Study of unsteady flow past bodies behind shock waves

Several theoretical and experimental studies have been devoted to the problem of the nonstationary action of the stream behind a shock wave on bodies of varied shape. In particular, in [1], the pressure and density are calculated for flow about bodies of the more typical shapes in the initial stage of the process. The basic relations which accompany the interaction of shock waves are considered in [2, 3]. The analysis of the phenomena of diffraction of shock waves on the sphere, cylinder, and cone is presented in [4]. Problems of unsteady flow about a wing are examined in [5, 6]. A detailed review of the foreign studies on unsteady flow is given in [7]. Of great practical interest is the question of the time for flow formation and the magnitudes of the unsteady loads during this period. Experimental investigations have been made recently [8, 9] in which some criteria are presented for estimating the bow shock formation time for supersonic flow about the sphere and cylinder with flat blunting. However the question of the formation time of the stationary pressure on the body surface is not referred to in these studies and no relationship is shown between the transient position of the reflected wave and the corresponding unsteady pressure on the surface. Moreover, in [8] the dimensionless time criterion is determined very approximately, independently of the Mach number of the shock wave. The present study was undertaken with the object of determining the basic criteria which characterize unsteady flow about bodies behind a plane shock wave which has time-independent parameters, and clarification of the shock wave reflected from the body and the pressure on the surface of the body during the transient period. The most typical body shapes were studied: 1) a cylinder with flat face aligned with the stream; 2) a spherically-blunted cylinder; and 3) a cylinder transverse to the stream. The experiments were conducted in a conventional shock tube using the single-diaphragm scheme. The measurements of the pressure on the models and the velocity of the incident shock wave were made using the technique analogous to that of [10, 11]. A highspeed movie camera was used to record the pattern of the wave diffraction on the body. The Mach number of the incident shock wave varied in the range from M=1.5 to M≈6.0, which corresponded to a range of Mach numbers M_∞ of the stream behind the shock wave from 0.6 to 2.1. The calculations of the required gas dynamic parameters for high temperatures were made with account for equilibrium dissociation of the air on the basis of the data of [10, 12, 13]. The magnitude of the relative maximal shock wave standoff Δ at the stagnation point obtained in the present experiments was compared with the values of Δ from other studies. In the case of the flat-blunted cylinder it was in good agreement with the results of [8–14], and in the case of the spherically-blunted cylinder and the transverse cylinder it was in agreement with the results of [15].     

Flow of viscous gas out of a cylindrical channel into vacuum

A study is made of the exhausting of a jet of viscous gas from a cylindrical channel into vacuum in the presence of a flat bounding surface outside the channel in the plane of its exit section. The problem is solved numerically using the complete system of Navier—Stokes equations. The developed flow model makes it possible to take into account the influence of an external medium into which the jet exhausts on the structure of the flow in the exit section of the channel, and also the influence of the subsonic part of the boundary layer in the channel on the flow field of the jet.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 122–128, January–February, 1981.     

Formation of a pulsed jet behind a supersonic nozzle when the gas can relax at the entrance to the nozzle

The formation of a pulsed jet behind supersonic nozzles is considered when relaxation processes take place in the gas entering the nozzle. In a general formulation, the problem of the motion of the front of the exhausting matter and the disturbances accompanying it in the process of formation of a pulsed jet is determined by a large number of parameters, which characterize the exhausting gas and the residual gas of the pressure chamber and also the geometry of the flow conditions. A reliable computational model of a pulsed jet does not exist. To construct such a model, experiments are required in a wide range of boundary and initial conditions. An investigation was made into flow of shockheated argon, nitrogen, and carbon dioxide out of nozzles set up at the end of a shock tube. Generalized dependences were obtained for describing the motion of the front of the nonstationary jet and the wave in front of it in a wide range of the initial pressure-difference parameters and variation of the stagnation temperatures. The choice of the generalized parameters when relaxation of the excited internal degrees of freedom of the molecules of the gas can occur at the entrance to the nozzle is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 129–135, November–December, 1980.     

Solution of Riemann problem for dusty gas flow

A direct approach is used to solve the Riemann problem for a quasilinear hyperbolic system of equations governing the one dimensional unsteady planar flow of an isentropic, inviscid compressible fluid in the presence of dust particles. The elementary wave solutions of the Riemann problem, that is, shock waves, rarefaction waves and contact discontinuities are derived and their properties are discussed for a dusty gas. The generalised Riemann invariants are used to find the solution between rarefaction wave and the contact discontinuity and also inside rarefaction fan. Unlike the ordinary gasdynamic case, the solution inside the rarefaction waves in dusty gas cannot be obtained directly and explicitly; indeed, it requires an extra iteration procedure. Although the case of dusty gas is more complex than the ordinary gas dynamics case, all the parallel results for compressive waves remain identical. We also compare/contrast the nature of the solution in an ordinary gasdynamics and the dusty gas flow case.     

