Two-dimensional vortex pair interaction with the wedge
Keywords:
vortex dipole, two-dimensional flow, sharp edge, Kutta-Zhukovsky condition, numerical modeling, experimental investigation
Abstract
The problem of interaction of two-dimensional localized vortex dipole with an edge of the wedge, which moving perpendicular to one of the wedge surfaces at the initial moment is considered. Experimental studies have shown that the vortex dipoles at the approach to the solid surface are separated, and vortices moved in opposite directions. Vortex structure, when interacting with a sharp edge, generates secondary vortices that may form new vortex dipoles. The numerical model for the interaction of vortex dipole with an edge of the wedge, based on the model of point vortices in the approximation of an ideal incompressible fluid is formed. To avoid infinite velocities at the sharp edge model used Kutta-Zhukovsky condition, which adequately describes generating process of the secondary vorticity near the sharp edge. Comparative analysis of transferring processes of fluid forming at the initial moment “vortex atmosphere” shows good agreement of numerical results and experimental data.
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References
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Christiansen J.P. Numerical solution of hydrodynamics by the method of point vortices / J.P.Christiansen // Communications on Pure and Applied Mathematics. – 1973. – V.13. – p.363-379.
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Prandtl L. Hydro- und aeromechanik. V.1 / L.Prandtl, O.Tietjens // Berlin: Springer-Verlag, 193. – 224p.
Meleshko V.V. Stirring of an inviscid fluid by interacting point vortices / V.V.Meleshko. A.A.Gourjii // In "Modelling of Oceanic Vortices" (ed. by G.J.F. van Heijst), Proceeding Colloquium Royal Netherlands Academy of Arts and Science, Amsterdam, Noth-Holland, 1994. – p.271-281.
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Velasco Fuentes O.U. Experimental study of dipolar vortices on a topographic β-plane / O.U. Velasco Fuentes, van Heijst G.J.F. // Journal of Fluid Mechanics. – 1994. – Vol.259. – p.79-106.
Meleshko V.V. Advection of a vortex pair atmosphere in a velocity field of point vortices / V.V.Meleshko, M.Yu.Konstantinov, A.A.Gurzhi, T.P.Konovaljuk // Physics of Fluids. – 1992. – V.A4, N.12. – p.2779-2797.
Ottino J.M. The kinematics of mixing: stretching, chaos, and transport / J.M.Ottino // Cambridge: Cambridge University Press, 1989. – XXXp.
Gourjii A.A. Method of piece spline interpolation in the advection problem of passive impurity in known velocity field / A.A.Gourjii, V.V.Meleshko, G.J.F. van Heijst // Reports of AS of Ukraine. – 1996. – N.8. – p.48-54 (Ukraine) Original text: Гуржiй О.А., Мелешко В.В., ван Хейст Г.Я.Ф. Метод кускової сплан-iнтерполяцiї в задачi про адвекцiю пасивної домiшки у вiдомому полi швидкостi / О.А.Гуржiй, В.В.Мелешко, Г.Я.Ф.ван Хейст // Доповеди АН України. – 1996. – N8. – c.48-54.
Meleshko V.V. On Chaplygin's investigations of two-dimensional vortex structures in an inviscid fluid / V.V.Meleshko, van Heijst G.J.F.// Journal of Fluid Mechanics. – 1994. – V.272. – p.157-182.
Trieling R.R. Monopolar vortices in an irrotational annular shear flow / R.R. Trieling, A.H. Linssen, G.J.F. van Heijst // Journal of Fluid Mechanics. – 1998. – Vol.360. – p.273-294.
Tubes, sheets and singularities in fluid dynamics // ed. by Bayer K., and Moffatt H.K., Proceeding of the NATO ARW held in Zakopane, Poland, 2-7 September 2001. – London: Kluwer Acad. Publ., 2002. – 384p.
