The criterion of the existence or inexistence of transverse shock wave at wedge supported oblique detonation wave

A simplified theoretic method and numerical simulations were carried out to investigate the characterization of propagation of transverse shock wave at wedge supported oblique detonation wave.After solution validation,a criterion which is associated with the ratio Φ (u 2 /u CJ) of existence or inexistence of the transverse shock wave at the region of the primary triple was deduced systematically by 38 cases.It is observed that for abrupt oblique shock wave (OSW)/oblique detonation wave (ODW) transition,a transverse shock wave is generated at the region of the primary triple when Φ < 1,however,such a transverse shock wave does not take place for the smooth OSW/ODW transition when Φ > 1.The parameter Φ can be expressed as the Mach number behind the ODW front for stable CJ detonation.When 0.9 < Φ < 1.0,the reflected shock wave can pass across the contact discontinuity and interact with transverse waves which are originating from the ODW front.When 0.8 < Φ < 0.9,the reflected shock wave can not pass across the contact discontinuity and only reflects at the contact discontinuity.The condition (0.8 < Φ < 0.9) agrees well with the ratio (D ave /D CJ) in the critical detonation.     

Wave angle for oblique detonation waves

The flow field associated with a steady, planar, oblique detonation wave is discussed. A revision is provided for- diagrams, where is the wave angle and is the ramp angle. A new solution is proposed for weak underdriven detonation waves that does not violate the second law. A Taylor wave, encountered in unsteady detonation waves, is required. Uniqueness and hysteresis effects are also discussed.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Modification of Whitham's ray shock theory for analyzing the reflection of weak shock waves over small wedge angles

The weak Mach reflection phenomenon has been analyzed by applying both the shock dynamics approach and the disturbance propagation concept. The analysis which is based on modified Whitham's ray shock theory results in analytical expressions for the triple point trajectory angle,, and the shape of the curved Mach stem, which are functions of the incident shock wave Mach number,M _i, and the reflecting wedge angle, _w. The analytical results were found to be in good agreement with experimental results.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Numerical simulation of boundary-layer transition at Mach two

The linear and early nonlinear stages of boundary-layer transition at free-stream Mach numberM ==2.0 are investigated by direct numerical simulation of the compressible Navier-Stokes equations. Results from simulations with a large computational box and small-amplitude random initial conditions are compared with linear stability theory. The growth rates of oblique waves are reproduced correctly. Two-dimensional waves show a growth that is modulated in time, indicating the presence of an extra unstable mode which moves supersonically relative to the free stream. Further simulations are conducted to investigate the nonlinear development of two- and three-dimensional disturbances The transition due to oblique disturbance waves is the most likely cause of transition at this Mach number, and is found to lead to the development of strong streamwise vortices.     

Effect of the flow regime in an underexpanded jet issuing into a subsonic counterstream on the pressure distribution on a body

The pressure distribution on a cone with a half-angle =75°, from which a single central underexpanded jet issues into a subsonic counterstream, has been experimentally investigated. The effect of the flow regime in the jet on the pressure distribution is demonstrated. Generalized relations for the pressure on the body are obtained for various jet-flow momentum ratios J and flow Mach numbers M = 0.35–0.9; the Mach number M_a at the exit of the conical nozzle with half-angle _a=10° was equal to 2.9. The working medium of the jet and the flow was air with stagnation temperatures T_0a = T₀ 260–265°K. The ratio of the nozzle outlet radius to the radius of the maximum cross section of the cone _a/R_M=0.1.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 182–185, January-February, 1988.     