Some characteristics of supersonic flow of an electrically conductive gas in an MHD-channel

We study supersonic flows of an electrically conductive gas in crossed electric and magnetic fields [1] in the presence of shock waves. It is shown that three steady flow regimes can exist, and that these are defined by the electrical conductivity of the gas as a function of temperature and density.

1. The normal regime is characterized by a tendency for the shock to move toward the channel entrance on increase of the static pressure at the channel exit. The steady regime of this type exists and is stable.
2. The anomalous regime (formally constructed) is characterized by a tendency for the shock to move toward the exit on increase of the static pressure at the channel exit. This regime is unstable and the flow in the MHD-channel may be either entirely supersonic or entirely subsonic.
3. The limiting (boundary) regime is intermediate between the normal and anomalous regimes and is characterized by the fact that the stationary position of the shock wave and its amplitude are not uniquely defined. Steady flow in this case is not unique.

This study involves formal construction both of the solution to the steady-state problem and the corresponding nonsteady-state problem [4]. The establishment of a steady regime in the solution of the unsteady problem, is at the same time, a verification of its stability.     

带喷流激波针流动特性实验研究

采用动态测力、动态测压和纹影等风洞实验技术,对加装了带喷流激波针的钝头体的绕流特性、稳定和非稳模态的形成条件和机理进行了研究.结果表明:带喷流激波针流场存在稳态和非稳态两种模态,超声速喷流的压比大于临界压比时流动处于稳定模态,反之则为非稳模态;增大激波针长度可减小钝头体阻力,但达到一定长度后,进一步减阻的效果不再显著;增大喷流压比能够有效减弱再附激波强度,有利于缓解单独激波针的肩部热斑问题;非稳模态下波系自激振荡对再附激波在钝头体表面所围的区域影响剧烈,振荡是周期性的,且存在确定的主导频率,主导频率随喷流压力比增大而减小;自激振荡的产生是由于喷流出口周围的反压在喷流压比小于临界压比时无法获得持续的平衡而导致.      

Qualitative gas dynamic model of the laser-electric metallization of holes in dielectrics

A gas dynamic model of the laser-electric hole metallization process is considered. The process consists of three stages: 1) laser beam piercing of holes in a dielectric with subsequent breakdown of the interelectrode medium and the formation of metal vapor at high pressure and temperature; 2) unsteady flow of metal vapor through the hole characterized by the passage of a shock wave and a surface of contact discontinuity; and 3) steady flow of metal vapor with deposition of metal particles on the channel walls. On the basis of a comparison of the theoretical and experimental results recommendations are given concerning the choice of the optimum parameters of the discharge circuit elements.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 105–110, November–December, 1991.The authors wish to express their gratitude to A. L. Itkin, O. V. Kulagina, O. I. Firsov, I. B. Vargaftik, and V. S. Yargin for their valuable assistance.     

Interaction of a supersonic jet exhausting into vacuum with an obstacle

A study is made of the interaction between an axisymmetric supersonic jet exhausting into vacuum and an obstacle of a fairly complicated configuration and positioned relative to the nozzle in such a way that in the interaction region behind the detached shock wave there is a three-dimensional flow possessing a symmetry plane. The flow in the interaction region is described by the system of equations of motion of an inviscid perfect gas with boundary conditions on the shock wave (Rankine-Hugoniot relation) and on the surface of the obstacle (no-flow condition). The other boundaries of the region are the symmetry plane of the flow and an arbitrarily chosen surface in the supersonic part of the flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti Gaza, No. 1, pp. 156–161, January–February, 1981.     

Transition of plane overdriven detonation wave to the Chapman-Jouguet regime

The asymptotic laws of behavior for plane, cylindrical, and spherical infinitely thin detonation waves were found in [1, 2] for increasing distance from an igniting source in those cases in which the waves changed into Chapman-Jouguet waves as they decayed. It was shown that the plane overdriven detonation wave approaches the Chapman-Jouguet regime asymptotically, while the transition of the cylindrical or spherical strong detonation wave into the Chapman-Jouguet wave may occur at a finite distance from the initiation source.Similar conclusions are valid for the propagation of stationary steadystate detonation waves which arise with flow of combustible gas mixtures past bodies.However, numerous experiments [3, 4] on firing bodies in a detonating gas show that the overdriven detonation wave which forms ahead of the body decays and decomposes into an ordinary compression shock and a slow combustion front. To establish why the wave does not make the transition to the Chapman-Jouguet regime, in the following we consider the propagation of a plane detonation wave and account for finite chemical reaction rates. We use the very simple two-front model (ordinary shock wave and following flame front). Conditions are found for which transition to the Chapman-Jouguet regime does not occur. We first consider the propagation of an unsteady plane wave and then the steady plane wave. It is found that for all the mixtures used in these experiments transition to the Chapman-Jouguet regime is not possible within the framework of the assumed model.     

