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.     

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 VON NEUMANN PARADOX FOR THE EULER EQUATIONS

The reflection of a weak shock wave is considered using a shock polar.We present a sufficient condition under which the von Neumann paradox appears for the Eule...     

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.     

MECHANISM FOR THE TRANSITION FROM A REGULAR REFLECTION TO A MACH REFLECTION OR A VON NEUMANN REFLECTION

In this paper,by taking into account the thickness of the incident shock as well as the influence of the boundary layer,we point out that even in a regular reflection there should be present a contact discontinuity.By using the smallest energy criterion,the inclined angle of this contact discontinuity can be determined for differen incident angle.Then,with this inclined contact discontinuity,together with the law of conservation of mass,the mechanism for the transition from a regular reflection to a Mach reflection or a von Neumann reflection becomes clear.The important roles played by the leftest point in the reflected shock polar are identified.     

Elliptic problems with discontinuities

In this paper we prove two existence theorems for elliptic problems with discontinuities. The first one is a noncoercive Dirichlet problem and the second one is a Neumann problem. We do not use the method of upper and lower solutions. For Neumann problems we assume that f is nondecreasing. We use the critical point theory for locally Lipschitz functionals.     

Nonlinear geometric optics for contact discontinuities and shock waves in one-dimensional Euler system for gas dynamics

The aim of this paper is to study the rigorous theory of nonlinear geometric optics for a contact discontinuity and a shock wave to the Euler system for one-dimensional gas dynamics. For the problem of a contact discontinuity and a shock wave perturbed by a small amplitude, high frequency oscillatory wave train, under suitable stability assumptions, we obtain that the perturbed problem has still a shock wave and a contact discontinuity, and we give their asymptotic expansions.     

Features of the reflection of waves with strong discontinuities from thin compressible obstacles

Using the example of the plane contact problem of hydroelasticity theory, the multiple reflection of waves with strong discontinuities, propagating in an ideally elastic liquids, from thin films having a finite acoustic impedance, is investigated analytically. The wave solution is presented in the form of the sum of a basic component (no film) and a perturbed component. An algorithm is developed for the successive analytical calculation of the perturbed components after multiple reflections from an obstruction.     

Riemann problem in non-ideal gas dynamics

In this paper we consider the Riemann problem for gas dynamic equations governing a one dimensional flow of van der Waals gases. The existence and uniqueness of shocks, contact discontinuities, simple wave solutions are discussed using R-H conditions and Lax conditions. The explicit form of solutions for shocks, contact discontinuities and simple waves are derived. The effects of van der Waals parameter on the shock and simple waves are studied. A condition is derived on the initial data for the existence of a solution to the Riemann problem. Moreover, a necessary and sufficient condition is derived on the initial data which gives the information about the existence of a shock wave or a simple wave for a 1-family and a 3-family of characteristics in the solution of the Riemann problem.     

The Interaction of Shocks with Dispersive Waves I. Weak Coupling Limit

We introduce and analyze a model for the interaction of shocks with a dispersive wave envelope. The model mimicks the Zakharov system from weak plasma turbulence theory but replaces the linear wave equation in that system by a nonlinear wave equation allowing the formation of shocks. This paper considers a weak coupling in which the nonlinear wave evolves independently but appears as the potential in the time-dependent Schrodinger equation governing the dispersive wave. We first solve the Riemann problem for the system by constructing solutions to the Schrodinger equation that are steady in a frame of reference moving with the shock. Then we add a viscous diffusion term to the shock equation and by explicitly constructing asymptotic expansions in the (small) diffusion coefficient, we show that these solutions are zero diffusion limits of the regularized problem. The expansions are unusual in that it is necessary to keep track of exponentially small terms to obtain algebraically small terms. The expansions are compared to numerical solutions. We then construct a family of time-dependent solutions in the case that the initial data for the nonlinear wave equation evolves to a shock as t → t* < ∞. We prove that the shock formation drives a finite time blow-up in the phase gradient of the dispersive wave. While the shock develops algebraically in time, the phase gradient blows up logarithmically in time. We construct several explicit time-dependent solutions to the system, including ones that: (a) evolve to the steady states previously constructed, (b) evolve to steady states with phase discontinuities (which we call phase kinked steady states), (c) do not evolve to steady states.     

