Journal of Ocean Engineering and Technology (한국해양공학회지)

Korean Society of Ocean Engineers (한국해양공학회)
권호 표지
DOI QR코드

DOI QR Code

Analysis of Steady Vortex Rings Using Contour Dynamics Method for Fluid Velocity

  • Choi, Yoon-Rak (School of Naval Architecture and Ocean Engineering, University of Ulsan)
  • Received : 2021.09.23
  • Accepted : 2022.02.12
  • Published : 2022.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

Most studies on the shape of the steady vortex ring have been based on the Stokes stream function approach. In this study, the velocity approach is introduced as a trial approach. A contour dynamics method for fluid velocity is used to analyze the Norbury-Fraenkel family of vortex rings. Analytic integration is performed over the logarithmic-singular segment. A system of nonlinear equations for the discretized shape of the vortex core is formulated using the material boundary condition of the core. An additional condition for the velocities of the vortical and impulse centers is introduced to complete the system of equations. Numerical solutions are successfully obtained for the system of nonlinear equations using the iterative scheme. Specifically, the evaluation of the kinetic energy in terms of line integrals is examined closely. The results of the proposed method are compared with those of the stream function approaches. The results show good agreement, and thereby, confirm the validity of the proposed method.

Keywords

References

  1. Akhmetov, D.G. (2009). Vortex Rings. New York, USA: Springer.
  2. Batchelor, G.K. (1967). An Introduction to Fluid Dynamics. Cambridge, UK: Cambridge University Press.
  3. Choi, Y.-R. (2020). Analysis of Steady Vortex Rings Uising Contour Dynamics Method for the Stream Function. Journal of Ocean Engineering and Technology, 34(2), 89-96. https://doi.org/10.26748/KSOE.2020.010
  4. Fraenkel, L.E. (1970). On Steady Vortex Rings of Small Cross-Section in an Ideal Fluid. Proceedings of the Royal Society A, 316(1524), 29-62. https://doi.org/10.1098/rspa.1970.0065
  5. Fraenkel, L.E. (1972). Examples of Steady Vortex Rings of Small Cross-Section in an Ideal Fluid. Journal of Fluid Mechanics, 51(1), 119-135. https://doi.org/10.1017/S0022112072001107
  6. Gradshteyn, I.S., & Ryzhik, I.M. (2000). Table of Integrals, Series, and Products (6th ed.). San Diego, USA: Academic Press.
  7. Helmholtz, H. (1867). On Integrals of the Hydrodynamical Equations, Which Express Vortex-Motion. Philosophical Magazine and Journal of Science (4th series), 33(226), 485-512. https://doi.org/10.1080/14786446708639824
  8. Hill, M.J.M. (1894). On a Spherical Vortex. Philosophical Transactions of the Royal Society A, 185, 213-245. https://doi.org/10.1098/rsta.1894.0006
  9. Lamb, H. (1932). Hydrodynamics (6th ed.). Cambridge, UK: Cambridge University Press.
  10. Krueger, P.S., Moslemi, A.A., Nichols, J.T., Bartol, I.K., & Stewart, W.J. (2008). Vortex Rings in Bio-inspired and Biological Jet Propulsion. Advances in Science and Technology, 58, 237-246. https://doi.org/10.4028/www.scientific.net/AST.58.237
  11. Norbury, J. (1973). A Family of Steady Vortex Rings. Journal of Fluid Mechanics, 57(3), 417-431. https://doi.org/10.1017/S0022112073001266
  12. Pozrikidis, C. (1986). The Nonlinear Instability of Hill's Vortex. Journal of Fluid Mechanics, 168, 337-367. https://doi.org/10.1017/S002211208600040X
  13. Press, W.H., Teukolsky, S.A., Vetterling, W.T., & Flannery, B.P. (1992). Numerical Recipes in Fortran 77 (2nd ed.). Cambridge, UK: Cambridge University Press.
  14. Pullin, D.I. (1992). Contour Dynamics Method. Annual Review of Fluid Mechanics, 24, 89-115. https://doi.org/10.1146/annurev.fl.24.010192.000513
  15. Shariff, K., Leonard, A., & Ferziger, J.H. (1989). Dynamics of a Class of Vortex Rings (NASA Technical Memorandum 102257). Moffett Field, USA: NASA.
  16. Shariff, K., Leonard, A., & Ferziger, J.H. (2008). A Contour Dynamics Algorithm for Axisymmetric Flow. Journal of Computational Physics, 227(10), 9044-9062. https://doi.org/10.1016/j.jcp.2007.10.005
  17. Smith, S.G.L., Chang, C., Chu, T., Blyth M., Hattori, Y., & Salman, H. (2018). Generalized Contour Dynmics: A Review. Regular and Chatic Dynamics. 23(5), 507-518. https://doi.org/10.1134/S1560354718050027
  18. Zabusky, N.J., Hughes, M.H., & Roberts, K.V. (1979). Contour Dynamics for the Euler Equations in Two Dimensions. Journal of Computational Physics, 30, 96-106. https://doi.org/10.1016/0021-9991(79)90089-5