Transonic Shock Problem for the Euler System in a Nozzle |
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Authors: | Zhouping Xin Wei Yan Huicheng Yin |
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Institution: | (1) Department of Mathematics and IMS, CUHK, Shatin, NT, Hong Kong;(2) Department of Mathematics and IMS, Nanjing University, 210093 Nanjing, China;(3) The Institute of Mathematical Sciences, CUHK, Shatin, NT, Hong Kong |
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Abstract: | In this paper, we study the well-posedness problem on transonic shocks for steady ideal compressible flows through a two-dimensional
slowly varying nozzle with an appropriately given pressure at the exit of the nozzle. This is motivated by the following transonic
phenomena in a de Laval nozzle. Given an appropriately large receiver pressure P
r
, if the upstream flow remains supersonic behind the throat of the nozzle, then at a certain place in the diverging part of
the nozzle, a shock front intervenes and the flow is compressed and slowed down to subsonic speed, and the position and the
strength of the shock front are automatically adjusted so that the end pressure at exit becomes P
r
, as clearly stated by Courant and Friedrichs Supersonic flow and shock waves, Interscience Publishers, New York, 1948 (see
section 143 and 147)]. The transonic shock front is a free boundary dividing two regions of C
2,α flow in the nozzle. The full Euler system is hyperbolic upstream where the flow is supersonic, and coupled hyperbolic-elliptic
in the downstream region Ω+ of the nozzle where the flow is subsonic. Based on Bernoulli’s law, we can reformulate the problem by decomposing the 3 ×
3 Euler system into a weakly coupled second order elliptic equation for the density ρ with mixed boundary conditions, a 2 × 2 first order system on u
2 with a value given at a point, and an algebraic equation on (ρ, u
1, u
2) along a streamline. In terms of this reformulation, we can show the uniqueness of such a transonic shock solution if it
exists and the shock front goes through a fixed point. Furthermore, we prove that there is no such transonic shock solution
for a class of nozzles with some large pressure given at the exit.
This research was supported in part by the Zheng Ge Ru Foundation when Yin Huicheng was visiting The Institute of Mathematical
Sciences, The Chinese University of Hong Kong. Xin is supported in part by Hong Kong RGC Earmarked Research Grants CUHK-4028/04P,
CUHK-4040/06P, and Central Allocation Grant CA05-06.SC01. Yin is supported in part by NNSF of China and Doctoral Program of
NEM of China. |
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Keywords: | |
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