Transonic Shocks in Compressible Flow Passing a Duct for Three-Dimensional Euler Systems

Chen, Shuxing; Yuan, Hairong

doi:10.1007/s00205-007-0079-z

Cited by 88 publications

(64 citation statements)

References 18 publications

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“…However, since the flow fields of the transonic shock we studied here are relatively simple, it may help us gain some insight into understanding those more complicated transonic shocks appearing, for example, in de Laval nozzles. Note that all the works cited above on transonic shocks and [8], [7], [29] are devoted to the study of the class of transonic shocks we investigate here.…”

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confidence: 99%

“…The author [29] has also investigated the case for flows in a cylinder with cylindrical symmetry by a different method from [8], and the radial velocity vanishing condition was posed at the exit. In [7] Chen and Yuan developed the methods initiated in [29] and solved the transonic shock problem for a three-dimensional steady full Euler system under the periodic conditions on the lateral boundary of the duct, and hence obtained the solution of the transonic shock problem in a three-dimensional duct with a constant square section under certain assumptions on the symmetry of the coming flow. The ill-posedness for given pressure at the exit is also demonstrated in detail there.…”

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On Transonic Shocks in Two‐Dimensional Variable‐Area Ducts for Steady Euler System

Yuan¹

2006

SIAM J. Math. Anal.

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Abstract. This paper concerns transonic shocks in compressible inviscid flow passing a twodimensional variable-area duct for the complete steady Euler system. The flow is supersonic at the entrance of the duct, whose boundaries are slightly curved. The condition of impenetrability is posed on the boundaries. After crossing a nearly flat shock front, which passes through a fixed point on the boundary of the duct, the flow becomes subsonic. We show that to ensure the stability of such shocks, pressure should not be completely given at the exit: it only should be given with freedom one, that is, containing an unknown constant to be determined by the upstream flow and the profile of the duct. Careful analysis shows that this is due to the requirement of conservation of mass in the duct. We used Lagrangian transformation and characteristic decomposition to write the Euler system as a 2 × 2 system, which is valid for general smooth flows. Due to such a simplification, we can employ the theory of boundary value problems for elliptic equations to discuss well-posedness or ill-posedness of transonic shock problems in variable-area duct for various conditions giving at the exit.

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On Transonic Shocks in Two‐Dimensional Variable‐Area Ducts for Steady Euler System

Yuan¹

2006

SIAM J. Math. Anal.

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“…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.…”

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confidence: 99%

Three dimensional steady subsonic Euler flows in bounded nozzles

Chen

Xie

2014

Journal of Differential Equations

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Abstract. In this paper, we study the existence and uniqueness of three dimensional steady Euler flows in rectangular nozzles when prescribing normal component of momentum at both the entrance and exit. If, in addition, the normal component of the voriticity and the variation of Bernoulli's function at the exit are both zero, then there exists a unique subsonic potential flow when the magnitude of the normal component of the momentum is less than a critical number. As the magnitude of the normal component of the momentum approaches the critical number, the associated flows converge to a subsonic-sonic flow. Furthermore, when the normal component of vorticity and the variation of Bernoulli's function are both small, the existence of subsonic Euler flows is established. The proof of these results is based on a new formulation for the Euler system, a priori estimate for nonlinear elliptic equations with nonlinear boundary conditions, detailed study for a linear div-curl system, and delicate estimate for the transport equations.

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“…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.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

“…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.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

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

Li¹,

Xin²,

Yin³

2011

Pacific J. Math.

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We focus on the uniqueness problem of a 3D transonic shock solution in a conic nozzle when the variable end pressure in the diverging part of the nozzle lies in an appropriate scope. By establishing the monotonicity of the position of shock surface relative to the end pressure, we remove the nonphysical assumptions on the transonic shock past a fixed point made in previous studies and further obtain uniqueness.

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Transonic Shocks in Compressible Flow Passing a Duct for Three-Dimensional Euler Systems

Cited by 88 publications

References 18 publications

On Transonic Shocks in Two‐Dimensional Variable‐Area Ducts for Steady Euler System

On Transonic Shocks in Two‐Dimensional Variable‐Area Ducts for Steady Euler System

Three dimensional steady subsonic Euler flows in bounded nozzles

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

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