Mechanisms of detonation transmission in layered H2-O2 mixtures

When a plane detonation propagating through an explosive comes into contact with a bounding explosive, different types of diffraction patterns, which may result in the transmission of a detonation into the bounding mixture, are observed. The nature of these diffraction patterns and the mode of detonation transmission depend on the properties of the primary and bounding explosives. An experimental and analytical study of such diffractions, which are fundamental to many explosive applications, has been conducted in a two channel shock tube, using H₂-O₂ mixtures of different equivalence ratios as the primary and bounding or secondary explosive. The combination of mixtures was varied from rich primary / lean secondary to lean primary / rich secondary since the nature of the diffraction was found to depend on whether the Chapman-Jouguet velocity of the primary mixture,D _p, was greater than or less than that of the secondary mixture,D _s. Schlieren framing photographs of the different diffraction patterns were obtained and used to measure shock and oblique detonation wave angles and velocities for the different diffraction patterns, and these were compared with the results of a steady-state shock-polar solution of the diffraction problem. Two basic types of diffraction and modes of detonation reinitiation were observed. WhenD _p>D _s, an oblique shock connecting the primary detonation to an oblique detonation in the secondary mixture was observed. WithD _p<D _s, two modes of reinitiation were observed. In some cases, ignition occurs behind the Mach reflection of the shock wave, which is transmitted into the secondary mixture when the primary detonation first comes into contact with it, from the walls of the shock tube. In other cases, a detonation is initiated in the secondary mixture when the reflected shock crosses the contact surface behind the incident detonation. These observed modes of Mach stem and contact surface ignition have also been observed in numerical simulations of layered detonation interactions, as has the combined oblique-shock oblique-detonation configuration whenD _p>D _s. WhenD _p>D _s, the primary wave acts like a wedge moving into the secondary mixture with velocityD _p after steady state has been reached, a configuration which also arises in oblique-detonation ramjets and hypervelocity drivers.     

Stress analysis of plates with a circular hole reinforced by flange reinforcing member

This paper deals with the problem of stress analysis of plates with a circular hole reinforced by flange reinforcing member. The so called flange reinforcing member here means that the reinforcing member is built up by setting shapes or bars with any section shape on both sides of the plates along the edge of the hole. Two cases of external loads are considered. In one case the external loads are stressesσ_X^(∞),σ_Y^(∞),and τ_XY^(∞) acting at infinite point of the plate, and in the other the external loads are linear distributed normal stresses. The procedure of solving the problems mentioned above consists of three steps. Firstly, the reinforcing member is taken out from the plates and considered to be a circular bar being solved to determine its deformation under the action of radial force q₀(θ) and tangential force t₀(θ) which are forces acting upon each other between reinforcing member and plate. Secondly, the displacements of plate with a circular hole under the action of q₀(θ) and t₀(θ) and external loads are determined. Finally, forces q₀(θ) and t₀(θ) are obtained by the compatibility of deformations between reinforcing member and plate. Then the internal forces and displacements of reinforcing member and plate are deduced from q₀(θ) and t₀(θ) obtained.     

Discrete vortices in the wake of various profiles in subsonic and supersonic two-dimensional gas flow

The Mach number dependence of the Strouhal number, the frequency of discrete vortices, the vortex velocity, and other parameters are determined in the wake of wedges and flat plates for low angles of attack. The studies were made using high-speed motion-picture photography through a Schlieren system and with photomultipliers. The results are presented in tabular and graphical form.Notation h transverse distance - l longitudinal distance between vortices - V freestream velocity in m/sec - n_v vortex frequency for one row of vortex street in sec - M freestream Mach number - S₁ Strouhal number based on projection of the model onto the plane perpendicular to the freestream direction - S₂ Strouhal number calculated from the wake neck width d₂ for M>1 - R Reynolds number calculated from d - R_* critical Reynolds number - model apex angle - angle of attack - L length in flow direction in mm The author wishes to thank G. I. Petrov for his interest in the study and his advice.     

Detonation combustion of hydrogen in an axisymmetric channel with a central body

The problem of initiation and stabilization of detonation combustion of a hydrogen–air mixture injected into an axisymmetric channel with a finite-length central body in a flow with a Mach number M₀ = 5–9 is solved. It is numerically demonstrated that the presence of the central body both in a convergent–divergent nozzle and in an expanding channel leads to stabilization of detonation combustion of a stoichiometric hydrogen–air mixture at free-stream Mach numbers M₀ > 7. Various channel configurations that ensure different values of thrust generated by detonation combustion of a stoichiometric hydrogen–air mixture are compared.     

Interaction of a shock wave with a two-phase interface

The acceleration by an incident shock of a planar interface between a gas and a particle-gas mixture has been investigated experimentally and numerically. The experiments were conducted in a newly developed vertical shock tube in which the planar interface of the particle-gas mixture was generated and its particle concentration history was measured. Polydisperse corn starch particles with a mean diameter of 10m were used. We recorded the motion of the interface, as well as of the incident and reflected shock by using a 4 channel spark shadowgraph. The experimental conditions were Mach numberM _s=5.15 and initial pressurep ₁=50kPa for various particle concentrations in nitrogen. The reflected shock appears with a delay after the incident shock enters the particle-gas mixture. Numerical methods were employed to solve the two-phase governing equations. Experiments and numerical solutions are in good agreement.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Flow of rarefied gas past a sweptback cylinder

A numerical investigation has been made of the hypersonic flow of a rarefied monatomic gas past the windward part of the side surface of an infinite circular cylinder. The calculation was made by direct statistical Monte Carlo modeling for freestream Mach number M_t8=20, ratio of the surface temperature of the body to the stagnation temperature equal to t_tw =T _tw/T _t0 = 0.03, sweep angle 75°, and Reynolds number Re_t0 30.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 146–154, January–February, 1992.     

