Three-Dimensional Steady Supersonic Euler Flow Past a Concave Cornered Wedge with Lower Pressure at the Downstream

In this paper, we study the stability of the three-dimensional jet created by a supersonic flow past a concave cornered wedge with the lower pressure at the downstream. The gas beyond the jet boundary is assumed to be static. It can be formulated as a nonlinear hyperbolic free boundary problem in a cornered domain with two characteristic free boundaries of different types: one is the rarefaction wave, while the other one is the contact discontinuity, which can be either a vortex sheet or an entropy wave. A more delicate argument is developed to establish the existence and stability of the square jet structure under the perturbation of the supersonic incoming flow and the pressure at the downstream. The methods and techniques developed here are also helpful for other problems involving similar difficulties.

[1]  Beixiang Fang,et al.  The uniqueness of transonic shocks in supersonic flow past a 2-D wedge , 2016 .

[2]  Volker Elling,et al.  Supersonic flow onto a solid wedge , 2007, 0707.2108.

[3]  G. Métivier.  Stability of Multi-Dimensional Weak Shocks , 1990 .

[4]  Yi Wang,et al.  Fluid dynamic limit to the Riemann Solutions of Euler equations: I. Superposition of rarefaction waves and contact discontinuity , 2010, 1011.1990.

[5]  Gui-Qiang G. Chen,et al.  Stability of Transonic Characteristic Discontinuities in Two-Dimensional Steady Compressible Euler Flows , 2012, 1208.5183.

[6]  Jean-François Coulombel,et al.  NONLINEAR COMPRESSIBLE VORTEX SHEETS IN TWO SPACE DIMENSIONS , 2008 .

[7]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[8]  Tatsien Li,et al.  Boundary value problems for quasilinear hyperbolic systems , 1985 .

[9]  Gui-Qiang G. Chen,et al.  Stability of Transonic Shock-Fronts in Three-Dimensional Conical Steady Potential Flow past a Perturbed Cone , 2007, 0711.3666.

[10]  Richard Courant,et al.  Supersonic Flow And Shock Waves , 1948 .

[11]  Kurt Friedrichs,et al.  Symmetric positive linear differential equations , 1958 .

[12]  Yongqian Zhang.  Steady supersonic flow past an almost straight wedge with large vertex angle , 2003 .

[13]  Z. Xin,et al.  GLOBAL MULTIDIMENSIONAL SHOCK WAVE FOR THE STEADY SUPERSONIC FLOW PAST A THREE-DIMENSIONAL CURVED CONE , 2006 .

[14]  Beixiang Fang,et al.  Stability of transonic shocks in supersonic flow past a wedge , 2007 .

[15]  Xuemei Deng,et al.  Shock Diffraction by Convex Cornered Wedges for the Nonlinear Wave System , 2012, 1202.0856.

[16]  Shuxing Chen,et al.  Cauchy problem with general discontinuous initial data along a smooth curve for 2-d Euler system , 2014 .

[17]  Zhan Wang,et al.  Local Structural Stability of a Multidimensional Centered Rarefaction Wave for the Three-Dimensional Steady Supersonic Euler Flow around a Sharp Corner , 2010, SIAM J. Math. Anal..

[18]  Gang Xu,et al.  Stability of spherically symmetric subsonic flows and transonic shocks under multidimensional perturbations , 2015, 1505.02998.

[19]  Tai-Ping Liu,et al.  Well‐posedness theory for hyperbolic conservation laws , 1999 .

[20]  L. Hörmander,et al.  The boundary problems of physical geodesy , 1976 .

[21]  A. Majda.  The stability of multi-dimensional shock fronts , 1983 .

[22]  Gang Xu,et al.  Global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone , 2008 .

[23]  Shuxing Chen.  Existence of local solution to supersonic flow past a three-dimensional wing , 1992 .

[24]  S. Alinhac,et al.  Existence d'ondes de rarefaction pour des systems quasi‐lineaires hyperboliques multidimensionnels , 1989 .

[25]  Mikhail Feldman,et al.  Global Solutions of Shock Reflection by Large-Angle Wedges for Potential Flow , 2007, 0708.2540.

[26]  Gui-Qiang G. Chen,et al.  Well-posedness of transonic characteristic discontinuities in two-dimensional steady compressible Euler flows , 2012, 1209.3806.

[27]  Fang Yu,et al.  Stability of contact discontinuities in three-dimensional compressible steady flows , 2013 .

[28]  Shuxing Chen.  Existence of Stationary Supersonic Flows Past a Pointed Body , 2001 .

[29]  Jeffrey Rauch,et al.  Symmetric positive systems with boundary characteristic of constant multiplicity , 1985 .

[30]  Ya-Guang Wang,et al.  Weak stability of transonic contact discontinuities in three-dimensional steady non-isentropic compressible Euler flows , 2015 .