Derivation of a linear theory for nonequilibrium and equilibrium flows

Conventional linear theory of nonequilibrium and equilibrium gas flows yields correct results only for very small deviations of the stream parameters from the unperturbed values. Moreover, if in linearization we take the coordinates in planar flow as independent variables, then the flow past concave and convex corners is described in exactly the same fashion. In this case the characteristic emanating from the corner is (depending on the type of corner) a compression or rarefaction shock. In the case of a break in the wall of an axisymmetric channel the shock intensity approaches infinity with approach to the centerline, which indicates a deficiency of this type of linear theory. In the following we use a modification which eliminates the deficiencies noted above. This involves conversion to new independent and dependent variables such that the coefficients of the exact equations being linearized become weakly varying functions of the unknown parameters, the linearized boundary conditions coincide with the exact conditions at all or part of the boundaries, and the rarefaction shocks become rarefaction wave bundles of finite width. The last condition is achieved as a result of the fact that, in accordance with the Lighthill method of deformable coordinates [1], we take as one of the independent variables a quantity which maintains a constant value on each characteristic of the bundle of characteristics emanating from the break point [for equilibrium flows the semicharacteristic (or characteristic) independent variables were used in deriving the linear theory, for example, in [2–4]]. The study was based on the example of two-dimensional stationary nonequilibrium flow of an inviscid and nonheatconducting gas. In this case we find that boththelinear equations at a finite distance from the walls and the boundary conditions for determining the potential and nonequilibrium parameters outside the rarefaction wave bundles coincide with the equations and the conditions of conventional linear theory [5], while the relations associating the values of the parameters on the closing characteristics of each bundle (outside the bundles the same value of the characteristic variable corresponds to these characteristics) at some distance from the axis or from some reflecting surface are identical to the conditions on the rarefaction shocks. This fact makes it possible to use several results of conventional linear theory.     

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.     

Slender bodies of revolution with minimum wave drag in nonequilibrium supersonic flow

We examine the problem of finding the generatrix shape of a body of revolution which travels at supersonic speed and has minimum wave drag. We assume that any number of nonequilibrium processes can take place in the flow. The pressure distribution over the body surface is taken in the linear approximation [1, 2]. A survey of studies using linear theory to find bodies of revolution of optimal form in supersonic perfect gas flow can be found in [3]. The solution of the problem of finding the form of two-dimensional slender bodies of minimum wave drag in nonequilibrium supersonic flow was obtained in [4]. In the following we examine the optimization of only those bodies of revolution for which the leading point lies on the axis of symmetry.The author wishes to thank A. N. Kraiko for his helpful comments.     

Chemical relaxation in a viscous shock

We consider the flow of a nonequilibrium dissociating diatomic gas in a normal compression shock with account for viscosity and heat conductivity. The distribution of gas parameters in the flow is found by numerically solving the Navier-Stokes and chemical kinetics equations. The greatest difficulty in numerical integration comes from the singular points of this system at which the initial conditions are given. These points lead to instability of the numerical results when the problem is solved by standard numerical methods. An integration method is proposed that yields stable numerical results-continuous profiles of the distribution of the basic gas parameters in the shock are obtained.We consider steady one-dimensional flow in which the gas passes from equilibrium state 1 to another equilibrium state 2, which has higher values for temperature, density, and pressure. Such a flow is termed a normal compression shock.The parameter distribution in normal shock for nonequilibrium chemical processes has usually been calculated [1–3] without account for the transport phenomena (viscosity, heat conduction, and diffusion). The presence of an infinitely thin shock front perpendicular to the flow velocity direction was postulated. It was assumed that the flow is undisturbed ahead of the shock front. The gas parameters (velocity, density, and temperature) change discontinuously across the shock front, but the gas composition does not change. The composition change due to reactions takes place behind the shock front. The gas parameter distribution behind the front was calculated by solving the system of gasdynamic and chemical kinetics equations using the initial values determined from the Hugoniot conditions at the front to state 2 far downstream.Several studies (for example, [4, 5]) do account for transport phenomena in calculating parameter distribution in a compression shock, but not for nonequilibrium chemical reactions. These problems are solved by integrating the Navier-Stokes equations continuously from state 1 in the oncoming flow to state 2 downstream.We present a solution to the problem of normal compression shock in nonequilibrium dissociating oxygen with account for viscosity and heat conduction using the Navier-Stokes equations.     

Dynamics of the cavitation zone during an underwater explosion near a free surface

The problem of the development of the cavitation zone and the rarefaction wave profile in the region of regular reflection of the spherical shock wave of an underwater explosion from a free surface is analyzed for the axisymmetric formulation within the framework of a model of the two-phase medium consisting of a liquid with cavitation nuclei of the free gas uniformly distributed in it. An example of the calculation of the rarefaction wave profile and the zone of visible cavitation at different times is given for the case of the explosion of 1-g charge at depths of 3 and 5.3 cm for an initial volumetric gas concentration of 10^?11 and an initial cavitation nucleus radius of 5 · 10^?5 cm. The results of the calculation are compared with experiment.     

