Numerical modeling of three-dimensional viscous flows using a space-marching method with pressure iterations

A method of calculating steady three-dimensional compressible and incompressible flows on the basis of simplified Navier-Stokes equations is examined. Some calculation results are presented. These relate to the problem of supersonic flow past blunt bodies and to incompressible channel flows.Based on paper read at the Seventh Congress on Theoretical and Applied Mechanics, Moscow, August I991. Presented by V. I. Polezhaew.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.5, pp. 132–147, September–October, 1992.     

Boundary layer development in supersonic MHD generators

A numerical model of the turbulent boundary layers in the gas dynamic channel of a supersonic MHD generator is constructed. This model describes the development and structure of the layers in the nozzle, on the electrode and insulating walls of the duct, in the two-dimensional approximation. The characteristics of the boundary layers in various generator operating regimes are investigated numerically. The integral boundary layer thicknesses characterizing the nonuniformity of the gas dynamic and electrodynamic quantities are calculated. The limits of applicability of the integral calculation method are determined for typical MHD generator operating conditions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 35–41, May–June, 1990.The authors wish to thank A. B. Vatazhin, V. A. Bityurin, and V. A. Zhelnin for discussing the formulation of the problem, A. A. Yakushev for participating in the discussion of the results, and Yu. V. Rakseeva and L. V. Yashina for preparing the article.     

Two-layer flows of gas in supersonic axisymmetric nozzles

This article describes two methods for calculating two-layer flows. The first is a generalization of a numerical method for solving the inverse problem [1] for the case of two-layer flows, without taking mixing into account. The second is a method of characteristics, for calculating a two-layer flow in a supersonic nozzle. In this case, the usual method of characteristics is changed in such a way that it is possible to calculate a point on the interface between two layers having different adiabatic indices, and different total pressures and temperatures. This article also gives the results of calculation of two-layer flows in nozzles with different adiabatic indices and different ratios of the mass flow rates of the gas in the layers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 76–81, July–August, 1970.The calculations were programmed and carried out by G. D. Vladimirova and M. F. Tamarovskii, to whom the author expresses his thanks.     

Influence of the mixing of two flows with different total parameters in a laval nozzle on its integral characteristics

A calculation is made of the turbulent zone of mixing of two flows of viscous and heat conducting gas in a Laval nozzle. For such a nozzle of given geometry, a comparison is made of calculations of the integrated characteristics of flows that are nonuniform with respect to the total parameters in the framework of various models: laminar hydraulics, viscous laminar hydraulics, and total mixing without hydraulic losses. The calculations are made for a stationary, nonswlrling flow of a viscous heat conducting gas with nearly discontinuous step distribution of the total parameters at the entrance to an axisymmetric Laval nozzle of given geometry. In this situation, the gas flows with different total parameters at the entrance to the nozzle are separated by a surface near which the profiles of the flow parameters are specified on the basis of boundary-layer theory. In the blocked regime investigated here, the flow in the part where the nozzle becomes narrower and at least at the beginning of the expanding part does not depend on the pressure of the surrounding medium. The integrated characteristics of the nozzle (gas flow rate G, impulse I, specific impulse i = I/G, etc.) depend on the parameter distributions at the entrance to the nozzle, and also on the turbulent mixing of the flows in the mixing zone. To analyze the dependence of the integrated characteristics on the turbulent mixing, the values of these characteristics calculated in the framework of the three models are compared. The model of mixing without hydraulic losses presupposes complete equalization of the parameters of the original inhomogeneous flow in the constant-area chamber in front of the nozzle with conservation of the mass, energy, and momentum fluxes. The model of laminar hydraulics is described in detail in [1, 2]. The model of viscous laminar hydraulics will be described in Sec. 1.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 114–119, July–August, 1979.I thank A. N. Kraiko for supervising the work, A. N. Sekundov for helpful discussions, and I. P. Smirnova and A. B. Lebedev for making available the computer program.     

Solution of the direct problem of the mixed flow of a conducting gas in a laval nozzle with a meridional magnetic field

