Asymptotic solution to the problem of hypersonic flow over bodies in the neighborhood of the separation point of a thin shock layer

A study is made of the problem of hypersonic flow of an inviscid perfect gas over a convex body with continuously varying curvature. The solution is sought in the framework of the asymptotic theory of a strongly compressed gas [1–4] in the limit M when the specific heat ratio tends to 1. Under these assumptions, the disturbed flow is situated in a thin shock layer between the body and the shock wave. At the point where the pressure found by the Newton-Buseman formula vanishes there is separation of the flow and formation of a free layer next to the shock wave [1–4]. The singularity of the asymptotic expansions with respect to the parameter ₁ = ( –1)/( + 1) associated with separation of the strongly compressed layer has been investigated previously by various methods [3–9]. Local solutions to the problem valid in the neighborhood of the singularity have been obtained for some simple bodies [3–7]. Other solutions [7, 9] eliminate the singularity but do not give the transition solution entirely. In the present paper, an asymptotic solution describing the transition from the attached to the free layer is constructed for a fairly large class of flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 99–105, January–February, 1982.     

Supersonic air flow past a sphere with account for vibrational relaxation

A numerical solution is obtained for the problem of air flow past a sphere under conditions when nonequilibrium excitation of the vibrational degrees of freedom of the molecular components takes place in the shock layer. The problem is solved using the method of [1]. In calculating the relaxation rates account was taken of two processes: 1) transition of the molecular translational energy into vibrational energy during collision; 2) exchange of vibrational energy between the air components. Expressions for the relaxation rates were computed in [2]. The solution indicates that in the state far from equilibrium a relaxation layer is formed near the sphere surface. A comparison is made of the calculated values of the shock standoff with the experimental data of [3].Notation uV_max, vV_max velocity components normal and tangential to the sphere surface - V_max maximal velocity - P V _max ² pressure - density - TT temperature - e_viRT vibrational energy of the i-th component per mole (i=–O₂, N₂) - =rb–1 shock wave shape - a _f the frozen speed of sound - HRT/m gas total enthalpy     

Two-Phase Models of Formation of Cavitating Spalls in a Liquid

The dynamics of the structure of a liquid layer structure (with microbubbles of a free gas) behind a rarefaction wave front is studied numerically using the two-phase Iordansky–Kogarko–van Wijngaarden model and the frozen mass-velocity field model. An analysis of the initial stage of cavitation by the Iordansky–Kogarko–van Wijngaarden model showed that tensile stresses behind the rarefaction wave front relax quickly and the mass-velocity field in the cavitation zone turns out to be frozen. This effect is used to describe the late stage of the development of the cavitation zone. These models were combined to study the formation of cavitating spalls in a free-surface liquid under shock-wave loading.     

Investigation of the flow past wings of infinite span with a blunt leading edge in the vorticity interaction regime

An asymptotic analysis of the Navier-Stokes equations is carried out for the case of hypersonic flow past wings of infinite span with a blunt leading edge when 0, Re , and M . Analytic solutions are obtained for an inviscid shock layer and inviscid boundary layer. The results of a numerical solution of the problems of vorticity interaction at the blunt edge and on the lateral surface of the wing are presented. These solutions are compared with the solution of the equations of a thin viscous shock layer and on the basis of this comparison the boundaries of the asymptotic regions are estimated.deceasedTranslated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 120–127, November–December, 1987.     

Spatial resolution and measurement of turbulence in the viscous sublayer using subminiature hot-wire probes

Measurements in the viscous sublayer of a flat-plate turbulent boundary layer in air, using single hot-wire sensors with lengths from 1–60 viscous length scales show that, at a given distance from the surface, the turbulence intensity, flatness factor, and skewness factor of the longitudinal velocity fluctuation are nearly independent of wire length when the latter is less than 20–25 times the viscous length scale (i.e. 20–25 wall units), and decrease significantly and abruptly for larger wire lengths. This conclusion is consistent with other workers' probability density functions of streak spacing: the lateral spacing of streaks in the viscous sublayer is 80–100 wall units on average with minimum spacing of 20–25 wall units, which implies that signals would be strongly attenuated by wires whose length exceeds 20–25 wall units. To achieve wire lengths of less than 20–25 wall units, subminiature hot wire probes like those described by Ligrani and Bradshaw (1987), having lengths as small as 150 m, are necessary for sublayer measurements in typical laboratory wind tunnels. As well as the measurements mentioned above, dissipation spectra are presented, to show the effect of spanwise averaging on the high-frequency motions, which is necessarily more severe than the effect on overall intensities.     

Hypersonic flow past blunt-edged delta wings

A method is proposed for calculating hypersonic ideal-gas flow past blunt-edged delta wings with aspect ratios = 100–200. Systematic wing flow calculations are carried out on the intervals 6 M 20, 0 20, 60 80; the results are analyzed in terms of hypersonic similarity parameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 175–179, September–October, 1990.     