Anisotropic propagation of weak discontinuities in flows of thermally conducting and dissociating gases

The object of the present investigation is to study the anisotropic propagation of weak discontinuities in flows of thermally conducting and dissociating gases. The velocity of propagation of the wave frcnt is determined. A set of differential equations governing the growth and decay of weak discontinuities are obtained and solved. It is found that if the sonic wave is a compressive wave of order 1, then it terminates into a shock wave after a critical timet _c which has been determined. It is also observed that the effects of heat conduction and dissociation are to decrease the duration of time by which a weak discontinuity will generate into a shock wave.     

Water wave scattering by two sharp discontinuities in the surface boundary conditions

^** Corresponding author. Email: biren{at}isical.ac.in The problem of water wave scattering by two sharp discontinuitiesin the surface boundary conditions involving infinitely deepwater is examined here by reducing it to two coupled Carleman-typesingular integral equations. The discontinuities arise due tothe presence of two types of non-interacting materials floatingon the surface, one type being in the form of an infinite stripof finite width sandwiched between another type. The non-interactingmaterials form an inertial surface which is a mass-loading modelof floating ice and is regarded as a material of uniform surfacedensity having no elastic property. The two integral equationsare solved approximately by assuming the two discontinuitiesto be widely separated, and approximate analytical expressionsfor the reflection and transmission coefficients are also obtained.This problem has applications in wave propagation through stripsof frazil or pancake ice modelled as floating inertial surfaces.Numerical results for the reflection coefficient are depictedgraphically against the wave number for different values ofthe surface densities of the two types of floating materials.The main feature of the graphs is the oscillatory nature ofthe reflection coefficient and occurrence of zero reflectionfor an increasing sequence of discrete values of the wave number.A direct analytical treatment to solve the integral equationsnumerically, when the separation length between the two discontinuitiesis arbitrary, is also indicated. For the case of more than twodiscontinuities the solution methodology of the correspondingscattering problem is described briefly.     

On weak discontinuities through thermally conducting and dissociating gases

Using singular surface theory, the phenomena associated with the uniform and nonuniform propagation of weak discontinuities through thermally conducting and dissociating gases is studied. The basic differential equations governing the criteria for decay or blow up of these discontinuities is obtained. It turns out that growth and decay of weak discontinuities are derived and solved completely. The the thermal conduction and dissociation allow the existence of a singular surface carrying a weak discontinuity which grows into a shock and the role of dissociation and thermal conduction is to cause rapid damping in the formation of this shock.     

Diffraction of normal shock wave by yawed wedges

The diffraction of normal shock with yawed wedges of small angles have been considered in this paper. Vorticity distribution of the fluid particle over the diffracted shock has been determined for several Mach numbers of the shock wave. The Mach reflection effects have also been investigated when the bend is concave to the flow.     

Propagation of discontinuities along bicharacteristics in the unsteady flow of a relaxing gas

Growth and decay of weak discontinuities headed by wave front of arbitrary shape in three dimensions are investigated in an unsteady flow of a relaxing gas. The transport equations representing the rate of change of discontinuities in the normal derivatives of the flow variables are obtained and it is found that the nonlinearity in the governing equations plays an important role in the interplay of damping and steepening tendencies of the wave. An explicit criterion for the growth and decay of weak discontinuities along bicharacteristic curves in the characteristic manifold of the governing differential equations is given and special reference is made of diverging and converging waves under different thermodynamical situations. It is shown that there is a special case of a compressive converging wave, irrespective of the thermodynamical state whether weak or strong, in which the effects of thermodynamical influences and that of wave front curvature are unable to overcome the tendency of the wave to grow into a shock.     