Interaction of a jet exhausting from a body with an opposing supersonic stream of rarefied gas

The experimental investigation of supersonic flow past a sphere with a jet exhausting from the front point of the sphere into the flow at large [1] and moderate [2] Reynolds numbers Re has revealed an effect of shielding from the oncoming stream, this leading to a decrease in the drag coefficient of the sphere and of the energy flux to it. A numerical simulation of the flow has been made in the case of supersonic flow past a sphere with a sonic jet from a nozzle situated on the symmetry axis in the continuum regime [3]. In the present paper, this problem is investigated for flow of a rarefied gas on the basis of numerical solution of a model kinetic equation for a monatomic gas.     

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.     

The propagation of shock waves in a semibounded volume

When a shock wave emerges from a pipe situated in a semibounded volume a system of waves arises between the end of the pipe and the bottom of the volume, and also in the gap between the pipe and the side walls of the volume. Paper [1] considers the propagation of a shock wave after emerging from the pipe until touching the side walls of the volume. The present paper considers the gas motion in a semibounded volume after the shock wave has traversed the volume and made contact with the side walls. In part 1 a physical model is constructed of the gas motion up to the time when the primary shock wave reaches the bottom of the volume. In part 2 relations are found which enable us to determine the stream parameters in the semibounded volume up to the time when the primary shock wave arrives at the bottom of the volume. Section 3 considers the motion of the reflected shock wave between the pipe and the side walls of the volume.     

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.     

Characteristics of unsteady type IV shock/shock interaction

Characteristics of the unsteady type IV shock/shock interaction of hypersonic blunt body flows are investigated by solving the Navier–Stokes equations with high-order numerical methods. The intrinsic relations of flow structures to shear, compression, and heating processes are studied and the physical mechanisms of the unsteady flow evolution are revealed. It is found that the instantaneous surface-heating peak is caused by the fluid in the “hot spot” generated by an oscillating and deforming jet bow shock (JBS) just ahead of the body surface. The features of local shock/boundary layer interaction and vortex/boundary layer interaction are clarified. Based on the analysis of flow evolution, it is identified that the upstream-propagating compression waves are associated with the interaction of the JBS and the shear layers formed by a supersonic impinging jet, and then the interaction of the freestream bow shocks and the compression waves results in entropy and vortical waves propagating to the body surface. Further, the feedback mechanism of the inherent unsteadiness of the flow field is revealed to be related to the impinging jet. A feedback model is proposed to reliably predict the dominant frequency of flow evolution. The results obtained in this study provide physical insight into the understanding of the mechanisms relevant to this complex flow.     

Aspects of self-similar diffraction gas flows and their numerical simulation

A large number of papers has been devoted to the investigation of the interaction of a plane shock wave with bodies of various geometric shapes, and they have been generalized and classified for a stationary body in [1, 2]. Separate results of experimental and theoretical investigations of the interaction of a shock wave with a wedge, cone, sphere, and cylinder moving with supersonic velocities are contained in [3–9]. Analysis of the available results shows that the features of the unsteady gas flows formed in this case largely depend on the nature of the boundary-value problem that arises for the system of differential gas dynamic equations. The question of the wave structure of the unsteady gas flow and the accuracy of the obtained solution is central to the numerical investigation of the present class of problems. The most characteristic types of unsteady self-similar gas flows that arise on the interaction of a plane shock wave with bodies of a wedge or convex corner type are calculated on the basis of an explicit numerical continuous calculation method of the second order of accuracy. The accuracy of the numerical solutions is discussed on the basis of a comparison with the experimental data. The case of the interaction of a shock wave with the rarefaction wave that arises in a supersonic flow past a convex corner is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 146–152, July–August, 1986.     

Turbulent bore in a supercritical flow over an irregular bed

Breaking waves in a free-surface homogeneous fluid flow in the neighborhood of a local variation in the channel depth are studied experimentally and theoretically. The structure of both a steady-state hydraulic jump generated by a local obstacle in the channel and an unsteady wave configuration consisting of two turbulent bores in the problem of lock failure is studied. Using the turbulent bore model [1], analytic profiles of breaking waves are obtained and the time-dependent problem is numerically investigated and compared with experimental data. It is shown that the model [1] with a hydrostatic pressure distribution over the depth adequately describes both the location and the structure of the steady-state and unsteady wave fronts.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 62–70. Original Russian Text Copyright © 2005 by Gusev and Lyapidevskii.     

