Abstract
In this paper we study the transonic shock in steady compressible flow passing a duct. The flow is a given supersonic one at the entrance of the duct and becomes subsonic across a shock front, which passes through a given point on the wall of the duct. The flow is governed by the three-dimensional steady full Euler system, which is purely hyperbolic ahead of the shock and is of elliptic–hyperbolic composed type behind the shock. The upstream flow is a uniform supersonic one with the addition of a three-dimensional perturbation, while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference. The problem of determining the transonic shock and the flow behind the shock is reduced to a free-boundary value problem. In order to solve the free-boundary problem of the elliptic–hyperbolic system one crucial point is to decompose the whole system to a canonical form, in which the elliptic part and the hyperbolic part are separated at the level of the principal part. Due to the complexity of the characteristic varieties for the three-dimensional Euler system the calculus of symbols is employed to complete the decomposition. The new ingredient of our analysis also contains the process of determining the shock front governed by a pair of partial differential equations, which are coupled with the three-dimensional Euler system.
Similar content being viewed by others
References
Cole J.D., Cook L.P. Transonic Aerodynamics, vol. 30. North-Holland, Amsterdam, 1986
Chen G.Q., Feldman M. (2003) Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. J. Am. Math. Soc. 16, 461–494
Chen G.Q., Feldman M. (2004) Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations. Commun. Pure Appl. Math. 57, 310–356
Čanić S., Keyfitz B.L., Lieberman G.M. (2000) A proof of existence of perturbed steady transonic shocks via a free boundary problem. Commun. Pure Appl. Math. 53, 484–511
Canic S., Kerfitz B., Kim E.H. (2000) A free boundary problems for unsteady transonic small disturbance equation: transonic regular reflection. Methods Appl. Anal. 7, 313–336
Canic S., Kerfitz B., Kim E.H. (2002) A free boundary problems for a quasilinear degenerate elliptic equation: transonic regular reflection. Commun. Pure Appl. Math. 55, 71–92
Chen S. (1980) On the initial-boundary value problem for quasilinear symmetric hyperbolic system and applications. Chin. Ann. Math. 1, 511–522
Chen S. (2002) Stability of oblique shock fronts. Sci. China 45, 1012–1019
Chen S. (2005) Stability of transonic shock fronts in two-dimensional Euler systems. Trans. Am. Math. Soc. 357, 287–308
Courant R., Friedrichs K.O. (1948) Supersonic Flow and Shock Waves. Interscience, Publishers Inc., New York
Glaz H.M., Liu T.P. (1984) The asymptotic analysis of wave interactions and numerical calculations of transonic flow. Adv. Appl. Math. 5, 111–146
Gamba I.M., Morawetz C.S. (1996) A viscous approximations for a 2-D steady semiconductor or transonic gas dynamic flow: existence theorem for potential flow. Commun. Pure Appl. Math. 49, 999–1049
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equation of Second Order, 2nd edn. Grundlehren der Mathematischen Wissenschaften, 224. Springer, Berlin, 1983
Hörmander L. (1985) The Analysis of Linear Partial Differential Operators, 3. Springer, Berlin
Kuz’min A.G. (2002) Boundary Value Problems for Transonic Flow. John Wiley, London
Liu T.P. (1982) Nonlinear stability and instability of transonic flows through a nozzle. Commun. Math. Phys. 83, 243–260
Mcowen R.C. (2003) Partial Differential Equations, Methods and Aplications, 2nd edn. Pearson Education, Upper Saddle River, NJ
Morawetz C.S. (1956) On the non-existence of continuous transonic flows past profiles. I, II, III. Commun. Pure Appl. Math. 9, 45–68
Morawetz C.S. (1957) On the non-existence of continuous transonic flows past profiles. I, II, III. Commun. Pure Appl. Math. 10, 107–131
Morawetz C.S. (1958) On the non-existence of continuous transonic flows past profiles. I, II, III. Commun. Pure Appl. Math. 11, 129–144
Morawetz C.S. (1964) Non-existence of transonic flows past profiles. Commun. Pure Appl. Math. 17, 357–367
Showalter R.E. (1977) Hilbert Space Methods for Partial Differential Equations. Pitman, London–San Francisco–Melbourne
Smoller J. (1994) Shock Waves and Reaction–Diffusion Equations, 2nd edn. Springer, New York,
Xin Z.P., Yin H.C. (2005) Transonic shock in a nozzle I: two-dimensional case. Commun. Pure Appl. Math. 58, 999–1050
Zeidlerm E. (1986) Nonlinear Functional Analysis and its Applications. Fixed-Point Theorems, vol. 1. Springer, New York
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by T.-P. Liu
The paper is partially supported by National Natural Science Foundation of China 10531020, the National Basic Research Program of China 2006CB805902, and the Doctorial Foundation of National Educational Ministry 20050246001.
Rights and permissions
About this article
Cite this article
Chen, S., Yuan, H. Transonic Shocks in Compressible Flow Passing a Duct for Three-Dimensional Euler Systems. Arch Rational Mech Anal 187, 523–556 (2008). https://doi.org/10.1007/s00205-007-0079-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-007-0079-z