Anderson J.D.Jr. Computational fluid dynamics / J.D.Anderson Jr., Degroote J., Degrez G., at al. // Berlin: Springer-Verlag, 2009. – 332p.
Truesdell C. The kinematics of vorticity / C.Truesdell // Bloomington: Indiana Univ., 1954 – 232p.
Schlichting H. Boundary-layer theory / H. Schlichting // New York: McGraw-Hill, 1979. – 817p.
Robinson S.K. Coherent motions in the turbulent boundary layer / S.K.Robinson // Annual. Review of Fluid Mechanis. – 1991. – V.23. – p.601-639.
Casciola C.M. Vorticity generation on a flat surface in 3D flows / C.M.Casciola, R.Piva, P.Bassanini // Journal of Computational Physics. – 1996. – V.129. – p.345-356.
Belotserkovsky S.M. Separated and unseparated overflow around thin wings by an ideal fluid / S.M. Belotserkovsky, Nisht M.I. // Moscow: Nauka, 1978. – 352p. (Russian) Original text: Белоцерковский С.М. Отрывное и безотрывное обтекание тонких крыльев идеальной жидкостью / С.М.Белоцерковский, М.И.Ништ // М.: Наука. 1978. – 352с.
Leonard A. Vortex simulation of forced/unforced mixing layers / A.Leonard, A.Inoue // AIAA journal. – 1987. – V.288. – p.1-19.
Lamb H. Hydrodynamics / H.Lamb // Cambridge: Cambridge university press, 1932. – 755p.
Meleshko V.V. Dynamics of vortex structures / V.V.Meleshko, M.Yu.Konstantinov // Kiev: Dumka, 1993. – 279p. (Russian) Original text: Мелешко В.В. Динамика вихревых структур / В.В.Мелешко, М.Ю.Константинов // Киев: Наукова думка, 1993. – 279c.
Villat H. Vortex theory / H/Villat // Moscow: ONTI, 1936. – 238p. (Russian) Original text: Вилля Г. Теория вихрей / Г.Вилля // Л.,М.: ОНТИ, 1936. – 238с.
Barker S.J. The motion of two-dimensional vortex pairs / S.J.Barker, S.C.Crow // Journal of Fluid Mechanics. – 1977. – V.82. – p.659-671.
Orlandi P. Vortex dipole rebound from a wall / P.Orlandi // Physics of Fluids A: Fluid Dynamics. – 1990. – Vol.2, N.8. – p.1429-1436.
van Dyke M. An album of fluid motion / M. van Dyke // Stanford: Parabolic press, 1982, – 180p.
Landau L.D. Hydrodynamics / L.D.Landau, E.M.Lifshits // Moscow^ Nauka, 1986. – 736p. (Russian) Original text: Ландау Л.Д. Гидродинамика. Т.6 / Л.Д.Ландау, Е.М.Лифшиц // М.: Наука, 1986. – 736с.
Rouch P. Computational fluid dynamics / P.Rouch // Moscow: Mir, 1980. – 616p. (Russian) Original text: Роуч П. Вычислительная гидродинамика / П.Роуч // М.: Мир. 1980. – 616с.
Fletcher C.A.J. Computational techniques for fluid dynamics. V.1 / C.A.J.Fletcher // Berlin: Springer-Verlag, 1988. – 409с.
Baker A.J. Finite element computational fluid mechanics / A.J.Baker // New York: McGraw-Hill. – 482p.
Peyret R. Computational methods for fluid flow / R.Peyret, T.D.Taylor // Berlin: Springer, 1983. – 548p.
Goodman J. Convergence of the point vortex method for the 2D Euler equations / J.Goodman, T.Y.Hou, J.Lowengrub // Communications on Pure and Applied Mathematics. – 1990. – V.43. – p.415-430.
Sarpkaya T. Computational methods with vortices / T.Sarpkaya // Journal of Fluid Mechanics. – 1989. – V.11. – p.15-52.