Approximate relationships for the determination of the pressure on the surface of a sphere or cylinder with an arbitrary mach number in the free stream

Numerical calculations have been made [1–4] of the pressure distribution over the surface of a sphere or cylinder during transverse flow in the range 0 /2, where is the angle reckoned from the stagnation point along the meridional plane, and on the basis of these results simple analytical equations have been proposed in order to determine the pressure for arbitrary Mach numbers M in the free stream. The gas is assumed to be ideal and perfect.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 185–188, March–April, 1985.     

Application of dispersion theory to time domain reflectometry in soils

With time domain reflectometry (TDR) two dispersive parameters, the dielectric constant, , and the electrical conductivity, can be measured. Both parameters are nonlinear functions of the volume fractions in soil. Because the volume function of water ( _w) can change widely in the same soil, empirical equations have been derived to describe these relations. In this paper, a theoretical model is proposed based upon the theory of dispersive behaviour. This is compared with the empirical equations. The agreement between the empirical and theoretical aproaches was highly significant: the ( _w) relation of Topp et al. had a coefficient of determination r ² = 0.996 and the (_u) relation of Smith and Tice, for the unfrozen water content, _u, at temperatures below 0°C, had an r ² = 0.997. To obtain ( _w) relations, calibration measurements were performed on two soils: Caledon sand and Guelph silt loam. For both soils, an r ² = 0.983 was obtained between the theoretical model and the measured values. The correct relations are especially important at low water contents, where the interaction between water molecules and soil particles is strong.     

Investigation of the influence of the unit Reynolds number on the transition of a boundary layer on a sharp cone

To establish the influence of the unit Reynolds number on the transition of a boundary layer on the side surface of a cone, the transition was investigated on a model of a sharp cone with half-angle = 7.5 ° and lengths from 150 to 400 mm. The experiments were made in a shock tube at Mach number M = 6.1 in the wide range of Reynolds numbers Re_eL = 1.3·10⁶-5.5·10⁷. The position of the transition region was determined from the results of measurement of the local heat flux by calorimetric thermocouple converters. Data were obtained on the influence on the transition of the unit Reynolds number at large values. It was also shown that under the investigated conditions the base region does not influence the transition of the boundary layer on the surface of the cone.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 32–38, July–August, 1982.     

Flow about a circular cylinder with a single large-scale surface perturbation

An experimental study of the flow around a cylinder with a single straight perturbation was conducted in a wind tunnel. With this bluff body, positioned in a uniform crossflow, the vortex shedding frequency and other flow characteristics could be manipulated.The Strouhal number has been shown to be a function of the perturbation angular position, _p , as well as the perturbation size and Reynolds number. As much as a 50% change in Strouhal number could be achieved, simply by changing _p by 1°. The perturbation size compared to the local boundary layer thickness, , was varied from approximately 1 to about 20 . The Reynolds number was varied from 10,000 to 40,000. For perturbation sizes approximately 5 to 20 and Reynolds numbers of 20,000 to 40,000, a consistent Strouhal number variation with _p was observed.A detailed investigation of the characteristic Strouhal number variation has shown that varying _p had a significant influence on the boundary layer separation and transition to turbulence. These significant changes occurring in the boundary layer have been shown to cause variations in the spacing between the shear layers, base pressure, drag, lift, and the longitudinal spacing between the vortices in the vortex street.List of Symbols a longitudinal spacing of vortices in the vortex street - C _d drag coefficient - C _dc drag coefficient corrected for blockage effect - C _l lift coefficient - C _p pressure coefficient, p _i –p /q - C _pb base pressure coefficient - C _pbc base pressure coefficient corrected for blockage effect - d perturbation diameter - d ^* spacing between the shear layers; defined as conditionally averaged spacing between points in the shear layers corresponding to 0.99u _max/U - D cylinder diameter; diameter of the circumscribing circle for a cable - f _v vortex shedding frequency - H wind tunnel test section cross-sectional width - L spanwise length of the cylinder - p _i tap pressure - p free stream static pressure - q free stream dynamic pressure - Re Reynolds number based on cylinder diameter - rms root-mean-square - S Strouhal number, f _v D/U - S _max maximum value of S - S _min minimum value of S - t time - u _c vortex convection velocity - u _max maximum velocity in the shear layer - U free stream velocity - U _c free stream velocity, corrected for blockage effect - x streamwise dimension referenced from the back of the cylinder - z lateral wake dimension, i.e., perpendicular to the free stream velocity vector and cylinder axis, referenced from the cylinder axis - x spacing between two hot wire probes aligned streamwise - phase difference between two hot wire probes aligned streamwise - boundary layer thickness - angle from stagnation point in degrees - _p perturbation angular position - _{b p} where S drops back to about the S of a cylinder - _c critical angle, angular position where S drops steeply with 1° change in - _{m p} where S was minimum - _{r p} after S recovers from drop in value - _{t p} where S starts to increase from about the S of a cylinder     