Nonlinear wave motions on liquid surfaces

A study is made of the formation of a shock wave (bore), produced by the movement of an initially weak discontinuity in the spatial derivatives of velocity and liquid depth in an area of stationary current in a channel of constant inclination. The formation of shock waves from compression waves was first studied by Riman [1]. Frictional resistance was considered in the Chezy form. The equations obtained therein for determination of the moment in time and spatial coordinates of the point at which the shock wave is formed, as well as the laws for propagation of shock waves are applicable to the problem of one-dimensional transient motion in a gas, the pressure of which is dependent on density. Instantaneous collapse of waves, as well as formation and movement of bores in rivers for an idealized flow model in a channel with horizontal bottom, neglecting friction, were described by Khristianovich, Mikhlin, and Devison [2], and Stoker [3]. Recently in the work of Sachdev and Bhatnagar [4], using numerical integration of the equation for bore intensity, the problem of shock wave propagation in a channel of constant inclination with consideration of fluid resistance in the Chezy form was studied. Gradual wave collapse and the bore formation mechanism were studied by Stoker [3] on the basis of the shallow-water theory. Neglecting friction on the horizontal channel bottom, he calculated the moment of time and coordinates of the point at which the shock wave is formed in the case where the initial disturbance is sinusoidal. The dependence of these values on wave amplitude for a channel of constant inclination was obtained by Jeffrey [5], who also neglected friction on the channel bottom and considered the initial disturbance to be sinusoidal. Lighthill and Whitham [6] discovered that for Froude numbers greater than two, the linear theory led to unlimited growth in the intensity of the flood wave. We note that the studies of flood-wave motion in the region of the first characteristic, performed in [3, 6], differ only in the forms of the resistance laws and dependences of the unknown functions on the variables. Physical peculiarities of various liquid wave motions were also examined by Lighthill in [7].Saratov. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 62–66, March–April, 1972.     

Supersonic flow of a combustible gas mixture past a sphere

In recent years considerable interest has developed in the problems of steady-state supersonic flow of a mixture of gases about bodies with the formation of detonation waves and slow combustion fronts. This is due in particular to the problem of fuel combustion in a supersonic air stream.In [1] the problem of supersonic flow past a wedge with a detonation wave attached to the wedge apex is solved. This solution is based on using the equation of the detonation polar obtained in [2]-the analog of the shock polar for the case of an exothermic discontinuity. In [3] a solution is given of the problem of cone flow with an attached detonation wave, and [4] presents solutions of the problems of supersonic flow past the wedge and cone with the formation of attached adiabatic shocks with subsequent combustion of the mixture in slow combustion fronts. In the two latter studies two different solutions were also found for the problem of flow past a point ignition source, one solution with gas combustion in the detonation wave, the other with gas combustion in the slow combustion front following the adiabatic shock. These solutions describe two different asymptotic pictures of flow of a combustible gas mixture past bodies.In an experimental study of the motion of a sphere in a combustible gas mixture [5] it was found that the detonation wave formed ahead of the sphere splits at some distance from the body into an ordinary (adiabatic) shock and a slow combustion front. Arguments are presented in [6] which make it possible to explain this phenomenon and in certain cases to predict its occurrence.The present paper presents examples of the calculation of flow of a combustible gas mixture past a sphere with a detonation wave in the case when the wave does not split. In addition, the flow near the point at which the detonation wave splits is analyzed for the case when splitting occurs where the gas velocity behind the wave is greater than the speed of sound. This analysis shows that in the given case the flow calculation may be carried out without any particular difficulties. On the other hand, the calculation of the flow for the case when the point of splitting is located in the subsonic portion of the flow behind the wave (or in the region of influence of the subsonic portion of the flow) presents difficulties. This flow case is similar to the problem of the supersonic jet of finite width impacting on an obstacle.     

Amplification of shock waves in a bubble-filled liquid with gas concentration gradient

The properties are studied of the propagation of unsteady shock waves in a gas-liquid system of bubble structure in the case when the volume concentration of the gas changes in the direction of motion of the shock wave. It is established that when there is a sufficiently rapid drop in the gas content, an effect of amplification of the shock wave is observed which is due to the deceleration of the medium behind the shock wave. A study is made of the laws of the evolution of long- and short-wave pulsed perturbations in such systems. The authors consider processes of reflection of waves from obstacles and their passage from a gas into a bubble liquid, from a two-phase mixture into a pure liquid. The contribution is determined of nonequilibrium effects to the process of amplification of a wave.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 49–54, January–February, 1988.The authors wish to express gratitude to R. I. Nigmatulin for his interest in the study and for useful discussions.