Rotation-vibration-dissociation interaction in a multicomponent nonequilibrium viscous shock layer

A new, simple and physically adequate method of calculating vibrationally nonequilibrium dissociation constants is proposed on the basis of a dissociation model which takes into account the equilibrium excitation of the rotational degrees of freedom of the molecules and the nonequilibrium excitation of vibrational quantum states. This rotation-vibration-dissociation interaction model contains only the indeterminacy associated with the indeterminacy of the experimental data on the interaction potentials and the collision cross sections of the components. In the case of thermodynamic equilibrium the model gives values of the dissociation constants close to those generally accepted. The use of this model in multicomponent nonequilibrium total viscous shock layer calculations gives values for the shock detachment distance within 5% of the experimental values. The indeterminacy in the values of the vibrational energy lost by air molecules during dissociation and recovered during recombination does not lead to serious errors in the macrocharacteristics of the flow. The nonequilibrium excitation of vibrational degrees of freedom proves to be not so important in computing the macrocharacteristics of the flow as previously assumed and the existing algorithms for calculating chemically nonequilibrium flows on the assumption of thermodynamic equilibrium can be used with satisfactory accuracy for calculating the values of the heat flux, the position of the shock wave, and the temperature and pressure in the shock layer for partially dissociated and ionized air.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 166–180, November–December, 1994.     

Calculation of a nonequilibrium viscous shock layer with allowance for coupled heat transfer

An investigation is made into the characteristics of coupled heat and mass transfer using the theory of a nonequilibrium viscous shock layer in the case of an axisymmetric blunt body moving along a given trajectory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 146–153, March–April, 1984.     

A hypersonic chemically nonequilibrium viscous shock layer oh wings with a catalytic surface

Within the framework of the theory of a hypersonic viscous shock layer a study is made of flow round wings of infinite span with blunt leading edges at various angles of attack and slip. Account is taken of multicomponent diffusion, and homogeneous chemical reactions, including dissociation-recombination reactions and exchange reactions. On the shock wave the generalized Rankine-Hugoniot conditions are given, and on the surface of the body conditions which allow for heterogeneous catalytic reactions of the first order with reaction rate constants depending [1] or not depending [2] on the temperature. The cases of an ideally catalytic and a noncatalytic surface are also considered. The surface of the body is assumed to be heatinsulated. A numerical study was made of the problem in a broad range of variation in the angles of attack and slip for different cases of prescribed constants representing the rates of the heterogeneous reactions. The conditions of the flow corresponded to the motion of a body which possess a lifting force along the trajectory of entry into the Earth's atmosphere [3]. The dependences are given of the equilibrium temperature of the surface along the stagnation line of the wing on the height of the flight and the distribution of this temperature along the surface of wings with parabolic and hyperbolic contours. It is shown that for flow regimes with a relatively high degree of dissociation in cases when the proportion of atoms recombined on the surface of the body is small, the dependences of the heat flow and the temperature of the surface on the angle of slip are of a nonmonotonic nature.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhldkosti i Gaza., No. 6, pp. 127–135, November–December, 1984.     

Three-dimensional chemically nonequilibrium viscous shock layer on a catalytic surface with allowance for associated heat transfer

A three-dimensional flow of dissociating air past blunt bodies is investigated in the framework of the thin viscous shock layer theory. Multicomponent diffusion and homogeneous chemical reactions, including dissociation, recombination, and exchange reactions, are taken into account. The generalized Rankine-Kugoniot conditions are specified on the shock wave and the conditions which take into account the heterogeneous catalytic reactions, on the surface of the body. The viscous shock layer equations are solved together with the heat equations inside the coating, which is carbon with a deposited thin film of SiO₂, or quartz. The case of a thermally insulated surface is also considered. The problem for the case of the motion of a body along the re-entry trajectory into Earth's atmosphere is investigated numerically. The temperature of the surface and the heat flux toward it are given as a dependence on the height (tine) of the flight for different cases of the specification of the catalytic reactions. It is shown that the difference between the heat fluxes towards the thermally insulated surface and the fluxes toward the heat-conducting surface in the neighborhood of the stagnation point is of the order of 6–12% for all the cases considered. This makes it possible to decouple the solution of the problem of heat conduction in the body.Translated fron Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 140–146, November–December, 1985.deceased     

