Free Boundary Problems for Nonlinear Wave Systems: Mach Stems for Interacting Shocks
نویسندگان
چکیده
We study a family of two-dimensional Riemann problems for compressible flow modeled by the nonlinear wave system. The initial constant states are separated by two jump discontinuities, x = ±κay, which develop into two interacting shock waves. We consider shock angles in a range where regular reflection is not possible. The solution is symmetric about the y-axis and on each side of the y-axis consists of an incident shock, a reflected compression wave, and a Mach stem. This has a clear analogy with the problem of shock reflection by a ramp. It is well known that no triple point structure exists in which incident, reflected, and Mach stem shocks meet at a point. In this paper, we model the reflected wave by a continuous function with a singularity in the derivative. This fails to be a weak solution across the sonic line. We show that a solution to the free boundary problem for the Mach stem exists, and we conjecture that the global solution can be completed by the construction of a reflected shock, by a similar free boundary technique. The point of our paper is the capability to deal analytically with a Mach stem by solving a free boundary problem. The difficulties associated with the analysis of solutions containing Mach stems include (1) loss of obliqueness in the derivative boundary condition corresponding to the jump conditions across the Mach stem, and (2) loss of ellipticity at the formation point of the Mach stem. We use barrier functions to show that for sufficiently large values of κa the subsonic solution is continuous up to the sonic line at the Mach stem.
منابع مشابه
A Semi-empirical Model to Predict the Attached Axisymmetric Shock Shape
In this work, a simple semi-empirical model is proposed, based on Response Surface Model, RSM, to determine the shape of an attached oblique shock wave emanating from a pointed axisymmetric nose at zero angle of attack. Extensive supersonic visualization images have been compiled from various nose shapes at different Mach numbers, along with some others performed by the author for the present p...
متن کاملGlobal Stability for Thermal Convection in a Couple Stress Fluid Saturating a Porous Medium with Temperature-Pressure Dependent Viscosity: Galerkin Method
A global nonlinear stability analysis is performed for a couple-stress fluid layer heated from below saturating a porous medium with temperature-pressure dependent viscosity for different conducting boundary systems. Here, the global nonlinear stability threshold for convection is exactly the same as the linear instability boundary. This optimal result is important because it shows that lineari...
متن کاملShock Reflection-Diffraction and Nonlinear Partial Differential Equations of Mixed Type
We present our recent results on the existence and regularity of shock reflectiondiffraction configurations by a wedge up to the sonic wedge angle, which is the von Neumann sonic conjecture. The problem is first formulated as a boundary value problem for a second-order nonlinear partial differential equation of mixed hyperbolic-elliptic type in an unbounded domain. Then the boundary value probl...
متن کاملStudy of Parameters Affecting Separation Bubble Size in High Speed Flows using k-ω Turbulence Model
Shock waves generated at different parts of vehicle interact with the boundary layer over the surface at high Mach flows. The adverse pressure gradient across strong shock wave causes the flow to separate and peak loads are generated at separation and reattachment points. The size of separation bubble in the shock boundary layer interaction flows depends on various parameters. Reynolds-averaged...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- SIAM J. Math. Analysis
دوره 37 شماره
صفحات -
تاریخ انتشار 2006