Numerical solution of the problem of the flow of a supersonic gas stream over the upper surface of a delta wing in the expansion region

A numerical method is described for the calculation of supersonic flow over the arbitrary upper surface of a delta wing in the expansion region. The shock wave must be attached everywhere to the leading edge of this wing from the side of the lower surface. The stream flowing over the wing is assumed to be nonviscous. A problem with initial conditions at some plane and with boundary conditions at the wing surface and the characteristic surface is set up for the nonlinear system of equations of gas dynamics. The difference system of equations, which approximates the original system of differential equations on a grid, has a second order of accuracy and is solved by the iteration system proposed in [1]. The initial conditions are determined by the method of establishment of self-similar flow. A number of examples are considered. Comparison is made with the solutions of other authors and with experiment.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 76–81, November–December, 1973.The author thanks A. S. II'ina who conducted the calculations and V. S. Tatarenchik for advice.     

Effect of supersonic stream parameters on the characteristic time of transient step flow

There have been many studies of viscous compressible gas flow in wakes and behind steps [1–6] in which attention has been focused on the steady-state flow regime. The problem of the supersonic flow of a viscous compressible heat-conducting gas past a plain backward-facing step is considered. The problem is solved numerically within the framework of the complete system of Navier-Stokes equations. The passage of the solution from the initial data to the steady-state regime and the effect of the gas dynamic parameters of the external flow on the characteristic flow stabilization time are investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–140, July–August, 1989.     

Stability of the motion of a rigid body in an unsteady flow

The problem of rigid-body motion in an unsteady gas flow is considered using a flow model [1] in which the motion of the body is described by a system of integrodifferential equations. The case in which among the characteristic exponents of the fundamental system of solutions of the linearized equations there are not only negative but also one zero exponent is analyzed. The instability conditions established with respect to the second-order terms on the right sides of the equations are noted. The problem may be regarded as a generalization of the problem of the lateral instability of an airplane in the critical case solved by Chetaev [2], pp. 407–408.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 18–22, May–June, 1989.     

Universal equations for the laminar boundary layer on a solid of revolution in an oblique gas flow

We consider the problem of laminar gas motion in the boundary layer on a solid of revolution oriented at an angle of attack. The parametric method of L. G. Loitsyanskii is used for the solution. The effect of the external current and the form of the body are considered by introduction of three series of parameters. A corresponding system of universal equations is obtained, which is then numerically integrated over a wide range of parameters and their combinations. The results permit evaluation of the general principles of flow in a boundary layer on a solid of revolution in an oblique gas flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 32–41, July–August, 1973.     

Chemical relaxation in a viscous shock

We consider the flow of a nonequilibrium dissociating diatomic gas in a normal compression shock with account for viscosity and heat conductivity. The distribution of gas parameters in the flow is found by numerically solving the Navier-Stokes and chemical kinetics equations. The greatest difficulty in numerical integration comes from the singular points of this system at which the initial conditions are given. These points lead to instability of the numerical results when the problem is solved by standard numerical methods. An integration method is proposed that yields stable numerical results-continuous profiles of the distribution of the basic gas parameters in the shock are obtained.We consider steady one-dimensional flow in which the gas passes from equilibrium state 1 to another equilibrium state 2, which has higher values for temperature, density, and pressure. Such a flow is termed a normal compression shock.The parameter distribution in normal shock for nonequilibrium chemical processes has usually been calculated [1–3] without account for the transport phenomena (viscosity, heat conduction, and diffusion). The presence of an infinitely thin shock front perpendicular to the flow velocity direction was postulated. It was assumed that the flow is undisturbed ahead of the shock front. The gas parameters (velocity, density, and temperature) change discontinuously across the shock front, but the gas composition does not change. The composition change due to reactions takes place behind the shock front. The gas parameter distribution behind the front was calculated by solving the system of gasdynamic and chemical kinetics equations using the initial values determined from the Hugoniot conditions at the front to state 2 far downstream.Several studies (for example, [4, 5]) do account for transport phenomena in calculating parameter distribution in a compression shock, but not for nonequilibrium chemical reactions. These problems are solved by integrating the Navier-Stokes equations continuously from state 1 in the oncoming flow to state 2 downstream.We present a solution to the problem of normal compression shock in nonequilibrium dissociating oxygen with account for viscosity and heat conduction using the Navier-Stokes equations.     

Numerical investigation of supersonic viscous gas flow over long blunt cones with allowance for equilibrium physicochemical effects

Supersonic axisymmetric viscous heat-conducting gas flow over long spherically biunted cones is considered over a broad range of Reynolds numbers on the basis of the complete system of viscous shock layer equations. An economical numerical method based on global iterations is used to solve the viscous shock layer equations. The general influence of the second-approximation effects of boundary layer theory and the influence of equilibrium physicochemical processes on the heat loads are determined for bodies with a large aspect ratio.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 202–205, January–February, 1993.The author wishes to thank S. A. Vasil'evskii and I. A. Sokolova for providing the tables used to calculate the transport coefficients and G. A. Tirskii for his constant interest and useful discussions.     

