Stability condition for strong shock waves in the problem of flow around an infinite plane wedge

We consider the thermodynamical equilibrium state flow of an inviscid non-heat-conducting gas flowing around a plane infinite wedge, and study the stationary solution to this problem–the so-called strong shock wave; the flow behind the shock front is subsonic.We find the solution to the linear analog of the original mixed problem, prove that the solution trace on the shock wave is the superposition of the direct and reflected waves, and (the main point) justify the Lyapunov asymptotical stability of the strong shock wave provided that the uniform Lopatinsky condition is fulfilled. The initial data have a compact support, and the solvability conditions occur.     

Justification of the Courant-Friedrich's hypothesis in the case of a weak shock. Part I. Presentation of solution to linear problem

As is well known, two solutions of the problem of a supersonic stationary inviscid nonheatconducting gas flow onto a planar infinite wedge are theoretically possible: the solution with a strong shock (the flow speed behind the shock is subsonic) and the solution with a weak shock (the flow speed behind the shock is supersonic). Unlike the well-studied case of a strong shock that is generically unstable [A.M. Blokhin, D.L. Tkachev, L.O. Baldan, Study of the stability in the problem on flowing around a wedge. The case of strong wave, J. Math. Anal. Appl. 319 (2006) 248-277; A.M. Blokhin, D.L. Tkachev, Yu.Yu. Pashinin, Stability condition for strong shock waves in the problem of flow around an infinite plane wedge, Nonlinear Anal. Hybrid Syst. 2 (2008) 1-17], R. Courant and K.O. Friedrichs [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, New York, 1948] assumed that the solution with a weak shock is asymptotically stable by Lyapunov. Presentation of classical solution to the corresponding problem which is found in the present paper is the first step on the way to justification of Courant-Friedrichs hypothesis on linear level.     

Stability of a supersonic flow past a wedge with adjoint weak neutrally stable shock wave

We study the classical problem of a supersonic stationary flow of a nonviscous nonheat-conducting gas in local thermodynamic equilibrium past an infinite plane wedge. Under the Lopatinski? condition on the shock wave (neutral stability), we prove the well-posedness of the linearized mixed problem (the main solution is a weak shock wave), obtain a representation of the classical solution, where, in this case (in contrast to the case of the uniform Lopatinski? condition—an absolutely stable shock wave), plane waves additionally appear in the representation. If the initial data have compact support, the solution reaches the given regime in infinite time.     

Continuous compression waves in the two-dimensional Riemann problem

The interaction between a plane shock wave in a plate and a wedge is considered within the framework of the nondissipative compressible fluid dynamic equations. The wedge is filled with a material that may differ from that of the plate. Based on the numerical solution of the original equations, self-similar solutions are obtained for several versions of the problem with an iron plate and a wedge filled with aluminum and for the interaction of a shock wave in air with a rigid wedge. The behavior of the solids at high pressures is approximately described by a two-term equation of state. In all the problems, a two-dimensional continuous compression wave develops as a wave reflected from the wedge or as a wave adjacent to the reflected shock. In contrast to a gradient catastrophe typical of one-dimensional continuous compression waves, the spatial gradient of a two-dimensional compression wave decreases over time due to the self-similarity of the solution. It is conjectured that a phenomenon opposite to the gradient catastrophe can occur in an actual flow with dissipative processes like viscosity and heat conduction. Specifically, an initial shock wave is transformed over time into a continuous compression wave of the same amplitude.     