Analysis of unsteady wave processes in a rotating channel

The impact of passage rotation on the gasdynamic wave processes is analyzed through a numerical simulation of ideal shock-tube flow in a closed rotating-channel containing a gas in an initial state of homentropic solid-body rotation. Relevant parameters of the problem such as wheel Mach number, hub-to-tip radius ratio, length-to-tip radius ratio, diaphragm temperature ratio, and diaphragm pressure ratio are varied. It is shown that for a fixed geometry and initial conditions, the contact interface acquires a distorted three-dimensional time-dependent orientation at non-zero wheel Mach numbers. At a fixed wheel Mach number, the level of distortion depends primarily on the density ratio across the interface and also the hub-to-tip radius ratio. The nature of the rarefaction and shock wave propagation is one-dimensional, although the acoustic waves are diffracted due to the radially varying propagation speed. Under conditions of initially homentropic solid-body rotation, a degree of similarity exists between rotating and stationary shock-tube flows. This similarity is exploited to arrive at an approximate analytical solution to the Riemann problem in a rotating shock-tube.     

Wave structure induced by fluid-dynamic limits in the Broadwell model

Consider the fluid-dynamic limit problem for the Broadwell system of the kinetic theory of gases, for Maxwellian Riemann initial data. The formal limit is the Riemann problem for a pair of conservation laws and is invariant under dilations of coordinates. The approach of self-similar fluid-dynamic limits consists in replacing the mean free path in the Broadwell model so that the resulting problem preserves the invariance under dilations. The limiting procedure was justified in [ST]. Here, we study the structure of the emerging solutions. We show that they consist of two wave fans separated by a constant state. Each wave fan is associated with one of the characteristic fields and is either a rarefaction wave or a shock wave. The shocks satisfy the Lax shock conditions and have the internal structure of a Broadwell shock profile.     

A multi-dimensional time-marching method of characteristics for viscous and heat-conducting flows

Summary A numerical scheme is presented which employs the characteristic surfaces in space-time for solving Navier-Stokes equations for compressible fluid flow. We consider the general case of a three-dimensional flow, a simplification of which yields the equations of the two-dimensional case. Emphasis is put on the method itself. We apply it to simulate a laminar hypersonic flow around a circular cylinder of a five-components gas mixture of nitrogen and oxygen with thermally perfect constituents and at chemical nonequilibrium. First, the partial differential equations are transformed into a standard form with directional derivatives, enabling to attain the compatibility conditions, including the viscosity terms. These conditions are discretized by approximating their integrals along the corresponding characteristic surfaces. The result is an explicit time-marching numerical scheme. Using a grid fitted between the shock and the cylinder, and starting from roughly estimated initial conditions, a steady solution is searched. A comparison is made with the solution obtained under the assumption of a perfect gas. Received 6 April 1999; accepted for publication 13 May 1999     

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.     

Regular reflection of a magnetohydrodynamic shock wave from a conducting wall

In two-dimensional supersonic gasdynamics, one of the classical steady-state problems, which include shock waves and other discontinuities, is the problem concerning the oblique reflection of a shock wave from a plane wall. It is well known [1–3] that two types of reflection are possible: regular and Mach. The problem concerning the regular reflection of a magnetohydrodynamic shock wave from an infinitely conducting plane wall is considered here within the scope of ideal magnetohydrodynamics [4]. It is supposed that the magnetic field, normal to the wall, is not equal to zero. The solution of the problem is constructed for incident waves of different types (fast and slow). It is found that, depending on the initial data, the solution can have a qualitatively different nature. In contrast from gasdynamics, the incident wave is reflected in the form of two waves, which can be centered rarefaction waves. A similar problem for the special case of the magnetic field parallel to the flow was considered earlier in [5, 6]. The normal component of the magnetic field at the wall was equated to zero, the solution was constructed only for the case of incidence of a fast shock wave, and the flow pattern is similar in form to that of gasdynamics. The solution of the problem concerning the reflection of a shock wave constructed in this paper is necessary for the interpretation of experiments in shock tubes [7–10].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 102–109, May–June, 1977.The author thanks A. A. Barmin, A. G. Kulikovskii, and G. A. Lyubimov for useful discussion of the results obtained.     

Nonequilibrium hypersonic flow of gas past a wing of small aspect ratio

It is well known [1] that nonequilibrium physicochemical processes taking place in gases at high temperature influence the gas-dynamic parameters and aerodynamic characteristics of bodies in hypersonic flight. In the present paper, the thin shock layer method [2–4] is used to consider the problem of nonequilibrium hypersonic flow of gas past a wing of small aspect ratio at an angle of attack. It is shown that the flow component of the vorticity is conserved along the streamlines, and this property is exploited to obtain an analytic solution of the equations of the three-dimensional nonequilibrium shock layer. The influence of the disequilibrium on the thickness of the shock layer and the pressure distribution is investigated.     

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.     

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.     