Calculation of nonstationary viscous flows

The present study will consider the problem of formation of a nonstationary flow of a viscous, thermally conductive gas around a semiinfinite plate, with the gas being set in motion instantaneously from a state of rest (problem I) and behind a shock wave incident upon the gas (problem H). A numerical method will be used, based on integration of the complete system of Navier-Stokes equations. Special attention is given in the solution to these times when the simplifying assumptions of boundary layer theory are invalid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 167–171, November–December, 1978.The authors thank Yu. A. Dem'yanov for evaluating the results of the study.     

Motion of triple configuration of shock waves with formation of wake behind branching point

If three developing shock-wave fronts come together at one point the laws of conservation connecting the parameters of the gas in the vicinity of this point give an overdetermined system of equations. To remove the possible contradiction it is necessary to increase the number of initial parameters. As a rule, the assumption of the presence of a contact discontinuity emerging from the branching point is sufficient. It is also possible for two contact discontinuities to develop which form two shock waves with respect to the branching point opposite the boundary of the isobaric region filled with gas in a state of rest. Such a region is called the wake of the triple point by analogy with the aerodynamic wake for flow around bodies with flow separation. A closed system of five simultaneous differential equations which describes approximately the dynamics of a stream containing a branched system of shock waves with a developing wake behind the triple point is derived and discussed in the report.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 67–75, November–December, 1973.     

Numerical modeling of turbulent boundary layer suction

For the problem of gas suction from the turbulent flow over a plate, the results of numerically solving the Navier-Stokes equations complemented by a differential model of turbulence are presented. On the range of Mach numbers from 0.8 to 1, the effect of suction of various intensities on the flow parameters in the neighborhood of the suction region and downstream is considered.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 45–49, November–December, 1996.     

Influence of second-order boundary layer effects in hypersonic flow over slender blunt cones

The aim of this study is to determine the influence of second-order effects in the aggregate on supersonic axisymmetric flow over slender blunt cones and also to determine the region of applicability of approximate methods of taking into account the strongest of these second-order effects — entropy layer absorption. A system of complete viscous shock layer equations containing all the terms of the gas dynamic Euler equations and all the second-order terms of asymptotic boundary layer theory is chosen as the gas-dynamic model. Within the calculation domain the problem is solved in a unified manner.Translated from Izvestiya Rossiiskoi Akademii Nauk, Meknanika Zhidkosti i Gaza, No.4, pp. 129–134, July–August, 1992.     

Numerical investigation of flow separation ahead of a step

A numerical investigation by means of the Navier-Stokes equations is used to solve the problem of the flow of a viscous perfect gas over a step at subsonic and supersonic velocities of the undisturbed flow and different Reynolds numbers. The dependence of the resulting separation flow on the parameters of the undisturbed flow is investigated. The gas-dynamic characteristics and the size of the separated flow are determined. The results of the calculations are compared with the experimental data of Chapman et al. [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No 5, pp. 72–79, September–October, 1979.     

Gas-dynamic calculation of pulsating flow in pipelines

A numerical solution is examined for a system of equations of one-dimensional isothermal flow of a perfect gas in a horizontal pipe with a periodically varying function of the flow rate at the boundary. The numerical solution is compared with the solution of the linearized problem. The results can be used to calculate the pulsating motion of gas in the pipeline systems of piston compressors [1].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 85–88, July–August, 1972.     

Modeling of retrograde condensation in the process of steady radial flow through a porous medium

The problem of the steady axisymmetric two-phase flow of a multicomponent mixture through a porous medium with phase transitions is considered. It is shown that the system of equations for the two-phase multicomponent flow process, together with the equations of phase equilibrium, reduces to a system of two ordinary differential equations for the pressures in the gas and liquid phases. A family of numerical solutions is found under certain assumptions concerning the pressure dependence of the molar fraction of the liquid phase.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 92–97, November–December, 1994.     

Calculation of two-dimensional dispersion in a gas in unsteady flow in a porous medium

A two-dimensional problem of the flow of a gas containing an impurity through a porous medium is considered. At the initial time, the gas containing a uniformly distributed impurity is at a high pressure in a spherical cavity in a porous medium at a certain distance from a flat surface. It is assumed that for t > the motion of the carrier gas is described by the system of equations for flow in a porous medium and the dispersion of the impurity is described by the equations of convective diffusion and nonequilibrium adsorption. A numerical method for solving the problem is discussed. Some results of calculations are given. The influence of the flat surface on the flow of the gas and the dispersion of the impurity is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 61–67, September–October, 1982.We thank V. N. Nikolaevskii for comments which permitted a significant improvement in the paper.     