Supersonic flow onto a solid wedge

We consider the problem of two‐dimensional supersonic flow onto a solid wedge, or equivalently in a concave corner formed by two solid walls. For mild corners, there are two possible steady state solutions, one with a strong and one with a weak shock emanating from the corner. The weak shock is observed in supersonic flights. A longstanding natural conjecture is that the strong shock is unstable in some sense. We resolve this issue by showing that a sharp wedge will eventually produce weak shocks at the tip when accelerated to a supersonic speed. More precisely, we prove that for upstream state as initial data in the entire domain, the time‐dependent solution is self‐similar, with a weak shock at the tip of the wedge. We construct analytic solutions for self‐similar potential flow, both isothermal and isentropic with arbitrary γ ≥ 1. In the process of constructing the self‐similar solution, we develop a large number of theoretical tools for these elliptic regions. These tools allow us to establish large‐data results rather than a small perturbation. We show that the wave pattern persists as long as the weak shock is supersonic‐supersonic; when this is no longer true, numerics show a physical change of behavior. In addition, we obtain rather detailed information about the elliptic region, including analyticity as well as bounds for velocity components and shock tangents. © 2007 Wiley Periodicals, Inc.     

The flow of a supersonic ideal gas with “weak” and “strong” shocks over a wedge

The problem of the flow of a uniform supersonic ideal (inviscid and non-heat-conducting) gas over a wedge is considered. If the turning angle of the flow, which is equal to the angle of inclination of the generatrix of the wedge, is less than the maximum value, the problem has two solutions. In the solution with an oblique low-intensity (“weak”) shock, the uniform flow between the shock and the wedge is almost always supersonic. One exception is a small vicinity of the maximum turning angle. For an ideal gas this vicinity does not exceed a fraction of a degree at all Mach numbers. Behind a high-intensity (“strong”) shock, the flow of an ideal gas is always subsonic. “Weak” shocks are observed in all experiments with finite wedges. Some researchers attribute this preference to the “downstream” boundary conditions (“on the right at infinity” for a flow incident on the wedge from the left), and others attribute it to the instability (“Lyapunov” instability) of a flow with a strong shock when it flows over the wedge and to the stability of flow with a weak shock. The results presented below from calculations of the flows that occur for finite wedges within the two-dimensional unsteady Euler equations, when the parameters behind the strong shock are specified on the right-hand boundary, i.e., on the arc of a circle between the wedge and the shock, demonstrate the correctness of the conclusion of the first group of researchers and the incorrectness of the conclusion of the other group. In these calculations, after both small and fairly large perturbations, the flows investigated (which are, in fact, Lyapunov unstable!) return to the solution with a strong shock. In addition, the problem of steady flow over a wedge was regarded as the limit of the two-dimensional non-steady problems at infinite time. Simplification of one of them leads to the problem of the submerged over-expanded supersonic steady outflow. In the ideal gas model this problem is equivalent to flow over a wedge with both weak and strong shocks. All the solutions considered are stable.     

Existence of Entropy Solutions to Two-Dimensional Steady Exothermically Reacting Euler Equations

We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function ?(T) is assumed to have a positive lower bound. We first consider the Cauchy problem (the initial value problem), that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is sufficiently small. Then we analyze the problem of steady supersonic, exothermically reacting Euler flow past a Lipschitz wedge, generating an additional detonation wave attached to the wedge vertex, which can be then formulated as an initial-boundary value problem. We establish the global existence of entropy solutions containing the additional detonation wave (weak or strong, determined by the wedge angle at the wedge vertex) when the total variation of both the slope of the wedge boundary and the incoming flow is suitably small. The downstream asymptotic behavior of the global solutions is also obtained.     

Stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow

We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than $1$, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.     

Steady supersonic flow past an almost straight wedge with large vertex angle

This paper studies the problem on the steady supersonic flow at the constant speed past an almost straight wedge with a piecewise smooth boundary. It is well known that if each vertex angle of the straight wedge is less than an extreme angle determined by the shock polar, the shock wave is attached to the tip of the wedge and constant states on both side of the shock are supersonic. This paper is devoted to generalizing this result. Under the hypotheses that each vertex angle is less than the extreme angle and the total variation of tangent angle along each edge is sufficiently small, a sequence of approximate solutions constructed by a modified Glimm scheme is proved to be convergent to a global weak solution of the steady problem. A sequence of the corresponding approximate leading shock fronts issuing from the tip is shown to be convergent to the leading shock front of the obtained solution. The regularity of the leading shock front is established and the asymptotic behaviour of the obtained solution at infinity is also studied.     

