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Numerical verification of the similarity laws for the formation of laminar vortex rings

Published online by Cambridge University Press:  15 October 2007

M. HETTEL ,
F. WETZEL ,
P. HABISREUTHER  and
H. BOCKHORN
M. HETTEL
Affiliation:
University of Karlsruhe, Engler-Bunte-Institute, Division of Combustion Technology, 76131 Karlsruhe, Engler-Bunte-Ring 1, [email protected].
F. WETZEL
Affiliation:
University of Karlsruhe, Engler-Bunte-Institute, Division of Combustion Technology, 76131 Karlsruhe, Engler-Bunte-Ring 1, [email protected].
P. HABISREUTHER
Affiliation:
University of Karlsruhe, Engler-Bunte-Institute, Division of Combustion Technology, 76131 Karlsruhe, Engler-Bunte-Ring 1, [email protected].
H. BOCKHORN
Affiliation:
University of Karlsruhe, Engler-Bunte-Institute, Division of Combustion Technology, 76131 Karlsruhe, Engler-Bunte-Ring 1, [email protected].
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Abstract

From analytical investigations it is well known that the roll-up of an inviscid plane vortex sheet which separates at the edge of a body is a self-similar process which can be described by scaling laws. Unlike plane vortices, ring vortices have a curved rotational axis. For this special vortex type experimental investigations as well as calculations in the literature suggest that the scaling laws are only partially valid. The main goal of this work is to clarify how far these similarity or scaling laws are also valid for the formation of viscid laminar vortex rings. Therefore, the formation process of laminar vortex rings was investigated numerically using a CFD (computational-fluid-dynamics) code. The calculations refer to an experimental setup for which detailed experimental data are available in the literature. In this setup, laminar ring vortices are generated by ejecting water from a circular tube into a quiescent environment by means of a piston. First, a case based on a constant piston velocity was investigated. Comparing calculated and measured data yields a very good agreement. Further calculations were made when forcing the velocity of the piston by three different time-dependent functions. The results of these calculations show that the formation laws for inviscid plane vortices are also valid for the formation process of viscid ring vortices. This applies to the normalized axial and radial position of the vortex centre as well as the normalized diameter of the vortex spiral. However, the similarity laws are valid only if the process is considered in a special frame of reference which moves in conjunction with the front of the jet and if the starting time of the formation process with respect to the starting time of the ejection is taken into account. Additionally, the formation of a ring vortex, which occurs during the start-up process of a free jet flow, was calculated. The results confirm a dependence for the motion of the jet front, which is known from analytical considerations and allows some interesting features to be identified.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 590 , 10 November 2007 , pp. 35 - 60
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Anton, L. 1939 Ausbildung eines Wirbelpaares an den Kanten einer Platte. Ing. Arch. 10, 411–427. (English trans. NACA Tech. Memo. 1398 (1956)).CrossRefGoogle Scholar
Auerbach, D. 1987 Experiments on the trajectory and circulation of the starting vortex. J. Fluid Mech. 183, 185–198.CrossRefGoogle Scholar
Brady, M., Leonard, A. & Pullin, D. I. 1998 Regularized vortex sheet evolution in three dimensions. J. Comput. Phys. 146, 520–545.CrossRefGoogle Scholar
Chorin, A. J., & Bernard, P. S. 1973 Discretization of a vortex sheet, with an example of Roll-up. J. Comput. Phys. 13, 423–429.CrossRefGoogle Scholar
Didden, N. 1977 Untersuchung laminarer, instabiler Ringwirbel mittels Laser-Doppler-Anemometrie. Mitteilungen aus dem Max-Planck-Institut für Strömungsforschung und der Aerodynamischen Versuchsanstalt, Herausgegeben von E.-A. Müller und H. Schlichting, Göttingen.Google Scholar
Didden, N. 1979 On the formation of vortex rings: rolling-up and production of circulation. Z. Angew. Math. Phys. 30, 101–116.CrossRefGoogle Scholar
Durst, F. & Fuchs, W. 1974 Die Bildung von Wirbelringen durch fallende Tropfen und die Dynamik der Wirbelringfortbewegung. Bericht SFB 80/ET/23, Sonderforschungsbereich 80, Ausbreitungs- und Transportvorgänge in Strömungen, Universität Karlsruhe (TH).Google Scholar
Fabris, D. & Liepmann, D. 1997 Vortex ring structure at late stages of formation. Phys. Fluids 9, 2801–2803.CrossRefGoogle Scholar
Glezer, A. 1988 The formation of vortex rings. Phys. Fluids 31, 3532.CrossRefGoogle Scholar
Gosman, A. D. & Ideriah, F. J. K. 1976 TEACH-2E: A general computer program for two-dimensional, turbulent, recirculating flows. Report, Department of Mechanical Engineering, Imperial College, London.Google Scholar
Gühler, M. & Sallet, D. W. 1979 The formation of vortex rings and their initial motion. Z. Flugwiss. Weltraumforsch. 3, 109–155.Google Scholar
