اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدGlobal Solutions of Shock Reflection by Large-Angle Wedges for Potential Flow
165
0
0.0
(
0
)
تأليف
Gui-Qiang Chen
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
When a plane shock hits a wedge head on, it experiences a reflection-diffraction process and then a self-similar reflected shock moves outward as the original shock moves forward in time. Experimental, computational, and asymptotic analysis has shown that various patterns of shock reflection may occur, including regular and Mach reflection. However, most of the fundamental issues for shock reflection have not been understood, including the global structure, stability, and transition of the different patterns of shock reflection. Therefore, it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection. On the other hand, there has been no rigorous mathematical result on the global existence and structural stability of shock reflection, including the case of potential flow which is widely used in aerodynamics. Such problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptic-hyperbolic type, free boundary problems, and corner singularity where an elliptic degenerate curve meets a free boundary. In this paper we develop a rigorous mathematical approach to overcome these difficulties involved and establish a global theory of existence and stability for shock reflection by large-angle wedges for potential flow. The techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties.
قيم البحث
اقرأ أيضاً
We are concerned with the suitability of the main models of compressible fluid dynamics for the Lighthill problem for shock diffraction by a convex corned wedge, by studying the regularity of solutions of the problem, which can be formulated as a fre
e boundary problem. In this paper, we prove that there is no regular solution that is subsonic up to the wedge corner for potential flow. This indicates that, if the solution is subsonic at the wedge corner, at least a characteristic discontinuity (vortex sheet or entropy wave) is expected to be generated, which is consistent with the experimental and computational results. Therefore, the potential flow equation is not suitable for the Lighthill problem so that the compressible Euler system must be considered. In order to achieve the non-existence result, a weak maximum principle for the solution is established, and several other mathematical techniques are developed. The methods and techniques developed here are also useful to the other problems with similar difficulties.
In this paper, we study the problem of shock reflection by a wedge, with the potential flow equation, which is a simplification of the Euler System. In the work of M. Feldman and G. Chen, the existence theory of shock reflection problems with the p
otential flow equation was established, when the wedge is symmetric w.r.t. the direction of the upstream flow. As a natural extension, we study non-symmetric cases, i.e. when the direction of the upstream flow forms a nonzero angle with the symmetry axis of the wedge. The main idea of investigating the existence of solutions to non-symmetric problems is to study the symmetry of the solution. Then difficulties arise such as free boundaries and degenerate ellipticity, especially when ellipticity degenerates on the free boundary. We developed an integral method to overcome these difficulties. Some estimates near the corner of wedge is also established, as an extension of G.Liebermans work. We proved that in non-symmetric cases, the ideal Lipschitz solution to the potential flow equation, which we call regular solution, does not exist. This suggests that the potential flow solutions to the non-symmetric shock reflection problem, should have some singularity which is not encountered in symmetric case.
For an upstream supersonic flow past a straight-sided cone in $R^3$ whose vertex angle is less than the critical angle, a transonic (supersonic-subsonic) shock-front attached to the cone vertex can be formed in the flow. In this paper we analyze the
stability of transonic shock-fronts in three-dimensional steady potential flow past a perturbed cone. We establish that the self-similar transonic shock-front solution is conditionally stable in structure with respect to the conical perturbation of the cone boundary and the upstream flow in appropriate function spaces. In particular, it is proved that the slope of the shock-front tends asymptotically to the slope of the unperturbed self-similar shock-front downstream at infinity.
We construct global weak solutions to isothermal quantum Navier-Stokes equations, with or without Korteweg term, in the whole space of dimension at most three. Instead of working on the initial set of unknown functions, we consider an equivalent refo
rmulation, based on a time-dependent rescaling, that we introduced in a previous paper to study the large time behavior, and which provides suitable a priori estimates, as opposed to the initial formulation where the potential energy is not signed. We proceed by working on tori whose size eventually becomes infinite. On each fixed torus, we consider the equations in the presence of drag force terms. Such equations are solved by regularization, and the limit where the drag force terms vanish is treated by resuming the notion of renormalized solution developed by I. Lacroix-Violet and A. Vasseur. We also establish global existence of weak solutions for the isothermal Korteweg equation (no viscosity), when initial data are well-prepared, in the sense that they stem from a Madelung transform.
Although local existence of multidimensional shock waves has been established in some fundamental references, there are few results on the global existence of those waves except the ones for the unsteady potential flow equations in n-dimensional spac
es (n > 4) or in special unbounded space-time domains with non-physical boundary conditions. In this paper, we are concerned with both the local and global multidimensional conic shock wave problem for the unsteady potential flow equations when a pointed piston (i.e., the piston degenerates into a single point at the initial time) or an explosive wave expands fast in 2-D or 3-D static polytropic gas. It is shown that a multidimensional shock wave solution of such a class of quasilinear hyperbolic problems not only exists locally, but it also exists globally in the whole space-time and approaches a self-similar solution as t goes to infinity.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
