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.     

Investigation of the hypersonic viscous shock layer around blunt bodies in a nonuniform flow

Unseparated viscous gas flow past a body is numerically investigated within the framework of the theory of a thin viscous shock layer [13–15]. The equations of the hypersonic viscous shock layer with generalized Rankine-Hugoniot conditions at the shock wave are solved by a finite-difference method [16] over a broad interval of Reynolds numbers and values of the temperature factor and nonuniformity parameters. Calculation results characterizing the effect of free-stream nonuniformity on the velocity and temperature profiles across the shock layer, the friction and heat transfer coefficients and the shock wave standoff distance are presented. The unseparated flow conditions are investigated and the critical values of the nonuniformity parameter a_k [10] at which reverse-circulatory zones develop on the front of the body are obtained as a function of the Reynolds number. The calculations are compared with the asymptotic solutions [10, 12].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 154–159, May–June, 1987.     

Viscous shock layer near the stagnation point on a rotating body with unsteady injection and surface cooling

Unsteady supersonic flow regimes in the neighborhood of a stagnation point are investigated on the basis of a system of viscous shock layer equations [10] containing all the terms of the Euler equations and the boundary layer equations. An analytic solution of the unsteady equations valid near the surface of the body is found in the case of strong injection. The unsteady equations of the viscous shock layer are solved numerically on the basis of a divergent implicit scheme of the second order of approximation across the shock layer, using Newtonian linearization and vector sweep methods with allowance for the boundary relations on the surface of the body and at the isolated bow shock. Certain calculation results illustrating the effect of injection, surface cooling, the swirl of the external flow and the angular velocity of the body on the structure of the steady and unsteady viscous shock layer are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 114–122, September–October, 1987.     

The blunt-body problem in a nonuniform hypersonic viscous gas flow

The laws of heat transfer associated with the interaction of underexpanded supersonic gas jets and obstacles or blunt bodies have been investigated, for example, in [1–3]. Similar problems of nonuniform flow occur when bodies move in the wake behind other bodies; however, in this case the laws of heat transfer have so far received little attention [4–8]. It has been established that for a certain Reynolds number and flow nonuniformity parameters a zone of reverse-circulatory flow develops near the front of the blunt body. However, the conditions of transition to separated flow have not been determined. This paper presents a self-similar solution of the equations of the viscous shock layer near the stagnation line in supersonic flow past an axisymmetric blunt body located behind another body. On the basis of this solution a separationless flow criterion is proposed. The effect of the nonuniformity and the Reynolds number on the shock standoff distance, the convective heat flux and the friction drag of the blunt body is investigated. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 120–125, November–December, 1986. In conclusion the authors wish to thank I. G. Eremeitsev for useful suggestions and G. A. Tirskii for discussing their work.     

Analytic investigation of friction and heat transfer in the neighborhood of a three-dimensional stagnation point at low and moderate Reynolds numbers

A study is made of hypersonic three-dimensional flow of a viscous gas past blunt bodies at low and moderate Reynolds numbers with allowance for the effects of slip and a jump of the temperature across the surface. The equations of the three-dimensional viscous shock layer are solved by an integral method of successive approximation and a finite-difference method in the neighborhood of the stagnation point. In the first approximation of the method an analytic solution to the problem is found. Analysis of the obtained solution leads to the proposal of a simple formula by means of which the calculation of the heat flux to a three-dimensional stagnation point is reduced to the calculation of the heat flux to an axisymmetric stagnation point. A formula for the relative heat flux obtained by generalizing Cheng's well-known formula [1] is given. The accuracy and range of applicability of the obtained expressions are estimated by comparing the analytic and numerical solutions. Three-dimensional problems of the theory of a supersonic viscous shock layer at small Reynolds numbers were considered earlier in [2–5] in a similar formulation but without allowance for the effects of slip.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 143–150, March–April, 1988.     