Heeg, R. S. & Riley, N. 1997 Simulations of the formation of an axisymmetric vortex ring. J. Fluid Mech. 339, 199–211.CrossRefGoogle Scholar
Hettel, M. 2006 Analytische und numerische Untersuchungen der Dynamik von Vormischflammen sowie deren Interaktion mit Ringwirbelstrukturen. PhD thesis, University of Karlsruhe.Google Scholar
Hettel, M., Büchner, H., Habisreuther, P., Bockhorn, H. & Zarzalis, N. 2004 Modeling of ring-vortices and their interaction with turbulent premixed flames. Combust. Sci. Tech. 176, 835–850.CrossRefGoogle Scholar
Hettel, M., Büchner, W., Weib, H., Habisreuther, P., Bockhorn, H. & Zarzalis, N. 2005 URANS – Modelling of pulsed turbulent jets and premixed jet flames. Prog. Comput. Fluid Dyn. 5, 386–397.CrossRefGoogle Scholar
Hirsch, C. 1995 Ein Beitrag zur Wechselwirkung von Turbulenz und Drall. PhD thesis, University of Karlsruhe.Google Scholar
James, S. & Madnia, C. K. 1996 Direct numerical simulation of a laminar vortex ring. Phys. Fluids 8, 2400–2414.CrossRefGoogle Scholar
Kaden, H. 1931 Aufwicklung einer unstabilen Unstetigkeitsfläche. Ing. Arch. 2, 140–169. (English trans. R. A. Lib. Trans. no. 403).CrossRefGoogle Scholar
Leonard, A. 1980 Vortex methods for flow simulation. J. Comput. Phys. 37, 289–335.CrossRefGoogle Scholar
Liess, C. 1978 Experimentelle Untersuchung des Lebenslaufes von Ringwirbeln. Max-Planck-Institut für Strömungsforschung, Göttingen, Bericht 1/1978.Google Scholar
Liess, C. & Didden, N. 1976 Experimente zum Einfluß der Anfangsbedingungen auf die Instabilität von Ringwirbeln. Z. Angew. Math. Mech. 56, 206–208.CrossRefGoogle Scholar
Lugt, H. J. 1983 Vortex flow in Nature and Technology, John Wiley & Sons. (Translation of: Wirbelströmungen in Natur und Technik, G. Braun.)Google Scholar
Maxworthy, T. 1972 The structure and stability of vortex rings. J. Fluid Mech. 51, 15–32.CrossRefGoogle Scholar
Maxworthy, T. 1976 Some experimental studies of vortex rings. J. Fluid Mech. 81, 465–495.CrossRefGoogle Scholar
Mohseni, K., Ran, H. & Colonius, T. 2001 Numerical experiments on vortex ring formation. J. Fluid Mech. 430, 267–282.CrossRefGoogle Scholar
Moore, D. W. 1974 A numerical study of the roll-up of a finite vortex sheet. J. Fluid Mech. 63, 225–235.CrossRefGoogle Scholar
Nitsche, M. 1996 Scaling properties of vortex ring formation. Phys. Fluids 8, 1848–1855.CrossRefGoogle Scholar
Nitsche, M. 2001 Self-similar shedding of vortex rings. J. Fluid Mech. 435, 397–407.CrossRefGoogle Scholar
Nitsche, M. & Krasny, R. 1994 A numerical study of vortex ring formation at the edge of a circular tube. J. Fluid Mech. 276, 139–161.CrossRefGoogle Scholar
Noll, B. 1992 Evaluation of a bounded high resolution scheme for combustor flow computations. AIAA J. 30, (1).CrossRefGoogle Scholar
Patankar, S. V. 1980 Numerical Heat Transfer and Fluid Flow. Hemisphere.Google Scholar
Prandtl, L. 1924 Über die Entstehung von Wirbeln in der idealen Flüssigkeit, mit Anwendung auf die Tragflügeltheorie und andere Aufgaben. Vorträge aus dem Gebiet der Hydro- und Aerodynamik, Herausgegeben v. Th. v. Kármán u. T. Levi-Civita. 19–34, Berlin.CrossRefGoogle Scholar
Pullin, D. I. 1978 The large-scale structure of unsteady self-similar rolled-up vortex sheets. J. Fluid Mech. 88, 401–430.CrossRefGoogle Scholar
Rosenfeld, M., Rambod, E. & Gharib, M. 1998 Circulation and formation number of laminar vortex rings. J. Fluid Mech 376, 292–318.CrossRefGoogle Scholar
Saffman, P. G. 1978 The number of waves on unstable vortex rings. J. Fluid Mech. 84, 625–639.CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Sallet, D. W. & Widmayer, R. S. 1974 An experimental Investigation of laminar and turbulent vortex rings in air. Z. Flugwiss. 22, 201–215.Google Scholar
Schneider, E. 1978 Werden, Bestehen, Instabilität, Regeneration, Vergehen eines Ringwirbels. Max-Planck-Institut für Strömungsforschung, Göttingen, Bericht 17/1978.Google Scholar
Schneider, E. 1980 Experimentelle Untersuchung der Instabilitätsphasen eines laminaren Ringwirbels im Hinblick auf vergleichbare Instabilitätsereignisse in turbulenten Scherschicht-, Grenzschicht- und Kanalströmungen. Max-Planck-Institut für Strömungsforschung, Göttingen, Bericht 17/1980.Google Scholar
Seno, T., Kageyama, S. & Ito, R. 1988 A modeling of vortex rings in an axisymmetric pulsed jet. J. Chem. Engng Japan 21, 1–5.CrossRefGoogle Scholar
Shariff, K. & Leonard, A. 1992 Vortex rings. Annu. Rev. Fluid. Mech. 24, 235.CrossRefGoogle Scholar
Southerland, K. B., Porter, J. R., Dahm, W. J. A. & Buch, K. A. 1991 An experimental study of the molecular mixing process in an axisymmetric laminar vortex ring. Phys. Fluids A 3, 1385–1392.CrossRefGoogle Scholar
Wedemeyer, E. 1961 Ausbildung eines Wirbelpaares an den Kanten einer Platte. Ing. Arch. 30, 187–200.CrossRefGoogle Scholar
Wille, R. 1952 Über Strömungserscheinungen im Übergangsgebiet von geordneter zu ungeordneter Bewegung. Jahrbuch der Schiffbautechnischen Gesellschaft 46, 176–187.Google Scholar