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Dynamics of vortex interaction with a density interface

Published online by Cambridge University Press:  26 April 2006

Werner J. A. Dahm,
Christine M. Scheil and
Grétar Tryggvason
Werner J. A. Dahm
Affiliation:
Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 48109–2140, USA
Christine M. Scheil
Affiliation:
Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 48109–2140, USA
Grétar Tryggvason
Affiliation:
Department of Mechanical Engineering and Applied Mechanics, The University of Michigan Ann Arbor, MI 48109–2125, USA
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Abstract

We present results from an experimental and numerical investigation into the dynamics of the interaction between a planar vortex pair or axisymmetric vortex ring with lengthscale a and circulation Γ encountering a planar interface of thickness δ across which the fluid density increases from ρ1 to ρ2. Similarity considerations indicate that baroclinic generation of vorticity and its subsequent interaction with the original vortex is governed by two dimensionless parameters, namely (a/δ) A and R, where A ≡ (ρρ2 − ρ1)/(ρ2 + ρ1) and R ≡ (a3g/Γ2). For thin interfaces (δ [Lt] a), the interaction is governed only by the parameters A and R. Furthermore, in the Boussinesq limit (A → 0), the dynamics are governed solely by the product AR and the interaction is entirely invertible with respect to the initial locations and direction of propagation of the vortices. We document details of the interaction dynamics in the Boussinesq limit over a range of the parameter AR. Results show that, for relatively small values of AR, rather than the vortex simply rebounding at the interface, its outermost layers are instead successively ‘peeled’ away by baroclinically generated vorticity and form a topologically complex backflow in which the ring fluid, the light fluid and the heavy fluid are intertwined. For larger values of AR the vortex barely penetrates the interface, and our results suggest that in the limit AR → ∞ the interaction with a density interface becomes similar to the interaction at a solid wall. We also present results for thick interfaces in the Boussinesq limit, as well as larger density jumps for which the density parameter A enters as a second similarity quantity. Comparison of the experimental and numerical results demonstrate that many of the features of such interactions can be understood within the context of inviscid fluids, and that inviscid vortex methods can be used to accurately simulate the dynamics of such interactions.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 205 , August 1989, pp. 1 - 43
Copyright
© 1989 Cambridge University Press