Construction of exact numerical solutions of the stationary traveling wave type for viscous thin films

An effective numerical procedure, based on the Galerkin method, for finding solutions of the stationary traveling wave type in the complete formulation is proposed for the case of viscous liquid films. Examples of a viscous film flowing freely down a vertical surface have been calculated. The calculations have been made for various values of the dimensionless surface tension , including =0. The method makes it possible to predict a number of bifurcations that occur as decreases. The existence of numerous families of stationary traveling waves when 1 was demonstrated in [6]. The present study shows that as 1 all but one of these families of wave solutions disappear. The shape of the periodic and solitary waves and the pressure distribution in the film are found for various . When =0 and the wave number is fairly small, the periodic solution has a singularity, as predicted in [14]: at the crest of the wave a corner point appears; the angle between the tangents at this point =140–150. The method proposed can be used to calculate other wavy film flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 94–100, May–June, 1990.     

Screening of selective radiation in a boundary layer

An analysis of numerous calculations of the flow and radiative-convective heat exchange in a hypersonic shock layer near a blunt body, both at an impermeable surface [1] and in the presence of ablation [1–4], made it possible to establish some relationships connected with the screening in the boundary layer of radiation from the high-temperature part of the shock layer. It was established that the dependence of the screening coefficient of the boundary layer, which consists of the ratio of the integral flux q⁺ _RW of radiation incident on the body to the flux q⁺ _RW arriving at the outer limit of the boundary layer from the high-temperature region behind the shock wave, on the radius has a nonmonotonic character. In this case the absolute value of the minimum _min is determined mainly by the temperature T_S behind the shock wave and by the injection velocityf _W = (v)_W/(v), while its position (R_min) is determined by the pressure P_S in the shock layer. An analysis of the spectral characteristics of the screening of the boundary layer showed that the region of wavelengths > 0.5 m is very important from the point of view of the radiative heating and destruction of the surface. Components which efficiently absorb high-temperature radiation in this spectral interval are absent from the gaseous products of the disintegration of specific heat-protection coatings. An analysis is made permitting an evaluation of the possibility of introducing into the coatings additives possessing those optical and thermochemical properties which would provide for screening of radiation in this part of the spectrum.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 71–77, September–October, 1978.The authors thank A. B. Karasev for advice and a discussion.     

Nonexistence of a shock layer in gas dynamics with a nonconvex equation of state

A classical result of Gilbarg states that a simple shock wave solution of Euler's equations is compressive if and only if a corresponding shock layer solution of the Navier-Stokes equations exists, assuming, among other things, that the equation of state is convex. An entropy condition appropriate for weeding out unphysical shocks in the nonconvex case has been introduced by T.-P. Liu. For shocks satisfying his entropy condition, Liu showed that purely viscous shock layers exist (with zero heat conduction). Dropping the convexity assumption, but retaining many other reasonable restrictions on the equation of state, we construct an example of a (large amplitude) shock which satisfies Liu's entropy condition but for which a shock layer does not exist if heat conduction dominates viscosity. We also give a simple restriction, weaker than convexity, which does guarantee that shocks which satisfy Liu's entropy condition always admit shock layers.     

Spherical shock wave reflection from a surface with a heated gas layer

The two-dimensional axisymmetric problem of the interaction between smallscale spherical shock waves initiated by a laser explosion and an absolutely rigid surface in the presence of a layer of hot gas is numerically investigated. A number of effects previously observed in physical and numerical experiments [5–8] are confirmed, in particular: the distortion of the reflected shock front and its acceleration on passage through the hot central zone of the laser explosion (lens effect), the strong deformation of this zone, and the formation of a precursor on the surface ahead of the shock wave interacting with the thermal layer. In addition, certain new anomalous effects are revealed: the formation of a pair of suspended shocks — one on the periphery of the hot central zone upon interaction with the reflected shock wave and the other behind the Mach stem in the triple point zone, etc.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 141–147, July–August, 1989.     

Contribution to the asymptotic theory of quasisteady viscous flows

The aim of this paper is to obtain equations which, as Re , describe quasisteady viscous incompressible flows with stream surfaces that are closed only in the limit as Re , and also to examine the simpler consequences of these equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 46–51, July–August, 1985.The author wishes to thank V. Ya. Neiland for taking an interest in his work and discussing the results.     

Shock Waves in a Fluid with Bubbles Containing Evaporating Drops of a Liquefied Gas

The propagation and reflection of one-dimensional plane unsteady waves and pulses in a mixture of a fluid with two-phase bubbles containing evaporating drops is investigated. A significant effect of unsteady evaporation of the drops in the zone ahead of the shock wave on the wave propagation is demonstrated. The evaporation of the drops results in a pressure increase ahead of the wave and the shock wave as it were climbs to increasing pressure level. In contrast to bubbly fluids with single-phase bubbles, in a fluid with two-phase bubbles, at a fixed phase volume fraction, a decrease in bubble size results in an increase rather than a decrease of the oscillation amplitude. The wave reflection from a solid wall is essentially nonlinear and the maximum pressure attained at the wall is several times greater than the incident-wave intensity.     

