Transonic Shocks for the Full Compressible Euler System in a General Two-Dimensional De Laval Nozzle

In this paper, we study the transonic shock problem for the full compressible Euler system in a general two-dimensional de Laval nozzle as proposed in Courant and Friedrichs (Supersonic flow and shock waves, Interscience, New York, 1948): given the appropriately large exit pressure p _e(x), if the upstream flow is still supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes p _e(x). We solve this problem completely for a general class of de Laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible Euler system. The problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic–elliptic system. One of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted Hölder space with low regularities for a second order quasilinear elliptic equation with a free parameter (the position of the shock curve at one wall of the nozzle) and non-local terms involving the trace on the shock of the first order derivatives of the unknown function.     

Transonic Shocks in Compressible Flow Passing a Duct for Three-Dimensional Euler Systems

In this paper we study the transonic shock in steady compressible flow passing a duct. The flow is a given supersonic one at the entrance of the duct and becomes subsonic across a shock front, which passes through a given point on the wall of the duct. The flow is governed by the three-dimensional steady full Euler system, which is purely hyperbolic ahead of the shock and is of elliptic–hyperbolic composed type behind the shock. The upstream flow is a uniform supersonic one with the addition of a three-dimensional perturbation, while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference. The problem of determining the transonic shock and the flow behind the shock is reduced to a free-boundary value problem. In order to solve the free-boundary problem of the elliptic–hyperbolic system one crucial point is to decompose the whole system to a canonical form, in which the elliptic part and the hyperbolic part are separated at the level of the principal part. Due to the complexity of the characteristic varieties for the three-dimensional Euler system the calculus of symbols is employed to complete the decomposition. The new ingredient of our analysis also contains the process of determining the shock front governed by a pair of partial differential equations, which are coupled with the three-dimensional Euler system. The paper is partially supported by National Natural Science Foundation of China 10531020, the National Basic Research Program of China 2006CB805902, and the Doctorial Foundation of National Educational Ministry 20050246001.     

Instabilities in a Two-Dimensional Combustion Model with Free Boundary

We prove instability of the planar travelling wave solution in a two-dimensional free boundary problem modelling the propagation of near- equidiffusional premixed flames in the whole plane. We reduce the problem to a fixed boundary fully nonlinear parabolic system. The spectrum of the linearized operator contains an interval [0,ω ^c ], ω ^c > 0, so we cannot construct backward solutions. We use an argument about stability of dynamical systems in Banach spaces in order to prove pointwise instability of the moving front. (Accepted: January 31, 2000)?Published online August 21, 2000     

温度对冲击相变波传播的影响

冲击加载下,相界面的传播是一热力耦合过程。相变波阵面不仅是力学和物质间断面,也是温度界面。为考虑温度对相变波传播的影响,本文首先建立了相界面上的热传导方程和热力耦合的相变本构方程,然后采用一维特征线理论和有限差分数值计算相结合的方法,分析了温度界面和相变波的基本相互作用规律,进而给出了连续温度梯度下和绝热冲击下相变波传播规律。结果表明,温度对相变波传播的作用主要体现在两个方面,一方面是作为温度界面将与各类间断面相互作用,另一方面冲击相变波阵面后区域热力学状态变化影响卸载波结构。其原因在于相变方式（可逆、不可逆）和相变阈值应力具有强烈的温度相关性。     

