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.     

Acoustic,entropy, and vorticity perturbations in a channel of variable cross section

A study is made of the problem of the propagation of infinitesimally small perturbations in a gas stream moving in a channel of variable cross section when the flow cannot be regarded as isentropic and irrotational. The solution is found in the framework of the linear theory of the flow of an ideal gas and the quasi-one-dimensional hydraulic approximation for the steady regime. For irrotational and isentropic perturbations in a nozzle, this problem was considered in [1–4]. In [1], the problem is generalized to take into account entropy perturbations in the nozzle for the case of longitudinal oscillations. The present paper treats arbitrary modes in a nozzle and takes into account not only entropy but also vorticity perturbations in the moving stream. For each of the three perturbation types — acoustic, entropy, and vorticity — the solutions are expanded in series in cylindrical functions. It is shown that in the considered approximation each oscillation mode can be analyzed independently of the others. In the special case of flow in a Laval nozzle, the concept of impedance (admittance), which is widely used in acoustics, is generalized to take into account entropy and vorticity perturbations. The contribution to the flow dynamics of the acoustic, entropy, and vorticity perturbations is estimated numerically for longitudinal and transverse modes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 91–98, January–February, 1982.     

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.     

The nonlinear evolution of two-dimensional and three-dimensional waves in mixing layers

The authors consider problems connected with stability [1–3] and the nonlinear development of perturbations in a plane mixing layer [4–7]. Attention is principally given to the problem of the nonlinear interaction of two-dimensional and three-dimensional perturbations [6, 7], and also to developing the corresponding method of numerical analysis based on the application to problems in the theory of hydrodynamic stability of the Bubnov—Galerkin method [8–14].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhldkosti i Gaza, No. 1, pp. 10–18, January–February, 1985.     

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.     

Asymmetric flow of subsonic and sonic jets over an infinite wedge

A study is made of the flow of subsonic or sonic jets over an infinite wedge when the stagnation streamline bifurcates at the tip of the wedge. This regime can be realized only for a definite (previously unknown) relationship between the geometrical parameters. The problem is solved in the hodograph plane by the numerical method of [1] developed for the problem of a profiled Laval nozzle. A solution to the asymmetric problem obtained in the hodograph plane can be realized physically only for a definite relationship between the boundary values for the flow function. This relationship (which generalizes Prandtl's well-known formula [2] derived for asymmetric flow of incompressible jets over a plate on the basis of the momentum theorem) is obtained by analyzing the asymptotic behavior of the solution near the stagnation point. Examples of calculations are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 137–141, March–April, 1980.     

Control of a submerged jet in a thin rectangular cavity

An innovative method is presented for control of an oscillatory turbulent jet in a thin rectangular cavity with a thickness to width ratio of 0.16. Jet flow control is achieved by mass injection of a secondary jet into the region above the submerged primary jet nozzle exit and perpendicular to the primary nozzle axis. An experimental model, a 2-D and a 3-D computational fluid dynamics (CFD) model are used to investigate the flow characteristics under various secondary injection mass flow rates and injection positions. Two-dimensional laser Doppler anemometry (LDA) measurements are compared with results from the CFD models, which incorporate a standard k–ε turbulence model or a 2-D and 3-D realisable k–ε model. Experimental results show deflection angles up to 23.3° for 24.6% of relative secondary mass flow are possible. The key to high jet control sensitivity is found to be lateral jet momentum with the optimum injection position at 12% of cavity width (31.6% of the primary nozzle length) above the primary nozzle exit. CFD results also show that a standard k–ε turbulence closure with nonequilibrium wall functions provides the best predictions of the flow.     

Three-dimensional instability of an elliptic Kirchhoff vortex

The instability of a Kirchhoff vortex [1–3] with respect to three-dimensional perturbations is considered in the linear approximation. The method of successive approximations is applied in the form described in [4–6]. The eccentricity of the core is used as a small parameter. The analysis is restricted to the calculation of the first two approximations. It is shown that exponentially increasing perturbations of the same type as previously predicted and observed in rotating flows in vessels of elliptic cross section [4–9] appear even in the first approximation. As distinct from the case of plane perturbations [1-3], where there is a critical value of the core eccentricity separating the stable and unstable flow regimes, instability is predicted for arbitrarily small eccentricity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 40–45, May–June, 1988.     

