The reflection of shock waves from an orifice at the end of a duct

Zusammenfassung Die Str?mungserscheinungen, die auftreten, wenn eine Stosswelle an ein mit einer Blende versehenes Rohrende gelangt, werden besprochen. üblicherweise werden sie unter der Annahme berechnet, dass man für stetige und unstetige Str?mungen die gleichen Randwertbedingungen in der Blende verwenden kann. Die reflektierte Welle ist dann entweder eine einfache Expansionswelle oder eine Stosswelle, je nach der St?rke des einfallenden Stosses und der Blenden?ffnung. Dieses Resultat stimmt nicht mit experimentellen Beobachtungen überein, die gezeigt haben, dass die reflektierte Welle immer aus einer Stossfront besteht, der eine Expansionswelle nachl?uft, bis der Druck genügend vermindert ist, um eine stetige Str?mung zu erm?glichen. Die überlagerung dieser Wellen erzeugt eine Druckspitze (?overshoot?), die den in der üblichen Weise berechneten Maximaldruck um einen erheblichen Bruchteil des Druckanstieges in der einfallenden Stosswelle übersteigen kann. Die Unzul?nglichkeit der üblichen Methode kann man qualitativ durch die Verz?gerung erkl?ren, die notwendig ist, um eine stetige Str?mung in der Blende herzustellen, nachdem die einfallende Stosswelle eine St?rung erzeugt hat. Die gegenw?rtige Untersuchung zeigt, dass man die überdruckspitze in Abh?ngigkeit von der Blendengr?sse, der Sto?st?rke und der Entfernung von der Blende auf Grund einiger einleuchtender Annahmen berechnen kann. Es ergibt sich, dass die überdruckspitze besonders dann bemerkbar wird, wenn die Druck?nderung über die gesamte reflektierte Welle verschwindet. Unter dieser Bedingung und für Stosswellen verschwindender St?rke wird sie anf?nglich genau so gross wie der Drucksprung der einfallenden Stosswelle. Mit wachsender St?rke des einfallenden Stosses verringert sich die relative Gr?sse der überdruckspitze, w?hrend ihre absolute Gr?sse bis zu einem Maximum von beinahe 40% des Druckes vor der einfallenden Stosswelle ansteigt. Dieses Maximum wird bei einem ungef?hren Druckverh?ltnis der einfallenden Stosswelle von 2,3 erreicht. Die überdruckspitze wird ziemlich unbedeutend, wenn das Druckverh?ltnis den Wert 3 überschreitet. Experimente mit einem Stosswellenrohr werden dann beschrieben, in denen die Druckver?nderungen der einfallenden und reflektierten Wellen für verschiedene Entfernungen von der Blende, Sto?st?rken und Blenden?ffnungen aufgezeichnet werden k?nnen. Die gemessenen überdruckwerte stimmen mit den gerechneten in allen F?llen gut überein. Es kann erwünscht sein, die überdruckspitze zu beseitigen, und die M?glichkeit einer speziellen Blendenkonstruktion wird gezeigt. Die Berechnung der überdruckspitze ist für eine einfallende Stosswelle abgeleitet, unter der Bedingung, dass das Gas vor der einfallenden Welle in Ruhe ist und dass sich die Blende am Ende des Rohres befindet. Erweiterungen der Methode auf beliebige Wellen, anf?ngliche Str?mungen und Blenden im Inneren des Rohres sind kurz besprochen.

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This work was sponsored by Project SQUID which is supported by the Office of Naval Research under Contract N6-ori-105 T.O.III, NR-098-038. Reproduction in full or in part is permitted for any use of the United States Government.     

The configuration of shock wave reflection for the TSD equation

In this paper, we mainly study the nonlinear wave configuration caused by shock wave reflection for the TSD (Transonic Small Disturbance) equation and specify the existence and nonexistence of various nonlinear wave configurations. We give a condition under which the solution of the RR (Regular reflection) for the TSD equation exists. We also prove that there exists no wave configuration of shock wave reflection for the TSD equation which consists of three or four shock waves. In phase space, we prove that the TSD equation has an IR (Irregular reflection) configuration containing a centered simple wave. Furthermore, we also prove the stability of RR configuration and the wave configuration containing a centered simple wave by solving a free boundary value problem of the TSD equation.     