Decay of a plane shock wave in a two-parameter medium with arbitrary equation of state

Special curves, called shock polars, are frequently used to determine the state of the gas behind an oblique shock wave from known parameters of the oncoming flow. For a perfect gas, these curves have been constructed and investigated in detail [1]. However, for the solution of problems associated with gas flow at high velocities and high temperatures it is necessary to use models of gases with complicated equations of state. It is therefore of interest to study the properties of oblique shocks in such media. In the present paper, a study is made of the form of the shock polars for two-parameter media with arbitrary equation of state, these satisfying the conditions of Cemplen's theorem. Some properties of oblique shocks in such media that are new compared with a perfect gas are established. On the basis of the obtained results, the existence of triple configurations in steady supersonic flows obtained by the decay of plane shock waves is considered. It is shown that D'yakov-unstable discontinuities decompose into an oblique shock and a centered rarefaction wave, while spontaneously radiating discontinuities decompose into two shocks or into a shock and a rarefaction wave.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 147–153, November–December, 1982.     

Normal interaction of shock waves in the presence of vibrational relaxation

The effect of nonequilibrium physicochemical processes on the flow resulting from the normal collision and reflection of shock waves is studied by the example of nonequilibrium excitation of molecular oscillations in nitrogen. It is shown that the thermal effect of vibrational relaxation is small and the problem can be linearized around a known solution [1]. A similar approach to the solution of the problem of flow around a wedge and certain one-dimensional non-steady-state problems was used earlier in [2–4]. The solution of these problems was constructed in an angular domain, bounded by the shock wave and a solid wall (or the contact surface) and was reduced to a well-known functional equation [6]. The solution of this problem, because of the presence of two angular domains divided by a tangential discontinuity, reduces to a functional equation of more general form than in [6]. The results are obtained in finite form. In the special case of shocks of equal intensity, the normal reflection parameters are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 90–96, July–August, 1976.     

Interference between a centered rarefaction wave and an inclined shock wave

The gas flow in the zone of interaction between an oblique shock and a centered isentropic rarefaction wave is studied using the direct statistical simulation method for solving the Boltzmann equation. The data of calculations of the shock and rarefaction wave structures, flow fields, and streamlines are given for the free-stream Mach number M_∞ = 6, 4 and 2. The formation of the interaction zone is simulated by a gas flow past a double-plane wedge in which the break of the generating line leads to formation of the centered isentropic rarefaction wave. The results of calculations of this flow in solving the Boltzmann equation are given in the Euler approximation.     

氢氧燃烧及爆轰驱动激波管

分析并观察了沿驱动段轴向分布多火塞燃烧驱动段的性能．提出主膜处同一管截面均匀分布三火花塞引燃的点火方法．用这种点火方法驱动产生的入射激波强度重复性较高,激波后气流速度、温度和压力的定常性亦大大改善,可满足气动试验实际要求．提出在驱动段尾端串接卸爆段来消除爆轰波反射高压,从而可使反向爆轰驱动段用来产生高焓高密度试验气流．这种反向爆轰驱动产生的入射激波重复性高,激波衰减弱．在主膜处的收缩段产生的反射波可缓解爆轰波后跟随的稀疏波的不利影响,从而使前向爆轰驱动具有实用性．在产生的入射激波强度相同条件下,前向爆轰驱动所需的爆轰驱动段可爆混合气初始压力可较反向爆轰低近一个量级．     

Flow in the neighborhood of the stagnation line on a blunt body traveling at supersonic speed through a zone with elevated temperature and vibrational energy of the molecules

The unsteady flow in the neighborhood of the stagnation line on a sphere traveling at supersonic speed through a plane layer of diatomic gas with elevated temperature and nonequilibrium excitation of the molecular vibrations is investigated. (The source of the inhomogeneity could be a gas discharge [1].) The problem is solved using the viscous shock layer model which makes it possible to take molecular transport processes into account and analyze the unsteady heat transfer. Such flows were previously calculated in [2] within the framework of the inviscid gas model.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i GazNo. 3, pp. 183–185, May–June, 1990.     

Blast wave propagation in air from a gaseous charge

In the point explosion problem it is assumed that an instantaneous release of finite energy causing shock wave propagation in the ambient gas occurs at a space point. The results of the solution of the problem of such blasts are contained in [1–4]. This point model is applied for the determination of shock wave parameters when the initial pressure in a sphere of finite radius exceeds the ambient air pressure by 2–3 orders of magnitude. The possibility of such a flow simulation at a certain distance from the charge is shown in papers [4, 5] as applied to the blast of a charge of condensed explosive and in [6, 8] as applied to the expansion of a finite volume of strongly compressed hot gas. In certain practical problems the initial pressure in a volume of finite dimensions exceeds atmospheric pressure by a factor 10–15 only. Such cases arise, for example, in the detonation of gaseous fuel-air mixtures. The present paper considers the problem of shock wave propagation in air, caused by explosion of gaseous charge of spherical or cylindrical shape. A numerical solution is obtained in a range of values of the specific energy of the charge characteristic for fuel-air detonation mixtures by means of the method of characteristics without secondary shock wave separation. The influence of the initial conditions of the gas charge explosion (specific energy, nature of initiation, and others) is investigated and compared with the point case with respect to the pressure difference across the shock wave and the positive overpressure pulse.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 110–118, May–June, 1986.