Effect of the acoustic regularization of initial perturbations on the development of ordered structures

Mechanisms are considered by which acoustic oscillations influence the structure of subsonic shear flows. Analysis of the experimental data [1–7] confirms the assumption made in [6] that the regularization of initial perturbations, which causes a higher degree of ordering and an increase in the life of vortices formed because of the development of instability waves or interaction of acoustic oscillations with the edge of the nozzle, is one of the mechanisms by which acoustics influences various shear flows. Photographs are given which show the regularizing effect of acoustics on the development of vortices in the wake behind the edge.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 171–174, January–February, 1986.     

Experimental investigation of laval nozzles with gas blown asymmetrically

An experimental determination was made of the transverse forces resulting from asymmetric blowing of a transverse gas jet into the supersonic part of a Laval nozzle. The experimental data are generalized on the basis of the analogy between blowing and flow over an equivalent body and using the generalized theory of one-dimensional flows. An approximating dependence is obtained for determining the gain as a result of blowing.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 188–193, March–April, 1981.     

Influence of acoustic reflectors on the discrete component in the noise spectrum of a supersonic off-design jet

An experimental investigation of the discrete component in the noise spectrum of an axisymmetrie off-design jet emerging from a sonic or supersonic nozzle into a submerged space is conducted. The influence of the diameter of the reflector, placed at the nozzle edge, on the level, frequency, and phase diagram of the discrete component is examined. The discrete component in the jet noise spectrum has been studied in [1–7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 157–160, November–December, 1977.     

Supersonic ideal-gas flow of in the throat region of a nozzle

The problem of profiling a supersonic nozzle with uniform outlet flow is considered when the contour is constructed from a point on a given convergent section of the nozzle. It is shown that there are contours such that the flow in the throat is supersonic in choked regimes (as distinct from common notions of mixed or uniform sonic flow). The influence of flow nonuniformity in the throat region on the thrust of a supersonic nozzle is analyzed.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 132–139, May–June, 1996.     

Heat transfer characteristics of a planar water jet impinging normally or obliquely on a flat surface at relatively low Reynolds numbers

The heat transfer characteristics of a planar free water jet normally or obliquely impinging onto a flat substrate were investigated experimentally. The planar jet issued from a rectangular slot nozzle with a cross section of 1.62 mm × 40 mm. The mean velocity at the nozzle exit ranged from 1.5 to 6.1 m s⁻¹. The corresponding Reynolds number range based on the nozzle gap and the mean velocity was 2200–8800. Constant heat-flux conditions were employed at the solid surface. Various impingement angles between the vertical planar jet and the inclined solid surface were investigated: 90° (normal collision), 70°, 60°, and 50°. In the case of normal collisions, the Nusselt number is high at the impingement line, and decreases with departures from it. The stagnation Nusselt numbers were compared to the predictions of several correlations proposed by other researchers. In oblique collisions, the profiles of the local Nusselt numbers are asymmetric. The locations of the peak Nusselt numbers do not coincide with the geometric center of the planar jet on the surface.     

Formation of active medium in a supersonic recombination-nonequilibrium plasma flow

A method of calculating a plasmodynamic laser from the given characteristics of the working fluid and the supersonic nozzle and from the stagnation parameters of the plasma in the nozzle forechamber is proposed. This method gives an analytic expression for the dependence of the inversion coefficient and the laser power on the above-mentioned parameters and serves mainly for the scientific planning of physical and detailed numerical experiments.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 109–114, January–February, 1988.     

Influence of the initial conditions on axisymmetric jets in a parallel flow

An experimental investigation was made of the initial-section flow of axisymmetric helium, air, and freon-12 jets in a parallel air flow for two different velocity profiles at the nozzle exit near the boundary of the jet. In one case, the velocity profile was determined by boundary layers on the nozzle walls; in the other case, it was produced artificially by means of a honeycomb of tubes of variable length. Measurements were made of the profiles of the mean and the pulsation velocity and the temperature. The flow was also photographed. The investigations showed that, depending on the initial conditions, the intensity of mixing of the jets in the initial section at Reynolds numbers Re 10⁴ (calculated using the jet diameter) can change from the level determined by molecular diffusion to the level characteristic of developed turbulent flow. The flow structure in the annular mixing layer also depends strongly on the initial conditions. The observed ordered structures in the mixing layer are related to a section of development of perturbations near the nozzle. The ordered structures are strongly influenced by the effect on the jet of acoustic vibrations from an external source. When the initial velocity profile is produced by the honeycomb, the transition to developed turbulence may be due to the development of long-wavelength perturbations or to the development of small-scale turbulence generated by the flow over the end of the honeycomb.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 18–24, July–August, 1980.We thank V. M. levlev and K. I. Artamonov for assistance and for discussing the work.     

