Transonic shocks in 3-D compressible flow passing a duct with a general section for Euler systems

Abstract

Abstract. This paper is devoted to the study of a transonic shock in threedimensional steady compressible flow passing a duct with a general section. The flow is described by the steady full Euler system, which is purely hyperbolic in the supersonic region and is of elliptic-hyperbolic type in the subsonic region. The upstream flow at the entrance of the duct is a uniform supersonic one adding 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 to determine the transonic shock and the flow behind the shock is reduced to a free boundary value problem of an elliptic-hyperbolic system. The new ingredients of our paper contain the decomposition of the elliptichyperbolic system, the determination of the shock front by a pair of partial differential equations coupled with the three-dimensional Euler system, and the regularity analysis of solutions to the boundary value problems introduced in our discussion.

Three dimensional steady subsonic Euler flows in bounded nozzles

“…As an important constituent of transonic flows, the existence of subsonic flows which are small perturbations of certain simple background flows was also studied in [22,23,10,11,32,33,34] and references therein. This paper studies general subsonic Euler flows when prescribing the normal component of the momentum on the boundary.…”

Monotonicity and uniqueness of a 3D transonic shock solution in a conic nozzle with variable end pressure

“…The transonic shock problem in a nozzle is a fundamental one in fluid dynamics and has been extensively studied by many authors under various assumptions, for example, that either the transonic flow is quasi-one-dimensional or that the transonic shock goes through some fixed point in advance; see [Liu 1982;Embid et al 1984;Chen et al 2007;Chen 2008;Chen and Yuan 2008;Xin and Yin 2008a;2008b;] and so on. However, Courant and Friedrichs [1948, p. 386] indicated that transonic shock in a nozzle can be formulated as follows: Given appropriately large end pressure p e (x), if the upstream flow is still supersonic behind the throat of the three-dimensional de Laval nozzle, then at a certain place in the diverging part of the nozzle, a shock front intervenes and the gas is compressed and slowed down to subsonic speed.…”

“…The steady transonic problem has been studied in great detail; see [Courant and Friedrichs 1948;Liu 1982;Gilbarg and Trudinger 1983;Embid et al 1984;Morawetz 1994;Čanić et al 2000;Kuz'min 2002;Zheng 2003;2006;Chen et al 2007;Chen 2008;Chen and Yuan 2008;Xin and Yin 2008a;2008b;Li et al 2010] and the references therein. However, most known results deal with 2D problems or 3D problems with special symmetries, or make additional a priori assumptions on shock positions.…”

The existence and monotonicity of a three-dimensional transonic shock in a finite nozzle with axisymmetric exit pressure

“…The transonic shock problem in a de Laval nozzle is a fundamental one in fluid dynamics and has been extensively studied by many authors under the assumption that the transonic flow is quasi-one-dimensional or the transonic shock goes through some fixed point in advance [Chen et al 2006;Chen et al 2007;Chen and Feldman 2003;Chen 2008;Courant and Friedrichs 1948;Embid et al 1984;Glaz and Liu 1984;Kuz'min 2002;Liu 1982a;1982b;Xin and Yin 2005;2008a;2008b;Yuan 2006]. Courant and Friedrichs [1948, page 386] proposed a physically more interesting transonic shock wave pattern in a de Laval nozzle as follows: Given an appropriately large end pressure p e (x), if the upstream flow is still 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 gas is compressed and slowed down to subsonic speed.…”

“…There has already been much work on the steady transonic problem; see [Bers 1950;1951;Čanić et al 2000;Chen et al 2006;Chen et al 2007;Chen and Feldman 2003;Chen 2008;Courant and Friedrichs 1948;Embid et al 1984;Glaz and Liu 1984;Kuz'min 2002;Li et al 2009a;2009b;2010a;2010b;Liu 1982a;1982b;Morawetz 1994;Xin and Yin 2005;2008b;2008a;Yuan 2006;Zheng 2003; and the references therein. In particular, for a threedimensional nozzle with a symmetric diverging part and a symmetric supersonic incoming flow near the diverging part of the nozzle, Xin and Yin [2008b] and Courant and Friedrichs [1948] have shown that there exist two constant pressures P 1 and P 2 with P 1 < P 2 such that if the exit pressure P e is in the interval (P 1 , P 2 ), then the transonic shock exists uniquely in the diverging part of the nozzle, and the position and the strength of the shock are completely determined by P e and the resulting ordinary differential equations.…”