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References

Barker, S. J. & Crow, S. C. 1977 The motion of two-dimensional vortex pairs in a ground effect. J. Fluid Mech. 82, 659–671.Google Scholar
Batchelor, G. K. 1967 Fluid Dynamics. Cambridge University Press.
Boldes, U. & Ferreri, J. C. 1973 Behavior of vortex rings in the vicinity of a wall. Phys. Fluids 16, 2005–2006.Google Scholar
Cerra, A. W. & Smith, C. R. 1983 Experimental observations of vortex ring interaction with the fluid adjacent to a surface. Report FM-4, Dept of Mech. Engng & Mech. Lehigh University.
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179.Google Scholar
Dahm, W. J. A. & Scheil, C. M. 1987 Dynamics of vortex ring interaction with a density interface. Bull. Am. Phys. Soc. 32, 2038 (abstract only).Google Scholar
Deem, G. S. & Zabusky, N. J. 1978 Stationary V-states: Interactions, recurrence and breaking. Phys. Rev. Lett. 40, 859.Google Scholar
Fraenkel, L. E. 1970 On steady vortex rings of small cross section in an ideal fluid. Proc. R. Soc. Lond. A 316, 29–62.Google Scholar
Harvey, J. K. & Perry, F. J. 1971 Flowfield produced by trailing vortices in the vicinity of the ground. AIAA J. 9, 1659–1660.Google Scholar
Hecht, A. M., Bilanin, A. J., Hirsh, J. E. & Snedeker, R. S. 1980 Turbulent vortices in stratified fluids. AIAA J. 18, 738–746.Google Scholar
Hecht, A. M., Bilanin, A. J. & Hirsch, J. E. 1981 Turbulent trailing vortices in stratified fluids. AIAA J. 19, 691–698.Google Scholar
Hill, F. M. 1975 A numerical study of the descent of a vortex pair in a stably stratified atmosphere. J. Fluid Mech. 71, 1–13.Google Scholar
Krasny, R. 1986 Desingularization of periodic vortex sheet roll-up. J. Comp. Phys. 65, 292–313.Google Scholar
Krauch, T. 1980 Die Bildung eines Ringwirbels an einer kreisförmigen Düsenmündung and seine Umbildung beim Aufstieg in einer Flüssigkeitsschicht und an der Flüssigkeitsoberfläche. Strömungsforschung Max-Planck Institut, Göttingen.
Krutzsch, C.-H. 1936 Über ein instabiles Gebiet bei Wirbelringen. Z. angew. Math. Mech. 16, 352–353.Google Scholar
Krutzsch, C.-H. 1939 Über eine experimentell beobachtete Erscheinung an Wirbelringen bei ihrer translatorischen Bewegung in wirklichen Flüssigkeiten. Ann. Phys. 35, 497–523.Google Scholar
Lamb, H. 1945 Hydrodynamics. Dover.
Leonard, A. 1980 Vortex methods for flow simulation. J. Comp. Phys. 37, 289–335.Google Scholar
Leonard, A. 1985 Computing three-dimensional incompressible flows with vortex elements. Ann. Rev. Fluid Mech. 17, 523–559.Google Scholar
Linden, P. F. 1973 The interaction of a vortex ring with a sharp density interface: a model for turbulent entrainment. J. Fluid Mech. 60, 467–480.Google Scholar
Macgarvey, R. H. & MacLatchy, C. S. 1964a The formation and structure of vortex rings. Can. J. Phys. 42, 678–683.Google Scholar
Macgarvey, R. H. & MacLatchy, C. S. 1964b The disintegration of vortex rings. Can. J. Phys. 42, 684–689.Google Scholar
Maxworthy, T. 1972 The structure and stability of vortex rings. J. Fluid Mech. 51, 15–32.Google Scholar
Maxworthy, T. 1977 Some experimental studies of vortex rings. J. Fluid Mech. 81, 465–495.Google Scholar
Moore, D. W. & Saffman, P. G. 1975 The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346, 413–425.Google Scholar
Saffman, P. G. 1970 The velocity of viscous vortex rings. Stud. Appl. Maths 49, 371–380.Google Scholar
Saffman, P. G. 1972 The motion of a vortex pair in stratified atmosphere. Stud. Appl. Maths 2, 107–119.Google Scholar
Saffman, P. G. 1978 The number of waves on unstable vortex rings. J. Fluid Mech. 84, 625–639.Google Scholar
Saffman, P. G. 1979 The approach of a vortex pair to a plane surface in inviscid fluid. J. Fluid Mech. 92, 497–503.Google Scholar
Saffman, P. G. & Baker, G. R. 1979 Vortex interactions. Ann. Rev. Fluid Mech. 11, 95–127.Google Scholar
Sarpkaya, T. 1983 Trailing vortices in homogeneous and density-stratified media. J. Fluid Mech. 136, 85–109.Google Scholar
Sullivan, J. P., Widnall, S. E. & Ezekiel, S. 1973 Study of vortex rings using a laser doppler velocimeter. AIAA J. 11, 1384–1389.Google Scholar
Tomokita, S. 1936 Instability and breaking up of a ring of an incompressible perfect fluid. Proc. Phys. Math. Soc. Japan 18, 535–549.Google Scholar
Tryggvason, G. 1987 Interaction of vorticity and density interfaces. Bull. Am. Phys. Soc. 32, 2073 (abstract only).Google Scholar
Tryggvason, G. 1988a Deformation of a free surface as a result of vortical flows. Phys. Fluids 31, 955–957.Google Scholar
Tryggvason, G. 1988b Numerical simulations of the Rayleigh-Taylor instability. J. Comput. Phys. 75, 253–282.Google Scholar
Tryggvason, G. 1989 Simulations of vortex sheet rollup by vortex methods. J. Comput. Phys. 80, 1–16.Google Scholar
Walker, J. D. A., Smith, C. R., Cerra, A. W. & Doligalski, T. L. 1987 The impact of a vortex ring on a wall. J. Fluid Mech. 181, 99–140.Google Scholar
Widnall, S. E. & Sullivan, J. P. 1973 On the stability of vortex rings. Proc. R. Soc. Lond. A 332, 335–353.Google Scholar
Widnall, S. E., Bliss, D. B. & Tsai, C.-Y. 1974 The instability of short waves on a vortex ring. J. Fluid Mech. 66, 35–47.Google Scholar
Yih, C.-S. 1975 Vortices and vortex rings of stratified fluids. SIAM J. Appl. Maths 28, 899–912.Google Scholar