Calculation of stationary inviscid flow over three-dimensional bodies

The results of the calculation of inviscid supersonic flow of a perfect gas over a blunt three-dimensional configuration are considered. An explicit finite-difference scheme of second order of accuracy [1] was used for the numerical integration of the hyperbolic system of equations, which was written in divergence form. The region of integration is situated between the body and the outer shock wave. The internal discontinuity surfaces were not separated out and the calculation was made through them. The points on the surface of the body were calculated using relations on characteristics written in a form that makes it possible to calculate flows with strong entropy layers. The results of calculation of flow over a three-dimensional configuration at an angle of attack are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 4, pp. 172–175, July–August, 1980.     

Method of computing the chemically nonequilibrium gas flow in a heated tube in almost equilibrium or frozen states

The solution of equations describing turbulent isobaric flow of a chemically reacting gas in a heated tube is investigated analytically. Solutions of the ordinary nonlinear differential equations are obtained for almost frozen flow by the perturbation method, and for almost equilibrium flow by an asymptotic method taking account of the zero and first approximations, Linear differential equations in variations are written down to find the subsequent approximations.Translated from Izvestiya Akademiya Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 8–14, July–August, 1973.     

Laminar boundary layer on swept-back wings of infinite span at an angle of attack

A study is made of the flow of a compressible gas in a laminar boundary layer on swept-back wings of infinite span in a supersonic gas flow at different angles of attack. The surface is assumed to be either impermeable or that gas is blown or sucked through it. For this flow and an axisymmetric flow an analytic solution to the problem is obtained in the first approximation of an integral method of successive approximation. For large values of the blowing or suction parameters, asymptotic solutions are found for the boundary layer equations. Some results of numerical solution of the problem obtained by the finite-difference method are given for wings of various shapes in a wide range of angles characterizing the amount by which the wings are swept back and also the blowing or suction parameters. A numerical solution is obtained for the equations of the three-dimensional mixing layer formed in the case of strong blowing of gas from the surface of the body. The analytic and numerical solutions are compared and the regions of applicability of the analytic expressions are estimated. On the basis of the solutions obtained in the present paper and studies of other authors a formula is proposed for the calculation of the heat fluxes to a perfectly catalytic surface of swept-back wings in a supersonic flow of dissociated and ionized air at different angles of attack. Flow over swept-back wings at zero angle of attack has been considered earlier (see, for example, [1–4]) in the theory of a laminar boundary layer. In [5], a study was made of flow over swept-back wings at nonzero angle of attack at small and moderate Reynolds numbers in the framework of the theory of a hypersonic viscous shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 27–39, May–June, 1980.We thank G. A. Tirskii for a helpful discussion of the results.     

Numerical simulation of flow of multicomponent mixtures in porous media

A study is made of the isothermal flow of multicomponent mixtures in a porous medium, accompanied by phase transitions, interphase mass exchange, and change in the physicochemical properties of the phases [1–3], It is assumed that at each point of the flow region, phase equilibrium is established instantaneously and the flow velocities of the separate phases conform to Darcy's law. Approximate solutions of problems of displacing oil by high-pressure gas were obtained in [1]. By generalizing the theory developed in [4], a study is made in [5] of the structure of the exact solutions of the problems of the flow of three-component systems which describe the displacement of oil by different reactants (gases, solvents, micellar solutions). The numerical solutions of the problems of multicomponent system flow are considered in [2, 3, 6, 7]. This paper presents a numerical method which is distinguished from the well-known ones [2, 3, 6, 7] by the following characteristics. The flow equations are approximated by a completely conservative finite-difference scheme of the implicit pressure-explicit saturation type, the calculation being carried out using Newton's method of iteraction with spect to both the pressure and the composition of the mixture. The minimum derivative principle [8] is used in the approximation of the divergence terms of the equations. The phase equilibrium is calculated using the equation of state.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 101–110, July–August, 1985.     

Nonlinear effects associated with the mutual displacement of two gases in a porous medium

The cyclic bidirectional process of isothermal flow of a binary singlephase compressible gas mixture in a porous medium accompanied by diffusion-dispersion mass transfer is considered. On the basis of the equations of multiphase multicomponent isothermal flow a system of two nonlinear partial differential equations with nonlinear boundary conditions corresponding to a given constant gas injection or takeoff rate is obtained and investigated. A numerical algorithm for solving the boundary-value problem obtained is proposed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 64–71, November–December, 1991.In conclusion, the author wishes to thank V. M. Maksimov for taking a constant interest in the work and discussing the results.     

Turbulent air flow over a moving curved surface

A numerical model of turbulent air flow over a curved surface is described. The model is based on two-dimensional nonlinear Reynolds equations and continuity equations written in a coordinate system moving with the profile of the curved surface. The Reynolds stresses are represented in the form of the product of the isotropic turbulent viscosity coefficient, which increases linearly with height, and the deformation tensor of the mean velocity field. Flow over a stationary sinusoidal surface and a sinusoidal gravity wave on water is simulated. The structure of the velocity and pressure wave fields is obtained. The differences in flow over stationary and moving surfaces are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 20–24, September–October, 1986.