We consider the direct problem in the theory of the axisymmetric Laval nozzle (including sonic transition) for the steady flow of an inviscid and nonheat-conducting gas of finite electrical conductivity. The problem is solved by numerical integration of the equations of unsteady gas flow using an explicit difference scheme that was proposed by Godunov [1,2], and was used to calculate steady and unsteady flows of a nonconducting gas in nozzles by Ivanov and Kraiko [3]. The subsonic and the supersonic flows of a conducting gas in an axisymmetric channel when there is no external electric field, the magnetic field is meridional, and the magnetic Reynolds numbers are small have previously been completely investigated. Thus, Kheins, Ioller and Élers [4] investigated experimentally and theoretically the flow of a conducting gas in a cylindrical pipe when there is interaction between the flow and the magnetic field of a loop current that is coaxial with the pipe. Two different approaches were used in the theoretical analysis in [4]: linearization with respect to the parameter S of the magnetogasdynamic interaction and numerical calculation by the method of characteristics. The first approach was used for weakly perturbed subsonic and supersonic flows and the solutions obtained in analytic form hold only for small S. This is the approach used by Bam-Zelikovich [5] to investigate subsonic and supersonic jet flows through a current loop. The numerical calculations of supersonic flows in a cylindrical pipe in [4] were restricted to comparatively small values of S since, as S increases, shock waves and subsonic waves appear in the flow. Katskova and Chushkin [6] used the method of characteristics to calculate the flow of the type in the supersonic part of an axisymmetric nozzle with a point of inflection. The flow at the entrance to the section of the nozzle under consideration was supersonic and uniform, while the magnetic field was assumed to be constant and parallel to the axis of symmetry. The plane case was also studied in [6]. The solution of the direct problem is the subject of a paper by Brushlinskii, Gerlakh, and Morozov [7], who considered the flow of an electrically conducting gas between two coaxial electrodes of given shape. There was no applied magnetic field, and the induced magnetic field was in the direction perpendicular to the meridional plane. The problem was solved numerically in [7] using a standard process. However, the boundary conditions adopted, which were chosen largely to simplify the calculations, and the accuracy achieved only allowed the authors [7] to make reliable judgments about the qualitative features of the flow. Recently, in addition to [7], several papers have been published [8–10] in which the authors used a similar approach to solve the direct problem in the theory of the Laval nozzle (in the case of a nonconducting gas).Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 5, pp. 14–20, September–October, 1971.In conclusion the author wishes to thank M. Ya. Ivanov, who kindly made available his program for calculating the flow of a conducting gas, and also A. B. Vatazhin and A. N. Kraiko for useful advice.     

Interaction of two supersonic radial gas flows

The flow resulting from the collision of two spherically symmetric supersonic gas streams generated by three-dimensional sources has been studied within the framework of the inviscid perfect gas model. When the characteristics of both sources are the same, the problem reduces to that of the interaction between a spherically symmetric flow and a plane barrier [1, 2]. By means of a certain passage to the limit, the flow from one of the sources can be reduced to a uniform supersonic gas flow. In this case the problem reduces to the problem of uniform gas flow past a source considered in [1, 3, 4]. The resulting flows are investigated with reference to all the parameters characterizing the two sources. The shock wave structure, both in the neighborhood of and at points remote from the axis of symmetry, is studied, together with the distributions of the gas dynamic quantities in the shock layers; certain similarity laws are established. The astrophysical applications of the problem associated, in particular, with a certain x-ray radiation mechanism in binary stellar systems [5] are of particular interest.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 159–165, July–August, 1990.The authors are grateful to V. B. Baranov for his constant interest and to S. A. Zhekov and V. V. Usov for useful discussions.     

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.     

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.     

Turbulent mixing of relaxing gases in a supersonic nozzle

Processes of turbulent mixing of hot nitrogen and carbon dioxide gas (CO₂) in a supersonic nozzle are investigated in connection with gasdynamic lasers with selective thermal excitation. The actual flow pattern in the nozzle and cavity is simulated by the system of equations of a turbulent boundary layer. The results of calculations of the gain profile, the CO₂ concentration, and the laser power agree satisfactorily with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 156–160, March–April, 1981.We are very grateful to B. S. Aleksandrov and V. K. Pozdyshev for numerous helpful discussions.     

Turbulent mixing and diffusion combustion of a jet in a channel

An attempt is made to study in some detail the turbulent mixing of reacting (propane) and inert jets (air and carbon dioxide) in a channel. The results are given of an experimental investigation into diffusion combustion in a channel, and these are compared with calculated data obtained using a semiempirical theory of turbulence. Such a comparison makes it possible to estimate the applicability of this theory for calculating the characteristics of diffusion combustion in a channel.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 25–33, July–August, 1980.We thank V. R. Kuznsatsov and A. N. Sekundov for great interest in the work and for discussing the results and V. I. Rasshchupkin for assisting in the experiments.     

Nonlinear waves in a liquid film entrained by a turbulent gas stream

In the long-wavelength approximation and on the basis of a simplified system of equations analogous to the one considered by Shkadov and Nabil' [1, 2], an investigation is made into waves of finite amplitude in thin films of a viscous liquid on the walls of a channel in the presence of a turbulent gas stream. A bibliography on the linear stability of such plane-parallel flows can be found in [3–5]. The nonlinear stability is considered in [6]. A stationary periodic solution is sought in the form of a Fourier expansion whose coefficients are found near the upper curve of neutral stability by Newton's method and near the lower branch of the stability curve by the method of Petviashvili and Tsvelodub [7, 8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 2, pp. 37–42, March–April, 1981.I thank V. Ya. Shkadov for supervising the work and all the participants of G. I. Petrov's seminar for a helpful discussion.     