一维绝热流动中初等波的相互作用

<正> 一维绝热流动的守恒律组(在拉格朗日座标下)为其中u——速度、p——压强、v——比容、E=e+u~2/2,而在多方气体的情形e=pv/(γ-1),绝热指数γ为常数,γ>1.人们称它的某些特解为初等波,其中包括前、后向激波S、S;前、后向中心疏散波R、R和上、下跳接触间断T、T.它们的相互作用一方面在     

Introduction to nonclassical shock solutions of the modified Burgers‐Korteweg‐de Vries equation

A general property of nonlinear hyperbolic equations is the eventual formation of discontinuities in the propagating signal. These discontinuities are not uniquely defined by the initial data for the problem and a central issue is the identification of acceptable weak solutions. Particular difficulties arise when the hyperbolic system ceases to be genuinely nonlinear in some of its characteristic fields. This equates in the case of a scalar law to the lack of convexity in the flux function. Here a representative example is provided by the modified Korteweg‐de Vries‐Burgers equation which exhibits a quadratic as well as a cubic nonlinear term and arises in a variety of engineering applications including weakly nonlinear waves in fluidized beds and two‐layer fluid flows. Its solutions have the distinguishing feature to generate undercompressive or nonclassical shocks in the hyperbolic limit with dispersion and dissipation balanced. The resulting rich variety of wave phenomena: shocks which emanate rather than absorb characteristics, compound shocks and shock fan combinations, which have no counterpart in classical shock theories is discussed.     

Ergodic actions of compact groups on operator algebras

Summary We prove that the only ergodic actions ofSU(2) are on Type I von Neumann algebras, using the theory developed in [8].     

Mach configuration in pseudo-stationary compressible flow

This paper is devoted to studying the local structure of Mach reflection, which occurs in the problem of the shock front hitting a ramp. The compressible flow is described by the full unsteady Euler system of gas dynamics. Because of the special geometry, the motion of the fluid can be described by self-similar coordinates, so that the unsteady flow becomes a pseudo-stationary flow in this coordinate system. When the slope of the ramp is less than a critical value, the Mach reflection occurs. The wave configuration in Mach reflection is composed of three shock fronts and a slip line bearing contact discontinuity. The local existence of a flow field with such a configuration under some assumptions is proved in this paper. Our result confirms the reasonableness of the corresponding physical observations and numerical computations in Mach reflection.

In order to prove the result, we formulate the problem to a free boundary value problem of a pseudo-stationary Euler system. In this problem two unknown shock fronts are the free boundary, and the slip line is also an unknown curve inside the flow field. The proof contains some crucial ingredients. The slip line will be transformed to a fixed straight line by a generalized Lagrange transformation. The whole free boundary value problem will be decomposed to a fixed boundary value problem of the Euler system and a problem to updating the location of the shock front. The Euler system in the subsonic region is an elliptic-hyperbolic composite system, which will be decoupled to the elliptic part and the hyperbolic part at the level of principal parts. Then some sophisticated estimates and a suitable iterative scheme are established. The proof leads to the existence and stability of the local structure of Mach reflection.

     

A criterion for the non-existence or non-uniqueness of solutions of self-similar problems in the mechanics of a continuous medium

Systems of hyperbolic partial differential equations expressing conservation laws are considered. A sufficient condition is formulated under which the self-similar problem of the disintegration of an arbitrary discontinuity (or the “piston” problem) either has no solution or the solution is not unique. This sufficient condition is determined by the existence of non-evolutionary discontinuities which may be considered as a sequence of two evolutionary discontinuities moving at the same velocity, if such a representation is unique. The condition is more general than that formulated previously, which was based on the existence of a non-proper Jouguet point. The new criterion is satisfied by weak quasitranverse shock waves in elastic media, whatever the sign of the coefficient of the non-linear deformation term. It also enables one to draw conclusions as to the non-existence or non-uniqueness of solutions of problems of the theory of elasticity in the case of finite-amplitude waves.