Determination of optimum nozzle parameters in a gasdynamic laser

In connection with the use of supersonic nozzles to create lasers, the question arises of the optimum parameters of the nozzle and the gas mixture from the aspect of obtaining the greatest population inversion of the energy levels of internal degrees of freedom of molecules of the working gas and the greatest output power of the lasers. A rather complete concept of the kinetic processes taking place during the escape of a relaxing gas mixture containing carbon dioxide through a supersonic nozzle has now been developed on the basis of calculated and experimental data. In [1–4] the problems of optimization of the parameters of a CO₂-N₂-H₂O-He mixture and of the shape of the nozzle were set up and solved in a one-dimensional steady-state formulation. The influence of the two-dimensionality of the stream in an optimum nozzle on the laser characteristics is studied in the present report. The method of through calculation suggested in [5] is used to calculate the two-dimensional flow of a relaxing gas.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 23–26, September–October, 1978.     

Study of vibrational deactivation of molecules of carbon dioxide gas during cooling of stream in a supersonic nozzle

The vibrational temperature of the antisymmetrical type of vibrations (v ₃) of the CO₂ molecule at the exit of a supersonic nozzle is measured in the present work using the method of recording the infrared emission. Freezing in of thev ₃-type vibrations was observed during the flow of undiluted carbon dioxide in a nozzle. In this case the vibrational temperature T₃ considerably exceeded the translational temperature. On the basis of a comparison of the experimental results with calculation it can be concluded that vibrational deactivation of CO₂ molecules occurs three to five times faster than the excitation of the vibrations during heating in a shock wave. All the experiments were conducted under the following conditions: maximum expansion of gas in nozzle A/A* = 115, temperature range 1900–2400 °K, pressure range 1–17.5 atm.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 32–40, November–December, 1973.The authors are grateful to U. G. Pirumov and É. A. Ashratov for the calculation of the nozzle profile and the distribution of streamlines as well as for a discussion of the results.     

Inverted population of Co2 molecules in expanding gas streams

The system of equations of hydrodynamics, which describes the process of escape of the mixtures CO₂ + N₂ + He, H₂O from a nozzle, is solved numerically in conjunction with the equations of the kinetics of the excitation of the vibrational degrees of freedom of the molecules. It is found that an inverted population of the CO₂ molecules with respect to the transition [00 °1] – [10 °0], is produced under certain conditions at the exit from the nozzle. The magnitude of the inversion depends both on the nozzle configuration and on the initial values of the gas temperature and pressure. It is shown that for a specified nozzle configuration there exist optimal values of these parameters, at which the inverted population of the CO₂ molecules reaches approximately 10¹⁵ cm^–3.Translated from Zhumal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 24–34, September–October, 1971.     

Supersonic gas flow over a semi-infinite plate with small harmonic spanwise disturbances

The method of matched asymptotic expansions is used to investigate the problem of supersonic perfect-gas flow over a semi-infinite surface with longitudinal ribbing formed by imposing small transverse harmonic perturbations on a flat plate. The ratio of the maximum amplitude of the surface perturbations to the thickness of the boundary layer is of the order of Re^–1/4. The problem is solved with allowance for four terms of the expansion.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 146–156, January–February, 1993.     

Theory of vortical helical ideal gas flows in laval nozzles

An asymptotic solution is found for the direct problem of the motion of an arbitrarily vortical helical ideal gas flow in a nozzle. The solution is constructed in the form of double series in powers of parameters characterizing the curvature of the nozzle wall at the critical section and the intensity of stream vorticity. The solution obtained is compared with available theoretical results of other authors. In particular, it is shown that it permits extension of the known Hall result for the untwisted flow in the transonic domain [1]. The behavior of the sonic line as a function of the vorticity distribution and the radius of curvature of the nozzle wall is analyzed. Spiral flows in nozzles have been investigated by analytic methods in [2–5] in a one-dimensional formulation and under the assumption of weak vorticity. Such flows have been studied by numerical methods in a quasi-one-dimensional approximation in [6, 7]. An analogous problem has recently been solved in an exact formulation by the relaxation method [8, 9]. A number of important nonuniform effects for practice have consequently been clarified and the boundedness of the analytical approach used in [2–7] is shown.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 126–137, March–April, 1978.The authors are grateful to A. N. Kraiko for discussing the research and for valuable remarks.