Transonic Shock Problem for the Euler System in a Nozzle

In this paper, we study the well-posedness problem on transonic shocks for steady ideal compressible flows through a two-dimensional slowly varying nozzle with an appropriately given pressure at the exit of the nozzle. This is motivated by the following transonic phenomena in a de Laval nozzle. Given an appropriately large receiver pressure P _r , if the upstream flow remains supersonic behind the throat of the nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the flow is compressed and slowed down to subsonic speed, and the position and the strength of the shock front are automatically adjusted so that the end pressure at exit becomes P _r , as clearly stated by Courant and Friedrichs [Supersonic flow and shock waves, Interscience Publishers, New York, 1948 (see section 143 and 147)]. The transonic shock front is a free boundary dividing two regions of C ^2,α flow in the nozzle. The full Euler system is hyperbolic upstream where the flow is supersonic, and coupled hyperbolic-elliptic in the downstream region Ω₊ of the nozzle where the flow is subsonic. Based on Bernoulli’s law, we can reformulate the problem by decomposing the 3 × 3 Euler system into a weakly coupled second order elliptic equation for the density ρ with mixed boundary conditions, a 2 × 2 first order system on u ₂ with a value given at a point, and an algebraic equation on (ρ, u ₁, u ₂) along a streamline. In terms of this reformulation, we can show the uniqueness of such a transonic shock solution if it exists and the shock front goes through a fixed point. Furthermore, we prove that there is no such transonic shock solution for a class of nozzles with some large pressure given at the exit. This research was supported in part by the Zheng Ge Ru Foundation when Yin Huicheng was visiting The Institute of Mathematical Sciences, The Chinese University of Hong Kong. Xin is supported in part by Hong Kong RGC Earmarked Research Grants CUHK-4028/04P, CUHK-4040/06P, and Central Allocation Grant CA05-06.SC01. Yin is supported in part by NNSF of China and Doctoral Program of NEM of China.     

一种简单的精确捕捉接触间断的黎曼求解器

基于Godunov型数值格式的有限体积法是求解双曲型守恒律系统的主流方法,其中用来计算界面数值通量的黎曼求解器在很大程度上决定了数值格式在计算中的表现。单波的Rusanov求解器和双波的HLL求解器具有简单、高效和鲁棒性好等优点,但是在捕捉接触间断时耗散太大。全波的HLLC格式能够精确捕捉接触间断,但是在计算中出现的激波不稳定现象限制了其在高马赫数流动问题中的应用。本文利用双曲正切函数和五阶WENO格式来重构界面两侧的密度值,并且结合边界变差下降算法来减小Rusanov格式耗散项中的密度差,从而提高格式对于接触间断的分辨率。研究表明,相比于全波的HLLC求解器,本文构造的黎曼求解器不仅具有更高的接触分辨率,而且还具有更好的激波稳定性。     

Focusing of a shock wave in a rarefied gas A numerical study

The process of focusing of a shock wave in a rarefied noble gas is investigated by a numerical solution of the corresponding two dimensional initial–boundary value problem for the Boltzmann equation. The numerical method is based on the splitting algorithm in which the collision integral is computed by a Monte Carlo quadrature, and the free flow equation is solved by a finite volume method. We analyse the development of the shock wave which reflects from a suitably shaped reflector, and we study influence of various factors, involved in the mathematical model of the problem, on the process of focusing. In particular, we investigate the pressure amplification factor and its dependence on the strength of the shock and on the accommodation coefficient appearing in the Maxwell boundary condition modelling the gas-surface interaction. Moreover, we study the dependence of the shock focusing phenomenon on the shape of the reflector, and on the Mach number of the incoming shock. Received 25 May 1998 / Accepted 4 January 2000     

Solution of a multidimensional impact deformation problem for an elastic half-space with curved boundary on the basis of a modified ray method

A generalization of the method for constructing approximate solutions of boundary value problems of impact deformation dynamics in the form of ray expansions for two-dimensional plane deformation problems is presented. For each shock wave, the solution near its front is determined on the basis of ray coordinates consistent with this wave. The nonlinear divergence of curvilinear rays is taken into account. A mechanism of transformation from one ray coordinate system to another, which is crucially important in the ray method, is described. The developed technique is illustrated by solving the impact deformation problem for a half-space with boundary of nonzero curvature.     