Cylindrically and spherically symmetrical rapid intense compression of an ideal perfect gas with adiabatic exponents from 1.001 to 3

The problem of the rapid intense cylindrically or spherically symmetrical compression of an ideal (non-viscous and non-heat-conducting) perfect gas with different adiabatic exponents is considered. We mean by rapid and intense a compression in a time much less than the time taken for the sound wave to propagate through the uncompressed target up to temperatures and densities as high as desired. It is found that the solution previously obtained with a focused non-self-similar compression wave at the point where the shock wave is reflected from the axis or centre of symmetry (henceforth the centre of symmetry) holds for adiabatic exponents not exceeding 1.9092 and 1.8698 respectively in the cylindrical and spherical cases. It was not possible to construct a complete solution with focusing at the centre of symmetry for gases with higher adiabatic exponents. On the other hand, one can focus the compression waves into a cylinder or sphere of as small, but finite, radius as desired at the instant of arrival on them, for example, of a special characteristic or reflected shock wave of the Guderley problem. It is shown that for high degrees of compression, the time dependences of the coordinates of the pistons which produce such focusing, and of the gas density on them are close to power laws.     

The balance of material momentum at a shock wave propagating in a thermo-elastic material

At a surface of discontinuity, the mechanical balance laws are represented by jump conditions. It is shown how the balance of material momentum at an adiabatic shock propagating in a thermo-elastic material is obtained from the discontinuous balances of physical momentum and energy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The decay of a magnetohydrodynamic shock wave

Résumé On utilise l'analyse linéaire pour étudier les faibles perturbations non-isentropiques des équations relatives aux écoulements unidimensionnels et non-stationnaires d'un fluide non visqueux idéal, parfaitement conducteur de l'électricité et compressible, soumis à l'action d'un champ magnétique transversal. On utilise la solution générale de la perturbation non-isentropique d'un écoulement par ondes simples centrées pour déterminer la perturbation qui se manifeste lorsqu'une onde de choc magnétohydrodynamique, tout d'abord uniforme et de force arbitraire, rencontre le régime d'onde simple.Dans le cas limité d'un champ magnétique nul, la solution se réduit exactement à celle du problème correspondant de la dynamique classique des gaz. C'est la une confirmation de la validité de la théorie.

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This research was supported by National Science Foundation grant GP 87190.     

Potential theory for regular and mach reflection of a shock at a wedge

If a plane shock hits a wedge, a self-similar pattern of reflected shocks travels outward as the shock moves forward in time. The nature of the pattern is explored for weak incident shocks (strength b) and small wedge angles 2θ_w through potential theory, a number of different scalings, some study of mixed equations and matching asymptotics for the different scalings. The self-similar equations are of mixed type. A linearization gives a linear mixed flow valid away from a sonic curve. Near the sonic curve a shock solution is constructed in another scaling except near the zone of interaction between the incident shock and the wall where a special scaling is used. The parameter β = c₁θ²_w(γ + 1)b ranges from 0 to ∞. Here γ is the polytropic constant and C₁ is the sound speed behind the incident shock. For β > 2 regular reflection (weak or strong) can occur and the whole pattern is reconstructed to lowest order in shock strength. For β < 1/2 Mach reflection occurs and the flow behind the reflection is subsonic and can be constructed in principle (with an open elliptic problem) and matched. The case β = 0 can be solved. For 1/2 < β < 2 or even larger β the flow behind a Mach reflection may be transonic and further investigation must be made to determine what happens. The basic pattern of reflection is an almost semi-circular shock issuing, for regular reflection, from the reflection point on the wedge and for Mach reflection, matched with a local interaction flow. Assuming their nature, choosing the least entropy generation, the weak regular reflection will occur for β sufficiently large (von Neumann paradox). An accumulation point of vorticity occurs on the wedge above the leading point. © 1994 John Wiley & Sons, Inc.     

Nonself-similar flow with a shock wave reflected from the center of symmetry and new self-similar solutions with two reflected shocks

In some problems concerning cylindrically and spherically symmetric unsteady ideal (inviscid and nonheat-conducting) gas flows at the axis and center of symmetry (hereafter, at the center of symmetry), the gas density vanishes and the speed of sound becomes infinite starting at some time. This situation occurs in the problem of a shock wave reflecting from the center of symmetry. For an ideal gas with constant heat capacities and their ratio γ (adiabatic exponent), the solution of this problem near the reflection point is self-similar with a self-similarity exponent determined in the course of the solution construction. Assuming that γ on the reflected shock wave decreases, if this decrease exceeds a threshold value, the flow changes substantially. Assuming that the type of the solution remains unchanged for such γ, self-similarity is preserved if a piston starts expanding from the center of symmetry at the reflection time preceded by a finite-intensity reflected shock wave propagating at the speed of sound. To answer some questions arising in this formulation, specifically, to find the solution in the absence of the piston, the evolution of a close-to-self-similar solution calculated by the method of characteristics is traced. The required modification of the method of characteristics and the results obtained with it are described. The numerical results reveal a number of unexpected features. As a result, new self-similar solutions are constructed in which two (rather than one) shock waves reflect from the center of symmetry in the absence of the piston.     

