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Compressible Flow and Transonic Shock in a Diverging Nozzle

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Abstract

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.

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Authors and Affiliations

  1. Institute of Mathematics, Fudan University, Shanghai, 200433, P.R.China

    Shuxing Chen

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  1. Shuxing Chen
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Correspondence to Shuxing Chen.

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Communicated by P. Constantin

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Chen, S. Compressible Flow and Transonic Shock in a Diverging Nozzle. Commun. Math. Phys. 289, 75–106 (2009). https://doi.org/10.1007/s00220-009-0811-7

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  • DOI: https://doi.org/10.1007/s00220-009-0811-7

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