Flow of a supersonic ideal gas over a planar cascade in the case of a subsonic normal component in regimes with attached shocks

The supersonic flow of an inviscid gas that does not conduct heat over a cascade of planar pointed profiles is considered in the case when the component of the velocity vector of the undisturbed flow normal to the cascade front is subsonic. The investigation is restricted to regimes without separation and shock waves attached to the leading edges of the profiles and fairly dense cascades, for which the characteristics or shock waves leaving the trailing edges do not enter the region in front of the cascade. In such cases, the conditions behind the cascade do not influence the flow in front of it. In this sense, the flow in the cascade, as in a Laval nozzle in the case of supercritical gradients is trapped, In the hodograph plane, trapped regimes of flow over the cascade correspond to velocity vectors of the undisturbed flow that lie on a certain line (see, for example, [1–3]), which is constructed in the process of solution of the problem. This property has been called the directing influence of the cascade on the oncoming flow. Regimes with detached shocks can also be trapped if the separation of the shocks is due to the profiles being blunt. A method is proposed that for regimes with attached shocks makes it possible to calculate the entire flow field, including the wave structure at large distances from the cascade front; some results obtained by the method are also given. The study of regimes with attached shocks, for which the analysis is simplest, is, first, of interest in its own right and, second, is a stage in the creation of methods of calculation and subsequent investigation of cascades with arbitrary regimes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 108–113, July–August, 1979.We are grateful to M. Ya. Ivanov for assistance in updating the supersonic flow calculation program of [7], to G. Yu. Stepanov for helpful comments, and to E. V. Buganov and V. A. Vostretsov for assistance in preparing the paper.     

Plate cascade in subsonic unsteady gas flow

Accounting for fluid compressibility creates serious difficulties in solving the problem of oscillations of a grid of thin, slightly curved profiles in a subsonic stream. The problem has been solved in [1–3] for a widely-spaced cascade without stagger whose profiles oscillate in phase opposition. The phenomenon of aerodynamic (acoustic) resonance, which may arise in a grid in the direction transverse to the stream for definite values of the stream velocity and profile oscillation frequency, was discovered in [2]. An approximate solution of the problem in which account is not taken of the effect of the vortex trails on the gas flow has been obtained in [4]. In [5, 6] Meister studied in the exact linear formulation the problem of unsteady gas motion through an unstaggered cascade of semi-infinite plates. In [7] Meister considered a grid of profiles with finite chords, but the problem solution was not completed. The problem of subsonic gas flow through a staggered lattice whose profiles oscillate following a single law with constant phase shift was solved most completely in the studies of Kurzin [8, 9] using the method of integral equations. A method of solving the problem for the case of arbitrary harmonic oscillation laws for the lattice profiles was indicated in [10]. The results of the calculation of the unsteady aerodynamic forces for the particular case of a plate cascade without stagger are presented in [9,11], and the possibility of the occurrence of aerodynamic resonance in the cascade in the directions transverse to and along the stream is indicated.Another method of solving the problem is given in [12], in which the more general problem of unsteady subsonic gas flow through a three-dimensional cascade of plates is solved. In the present study this method is applied to the solution of the problem of oscillations of staggered plate cascades in a two-dimensional subsonic gas flow. The results are presented of an electronic computer calculation of the unsteady aerodynamic characteristics of the cascade profiles, which show the essential influence of fluid compressibility on these characteristics. In particular, a sharp decrease of the aerodynamic damping in the acoustic resonance regimes is obtained.     

Gas dynamics with local energy release in subsonic and supersonic flow

The flows developing in the interaction of a supersonic gas stream with a continuously operating axisymmetric energy release source or as a result of the action of pulsed periodic energy injection on a subsonic gas stream are investigated numerically. For a continuously operating energy source two types of flow can be distinguished: with a shock wave detached from the source and with a shock attached to it. Approximate formulas for the gas density in the center of the energy release zone are obtained for the cases of constantly operating and periodic energy sources.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 142–148, March–April, 1995.     

Determining unsteady aerodynamic forces for three-dimensional plate cascade in subsonic gas flow

The general solution for the problem of three-dimensional gas flow through a cascade of plates was obtained in [1], It is shown below that in the considered direct and inverse aerodynamic problems there may exist nontrivial solutions for uniform boundary conditions. The flow regimes for which such solutions exist may be treated as aerodynamic resonance.The examples presented of the calculation for a plate cascade illustrate the behavior of the unsteady aerodynamic forces near the aerodynamic resonance regimes and make it possible to evaluate the limits of applicability of the hypothesis of plane sections. In addition, the calculation results show the possibility of self-induced vibrations of plate cascades with a single degree of freedom in a subsonic gas stream.     

