Compressible Flow and Transonic Shock in a Diverging Nozzle

The paper is devoted to the study of compressible flows and transonic shocks in diverging nozzles in the framework of the full compressible Euler system. Consider a nozzle having a shape as a diverging truncated sector with generic opening angle: if the upstream flow at the entrance is supersonic and is near to an axial symmetric flow, and if all parameters of the upstream flow and the receiver pressure at the exit are suitably assigned, then a transonic shock appears in the nozzle. To determine the transonic shock and the flow in the nozzle leads to a free boundary value problem for a nonlinear partial differential equation. We prove that the receiver pressure can uniquely determine the location of the transonic shock, as well as the flow behind the shock. Such a conclusion was conjectured by Courant and Friedrichs, and is confirmed theoretically in this paper for the divergent nozzles.The main advantage of this paper compared with the previous studies on this subject is that the section of the nozzle is allowed to vary substantially, while the transonic shock is not assumed to pass a fixed point. The situation coincides with the requirement in Courant-Friedrichs’ conjecture. To describe the compressible flow we use the full Euler system, which is purely hyperbolic in the supersonic region and is elliptic-hyperbolic in the subsonic region. Solving the free boundary value problem of an elliptic-hyperbolic problem forms the main part of this paper. In our demonstration some new approaches, including the introduction of a pseudo-free boundary problem and the corresponding relaxation, design of a delicate double iteration scheme, are developed to overcome the difficulties caused by the divergence of the nozzle.

[1]  Shuxing Chen,et al.  Transonic shocks in 3-D compressible flow passing a duct with a general section for Euler systems , 2008 .

[2]  Mikhail Feldman,et al.  MULTIDIMENSIONAL TRANSONIC SHOCKS AND FREE BOUNDARY PROBLEMS FOR NONLINEAR EQUATIONS OF MIXED TYPE , 2003 .

[3]  Eun Heui Kim,et al.  A free boundary problem for a quasi‐linear degenerate elliptic equation: Regular reflection of weak shocks , 2002 .

[4]  Transonic shocks for steady Euler flows with cylindrical symmetry , 2007 .

[5]  Gui-Qiang G. Chen,et al.  Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections , 2007 .

[6]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[7]  H. Glaz,et al.  The asymptotic analysis of wave interactions and numerical calculations of transonic nozzle flow , 1984 .

[8]  Andrew J. Majda,et al.  One Perspective on Open Problems in Multi-Dimensional Conservation Laws , 1991 .

[9]  S. Čanić,et al.  Free Boundary Problems for the Unsteady Transonic Small Disturbance Equation: Transonic Regular Reflection , 2000 .

[10]  Zhouping Xin,et al.  Transonic shock in a nozzle I: 2D case , 2005 .

[11]  Barbara Lee Keyfitz,et al.  A proof of existence of perturbed steady transonic shocks via a free boundary problem , 2000 .

[12]  Z. Xin,et al.  Three-dimensional transonic shocks in a nozzle , 2008 .

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

[14]  Ul’yan G. Pirumov,et al.  Gas Flow in Nozzles , 1986 .

[15]  Hairong Yuan.  Examples of steady subsonic flows in a convergent–divergent approximate nozzle , 2008 .

[16]  Tai-Ping Liu.  Nonlinear stability and instability of transonic flows through a nozzle , 1982 .

[17]  Hairong Yuan,et al.  On Transonic Shocks in Two-Dimensional Variable-Area Ducts for Steady Euler System , 2006, SIAM J. Math. Anal..

[18]  Gui-Qiang G. Chen,et al.  Free boundary problems and transonic shocks for the Euler equations in unbounded domains , 2004 .

[19]  Mikhail Feldman,et al.  Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations , 2004 .

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

[21]  Shuxing Chen,et al.  Stability of transonic shock fronts in two-dimensional Euler systems , 2004 .

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

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

[24]  Tai-Ping Liu.  Transonic gas flow in a duct of varying area , 1982 .

[25]  Hairong Yuan,et al.  Transonic Shocks in Compressible Flow Passing a Duct for Three-Dimensional Euler Systems , 2008 .

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

[27]  Z. Xin,et al.  The transonic shock in a nozzle, 2-D and 3-D complete Euler systems , 2008 .

[28]  Chen Shuxing.  Stability of oblique shock front , 2002 .

[29]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[30]  Gui-Qiang G. Chen,et al.  Transonic nozzle flows and free boundary problems for the full Euler equations , 2006 .

[31]  Tai-Ping Liu,et al.  Asymptotic states for hyperbolic conservation laws with a moving source , 1983 .