On global transonic shocks for the steady supersonic Euler flows past sharp 2-D wedges

In this paper, under certain downstream pressure condition at infinity, we study the globally stable transonic shock problem for the perturbed steady supersonic Euler flow past an infinitely long 2-D wedge with a sharp angle. As described in the book of Courant and Friedrichs [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948] (pages 317-318): when a supersonic flow hits a sharp wedge, it follows from the Rankine-Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the edge of the sharp wedge in terms of the different pressure states in the downstream region, which correspond to the supersonic shock and the transonic shock respectively. It has frequently been stated that the strong shock is unstable and that, therefore, only the weak shock could occur. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this open problem. More concretely, we will establish the global existence and stability of a transonic shock solution for 2-D full Euler system when the downstream pressure at infinity is suitably given. Meanwhile, the asymptotic state of the downstream subsonic solution is determined.     

An inverse problem for supersonic flow past a curved wedge

This paper studies an inverse problem for supersonic potential flow past a curved wedge, in which we design a suitable curved wedge such that the shock produced by the curved wedge can be controlled to the given position. Under suitable conditions, by characteristic method, we prove the existence of the global classical solution to this inverse problem and develop an optimal decay rate on the given shock’s second order derivatives. We finally construct a specific wedge such that the shock generated by the wedge is a convex combined one.     

Supersonic flow past a concave double wedge

The supersonic flow past a concave double wedge is discussed. Because of the interaction between the outer shock attached at the head of the wedge and the inner shock issued from the concave corner, there is a rarefaction wave issued from the intersection of the outer and inner shock. The rarefaction wave is reflected by the outer shock and the wedge infinitely, while the outer shock is also bent due to interaction. The global existence of the solution is proved under the assumptions that the outer shock is weak and the difference of two slopes of the double wedge is small. Meanwhile, a rigorous proof of the asymptotic behavior of the global solution is given. The property is often ap plied to numerical computation. Project partially supported by the National Natural Science Foundation of China and Doctoral Programme Foundation of IHEC.     

Asymptotic behavior of solutions for the 1-D isentropic Navier–Stokes–Korteweg equations with free boundary

In this paper, we consider the problem with a gas–gas free boundary for the one dimensional isentropic compressible Navier–Stokes–Korteweg system. For shock wave, asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary, and prove that if the initial data around the shifted viscous shock profile and its strength are sufficiently small, then the problem has a unique global strong solution, which tends to the shifted viscous shock profile as time goes to infinity. Also, we show the asymptotic stability toward rarefaction wave without the smallness on the strength if the initial data around the rarefaction wave are sufficiently small.     

Global Existence of a Shock for the Supersonic Flow Past a Curved Wedge

This note is devoted to the study of the global existence of a shock wave for the supersonic flow past a curved wedge. When the curved wedge is a small perturbation of a straight wedge and the angle of the wedge is less than some critical value, we show that a shock attached at the wedge will exist globally.     

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.     

Solutions of the Cauchy problem for the 2-D Euler system with fan-shaped structure

We study the Cauchy problem for the isentropic 2-D Euler system with initial data having discontinuity on a smooth curve. A local existence of a solution is established, which consists of shock wave, rarefaction wave and contact discontinuity.     

Stability of transonic shocks for the full Euler system in supersonic flow past a wedge

We study the stability of transonic shocks in steady supersonic flow past a wedge. It is known that in generic case such a problem admits two possible locations of the shock front, connecting the flow ahead of it and behind it. They can be distinguished as supersonic–supersonic shock and supersonic–subsonic shock (or transonic shock). Both these possible shocks satisfy the Rankine–Hugoniot conditions and the entropy condition. We prove that the transonic shock is conditionally stable under perturbation of the upstream flow or perturbation of wedge boundary. Copyright © 2005 John Wiley & Sons, Ltd.     

