the exit is close to that from a source, are constructed Channel profiles are obtained with a bend in the generator that suppress shock waves on the wall. The results are presented of profiling of exit channels with contact discontinuity, and also channels that realize discontinuities in the exit sections, these being made isentropic by means of compression characteristics focused at the points of the discontinuity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 112–118, July–August, 1983.We thank U. G. Pirumov, M. Ya. Ivanov, and A. N. Kraiko for helpful discussions.     

Computation of the base pressure in ejector nozzles of different length with zero coefficient of ejection

A simple method is proposed for the analysis of the base pressure in ejector nozzles with arbitrary cowling length by using an empirical criterion of attachment in a broad range of variation of the Mach number of the main nozzle.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 164–166, November–December, 1976.     

Limits of existence of shock-free extremal axisymmetric nozzles

A series of curves bounding the region of existence of shock-free extremal exhaust nozzles is constructed for various values of the specific heat ratio. Outside this region the extremal nozzles contain focused suspended shocks.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 175–177, March–April, 1991.     

Continua with self-rotation nuclei: evolution of asymmetric fields

We present a consistent theory of continua with defect distribution including the density of rotation nuclei. The elastic and self-fields of stresses and strains become asymmetric; the tensor of incompatibility also becomes asymmetric. We derive the dislocation–stress relations and the equations of motion related to the momentum and moment of momentum. Some applications important for earthquake engineering are presented.     

Calculation of the flow in ejector nozzles

A method is proposed for calculating the two-dimensional nonviscous flows in ejector nozzles of arbitrary shape, for two operating cycles: the subsonic flow cycle of a secondary stream and a cycle when the secondary stream attains critical velocity, i.e., it is cut off. In the second case, the possibility is allowed for the appearance of a direct compression shock in the supersonic part of the secondary stream.Translated from Mekhanika Zhidkosti i Gaza, No. 1, pp. 111–118, January–February, 1974.The authors thank A. N. Kraiko and M. Ya. Ivanov for useful discussions and assistance, V. V. Polyakov for interest in the project and L. P. Frolova for assistance in drawing up the task.     

Specific impulse losses due to friction and dispersion in a gas-film cooled liquid rocket engine nozzle

Conclusions The proposed method and program for calculating the compressible turbulent boundary layer in rocket engine nozzles with gas film cooling make it possible to determine the specific impulse losses due to friction, the heat fluxes and other characteristics of the flow. The calculations are based on the numerical solution of the equations of gas dynamics in the boundary layer approximation using a three-parameter differential turbulence model.The calculations for nozzles without film cooling showed that the contours occupying a narrow interval between the families of contours with uniform and variational characteristics have the minimum impulse losses due to friction and dispersion. In contrast to the known results, the loss minimum is displaced relative to nozzles with a variational characteristic (Rao nozzles) towards truncated nozzles with a uniform characteristic.The dependence of the maximum heat transfer to the wall in the critical throat section of the nozzle on the rate of flow of fuel into the film has been determined for nozzles with film cooling. It is shown that as the film flow rate increases, the friction losses decrease, and the minimum of the impulse losses due to friction and dispersion is shifted towards the contours with a variational characteristic, which have the minimum dispersion losses. The total impulse losses, which take into account the change in the fuel component ratio in the flow core due to the diversion of part of the fuel into the film, increase with increase in the film flow rate.The results of our numerical investigation of the effect of the contour shape and film flow rate indicate that the contour with a variational characteristic, which has near-minimum specific impulse losses due to friction and dispersion, should be used as the optimum contour for LRE nozzles.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 82–93, May–June, 1993.deceased.The authors wish to thank their colleagues at the Énergomash NPO L. P. Vereshchak and L. K. Danilyuk for assisting with the calculations, the participants in G. A. Lyubimov's seminar for discussing the results obtained, and D. A. Mel'nikov, U. G. Pirumov, and A. A. Sergienko for valuable advice.     

A boundary model for the thermal creep problem

A new integral boundary condition model for determining the particle velocity distribution function in a Knudsen layer, which makes it possible to improve the accuracy of the moment approximations, is proposed. The results for various models are compared with the exact numerical solution of the problem.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 160–163, November–December, 1993.This research was made possible by financial support from the Exploratory Research Program, US EPA and by a Fulbright grant for Professor I. N. Ivchenko.     

Design of universal supersonic nozzles with conical flow

The problem of designing the supersonic part of nozzles creating source type flows self-similar with respect to the isentropic exponent is solved. A function that approximates the geometry of nozzle channels with cone angles up to 15° is derived. The results of solving the direct nozzle problem, which confirm the accuracy of the approximation obtained, are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 183–186, July–August, 1989.The authors are grateful to A. N. Ganzhelo for supplying the supersonic flow calculation programs.     

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.     

Theoretical investigation of the combined flow of a two-dimensional,high-pressure stream and a one-dimensional,low-pressure stream of ideal gas in ejector nozzles

On the basis of [1] an improved method was developed which, within the framework of the model of an ideal gas, allows one to calculate the flow in ejector nozzles without a limit on the coefficient of ejection. During the development of the method it was established, on the basis of a preliminary analysis, that the difference equations which approximate the differential equations of the flow of coaxial streams in an ejector nozzle (high-pressure and low-pressure streams, treated in two-dimensional and one-dimensional approximations, respectively) have a singular point. Owing to the finiteness of the integration step the position of this singular point differs in the general case from the position of the singular point for the differential equations describing the flow under investigation. This difference is larger the smaller the coefficient of ejection. Now allowing for this fact in the existing methods of calculation in an analogous formulation [1–4] limits the possibilities of all these methods, as a rule, to cases of relatively large coefficients of ejection.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 110–116, November–December, 1978.The authors thank A. N. Kraiko for useful discussions and attention to the work.