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Topological structure of shock induced vortex breakdown

Published online by Cambridge University Press:  19 October 2009

SHUHAI ZHANG ,
HANXIN ZHANG  and
CHI-WANG SHU
SHUHAI ZHANG
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Centre, Mianyang, Sichuan 621000, China
HANXIN ZHANG
Affiliation:
China Aerodynamics Research and Development Centre, Mianyang, Sichuan 621000, China
CHI-WANG SHU*
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
*
Email address for correspondence: shu@dam.brown.edu
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Abstract

Using a combination of critical point theory of ordinary differential equations and numerical simulation for the three-dimensional unsteady Navier–Stokes equations, we study possible flow structures of the vortical flow, especially the unsteady vortex breakdown in the interaction between a normal shock wave and a longitudinal vortex. The topological structure contains two parts. One is the sectional streamline pattern in the cross-section perpendicular to the vortex axis. The other is the sectional streamline pattern in the symmetrical plane. In the cross-section perpendicular to the vortex axis, the sectional streamlines have spiral or centre patterns depending on a function λ (x, t) = 1/ρ(∂ρ/∂t+∂ρu/∂x), where x is the coordinate corresponding to the vortex axis. If λ > 0, the sectional streamlines spiral inwards in the near region of the centre. If λ < 0, the sectional streamlines spiral outwards in the same region. If λ = 0, the sectional streamlines form a nonlinear centre. If λ changes its sign along the vortex axis, one or more limit cycles appear in the sectional streamlines in the cross-section perpendicular to the vortex axis. Numerical simulation for two typical cases of shock induced vortex breakdown (Erlebacher, Hussaini & Shu, J. Fluid Mech., vol. 337, 1997, p. 129) is performed. The onset and time evolution of the vortex breakdown are studied. It is found that there are more limit cycles for the sectional streamlines in the cross-section perpendicular to the vortex axis. In addition, we find that there are quadru-helix structures in the tail of the vortex breakdown.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 639 , 25 November 2009 , pp. 343 - 372
Copyright
Copyright © Cambridge University Press 2009

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