Calculation of inviscid radiating gas flow past a blunt body

During hypersonic gas flow past a blunt body with a velocity on the order of the escape velocity or more, the gas radiation in the disturbed region behind the shock wave becomes the primary mechanism for aerodynamic heating and has a significant effect on the distribution of the gasdynamic parameters in the shock layer. This problem has been considered from different points of view by many authors. A rather complete review of these studies is presented in [1–4].In earlier studies [5, 6] the approximation of bulk emission was used. In this approximation, in order to account for the effect of radiative heat transfer a term is added in the energy equation which is equivalent to the body efflux, whose magnitude depends on the local thermodynamic state of the gas. However, the use of this assumption to solve the problem of inviscid flow past a blunt body leads to a singularity at the body [7, 8]. To eliminate the singularity, account is taken of the radiation absorption in a narrow wall layer [7], or the concept of a viscous and heat-conductive shock layer is used [8]. A further refinement was obtained by Rumynskii, who considered radiation selectivity and studied the flow of a radiating and absorbing gas in the vicinity of the forward stagnation point of a blunt body.In the present paper we study the distribution of the gasdynamic parameters in the shock layer over the entire frontal surface of a blunt body in a hypersonic flow of a radiating and absorbing gas with account for radiation selectivity.     

A method for calculating intense injection from surfaces of blunt bodies entering planetary atmospheres at hypersonic velocities

Flow past blunt bodies entering planetary atmospheres at hypersonic velocities is studied. A method for calculating the flowfield near the body nose is developed which allows for radiative heat transfer in the P ₁ approximation of the spherical harmonics method but does not take gas viscosity and heat conduction into account. The solution is constructed on the basis of a two-layer flow model, with account for intense injection of ablation products from the body nose due to radiative heat fluxes from the shock layer. The advantages of the method are that the multi-dimensional character of the radiation field is taken into account and the general problem of radiation gasdynamics is solved on the basis of a unified algorithm. The flow past a spherical segment and a spherically-blunted cone re-entering the Earth’s atmosphere at a velocity of 20 km/s and an entry angle of ?10° is calculated.     

Hypersonic viscous shock layer in a swirling gas flow on a permeable surface

A hypersonic swirling flow of viscous compressible gas past rotating axisymmetric blunt bodies is considered, its velocity vector being parallel to the axis of rotation of the body. The body surface is assumed permeable, while, in the general case, the gas is not injected (drawn off) along the normal to the body surface. An analytic solution of the problem, valid at small Reynolds numbers, is found in the first approximation of the integral method of successive approximations. On the basis of the results of the numerical solution, obtained in a wide range of variation of the determining parameters of the problem, we investigate the influence of the swirling of the free-stream flow, the angular velocity of rotation of the body, the Reynolds number and the injection (suction) parameter on the structure of the compressed layer, and the coefficients of friction and heat transfer on the body surface. The influence of the swirling of the flow on the nature of the asymptotic behavior of the viscous shock layer equations at large Reynolds numbers is studied. It is shown that the presence of a nonzero peripheral component for the velocity vector of the gas in the shock layer can lead to a qualitative change in the nature of the flow. Deceased Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 27–37, November–December, 1986. The authors thank G. G. Chernyi for his useful discussion of the results of the work.     

Strong mass injection at the surface of a blunt body in a supersonic flow

The case of supersonic flow over a blunt body when another gas is injected through the surface of the body in accordance with a given law is theoretically investigated. If molecular transport processes are neglected, the flow between the shock wave and the surface of the body should be regarded as two-layer, that is, as consisting of the flow in the shock layer between the shock wave and the contact surface and the flow in the layer of injected gas. A numerical solution of the problem is obtained near the front of the body and its accuracy is estimated. Approximate analytic solutions are obtained in the injected-gas layer: a constant-density solution and a solution of the boundary-layer type in the local similarity approximation. Near the flow axis the numerical and analytic solutions are fairly close, but at a distance from the axis the assumptions made reduce the accuracy of the approximate solutions. The flow in question can serve as a gas-dynamic model of a series of problems describing the radiant heating of blunt bodies in a hypersonic flow. In the presence of intense radiative heat transfer, vaporization is so great that the thickness of the vapor layer is comparable with the thickness of the shock layer. Moreover, the thermal shielding of various kinds of obstacles in channels through which a radiating plasma flows can be organized by means of the forced injection of a strong absorber. The formulation of a similar problem was reported in [1], but the results of the solution were not given. A two-layer model of the flow of an ideal gas over a blunt body was used in [2, 3] for the analysis of radiative heat transfer. In [2] the neighborhood of the stagnation point is considered. In [3] preliminary results relating to two-layer flow over blunt cones are presented. The solution is obtained by Maslen's approximate method.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 89–97, March–April, 1972.     