Stationary combustion modes with allowance for heat losses

An attempt is made to investigate the number of possible stationary combustion modes in a continuous-flow semi-infinite pipe with allowance for heat losses through the walls. Cases of a zero-order reaction in the reaction mixture or similarity of the concentration and temperature fields are considered. The equations are averaged with respect to the transverse coordinate . Within the framework of these approximations it is found that the number of stationary combustion modes is determined by the roots _n of some function. The roots _2k correspond only to trivial unstable solutions. The roots _2k–1 correspond to modes possible within broad regions of variation of the parameters characterizing the temperature of the mixture, the mixture feed rate, and the rate of heat removal. These regions intersect, forming zones where several stationary modes coexist. In these zones, apart from monotonic solutions there may also be solutions that initially make several oscillations. It is shown that the latter are obviously unstable and, in the last analysis, lead to one of the monotonic modes. The common case of not more than three roots is examined in detail.If the heat release function can change sign, then a similar picture is also observed in the absence of heat losses through the walls (the roots _2k–1 and _2k may change roles). In this case it is no longer necessary to average the equations with respect to , since there will not be any corresponding derivatives.     

Numerical Investigation of Off-Design Regimes of Flow Past Power-Law Waveriders Based on the Flows Behind Plane Shocks

The results of a numerical investigation of supersonic off-design flow past waveriders at the freestream Mach numbers M = 4 and 8 are presented. Flow regimes with M both greater and smaller than the design value M _d are analyzed. Configurations based on the flows behind plane shocks and described by power-law functions are considered. The results are obtained by the finite-volume solution of the Euler equations using higher-order TVD Runge-Kutta schemes.     

Experimental investigation of the asymptotically approached Mach reflection over the second surface in a double wedge reflection

The assumption that the Mach reflection which is formed over the second surface of a double wedge with angles _w ¹ and kw/2 approaches asymptotically the Mach reflection which would have been obtained by an identical incident shock wave over a single wedge with an angle _w = _w ² was verified experimentally. The verification of this assumption supports the shock polar analysis suggested by Ben-Dor et al. (1987) for the study of the reflection process of a planar shock wave over a double wedge. Measurements of the rate of approach to the asymptotic value are also provided.     

The critical angle of the anisotropic elastic wedge subject to uniform tractions

The classical solution for an isotropic elastic wedge loaded by a uniform pressure on one side of the wedge becomes infinite when the wedge angle 2₀ satisfies the equation tan 2₀ = 2₀. This is the critical wedge angle which also renders infinite solutions for other types of loadings. In this paper, we study the associated problem for the anisotropic elastic wedge. We first present uniform stress solutions which are possible for symmetric loadings. For antisymmetric loadings, a uniform stress solution is in general not possible and we present a non-uniform stress solution in which the stress depends on but not on r. The non-uniform stress solution breaks down at a critical angle. We present an equation for the critical angle which depends on the elastic constants. The Stroh formalism is employed in the analysis. An integral representation of the solution is obtained by using new identities which are derived in the paper.     

Behavior of sonic lines in a shock layer behind a detached shock wave

A numerical investigation is made into the formation of local supersonic zones in the subsonic flow region between a detached shock wave and the surface of the body in the case of supersonic three-dimensional flow over conical bodies with opening angle _k = 120 ° of the cone in the range of Mach numbers M = 2.5–15.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No.4 pp. 143–145, July–August, 1979.We thank G. I. Petrov for suggesting the problem and for helpful advice and O. M. Belotserkovskii for constant interest in the work.