A linear approximation for the regular reflection of a weak shock at a wedge

The problem of shock reflection by a wedge in the flow dominated by the unsteady potential flow equation is an important problem. In weak regular reflection, the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream. The reflected shock is transonic. Its position is a free boundary for the unsteady potential equation, which is degenerate at the sonic line in self-similar coordinates. Applying the special partial hodograph transformation used in [Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle I, 2-D case, Comm. Pure Appl. Math. LVII (2004) 1-51; Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle II, 3-D case, IMS, preprint, 2003], we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain. When the angle between incident shock and wedge is small, we can see the weak regular reflection as the disturbance of normal reflection as in [Chen Shuxing, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Differential Equations 21(78) (1996) 1103-1118]. By linearizing the resulted nonlinear equation and boundary conditions with the above viewpoint in [Chen Shuxing, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Differential Equations 21(78) (1996) 1103-1118], we obtain a linear degenerate elliptic equation with mixed boundary conditions in a curved quadrilateral domain. By means of elliptic regularization techniques, a delicate a priori estimate and compact arguments, we show that the solution of the linearized problem is smooth in the interior and Lipschitz continuous up to the degenerate boundary.     

The motion of a non-uniform hydromagnetic shock wave

Sommaire Nous considérons un écoulement, unidimensionnel, non stationaire, non isentropique, d'un fluide parfait, parfaitement conducteur de l'électricité, soumis à l'action d'un champ magnétique pour une orientation transversale du champ et le mouvement non continu d'une onde de choc magnetohydrodynamique.

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This research was supported by a National Aeronautics and Space Administration general research grant to the University of Wisconsin.     

Delayed reflection of the energy flow at a potential step for dispersive wave packets

We study Klein–Gordon equations with constant coefficients and different dispersion relations on two one‐dimensional semi‐infinite media coupled with transmission conditions. We obtain lower and upper bounds of the reflected part of the energy flow at the connecting point when the frequency band involved in the initial signal is sufficiently narrow. We detect a phenomenon of delayed reflection for low frequency wave packets, which is in accordance with the recent experiments of Haibel and Nimtz. The result is then generalized for a star‐shaped network of n semi‐infinite branches connected at one point. Copyright © 2004 John Wiley & Sons, Ltd.     

Solitary wave and shock wave solutions to a second order wave equation of Korteweg-de Vries type

This paper obtains the solitary wave as well as the shock wave solutions to the second order wave equation of Korteweg-de Vries type that was first proposed in 2002. The ansatz method is used to retrieve these solutions. The domain restrictions as well as the parameter regimes are all identified in the process of obtaining the solution.     

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations is performed. First, complete infinite-order approximate symmetry classification of the equation is obtained by means of the method originated by Fushchich and Shtelen. An optimal system of one-dimensional subalgebras is derived and used to construct general formulas of approximate symmetry reductions and similarity solutions. Second, we study approximate homotopy symmetry of the equation and construct connections between the two symmetry methods for the first-order and higher-order cases, respectively. The series solutions derived by the two methods are compared.     

Self-similar time-varying flows of an ideal gas with a change in the adiabatic exponent in a “reflected” shock wave

