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Circulation deposition on shock-accelerated planar and curved density-stratified interfaces: models and scaling laws

Published online by Cambridge University Press:  26 April 2006

Ravi Samtaney  and
Norman J. Zabusky
Ravi Samtaney
Affiliation:
Department of Mechanical and Aerospace Engineering, and CAIP Center, Rutgers University, Piscataway, NJ 08855, USA
Norman J. Zabusky
Affiliation:
Department of Mechanical and Aerospace Engineering, and CAIP Center, Rutgers University, Piscataway, NJ 08855, USA
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Abstract

Vorticity is deposited baroclinically by shock waves on density inhomogeneities. In two dimenslons, the circulation deposited on a planar interface may be derived analytically using shock polar analysis provided the shock refraction is regular. We present analytical expressions for Γ′, the circulation deposited per unit length of the unshocked planar interface, within and beyond the regular refraction regime. To lowest order, Γ′ scales as \[ \Gamma^\prime\propto (1-\eta^{-\frac{1}{2}})(\sin\alpha)(1+M^{-1}+2M^{-2})(M-1)(\gamma^{\frac{1}{2}}/\gamma + 1)\] where M is the Mach number of the incident shock, η is the density ratio of the gases across the interface, α is the angle between the shock and the interface and γ is the ratio of specific heats for both gases. For α ≤ 30°, the error in this approximation is less than 10% for 1.0 < M ≤ 1.32 for all η > 1, and 5.8 ≤ η ≤ 32.6 for all M. We validate our results by quantification of direct numerical simulations of the compressible Euler equations with a second-order Godunov code.

We generalize the results for total circulation on non-planar (sinusoidal and circular) interfaces. For the circular bubble case, we introduce a ‘near-normality’ ansatz and obtain a model for total circulation on the bubble surface that agrees well with results of direct numerical simulations. A comparison with other models in the literature is presented.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 269 , 25 June 1994 , pp. 45 - 78
Copyright
© 1994 Cambridge University Press

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