Smooth Transonic Flows Around Cones

Wen-Ching Lien; Yu-Yu Liu; Chen-Chang Peng; Wen-Ching Lien; Yu-Yu Liu; Chen-Chang Peng

doi:10.3934/nhm.2022028

Networks and Heterogeneous Media

2022, Volume 17, Issue 6: 827-845. doi: 10.3934/nhm.2022028

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Smooth Transonic Flows Around Cones

1.
Department of Mathematics, National Cheng Kung University, Taiwan
2.
Department of Applied Mathematics, National Chiayi University, Taiwan

Received: 01 March 2022 Published: 17 June 2022
Primary: 35L65, 35L67; Secondary: 76N10

We consider a conical body facing a supersonic stream of air at a uniform velocity. When the opening angle of the obstacle cone is small, the conical shock wave is attached to the vertex. Under the assumption of self-similarity for irrotational motions, the Euler system is transformed into the nonlinear ODE system. We reformulate the problem in a non-dimensional form and analyze the corresponding ODE system. The initial data is given on the obstacle cone and the solution is integrated until the Rankine-Hugoniot condition is satisfied on the shock cone. By applying the fundamental theory of ODE systems and technical estimates, we construct supersonic solutions and also show that no matter how small the opening angle is, a smooth transonic solution always exists as long as the speed of the incoming flow is suitably chosen for this given angle.
- Euler Equations,
- shock waves,
- conical flows,
- supersonic flows,
- hyperbolic conservation laws
Citation: Wen-Ching Lien, Yu-Yu Liu, Chen-Chang Peng. Smooth Transonic Flows Around Cones[J]. Networks and Heterogeneous Media, 2022, 17(6): 827-845. doi: 10.3934/nhm.2022028

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Abstract

We consider a conical body facing a supersonic stream of air at a uniform velocity. When the opening angle of the obstacle cone is small, the conical shock wave is attached to the vertex. Under the assumption of self-similarity for irrotational motions, the Euler system is transformed into the nonlinear ODE system. We reformulate the problem in a non-dimensional form and analyze the corresponding ODE system. The initial data is given on the obstacle cone and the solution is integrated until the Rankine-Hugoniot condition is satisfied on the shock cone. By applying the fundamental theory of ODE systems and technical estimates, we construct supersonic solutions and also show that no matter how small the opening angle is, a smooth transonic solution always exists as long as the speed of the incoming flow is suitably chosen for this given angle.

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