Calculation of supersonic viscid flow near stagnation line of a blunt body

A numerical study is made of supersonic flow of a viscous gas in the vicinity of the stagnation line of plane and axisymmetric blunt bodies (cylinder, sphere). As in [1–5], which consider the compressed layer of a viscous gas in the vicinity of the stagnation point, use is made of the locally self-similar approximation, which is used to transform the Navier-Stokes equations into a system of ordinary differential equations. In the present paper the solution is sought with the simplifications of [5] and with more general conditions, which makes it possible to study a broad class of flows. The proposed numerical algorithm permits obtaining the structure of the compressed layer near the stagnation line, including the shock wave and the boundary layer. The calculations made on a computer for different flow conditions are illustrated by graphs.The author wishes to thank G. I. Petrov, G. F. Telenin, and L. A., Chudov for their interest in the study and for their helpful discussions. discussions.     

Supersonic motion of bodies in a gas with shock waves

The authors consider the problem of supersonic unsteady flow of an inviscid stream containing shock waves round blunt shaped bodies. Various approaches are possible for solving this problem. The parameters in the shock layer on the axis of symmetry have been determined in [1, 2] by using one-dimensional theory. The authors of [3, 4] studied shock wave diffraction on a moving end plane and wedge, respectively, by the through calculation method. This method for studying flow around a wedge with attached shock was also used in [5]. But that study, unlike [4], used self-similar variables, and so was able to obtain a clearer picture of the interaction. The present study gives results of research into the diffraction of a plane shock wave on a body in supersonic motion with the separation of a bow shock. The solution to the problem was based on the grid characteristic method [6], which has been used successfully to solve steady and unsteady problems [7–10]. However a modification of the method was developed in order to improve the calculation of flows with internal discontinuities; this consisted of adopting the velocity of sound and entropy in place of enthalpy and pressure as the unknown thermodynamic parameters. Numerical calculations have shown how effective this procedure is in solving the present problem. The results are given for flow round bodies with spherical and flat (end plane) ends for various different values of the velocities of the bodies and the shock waves intersected by them. The collision and overtaking interactions are considered, and there is a comparison with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 141–147, September–October, 1984.     

Unsteady reflection of a shock wave from a body with a cylindrical recess

In a number of cases of supersonic flow past bodies with recesses pulsations in the flow arise [1–3]. Experiments [4, 5] indicate that stabilization of the steady supersonic flow past the body with a recess on which a shock wave is incident takes place after a series of oscillations of the bow wave. Numerical calculation of the interaction of a supersonic jet with a cylindrical cavity [6] reveals that damped pressure pulsations arise inside the cavity if the jet is homogeneous, and undamped pulsations it is inhomogeneous. The authors explain the damping of the pulsations by the influence of artificial viscosity. This paper investigates experimentally and theoretically (by numerical methods) the oscillations of the bow shock wave and the parameters of the flow behind it in the case of unsteady reflection of a shock wave from a body with a cylindrical recess turned towards the flow. The problem is posed as follows. A plane shock wave with constant parameters impinges on a cylinder with a cavity. The unsteady flow originating from this interaction is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 199–202, September–October, 1984.     

Supersonic nonuniform viscous gas flow past a blunt body with supply of gas from the surface

The problem of axisymmetric nonuniform gas flow past smooth blunt bodies at high Mach numbers is investigated. The approach stream is a parallel axisymmetric flow in which the velocity and temperature depend on the radial distance from the axis of symmetry and the pressure is constant. On the axis of symmetry the velocity has a minimum and the temperature a maximum. A characteristic feature of this flow is the existence of two qualitatively different flow regimes: separated [1-4], when in the shock layer on the front of the body there is a closed region of reverse-circulating flow, and unseparated [5, 6], when there is no such zone. In this study the case of unseparated flow is investigated. The equations of a thin viscous shock layer with generalized Rankine-Hugoniot conditions at the shock and boundary conditions on the body that take into account the supply of gas from the surface are solved numerically. The effect of the gas supply on the conditions of unseparated flow is analyzed in relation to the Reynolds number, and the critical values of the nonuniformity parameter a = a_k [5] are obtained. It is shown that at high Reynolds numbers the supply of gas from the surface has practically no effect on a_k, while at low and intermediate Reynolds numbers it reduces the region of unseparated flow. For high Reynolds numbers and an intense supply of gas from the surface an asymptotic solution of the problem is obtained for the neighborhood of the stagnation point. This is compared with the numerical solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 122–129, July–August, 1988.The authors wish to thank G. A. Tirskii for useful discussions of the results.     