Distinctive features of the flow past high-speed flight vehicles with air-breathing engines on subsonic,transonic, and low supersonic velocity ranges

The aerodynamic characteristics and distinctive features of the flow past hypersonic integral-layout flight vehicles with air-breathing engines intended for cruise flight in the atmosphere are experimentally investigated. The experiments were conducted on a simplified model designed with regard for the general principles of integration of vehicles of the class considered. The tests were performed in a wind tunnel over the Mach and Reynolds number ranges 0.6 ≤ M ≤ 4 and 6.3 × 10⁶ ≤ Re ≤ 16 × 10⁶, respectively. Balance testing was carried out, the pressure distributions over the vehicle surface were measured, and the flowfields on the model surface were photographed. The effects of mounting a nacelle and contouring the internal duct are considered. The effect of the corrections on the duct flow in the absence of jet modeling is estimated. The results obtained can be used as a basis for developing the aerodynamic configurations of integral-layout flight vehicles, for forming their thrust and aerodynamic parameters under full-scale flight conditions, and for testing computation methods.     

Interaction of an external supersonic flow with the subsonic layer near a wavy wall

A solution of the problem of supersonic flow past a wavy wall with an adjacent subsonic layer is obtained. The solution is a generalization of the well-known solutions [1] of the linear problem of purely subsonic and purely supersonic flow past a wavy wall and goes over into these solutions in the limiting cases of infinite and zero wall-layer thickness, respectively. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 97–103, January–February, 1997.     

Unsteady aerodynamic characteristics of a three-dimensional plate cascade in a subsonic gas flow

M. A. Lavrent'ev Institute of Hydrodynamics of the Siberian Division of the Russian Academy of Sciences, 630090 Novosibirsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 2, pp. 45–55, March–April, 1995.     

Calculation of the diffusion combustion of a subsonic jet in a co-current supersonic flow

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 91–95, January–February, 1988.     

Discrete vortices in the wake of various profiles in subsonic and supersonic two-dimensional gas flow

The Mach number dependence of the Strouhal number, the frequency of discrete vortices, the vortex velocity, and other parameters are determined in the wake of wedges and flat plates for low angles of attack. The studies were made using high-speed motion-picture photography through a Schlieren system and with photomultipliers. The results are presented in tabular and graphical form.Notation h transverse distance - l longitudinal distance between vortices - V freestream velocity in m/sec - n_v vortex frequency for one row of vortex street in sec - M freestream Mach number - S₁ Strouhal number based on projection of the model onto the plane perpendicular to the freestream direction - S₂ Strouhal number calculated from the wake neck width d₂ for M>1 - R Reynolds number calculated from d - R_* critical Reynolds number - model apex angle - angle of attack - L length in flow direction in mm The author wishes to thank G. I. Petrov for his interest in the study and his advice.     

Expansion solution for subsonic compressible flow

An expansion solution in the physical plane is developed for subsonic compressible fluid flow past an obstacle. Assuming that the stream is inviscid, isentropic, irrotational and steady, it is shown that the velocity potential may be expressed as a series of homogeneous Heun functions and radial distance terms. The basis of this analysis is Ludford's formal discussion of corresponding singularities in Bergman's Linear Integral Operator Method. A modification of these results permits reduction of the governing nonlinear partial differential equation to an ordinary, nonhomogeneous, linear differential equation. The expansion solution is compared with the Rayleigh-Janzen method and the Prandtl-Glauert theory. The comparison indicates that this expansion gives better results than other methods currently used. The simplicity and economy of this expansion solution facilitates direct practical application.     

Investigation of supersonic viscous flow past a sphere in the presence of subsonic and sonic injection

Supersonic flow past a sphere with a given rate of gas injection along the generator is investigated numerically on the range Re=10²–10⁴. Calculations have been made on the interval 0 90°, where is the angle between the axis of symmetry and the normal to the surface. It is shown that for high subsonic and sonic injection rates it is possible to observe qualitatively new features in the flow structure and in the distribution of the local supersonic flow characteristics around the perimeter of the sphere not previously noted in [9]. In the case of sonic injection the changes in flow structure occur only in the supersonic zone. In the neighborhood of the transition from a subsonic to sonic injection velocity the heat flux has a local maximum, which in absolute value does not exceed the heat flux in the absence of injection. It is shown that there may be qualitative differences in the pressure distribution over the surface of the body with increase in the injection parameter depending on the distribution and value of the injected gas flow rate and, moreover, the number Re.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 83–89, January–February, 1988.     

Integral equations for subsonic gas flow in turbines

Conclusions Integral equations provide an, exact formulation for the flow in a turbine lattice; solution of the integral equation by successive approximation gives the potential and flow speed in analytical form. If (3.16) and (4.7) are met, the process also converges to the exact value. This algorithm is comparatively readily, implemented with a medium-power computer.The result from the integral equations goes with standard results [3, 4, 7] to show that the technique is of some value in research on subsonic flows.Zaporozhe Engineering Institute. Translated from Prikladnaya Mekhanika, Vol. 14, No. 9, pp. 110–117, September, 1978.     

Plane-parallel supersonic flow with a discontinuity

This article considers a plant-parallel supersonic flow, with a shock wave terminating within the flow; the shock wave is regarded as a distortion. A line of discontinuity is located ahead of the shock wave in the supersonic zone. The problem is solved by the method of indeterminate coefficients.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 95–100, July–August, 1970.The authors thank S. V. Fal'kovich for his valuable advice and for his evaluation of the results obtained.