Quasi-classical approach to the inverse scattering problem for the KdV equation,and solution of the Whitham modulation equations

We consider an initial value problem for the KdV equation in the limit of weak dispersion. This model describes the formation and evolution in time of a nondissipative shock wave in plasma. Using the perturbation theory in power series of a small dispersion parameter, we arrive at the Riemann simple wave equation. Once the simple wave is overturned, we arrive at the system of Whitham modulation equations that describes the evolution of the resulting nondissipative shock wave. The idea of the approach developed in this paper is to study the asymptotic behavior of the exact solution in the limit of weak dispersion, using the solution given by the inverse scattering problem technique. In the study of the problem, we use the WKB approach to the direct scattering problem and use the formulas for the exact multisoliton solution of the inverse scattering problem. By passing to the limit, we obtain a finite set of relations that connects the space-time parameters x, t and the modulation parameters of the nondissipative shock wave.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 44–61, January, 1996.     

A difference scheme with anti-diffusion terms to improve the computation of contact discontinuities and the study of weak discontinuities in compressible flow

The investigation of Mach reflection formed after the impingement of a weak plane shock wave on a wedge with shock Mach number M_s near 1, is still an open problem[12]. It's difficult for shock tube experiments with interferometer to detect contact discontinuities if it is too weak; also difficult to catch with due accuracy the transition condition between Mach reflection and regular reflection. The interest to this phenomenon is continuing, especially for weak shocks, because there was systematic discrepancy between simplified three shock theory of von Neumann [8] and shock tube results [15] which was named by G. Birkhoff as “von Neumann Paradox on three shock theory” [18].In 1972, K.O.Friedrichs called for more computational efforts on this problem. Recently it is known that for weak impinging shocks it's still difficult to get contact discontinuities and curved Mach stem with satisfactory accuracy. Recent numerical computation sometimes even fails to show reflected shock wave[6]. These explain why von Neumann paradox of the three shock theory in case of weak discontinuities is still a problem of interesting [9,12,14]. In this paper, on one hand, we investigate the numerical methods for Euler's equation for compressible inviscid flow, aiming at improving the computation of contact discontinuities, on the other hand, a methodology is suggested to correctly plot flow data from the massive information in storage. On this basis, all the reflected shock wave , contact discontinuities and the curved Mach stem are determined. We get Mach reflection under the condition when over-simplified shock theory predicts no such configuration[5].     

On peculiarities of the application of Lagrangian variables in problems of time-dependent supersonic flow past bodies

The paper considers particularities in applying Lagrangian variables in problems of hypersonic flow past bodies. It is pointed out that, in problems with intense shock waves, it is advisable to introduce Lagrangian variables as the values of parameters that characterize a particle not on surface t = t ₀ (t ₀ = const) but on surface t = σ, where σ is the time instant at which the particle meets the surface of discontinuity. Considering examples of two-dimensional flow past two-dimensional and axisymmetric bodies moving with high time-dependent velocity, we show that the passage to Lagrangian variables enables us to obtain a system of equations describing the gas flow behind the front of an intense shock wave, which is suitable for the application of the thin-shock-layer flow method. A solution is constructed in the form of series in powers of a small parameter which characterizes the ratio of the gas densities at the front of the leading shock wave. We remark that all nonlinear effects of the problem are concentrated in the equation to determine the law of motion of a particle in zero-order approximation. The cases in which this equation can be integrated are pointed out. For the remaining unknowns, the solution is taken in quadratures. We study the rearrangement of the gas flow in the shock layer when the motion of the body is changed. The flow rearrangement zone is distinguished. Also, a condition to determine the life time of that region (the time of establishment of the new flow regime) is obtained. In a specific case of passing from a steady motion of a wedge to a uniformly accelerated motion, the time of establishment of the uniformly accelerated motion is determined from a quadratic equation.