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Transonic Shock Problem for the Euler System in a Nozzle

Xin, Zhouping ; Yan, Wei ; Yin, Huicheng

Archive for rational mechanics and analysis, 2009-10, Vol.194 (1), p.1-47 [Periódico revisado por pares]

Berlin/Heidelberg: Springer Berlin Heidelberg

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Título:
Transonic Shock Problem for the Euler System in a Nozzle
Autor: Xin, Zhouping ; Yan, Wei ; Yin, Huicheng
Assuntos: Classical Mechanics ; Complex Systems ; Compressible flows; shock and detonation phenomena ; Exact sciences and technology ; Flows in ducts, channels, nozzles, and conduits ; Fluid dynamics ; Fluid- and Aerodynamics ; Fundamental areas of phenomenology (including applications) ; General subsonic flows ; Mathematical and Computational Physics ; Mathematical methods in physics ; Numerical approximation and analysis ; Ordinary and partial differential equations, boundary value problems ; Physics ; Physics and Astronomy ; Supersonic and hypersonic flows ; Theoretical
É parte de: Archive for rational mechanics and analysis, 2009-10, Vol.194 (1), p.1-47
Descrição: 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.
Editor: Berlin/Heidelberg: Springer Berlin Heidelberg
Idioma: Inglês

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Transonic Shock Problem for the Euler System in a Nozzle

Xin, Zhouping ; Yan, Wei ; Yin, Huicheng

Berlin/Heidelberg: Springer Berlin Heidelberg

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