Second approximation in the problem of the strong viscous interaction on slender three-dimensional bodies

We consider the hypersonic flow of a perfect gas past a slender three-dimensional body in a regime of strong viscous interaction. We give equations which make it possible to reduce the problem of determining the aerodynamic characteristics of a body which is not axisymmetric to the problem of computing the flow past an equivalent body of rotation at zero angle of incidence. The second approximation for the heat transfer and drag coefficients is found by the method of external and internal combinations of asymptotic expansions. The region in which this method can be applied and the accuracy of the asymptotic theory are estimated by comparison with exact numerical computations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti, i Gaza, No. 5, pp. 107–113, September–October, 1970.     

Interference of oblique shock waves of one family in a hypersonic flow

A study is made of the irregular regime of interaction of two shock waves of the same direction when a hypersonic gas stream flows past bodies of complicated shape. It is shown that the rarefaction waves formed in the flow field significantly weaken the shock wave that approaches the body. This effect is confirmed by the results of an experiment and numerical calculations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 134–138, September–October, 1982.     

Hypersonic flow past a wing at large angles of attack with detached shock wave

The thin shock layer method [1–3] has been used to solve the problem of hypersonic flow past the windward surface of a delta wing at large angles of attack, when the shock wave is detached from the leading edge (but attached to the apex of the wing) and the velocity of the gas in the shock layer is of the same order as the speed of sound. A classification of the regimes of flow past a delta wing at large angles of attack has been made. A general solution has been obtained for the problem of three-dimensional hypersonic flow past the wing allowing for nonequilibrium physicochemical processes of thermal radiation of the gas at high temperatures.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 149–157, May–June, 1985.     

Investigation of a hypersonic viscous shock layer on rotating axisymmetric bodies in the presence of blowing

The hypersonic flow of a laminar stream of viscous compressible gas past blunt axisyrametric bodies rotating about the longitudinal axis is considered. It is assumed that gas blows from the surface of the body. The solution of the problem is obtained by a finite-difference method in a wide range of Reynolds numbers and blowing and rotation parameters. Some results of the calculations characterizing the effect of the rotation on the velocity and temperature profiles across the shock layer, on the friction and heat transfer coefficients, and the shock wave separation are given for the neighborhood of the stagnation point. For large Reynolds numbers and strong blowing an analytic solution of the problem is found in an approximation of two inviscid layers separated by a contact surface. The calculations are made for the flow past a sphere and a paraboloid and it is shown that in the presence of rotation the maximum of the heat flux is shifted from the stagnation point onto the side surface of the body. The dependence of the pressure distribution, the heat flux, and the friction coefficient is investigated for cases of constant and variable blowing over the contour of the body.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 106–114, January–February, 1986.     

Radiative transfer effects of an axisymmetric particulate jet into a cylindrical pipe

This paper discusses the radiative transfer effects of an axisymmetric gas-particles jet into a cylindrical pipe. The medium is gray and it participates to radiation by emission, absorption and scattering. The two-phase flow problem is solved numerically by the finite volume method. We investigate the radiative transfer through a sensitivity analysis which considers the effects of the particle radiative properties and the particle number density on the temperature field and on the radiative heat fluxes of the two-phase flow domain. Analysis of the temperature profile in the cylinder, without and then with particle radiation effects, shows a decrease in the medium temperatures and thus an important role of the radiative transfer. These results also show that the presence of scattering makes the medium temperature more uniform. Finally, analysis of the particle number density, through the variation of the injection velocity, shows that a decrease in the injection velocity decreases the temperatures of the gas and particles and rapidly equilibrates the gas and particle temperatures.     

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.     