Thin weakly curved wings with maximum aerodynamic quality

Lifting wings that only slightly disturb the supersonic gas flow are considered. The plan shape and thickness distribution of the wing and the free-stream parameters are given. The flow problem is solved within the framework of the Prandtl model. The outer potential flow is determined in accordance with the linear theory. The turbulent boundary layer is found by the method of plane sections with allowance for the three-dimensional inviscid flow pattern. A numerical model of the flow is constructed in the class of piecewise-constant functions on characteristic calculation grids [1]. The variational problem of finding the weakly curved middle surface of the wing giving maximum aerodynamic quality is reduced, by analogy with [2], to a problem of nonlinear programming and is solved by the gradient projection method.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 165–168, July–August, 1991.     

Numerical investigation of the flow of nitrogen past a sphere with allowance for rotational relaxation

The supersonic flow of nitrogen past a sphere is studied in the framework of the complete Navier-Stokes equations and the relaxation equation with allowance for rotational-translational relaxation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 170–174, March–April, 1980.I thank V. N. Gusev for interest in the work and valuable discussions and V. K. Molodtsov for assisting in the development of the numerical algorithm for solving the problem.     

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.     

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.     

The reflection of a plane shock wave from a body with groove

When a plane shock wave impinges on bodies with grooves and when a supersonic stream of gas flows past such bodies a complicated flow pattern develops. In a number of cases oscillations of the bow wave [1–3] and an anomalous heating of the gas in the groove [4–6] have been observed. Unsteady reflection of shock waves from bodies with grooves and the processes occurring inside the grooves have been investigated comparatively little.Translated from Izvestiya Akademii Nauk SSSR, Hekhanika Zhidkosti 1 Gaza, No. 5, pp. 180–186, September–October, 1935.The authors wish to thank V. I. Ivanov for carrying out the calculations.     

Flow in the neighborhood of the stagnation line on a blunt body traveling at supersonic speed through a zone with elevated temperature and vibrational energy of the molecules

The unsteady flow in the neighborhood of the stagnation line on a sphere traveling at supersonic speed through a plane layer of diatomic gas with elevated temperature and nonequilibrium excitation of the molecular vibrations is investigated. (The source of the inhomogeneity could be a gas discharge [1].) The problem is solved using the viscous shock layer model which makes it possible to take molecular transport processes into account and analyze the unsteady heat transfer. Such flows were previously calculated in [2] within the framework of the inviscid gas model.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i GazNo. 3, pp. 183–185, May–June, 1990.     

Computation of base flow characteristics with uniform blowing

Blowing at bluff body base was considered under different conditions and for small amount of blowing this problem was solved using dividing streamline model [1]. The effect of supersonic blowing on the flow characteristics of the external supersonic stream was studied in [2–4]. The procedure and results of the solution to the problem of subsonic blowing of a homogeneous fluid at the base of a body in supersonic flow are discussed in this paper. Analysis of experimental results (see, e.g., [5]) shows that within a certain range of blowing rate the pressure distribution along the viscous region differs very little from the pressure in the free stream ahead of the base section. In this range the flow in the blown subsonic jet and in the mixing zones can be described approximately by slender channel flow. This approximation is used in the computation of nozzle flows with smooth wall inclination [6, 7]. On the other hand, boundary layer equations are used to compute separated stationary flows with developed recirculation regions [8] in order to describe the flow at the throat of the wake. The presence of blowing has significant effect on the flow structure in the base region. An increasing blowing rate reduces the size of the recirculation region [9] and increases base pressure. This leads to a widening of the flow region at the throat, usually described by boundary-layer approximations. At a certain blowing rate the recirculation region completely disappears which makes it possible to use boundary-layer equations to describe the flow in the entire viscous region in the immediate neighborhood of the base section.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 76–81, January–February, 1984.     

Results of an experimental and numerical study of the aerodynamic heating of the undersurface of delta wings with sharp leading edges at mach numbers M∞=6.1 and 8

The heat transfer between a supersonic flow and the undersurface of delta wings with leading-edge sweep angles x=65 and 70° is investigated in a shock tunnel at angles of attack 15°. The supersonic inviscid flow over these wings in regimes in which the bow shock is attached to the sharp leading edges is calculated numerically. The compressible boundary layer problem is solved for the calculated inviscid flow fields in the laminar, transition and turbulent flow zones. The calculations and experimental values of the heat flux on the surface of the wings are compared. The calculations are in satisfactory agreement with the experimental data in the laminar and transition zones, but diverge significantly (by up to 20%) in the turbulent zone.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 183–188, July–August, 1991.The authors wish to thank A. A. Golubinskii for assisting with the solution of the problem of supersonic inviscid gas flow over a wing.     

A mixing layer in a homogeneous fluid

A mathematical model for the evolution of a mixing layer in shear flows is constructed. The problem of a mixing layer with pressure gradient is solved: in particular, the distributions of the velocity and basic characteristics of turbulent flow in the mixing layer are obtained. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 81–92, July–August, 2000.