Numerical simulation of shock wave intersections

A large number of papers, generalized and classified in [1, 2], have been devoted to unsteady gas flows arising in shock wave interaction. Experimental results [3–5] and theoretical analysis [6–9] indicate that the most interesting and least studied types of interaction arise in cases when there are several shock waves. At the same time, nonlinear effects, which depend largely on the nature of the shock wave intersections, become appreciable. Regions of existence of different types, of plane shock wave intersections have been analyzed in [10–13]. It has been shown that in a number of cases the simultaneous existence of different types of intersections is possible. The aim of the present paper is to study unsteady shock wave intersections in the framework of a numerical solution of the axisymmetric boundary-value problem that arises in the diffraction of a plane shock wave on a cone in a supersonic gas flow. Flow regimes that augment the experimental data of [3–5] and the theoretical analysis of [9] are considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 134–140, September–October, 1986.     

Effect of Gas Injection from the Surface of a Body on Its Interaction with a Temperature Inhomogeneity in a Supersonic Flow

The flow pattern in the shock layer and the aerodynamic characteristics of a hemisphere in unsteady axisymmetric interaction with a closed spherical hot-gas region embedded in the oncoming supersonic flow in the presence of intense injection of gas from the body surface into the shock layer are studied on the basis of the inviscid perfect gas model. Two cases are considered, namely, (1) when the radius of the permeable surface is greater than that of the temperature inhomogeneity and (2) when the injection is localized in the vicinity of the forward stagnation point and the permeable region is smaller the inhomogeneity.     

Unsteady thermocapillary convection in a nonuniformly heated fluid layer

In many technological processes, thin extended layers of nonuniformly heated fluid are used [1–3]. If they are sufficiently thin, thermocapillary forces have a decisive influence on the occurrence and development of motion of the fluid [4–6]. Investigation of convective motion in such a layer is of great interest for estimating the intensity of heat and mass transfer in technological processes. This paper is a study of unsteady thermocapillary motion in a layer of viscous incompressible fluid with free surface in which a thermal inhomogeneity is created at the initial time. Approximate expressions are obtained for the fields of the velocity, temperature, and pressure in the fluid, and also for the shape of the free surface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 17–25, May–June, 1991.     

Unsteady viscous shock layer near the leading edge of a wing of infinite span

The effect of unsteady injection and wall temperature variation on the parameters of the viscous shock layer near the stagnation line of a wing of infinite span at an angle of slip is investigated on the basis of the viscous shock layer model. An analytic solution of the nonstationary problem, valid near the surface of the wing for strong injection, is obtained. A numerical investigation is carried out and some results of calculating the unsteady viscous shock layer equations for various forms of the time dependence of the injection velocity and wing surface temperature are presented. The calculations are based on a finite-difference method of the second order of approximation in the space variable and the first order of approximation in time, which makes use of expression of the equations in divergence form, Newtonian linearization and vector sweeps across the shock layer. In the steady-state case the results of the calculations are in good agreement with the data of [7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 90–95, January–February, 1988.     

Generalization of the blunt-body problem in nonuniform supersonic flowfields

The properties of the steady-state gas flow in the shock layer associated with a blunt body immersed in a supersonic stream with an arbitrary shear-like nonuniformity are studied.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 164–170, March–April, 1996.     

Circulation flow at the front surface of a sphere in a supersonic wake

Several theoretical and experimental studies of supersonic flow past a blunt body located in the wake behind another body have been made [1–7]. It has been shown that a reverse-circulation flow can occur in the shock layer at the front surface. The possibility of such a flow forming depends on the nonuniformity of the freestream flow and the Reynolds number. This paper presents new results of the theoretical study of the structure of the shock wave at the front surface of such a sphere, obtained on the basis of numerical solution of Navier-Stokes equations. It is shown that for a fixed nonuniformity of the freestream flow, an increase in the Reynolds number and cooling of the surface of the body lead to the formation of a secondary vortex in the region where the contour of the body intersects the axis of symmetry. A study is made of the variations of the drag and heat transfer parameters over the front surface of a cooled and thermally insulated sphere. The possibility of numerical simulation of the flow on the basis of the Euler equations is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 143–148, May–June, 1985.     

