Existence of Stationary Supersonic Flows Past a Pointed Body

Chen, Shuxing

doi:10.1007/s002050100121

Cited by 55 publications

(38 citation statements)

References 17 publications

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“…The result in Li [41,42] was obtained without the additional hypothesis of the Hölder continuity employed in [56] via the Nash-Moser iterations. See also Chen [16,20] for the local static stability of supersonic shock waves past a three-dimensional wing and conical body for potential flow. For the first rigorous treatment of the local existence and stability of unsteady multidimensional shock fronts for nonlinear hyperbolic systems of conservation laws, see Majda [49][50][51] The global stability results we present here are originally motivated by these fundamental results, insights, and remarks mentioned above.…”

Section: 1)mentioning

confidence: 99%

Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems

Chen

2017

Sci. China Math.

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When an upstream steady uniform supersonic flow impinges onto a symmetric straight-sided wedge, governed by the Euler equations, there are two possible steady oblique shock configurations if the wedge angle is less than the detachment angle -the steady weak shock with supersonic or subsonic downstream flow (determined by the wedge angle that is less or larger than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which satisfy the entropy condition. The fundamental issue -whether one or both of the steady weak and strong shocks are physically admissible solutions -has been vigorously debated over the past eight decades. In this paper, we survey some recent developments on the stability analysis of the steady shock solutions in both the steady and dynamic regimes. For the static stability, we first show how the stability problem can be formulated as an initial-boundary value type problem and then reformulate it into a free boundary problem when the perturbation of both the upstream steady supersonic flow and the wedge boundary are suitably regular and small, and we finally present some recent results on the static stability of the steady supersonic and transonic shocks. For the dynamic stability for potential flow, we first show how the stability problem can be formulated as an initial-boundary value problem and then use the self-similarity of the problem to reduce it into a boundary value problem and further reformulate it into a free boundary problem, and we finally survey some recent developments in solving this free boundary problem for the existence of the PrandtlMeyer configurations that tend to the steady weak supersonic or transonic oblique shock solutions as time goes to infinity. Some further developments and mathematical challenges in this direction are also discussed.

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Section: 1)mentioning

confidence: 99%

Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems

Chen

2017

Sci. China Math.

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“…Remark 2.1 A conical coordinate system has been previously used in the study of conical shock waves for irrotational flow, see [8,9,10,11]. In these papers, it was assumed that the flow in x 1 direction is supersonic, i.e., v 1 > a, which is obviously a stronger condition than the downstream supersonic flow in Theorem 1.1.…”

Section: Transform and Formulationmentioning

confidence: 99%

“…Conical shock waves have been recently studied in the framework of isentropic irrotational flow [7,8,10,11]. The governing equation for isentropic irrotational flow is a second order quasi-linear wave equation for the velocity potential.…”

Section: Introductionmentioning

confidence: 99%

Conical shock waves for an isentropic Euler system

Chen

2005

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Conical shock waves are generated as sharp conical solid projectiles fly supersonically in the air. We study such conical shock waves in steady supersonic flow using isentropic Euler system. The stability of such attached conical shock waves for non-symmetrical conical projectile and non-uniform incoming supersonic flow are established. Meanwhile, the existence of the solution to the Euler system with such attached conical shock as free boundary is also proved for solid projectile close to a regular solid cone.

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“…In [4,5,6], we introduced two approaches to deal with multidimensional transonic shocks. Nontransonic shocks (hyperbolic-hyperbolic shocks) were analyzed in [7,25,33,40,45,49] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

Chen

Feldman

2006

Arch Rational Mech Anal

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Abstract. We establish the existence and stability of multidimensional transonic flows with transonic shocks through an infinite nozzle of arbitrary cross-sections, including a slowly varying de Lavel nozzle. The transonic flow is governed by the inviscid steady potential flow equation with supersonic upstream flow at the entrance, uniform subsonic downstream flow at the infinite exit, and the slip boundary condition on the nozzle boundary. The multidimensional transonic nozzle problem is reformulated into a free boundary problem, for which the free boundary is a transonic shock dividing two phases of C 1,α flow in the infinite nozzle, and the equation is hyperbolic in the supersonic upstream phase and elliptic in the subsonic downstream phase. We further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain and to solve the multidimensional transonic nozzle problem in a direct fashion. Our results indicate that, for the transonic nozzle problem, there exists a transonic flow such that the flow is divided into a C 1,α subsonic flow up to the nozzle boundary in the unbounded downstream region from the supersonic upstream flow by a C 1,α multidimensional transonic shock that is orthogonal to the nozzle boundary at every intersection point, and the uniform velocity state at the infinite exit in the downstream direction is uniquely determined by the supersonic upstream flow at the entrance which is sufficiently close to a uniform flow. The uniform velocity state at the exit can not be apriori prescribed from the corresponding pressure for such a flow to exist. We further prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance.

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Existence of Stationary Supersonic Flows Past a Pointed Body

Cited by 55 publications

References 17 publications

Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems

Supersonic flow onto solid wedges, multidimensional shock waves and free boundary problems

Conical shock waves for an isentropic Euler system

Existence and Stability of Multidimensional Transonic Flows through an Infinite Nozzle of Arbitrary Cross-Sections

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