Abstract
We establish the stability of 3-D axisymmetric transonic shock solutions of the steady full Euler system in divergent nozzles under small perturbations of an incoming radial supersonic flow and a constant pressure at the exit of the nozzles. To study 3-D axisymmetric transonic shock solutions of the full Euler system, we use a stream function formulation of the full Euler system for a 3-D axisymmetric flow. We resolve the singularity issue arising in stream function formulations of the full Euler system for a 3-D axisymmetric flow. We develop a new scheme to determine a shock location of a transonic shock solution of the steady full Euler system based on the stream function formulation.
Highlights
In [14, Chapter 147], the authors, using an approximate model, described a transonic shock phenomenon for a compressible invicid flow of an ideal polytropic gas in a convergent-divergent type nozzle called de Laval nozzle: if a subsonic flow accelerating as it passes through the convergent part of the nozzle reaches the sonic speed at the throat of the nozzle, it becomes a supersonic flow right after the throat of the nozzle
The stability of one-dimensional transonic shock solutions in flat nozzles was studied first. This subject was studied using the potential flow model in [7,8,9,26,27] and further studied using the full Euler system in [5,6,11,12,13,25,29,30]. These results showed that one-dimensional transonic shock solutions in flat nozzles are not stable under a perturbation of a physical boundary condition and, even if onedimensional transonic shock solutions in flat nozzles are stable, their shock locations are not uniquely determined unless there exists the assumption that a shock location passes through some point on the wall of the nozzle, as it can be expected from the behavior of one-dimensional transonic shock solutions in flat nozzles
We study the stability of radial transonic shock solutions in divergent nozzles under small perturbations of an incoming radial supersonic solution and a constant exit pressure using the full Euler system for the 3-D case for axisymmetric flows
Summary
In [14, Chapter 147], the authors, using an approximate model, described a transonic shock phenomenon for a compressible invicid flow of an ideal polytropic gas in a convergent-divergent type nozzle called de Laval nozzle: if a subsonic flow accelerating as it passes through the convergent part of the nozzle reaches the sonic speed at the throat of the nozzle, it becomes a supersonic flow right after the throat of the nozzle. In [3], the authors studied this subject using the non-isentropic potential model introduced in [3], and they obtained the stability result for radial transonic shock solutions in divergent nozzles. We study the stability of radial transonic shock solutions in divergent nozzles under small perturbations of an incoming radial supersonic solution and a constant exit pressure using the full Euler system for the 3-D case for axisymmetric flows. (We use this approach to prove the orthogonal completeness of eigenfunction of an associated Legendre problem of type m = 1 with a general domain (see Lemma 18).) Using the stream function formulation formulated by using the vector potential form of the stream function, we obtain the stability result for flows having C1,α interior and Cα up to boundary regularity.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