Two-phase mass injection from the stagnation zone of a blunt body in a hypersonic flow

A mathematical model of the hypersonic steady gas flow over the stagnation zone of an axisymmetric blunt body with given two-phase injection from the surface is proposed. The two-continuum model of a dusty gas [3] is used for describing the flow in the region of the wall. The problem is solved in the boundary layer and thin viscous shock layer approximations. On the basis of the numerical calculations the distribution of the parameters of the carrier and dispersed phases near the axis of symmetry is obtained. The similarity parameters determining the convective heat transfer are found. The stagnation point heat fluxes with and without particles are compared. The range of parameters on which particles can significantly reduce the heat transfer is determined.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 60–66, July–August, 1992.     

Multicomponent chemically-reacting turbulent viscous shock layer on a catalytic surface

Turbulent flows past blunt bodies at high supersonic speeds are mainly investigated within the framework of the boundary layer model. However, even at large Reynolds numbers owing to the strong entropy gradient on the lateral surface it becomes necessary to take boundary layer corrections into account in the higher approximations [1]. The use of viscous shock layer theory makes it possible to obtain fairly accurate results over a broad interval of variation of the Reynolds numbers without organizing iterations with respect to vorticity and displacement thickness. The nonequilibrium nature of both homogeneous and heterogeneous catalytic reactions is taken into account. The results obtained are compared with the experimental data [2, 3]. Previously, in [4, 5] turbulent flow was investigated within the framework of viscous shock layer theory in the case of equilibrium homogeneous reactions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 144–149, March–April, 1989.     

Disturbance evolution in the leading-edge region of a blunt flat plate in supersonic flow

The spatio-temporal dynamics of small disturbances in viscous supersonic flow over a blunt flat plate at freestream Mach number M_∞=2.5 is numerically simulated using a spectral approximation to the Navier–Stokes equations. The unsteady solutions are computed by imposing weak acoustic waves onto the steady base flow. In addition, the unsteady response of the flow to velocity perturbations introduced by local suction and blowing through a slot in the body surface is investigated. The results indicate distinct disturbance/shock-wave interactions in the subsonic region around the leading edge for both types of forcing. While the disturbance amplitudes on the wall retain a constant level for the acoustic perturbation, those generated by local suction and blowing experience a strong decay downstream of the slot. Furthermore, the results prove the importance of the shock in the distribution of perturbations, which have their origin in the leading-edge region. These disturbance waves may enter the boundary layer further downstream to excite instability modes.     

Hypersonic three-dimensional viscous shock layer in a nonuniform gas flow in the neighborhood of the stagnation point

Three-dimensional hypersonic viscous gas flow past smooth blunt bodies in the presence of injection or suction is considered. The effect of the nonuniformity of the approach stream on the shock-wave standoff, the flow structure and the friction and heat transfer coefficients is investigated with reference to the examples of flow from a supersonic spherical source and flow of the far wake type. It is shown that this effect depends importantly on the Reynolds number, the nature of the nonuniformity and the shape of the body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 136–145, November–December, 1987.     

Calculation of interaction of a turbulent near-wake behind a step with a supersonic jet

Existing computational methods [1–5] do not enable one to calculate complex flows behind steps, accounting for nonuniformity of the incident supersonic flow and the effect of compression and expansion waves arriving in the near-wake region. For example, computational methods based on the methods of [1] or [2] are used mainly in uniform supersonic flow ahead of the base edge and, for the most part, cannot be used to calculate flow in annular nozzles with irregular conditions. An exception is reference [6], which investigated flow in an annular nozzle behind a cylindrical center-body. The present paper suggests a method, based on references [7, 8] for calculating the base pressure behind two-dimensional and three-dimensional steps, washed by a supersonic jet.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 43–51, November– December 1977.