The influence of magnetic field upon the collapse of a cylindrical shock wave

In the present paper, the problem of propagation of collapsing cylindrical shock wave in an ideal gas permeated by a transverse magnetic field with infinite electrical conductivity is investigated. Here it is assumed that the medium ahead of the shock front is uniform and at rest. Also, its counter pressure concerning the motion of the wave front is neglected. This problem admits a self similar solution of second kind. The similarity exponent has been computed by solving a nonlinear eigenvalue problem and integrating numerically the self-similar equations for various values of adiabatic heat exponent and Cowling number. Numerical computations have been performed to determine the flow field behind the shock wave. The influence of magnetic field strength and adiabatic heat exponent on the flow parameters for various cases is presented.     

Unsteady and nonequilibrium airflow near a stagnation line

The propagation of shock waves in a medium with a nonuniform distribution of the parameters is the subject of recently published research [1–3]. The present paper deals with the problem of the gas flow ahead of the forward point of a blunt body moving at supersonic speed in air with variable parameters. The chemical reaction processes behind the shock front are taken into account. As a result of numerical calculations by the method of characteristics with isolation of the forward shock the time-dependent position of the shock front and the distributions of the composition and gas dynamic parameters in the shock layer are found. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 170–172, November–December, 1986.     

Analysis of shock wave reflection from fixed and moving boundaries using a stabilized particle method

In the present paper, the efficiency of an enhanced formulation of the stabilized corrective smoothed particle method （CSPM） for simulation of shock wave propagation and reflection from fixed and moving solid boundaries in compressible fluids is investigated. The Lagrangian nature and its accuracy for imposing the boundary conditions are the two main reasons for adoption of CSPM. The governing equations are further modified for imposition of moving solid boundary conditions. In addition to the traditional artificial viscosity, which can remove numerically induced abnormal jumps in the field values, a velocity field smoothing technique is introduced as an efficient method for stabilizing the solution. The method has been implemented for one- and two-dimensional shock wave propagation and reflection from fixed and moving boundaries and the results have been compared with other available solutions. The method has also been adopted for simulation of shock wave propagation and reflection from infinite and finite solid boundaries.     

Motion of a strong shock front in a two-dimensional gas layer with moving internal boundary

We examine the problem of planar one-dimensional motion of a strong shock wave with moving internal boundary in which the initial position of the front, its intensity, the mass of the gas involved in the motion, and the energy contained in this gas are known. The problem is not self-similar and its exact solution, which involves working with partial differential equations, presents serious difficulties. In the following we determine the law of shock-front motion in this problem via the method of [1], which makes it possible to find a system of ordinary differential equations for the problem. The method is based on an initial specification of the power-law coupling between the dimensionless Lagrangian and Eulerian variables and replacement of the energy equation by this coupling and the energy integral. The solution is sought in the first approximation.     

Propagation of converging spherical deformation waves in a heteromodular elastic medium

The unsteady one-dimensional boundary-value problem of shock deformation of a medium bounded by a sphere is solved. The propagation of converging deformation wave fronts in an elastic material with different tensile and compressive strengths is studied. A boundary condition is obtained that provides the formation of a converging spherical shock wave with constant velocity. The impact conditions on the boundary of the heteromodular sphere are determined that can lead to the formation of a transition zone (a spherical layer of constant density) between the compression and tension regions.     

Velocity field generated by wing vibrations propagating along the surface

The velocity field generated by wing vibrations propagating along an elastic wing surface with finite velocity is studied.The gasdynamic problem is reduced to a mixed boundary-value problem with a moving boundary for the three-dimensional wave equation. The solution is obtained in closed form when the wing travels at supersonic velocity following an arbitrary law, the vibration propagation front is an arbitrary curve displacing along the wing surface, and the wing edges are supersonic.     