Self-similar one-dimensional time-varying problems are considered under the assumption that there is a change in the adiabatic exponent in a shock wave (SW) running (“reflected”) from a centre or axis of symmetry (later from a centre of symmetry, CS) or from a plane. The medium is an ideal (inviscid and non-heat-conducting) perfect gas with constant heat capacities. In problems with strong SW, the change in the adiabatic exponent in a gas approximately simulates physicochemical processes such as dissociation and ionization and, in the problem of the collapse of a spherical cavity in a liquid, the conversion of liquid into vapour. In both cases, the adiabatic exponent decreases on passing across a reflected SW. Problems of the collapse of a spherical cavity, the reflection of a strong SW from a centre of symmetry and a simpler problem with a self-similarity index of one are examined. When it is assumed that there is an increase in the adiabatic exponent, the self-similar solutions of the first two problems are rejected due to the decrease in entropy from the instant when the SW is reflected. When it is assumed that there is a decrease in the adiabatic exponent, the solutions of these problems only become unsuitable after a finite time has elapsed for the same reason. Up to this time when the decrease in the adiabatic exponent has not reached a certain threshold, the structure of the self-similar solution does not undergo qualitative changes. When the above-mentioned threshold is exceeded, a self-similar solution is possible if a cylindrical or spherical piston expands according to a special law from the instant of SW reflection from the CS. When there is no piston, the flow behind the reflected wave becomes non-self- similar. In the case of the deceleration of a plane flow, conditions are possible with the joining of SW from different sides to a centred rarefaction wave.     

A linear approximation for the regular reflection of a weak shock at a wedge satisfying sonic condition

The problem of shock reflection by a wedge, which the flow is dominated by the unsteady potential flow equation, is a important problem. In weak regular reflection, the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream. The reflected shock is transonic. Its position is a free boundary for the unsteady potential equation, which is degenerate at the sonic line in self-similar coordinates. Applying the special partial hodograph transformation used in [Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle I, 2-D case, Comm. Pure Appl. Math. 57 (2004) 1-51; Zhouping Xin, Huicheng Yin, Transonic shock in a nozzle II, 3-D case, IMS, preprint (2003)], we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain. When the angle, which between incident shock and wedge, is small, we can see that weak regular reflection as the disturbance of normal reflection as in [Shuxing Chen, Linear approximation of shock reflection at a wedge with large angle, Comm. Partial Differential Equations 21 (78) (1996) 1103-1118]. By linearizing the resulted nonlinear equation and boundary conditions with above viewpoint, we obtain a linear degenerate elliptic equation with mixed boundary conditions and a linear degenerate elliptic equation with oblique boundary conditions in a curved quadrilateral domain. By means of elliptic regularization techniques, delicate a priori estimate and compact arguments, we show that the solution of linearized problem with oblique boundary conditions is smooth in the interior and Lipschitz continuous up to the degenerate boundary.     

The limiting steady rotations of a body on a string with a suspension point on the axis of symmetry

The steady motions of an axisymmetrical rigid body suspended from a fixed base by a weightless undeformable rod or a non-twisting inextensible string are investigated. The case when the rod is fastened to the body at a point situated on its axis of dynamic symmetry is considered. All types of limiting equilibrium configurations which are possible when there is an unlimited increase in the angular velocity of rotation of the system about the vertical are analysed. Domains in which each type of limiting regular precession and permanent rotation can exist are constructed in the space of dimensionless parameters, and the nature of their asymptotic behaviour when the angular velocity increases is determined. The limiting motions which are possible in the case of suspension on a rod and impossible in the case of suspension on a string are investigated.     

The dissociation of a pure diatomic gas behind a strong normal shock wave

Zusammenfassung Die Arbeit behandelt das Problem der Dissoziation eines reinen diatomischen Gases durch einen starken geraden Verdichtungsstoss. Mit Hilfe einiger vereinfachender Annahmen und eines theoretischen Ausdruckes für die Dissoziationsgeschwindigkeit wird eine analytische Darstellung der Resultate erzielt. Der Einfluss der Erhitzung und der Kompression durch den Stoss wird eingehend diskutiert.

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The research reported in this document has been sponsored in part by the Air Research and Development Command, United States Air Force, under contract AF 61(514)1124, through the European Office, A.R.D.C.     

The diffraction of a shock wave by an inclined body of revolution

Summary The interaction between a plane normal shock wave of arbitrary strength and a slender body of revolution, held with its axis at a small angle to the direction of propagation of the shock, is considered. The pressure field is determined by using the method of matched asymptotic expansions and the general results are applied to the problem of diffraction by an inclined circular cone.

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Zusammenfassung Es wird die Beeinflussung einer ebenen Stosswelle beliebiger Stärke durch einen schlanken Rotationskörper untersucht, dessen Achse einen kleinen Winkel mit der Ausbreitungsrichtung des Stosses bildet. Das Druckfeld wird mit Hilfe der abgestimmten asymptotischen Entwicklungen bestimmt, und die allgemeinen Resultate werden auf das Problem der Diffraktion durch einen angestellten Kreiskegel angewendet.