Method of mean-mass parameters for a three-dimensional boundary layer in a flow with vorticity

When a gas flows with hypersonic velocity over a slender blunt body, the bow shock induces large entropy gradients and vorticity near the wall in the disturbed flow region (in the high-entropy layer) [1]. The boundary layer on the body develops in an essentially inhomogeneous inviscid flow, so that it is necessary to take into account the difference between the values of the gas parameters on the outer edge of the boundary layer and their values on the wall in the inviscid flow. This vortex interaction is usually accompanied by a growth in the frictional stress and heat flux at the wall [2, 3]. In three-dimensional flows in which the spreading of the gas on the windward sections of the body causes the high-entropy layer to become narrower, the vortex interaction can be expected to be particularly important. The first investigations in this direction [4–6] studied the attachment lines of a three-dimensional boundary layer. The method proposed in the present paper for calculating the heat transfer generalizes the approach realized in [5] for the attachment lines and makes it possible to take into account this effect on the complete surface of a blunt body for three-dimensional laminar, transition, or turbulent flow regime in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 80–87, January–February, 1981.     

Asymptotic solution of the two-dimensional equations for a thin viscous rarefied shock layer on a cold surface

Hypersonic rarefied flow past blunt bodies is studied in the continuum-free-molecular transition regime. On the basis of an asymptotic analysis three rarefied gas flow patterns are established depending on the relation between the relevant parameters of the problem. In the first regime corresponding to a cold surface asymptotic solutions of the equations of a thin viscous shock layer are derived at low Reynolds numbers in the axisymmetric and plane cases. Simple analytical expressions for the pressure and the heat transfer and friction coefficients are obtained as functions of the freestream parameters and the body geometry. With decrease in the Reynolds number the coefficients approach the values corresponding to free-molecular flow. In this regime a similarity parameter for the hypersonic rarefied flow past bodies is determined. The asymptotic solutions are compared with numerical solutions and the results of direct statistical simulation by the Monte Carlo method.     

Investigation of the influence of blowing on the flow in a hypersonic viscous shock layer near the stagnation streamline on a blunt body

In the framework of the approximation of local similarity to the Navier-Stokes equations, an investigation is made of the axisymmetric flow of homogeneous gas in a hypersonic shock layer, this including the region of transition through the shock wave. Boundary conditions, which take into account blowing of gas, are specified on the surface of the body and in the undisturbed flow. A numerical solution to the problem is obtained in a wide range of variation of the Reynolds number and the blowing parameter. Expressions are found for the dependences on the blowing parameter usually employed in boundary layer theory of the coefficients of friction and heat transfer on the surface of the body, which are divided by their values obtained for blowing parameter equal to zero. It is shown that these dependences are universal and the same as the dependences obtained from the solution of the equations of a hypersonic viscous shock layer with modified Rankin-Hugoniot relations across the shock wave and from the solution of the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 199–202, January–February, 1980.     

Asymptotic solutions of the two-dimensional equations for a thin viscous shock layer in a rarefied gas

Two-dimensional hypersonic rarefied gas flow around blunt bodies is investigated for the continuum to free-molecular transition regime. In [1], as a result of an asymptotic analysis, three rarefied gas flow regimes, depending on the relationship between the problem parameters, were detected and one of these regimes was investigated. In the present study, asymptotic solutions of the thin viscous shock layer equations at small Reynolds numbers are obtained for the other two flow regimes. Analytical expressions for the heat transfer, friction and pressure coefficients are obtained as functions of the incident flow parameters and the body geometry and temperature. As the Reynolds number tends to zero, the values of these coefficients approach their values in free-molecular flow. The scaling parameters of hypersonic rarefied gas flow around bodies are determined for different regimes. The asymptotic solutions are compared with the results of direct Monte Carlo simulation.     

Slip boundary conditions on a catalytic surface in a multicomponent gas flow

The boundary conditions for the velocity slip and temperature and concentration jumps on the surface of a body in a rarefied multicomponent gas flow are obtained. The mathematical treatment is given in detail because of the need to examine critically some previous results which disagree with each other in spite of the fact that the initial premises and the methods of solution were the same. The results of this study, which are given in a convenient form, represent the boundary conditions for both the simplified and the complete Navier-Stokes equations in problems of hypersonic rarefied gas flow past bodies with a catalytically active surface.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 159–168, January–February, 1996.     

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.     

Investigation of hypersonic flow of rarefied gas past wings of infinite span

In the framework of the locally self-similar approximation of the Navier-Stokes equations an investigation is made of the flow of homogeneous gas in a hypersonic viscous shock layer, including the transition region through the shock wave, on wings of infinite span with rounded leading edge. The neighborhood of the stagnation line is considered. The boundary conditions, which take into account blowing or suction of gas, are specified on the surface of the body and in the undisturbed flow. A method of numerical solution of the problem proposed by Gershbein and Kolesnikov [1] and generalized to the case of flow past wings at different angles of slip is used. A solution to the problem is found in a wide range of variation of the Reynolds numbers, the blowing (suction) parameter, and the angle of slip. Flow past wings with rounded leading edge at different angles of slip has been investigated earlier only in the framework of the boundary layer equations (see, for example, [2], which gives a brief review of early studies) or a hypersonic viscous shock layer [3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 150–154, May–June, 1984.     