Supersonic nonequilibrium flow past segmental bodies of a mixture simulating the atmosphere of venus

Calculations of the flow of the mixture 0.94 CO₂+0.05 N₂+0.01 Ar past the forward portion of segmentai bodies are presented. The temperature, pressure, and concentration distributions are given as a function of the pressure ahead of the shock wave and the body velocity. Analysis of the concentration distribution makes it possible to formulate a simplified model for the chemical reaction kinetics in the shock layer that reflects the primary flow characteristics. The density distributions are used to verify the validity of the binary similarity law throughout the shock layer region calculated.The flow of a CO₂+N₂+Ar gas mixture of varying composition past a spherical nose was examined in [1]. The basic flow properties in the shock layer were studied, particularly flow dependence on the free-stream CO₂ and N₂ concentration.New revised data on the properties of the Venusian atmosphere have appeared in the literature [2, 3] One is the dominant CO₂ concentration. This finding permits more rigorous formulation of the problem of blunt body motion in the Venus atmosphere, and attention can be concentrated on revising the CO₂ thermodynamic and kinetic properties that must be used in the calculation.The problem of supersonic nonequilibrium flow past a blunt body is solved within the framework of the problem formulation of [4].Notation V body velocity - shock wave standoff - universal gas constant - ratio of frozen specific heats - hRt/m enthalpy per unit mass undisturbed stream P pressure - density - T temperature - m molecular weight - c_p specific heat at constant pressure - (X) concentration of component X (number of particles in unit mass) - R body radius of curvature at the stagnation point - _j rate of j-th chemical reaction shock layer P V ² pressure - density - TT temperature - mm molecular weight Translated from Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 5, No. 2, pp. 67–72, March–April, 1970.The author thanks V. P. Stulov for guidance in this study.     

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.     

Particle-in-a-box study of the evolution of an ion-acoustic shock wave

In an analysis of a one-dimensional numerical model of a nonisothermal plasma it is shown that an ion-acoustic shock wave of subcritical amplitude separates a soliton from the shock front after the reversing stage. This process is accompanied by turbulent flow behind the front and by trapping of ions in potential wells. The numerical particle-in-a-box method is being used widely to study plasma phenomena. One field in which this method has been found fruitful is in the study of a nonisothermal plasma, characterized by an ion-acoustic wave branch.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 3–5, May–June, 1971.The authors thank R. Z. Sagdeev for support and interest in this study.     

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.     

Hopf bifurcation in a three-dimensional system

In this paper,Liapunor-Schmidl reduction and singularity theory are employed to discuss Hopf and degenerate Hopf bifureations in global parametric region in a three-dimensional system x=-βx+y, y=-x-βy(1-kz), z=β[α(1-z)-ky²], The conditions on existence and stability are given.     

Investigation of a “hanging” shock wave near a generating point

The hodograph method is used to plot a hanging shock wave in the plane nonequilibrium supersonic flow of an ideal gas. This paper considers the general case of an analytical solution in the plane of the hodograph at the point of generation of the shock wave. A type of limiting line is established which makes it possible to plot a shock wave (it is found that the shock wave may not extend over the whole flow, with a convolution in the physical plane).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 30–37, November–December, 1971.     

Influence of rotation of the body and exterior vorticity on the heat transfer near the stagnation point of a blunt body in a supersonic stream

A study is made of the influence of longitudinal vorticity on the hypersonic viscous shock layer near an axisymmetric cooled surface that rotates about the longitudinal axis with angular velocity ₁ [1, 2]. The equations of the viscous shock layer for the neighborhood of the stagnation point are simplified on the basis of the theory of a thin three-dimensional shock layer [1, 2]. The results are given of some calculations of the influence of the parameters and ₁ on the heat transfer and the structure of the shock layer in the case of steady flow. An iterative numerical method proposed by the author [1] is used, and a modification to accelerate the convergence of the iterations is proposed. It is noted that the parameters and ₁ have characteristic ranges of variation, which depend on the Mach and Reynolds numbers, for which the distributions of the streamlines in the shock layer are qualitatively different.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 179–182, May–June, 1984.I thank V. Ya. Neiland and Yu. D. Shevelev for helpful discussions of this paper.     

Some problems concerning nonstationary currents in dense plasma systems confined by walls

Nonstationary currents are examined in a dense magnetized plasma with 1, in which energy release and heat loss by thermal conduction and radiation are possible. Solutions are found in two limiting cases: ¦f¦ ¦ div (T)¦ and ¦f¦ ¦ div(T)¦ (f is the radiation intensity, is the coefficient of heat conduction, and T is the temperature). In the first case a solution was obtained of some problems of the cooling and heating of a plasma illustrated in part by the evolution in time of the temperature profile in the boundary layer. In the second case an isomorphic solution was found for an arbitrary dependence of the coefficient of heat conduction on the temperature, pressure, and magnetic field.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 3–8, January–February, 1972.The author is grateful to G. I. Budker for formulating the problem.