Investigation of stationary flow of viscous gas in a thin three-dimensional shock layer

A three-dimensional shock layer near the blunt surface of a fairly smooth body is analyzed asymptotically. Equations of the first approximation are obtained and justified in various cases of the limit 1, 0, ( – 1)^–1M ^-2 0. These equations are simplified for the flow near the stagnation point of a body with double curvature and near the blunt leading edge of a sweptback wing. The results of some calculations are given and compared with the results of [17, 18] in the case of axisymmetric flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 115–126, September–October, 1980.     

Thermochemical nonequilibrium viscous shock layer studies of the orbital reentry experiment (OREX) vehicle

In 1994, the Orbital Reentry Experiment (OREX) was conducted by the National Aerospace Laboratory and National Space Development Agency of Japan. During this experiment aerodynamic heating and saturated ion current were successfully measured. In this paper, the thermochemical nonequilibrium shock layer of the OREX vehicle has been reconstructed using a VSL (Viscous Shock Layer) numerical simulation. In addition, heating to the OREX vehicle and electron density have been calculated and compared with the flight data. The numerical simulation has been made for the flight conditions at altitudes ranging from 96.7 km to 59.6 km, using a three temperature model composed of translational-rotational, vibrational and electron-electronic temperatures. The catalytic efficiency of the coating material on the nose of the vehicle was determined by comparing the flight data for heat flux with the numerical result for the finite catalytic wall. The calculated electron density values were in approximate agreement with the flight values. Received 17 June 2001 / Accepted 23 September 2001     

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.     

Wave-flow theory for a thin viscous liquid layer

On the basis of a simplified system of equations we study the process of development and stability of wave flows in a thin layer of a viscous liquid. Any unstable disturbance of the laminar flow grows and leads to the establishment of the wave regime. The time to establish the flow changes little for large flow rates, but increases sharply with reduction of the flow rate. Given the same amplitudes of the initial disturbances, the optimum regimes which provide the greatest flow rate in a layer of given average thickness develop more rapidly than the other regimes. All the wave regimes are unstable to disturbances in the form of traveling waves. With moderate flow rates, the optimum regimes will be most stable to near-by disturbances.Strictly periodic wave flows in a thin layer of a viscous liquid under the influence of the gravity force were calculated in [1], Various flow wave regimes which differ in wavelength can theoretically be established for a given liquid flow rate. In particular, there is a wavelength for which the flowing layer exhibits minimum average thickness (and maximum flow rate for a given average thickness). These optimum regimes correspond closely to the experimental data [2]; however, with regard to calculation technique these regimes are no different from the others. In the following we make a comparison of the wave regimes on the basis of the nature of their development and stability.     

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.     

Flow of a viscous gas in a shock layer with equilibrium chemical reactions

Steady flow of supersonic air over a sphere is examined, allowing for viscosity, heat conduction, and actual physical and chemical processes. Flow in the shock layer at flight speeds in the range 3 km/sec V10 km/sec (10⁴R10⁶) is investigated, under the assumption of local thermodynamic equilibrium. The flow is described by simplified Navier-Stokes equations, which are solved by a finite difference method. The case of a cooled surface is examined. The distribution of gasdynamic parameters is obtained in different flow regimes. The distribution of heat flux and friction coefficient is investigated as a function of the oncoming-stream parameters and the sphere radius. The shape and position of the shock wave are determined, and the stream lines and sonic lines are constructed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 150–153, July–August, 1970.The authors thank Yu. P. Lun'kin and F. D. Popov for their help in formulating the problem and their constant interest.     