Dynamic interaction of a magnetized solid body with a rarefied plasma flow

Dependences of the drag and lift coefficients of a magnetized sphere in a hypersonic rarefied plasma flow on the angle between the plasma flow velocity and the self-magnetic field induction vector of the body are obtained in a wide range of the ratio of the magnetic pressure to the plasma flow pressure. It is shown that changing the orientation of the magnetic field vector of the body and the incoming flow velocity can be used to control the dynamic interaction in the plasma–body system, namely, to decelerate and accelerate the magnetized sphere in a rarefied hypersonic plasma flow.     

Boundary layer development in a gas flow with stagnation enthalpy varying across the streamlines

We consider a laminar boundary layer for which the stagnation enthalpy specified in the initial section is variable with height. Such problems arise, for example, for bodies located in the wake behind another body, for hypersonic flow past slender blunted bodies (as a result of the large transverse entropy gradients in the highentropy layer), for stepwise variation of the temperature of a surface on which there is an already developed boundary layer, for sudden expansion of the boundary layer as a result of its flow past a corner of the surface, etc.Strictly, we should in such cases solve the boundary layer equations (if the longitudinal gradients are much smaller than the transverse) with the specified initial distribution of the quantities. However, from the physical point of view, the distributed region may be broken down into two regions, the near-wall boundary layer and an outer region which is a gas flow with constant velocity and the specified initial temperature profile, whose calculation yields the edge conditions for the boundary layer. The boundary between the regions is determined from the condition of adequately smooth matching of the solutions. This approach is much preferable to the first, since it permits avoiding (within the framework of boundary layer theory) the difficulties associated with the presence of a possible singularity at the initial point of the surface due to the discontinuity of the boundary conditions at this point, and also permits using conventional boundary layer theory if the effect of the viscosity in the outer region is not significant. However, this partition requires additional justifications of the possibility of independent determination of the solution in the outer region and the determination of the edge of the boundary layer, considered as the region of influence of the wetted surface. The boundary layer in a nonuniform flow has been considered in several works for a linear initial velocity or temperature profile [1–3].It should be noted that the linear initial enthalpy or velocity profiles for constant gas properties do not undergo changes under the influence of viscosity or thermal conductivity. Thus the fundamental characteristic features noted above which are associated with the presence of the two regions and their interaction in essence cannot be investigated using these examples.In this study we obtain and analyze the exact solutions of the equations of the compressible boundary layer for a power-law variation of the initial stagnation enthalpy profile as a function of the stream function for a constant initial velocity. Here it is shown that the influence of the boundary conditions at the wall are actually localized in the near-wall boundary layer, which is similar in dimensions to the conventional velocity or thermal boundary layers. In the region which is external with relation to this layer, in accordance with the physical picture described above, the solution coincides with the solution of the Cauchy problem for the heat conduction equation, which describes the development of the initial temperature profile in an infinite steady-state flow with constant velocity.It is shown that for the sufficiently smooth initial profiles which are of interest in practice the outer flow undergoes practically no changes until we reach the inner boundary layer, and it may be calculated using the perfect gas laws.     

Translationally nonequilibrium free jet boundary layer in the wake behind a wedge

Macroscopic equations obtained as a thin-layer version of the 13-moment Grad equations derived from kinetic considerations are used for describing the translationally nonequilibrium monatomic gas flow in a hypersonic free jet boundary layer formed in the wake behind a wedge. This model makes it possible to investigate flows with strong violations of equilibrium with respect to the translational degrees of freedom. A method of constructing the solution of this kinetically justified problem based on the solution of an analogous problem in the Navier-Stokes interpretation is proposed. It is established that for the kinetic variant of the problem considered the gas flow velocity distribution along the separating streamline in a plane orthogonal to the wedge generator coincides with the distribution obtained in solving the Navier-Stokes variant. It is found that taking into account the nonequilibrium nature of the flow with respect to the translational degrees of freedom of the gas particles has no effect on the base pressure and the wake angle.