Shock wave propagation through a model one dimensional heterogeneous medium

We study the problem of impact-induced shock wave propagation through a model one-dimensional heterogeneous medium. This medium is made of a model material with spatially varying parameters such that it is heterogeneous to shock waves but homogeneous to elastic waves. Using the jump conditions and maximal dissipation criteria, we obtain the exact solution to the shock propagation problem. We use it to study how the nature of the heterogeneity changes material response, the structure of the shock front and the dissipation.     

Thermo-acceleration waves and shock formation in Extended Thermodynamics of gravitational gases

The possibility of shock formation as degeneration of acceleration waves in a thermoviscous gravitational ideal gas is studied by exploiting the hyperbolic system of Extended Thermodynamics. The mathematical aspects of this problem are discussed by considering the different contributions of gravity and dissipative effects. In particular, we evaluate the critical time (i.e. the instant in which a shock wave starts) proving that it exists, in the usual physical situations, only for a sufficiently large critical initial amplitude of the acceleration jump. We show that an acceleration wave can never degenerate into a shock wave except in some limiting cases and so, since gravity force is overcome by dissipative effects, our results do not differ, qualitatively, from the case without gravity: this result implies the asymptotic stability (in the sense of [1]) of the static isothermal solutions.     

Diffraction of a sound wave by a slit

The diffraction of a sound wave by a slit in an unbounded plane is analyzed as an initial-boundary-value problem with a moving boundary for the two-dimensional wave equation. The initial-boundary-value problem is solved by the formation and inversion of Volterra integral equations. A solution is obtained in closed form in quadratures for an arbitrary angle of inclination of the incident wave front relative to the plane. The solution is presented in the form of recursion formulas, which take into account the influence of diffraction waves occurring in succession at the boundaries of the slit.     

Global Attractor for a Wave Equation with Nonlinear Localized Boundary Damping and a Source Term of Critical Exponent

The paper addresses long-term behavior of solutions to a damped wave equation with a critical source term. The dissipative frictional feedback is restricted to a subset of the boundary of the domain. This paper derives inverse observability estimates which extend the results of Chueshov et al. (Disc Contin Dyn Syst 20:459–509, 2008) to systems with boundary dissipation. In particular, we show that a hyperbolic flow under a critical source and geometrically constrained boundary damping approaches a smooth finite-dimensional global attractor. A similar result for subcritical sources was given in Chueshov et al. (Commun Part Diff Eq 29:1847–1876, 2004). However, the criticality of the source term in conjunction with geometrically restricted dissipation constitutes the major new difficulty of the problem. To overcome this issue we develop a special version of Carleman’s estimates and apply them in the context of abstract results on dissipative dynamical systems. In contrast with the localized interior damping (Chueshov et al. Disc Contin Dyn Syst 20:459–509, 2008), the analysis of a boundary feedback requires a more careful treatment of the trace terms and special tangential estimates based on microlocal analysis.      

A bending quasi-front generated by an instantaneous impulse loading at the edge of a semi-infinite pre-stressed incompressible elastic plate

A refined membrane-like theory is used to describe bending of a semi-infinite pre-stressed incompressible elastic plate subjected to an instantaneous impulse loading at the edge. A far-field solution for the quasi-front is obtained by using the method of matched asymptotic expansions. A leading-order hyperbolic membrane equation is used for an outer problem, whereas a refined (singularly perturbed) membrane equation of an inner problem describes a boundary layer, which smoothes a discontinuity predicted by the outer problem at the wave front. The inner problem is then reduced to one-dimensional by an appropriate choice of inner coordinates, motivated by the wave front geometry. Using the inherent symmetry of the outer problem, a solution for the quasi-front is derived that is valid in a vicinity of the tip of the wave front. Pre-stress is shown to affect geometry and type of the generated quasi-front; in addition to a classical receding quasi-front the pre-stressed plate can support propagation of an advancing quasi-front. Possible responses may even feature both types of quasi-front at the same time, which is illustrated by numerical examples. The case of a so-called narrow quasi-front, associated with a possible degeneration of contribution of singular perturbation terms to the governing equation, is studied qualitatively.