Asymptotic solution of the thin viscous shock layer equations at low Reynolds numbers for a cold surface

Hypersonic three-dimensional viscous rarefied gas flow past blunt bodies in the neighborhood of the stagnation line is considered. The question of the applicability of the gasdynamic thin viscous shock layer model [1] is investigated for the transition flow regime from continuum to free-molecular flow. It is shown that for a power-law temperature dependence of the viscosity coefficient T the quantity (Re)^1/(1+), where = ( – 1)/2 and is the specific heat ratio, is an important determining parameter of the hypersonic flow at low Reynolds numbers. In the case of a cold surface approximate asymptotic solutions of the thin viscous shock layer equations are obtained for noslip conditions on the surface and generalized Rankine-Hugoniot relations on the shock wave at low Reynolds numbers. These solutions give simple analytic expressions for the thermal conductivity and friction coefficients as functions of the determining flow parameters. As the Reynolds number tends to zero, the values of the thermal conductivity and friction coefficients determined by this solution tend to their values in free-molecular flow for an accommodation coefficient equal to unity. This tending of the thermal conductivity and friction coefficients to the free-molecular limit takes place for both two-and three-dimensional flows. The asymptotic solutions are compared with numerical calculations and experimental data.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 159–170. Original Russian Text Copyright © 2004 by Brykina.     

Unsteady effects in a thin viscous shock layer near a three-dimensional stagnation point of a body moving with given acceleration and deceleration

An unsteady viscous shock layer near a stagnation point is studied. The Navier-Stokes equations are analyzed in the limit 1, Re₀ , df/dt = ^n-mF(t/^m). The Reynolds number Re₀ is defined in the paper by Eq. (1.3) (df/dt is the velocity of the body with respect to an inertial frame of reference moving with the original steady velocity –V'_t8, ₂ = ( – 1)/( + 1)). Various flow regimes in the case 1, l, n max(2m, m + 1), m 0, where ² = 1/Re₀ are analyzed. Equations are derived that generalize the asymptotic analysis to the case of a viscous unsteady flow of gas in a thin three-dimensional shock layer. The problem of a thin unsteady viscous shock layer near the stagnation point of a body with two curvatures is formulated. Examples of numerical solution are given for different ratios of the principal curvatures of the body, the wall temperature, the parameters of the original steady flow, and the acceleration and deceleration regimes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 100–111, March–April, 1981.I thank Yu. D. Shevelev for a fruitful discussion of the work.     

Interaction of a shock wave with a loose dusty bulk layer

The interaction of a planar shock wave with a loose dusty bulk layer has been investigated both experimentally and numerically. Experiments were conducted in a shock tube. The incident shock wave velocity and particle diameters were measured with the use of pressure transducers and a Malvern particle sizer, respectively. The flow fields, induced by shock waves, of both gas and granular phase were visualized by means of shadowgraphs and pulsed X-ray radiography with trace particles added. In addition, a two-phase model for granular flow presented by Gidaspow is introduced and is extended to describe such a complex phenomenon. Based on the kinetic theory, such a two-phase model has the advantage of being able to clarify many physical concepts, like particulate viscosity, granular conductivity and solid pressure, and deduce the correlative constitutive equations of the solid phase. The AUSM scheme was employed for the numerical calculation. The flow field behind the shock wave was displayed numerically and agrees well with our corresponding experimental results.      

Sporadic interaction of a strong shock wave with a thin cone

The axisymmetrical problem concerning Mach reflection of an intense shock wave from the surface of a thin cone is considered in a formulation similar to that of [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 117–123, November–December, 1976.The authors express their appreciation to A. A. Grib for discussion of the work and for useful comments.     

Three-dimensional thin viscous shock layer in the absence of planes of symmetry in the flow

The basic laws of viscous homogeneous gas flow at high supersonic speeds past smooth blunt bodies with a permeable surface are investigated within the framework of the thin viscous shock layer model. An efficient numerical method of solving these equations, which makes it possible to consider cases of flow past bodies at angles of attack and slip, when there are no planes of symmetry in the flow, is proposed. Some results of calculating the flow past a triaxial ellipsoid with an axial ratio of 103n73 at angles of attack =0–45° and slip angles =0–45° over a broad interval of Reynolds numbers are presented as an example. The effect of the principal determining parameters of the problem on the flow structure in the shock layer and the surface friction and heat transfer coefficients is analyzed. An expression for calculating the heat fluxes to the impermeable surface of smooth blunt bodies in a supersonic homogeneous viscous gas flow over a broad interval of Reynolds numbers is proposed on the basis of the solutions obtained and the results of other authors.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 150–158, March–April, 1989.