Saltanov N.V. Vortex structures in fluid: analytical and numerical solutions / N.V.Saltanov, V.A.Gorban // Kiev: Naukova Dumka, 1993. – 244p. (Russsian) Original text: Салтанов Н.В. Вихревые структуры в жидкости: аналитические и численные решения / Н.В.Салтанов, В.А.Горбань // К: Наукова Думка, 1993. – 244с.
Sethian J. Validation study of vortex methods / J.Sethian, A.Ghoniem // Joural of Computational Physcs. – 1984. – V.54. – p.425-456.
Cottet G.-H. Vortex methods: theory and practice / G.-H.Cottet, P.D.Koumoutsakos // Cambridge: Cambridge Univ. Press. 2000. – 314p.
Brackbill J. FLIP: a method for low-dissipation particle in cell method for fluid flow / J.Brackbill, D.Khote // Journal of Computer Physics Communications. – 1986. – V.48. – p.25-38.
Christiansen J.P. Numerical solution of hydrodynamics by the method of point vortices / J.P.Christiansen // Communications on Pure and Applied Mathematics. – 1973. – V.13. – p.363-379.
Perlman M. On the accuracy of vortex methods / M.Perlman // Journal of Computational Physics. – 1985. – V.59. – p.200-223.
Conlisk A.T. Modelling of vortex-corner interaction using point vortices / A.T.Conlisk, D.Rockwell // Physics of Fluids. – 1981. – V.24. – p.2133-2142.
Belotserkovsky S.M Mathematical modelling of separated flow of plane bodies / S.M.Belotserkovsky, V.N.Kotovsky, Nisht M.I. // Moscow: Nauka, 1988. – 232p. (Russian) Original text: Белоцерковский С.М. Математическое моделирование плоскопараллельного отрывного обтекания тел / С.М.Белоцерковский, В.Н.Котовский, М.И.Ништ // М.: Наука. 1988. – 232p.
Chorin A.J. Numerical study of slightly viscous flow / A.J.Chorin // Computers & Fluids. – 1973. – V.17. – p.397-410.
Prandtl L. Hydro- und aeromechanik. V.1 / L.Prandtl, O.Tietjens // Berlin: Springer-Verlag, 193. – 224p.
Meleshko V.V. Stirring of an inviscid fluid by interacting point vortices / V.V.Meleshko. A.A.Gourjii // In "Modelling of Oceanic Vortices" (ed. by G.J.F. van Heijst), Proceeding Colloquium Royal Netherlands Academy of Arts and Science, Amsterdam, Noth-Holland, 1994. – p.271-281.
Krasnopolskaya T.S. Laminar stirring of fluids. Part 1. Methodological aspects of investigation / T.S.Krasnopolskaya, V.V.Meleshko // Applied Hydromechanics. – 2004. – Vol.4. – p.45-58.
Velasco Fuentes O.U. Experimental study of dipolar vortices on a topographic β-plane / O.U. Velasco Fuentes, van Heijst G.J.F. // Journal of Fluid Mechanics. – 1994. – Vol.259. – p.79-106.
Meleshko V.V. Advection of a vortex pair atmosphere in a velocity field of point vortices / V.V.Meleshko, M.Yu.Konstantinov, A.A.Gurzhi, T.P.Konovaljuk // Physics of Fluids. – 1992. – V.A4, N.12. – p.2779-2797.
Ottino J.M. The kinematics of mixing: stretching, chaos, and transport / J.M.Ottino // Cambridge: Cambridge University Press, 1989. – XXXp.
Gourjii A.A. Method of piece spline interpolation in the advection problem of passive impurity in known velocity field / A.A.Gourjii, V.V.Meleshko, G.J.F. van Heijst // Reports of AS of Ukraine. – 1996. – N.8. – p.48-54 (Ukraine) Original text: Гуржiй О.А., Мелешко В.В., ван Хейст Г.Я.Ф. Метод кускової сплан-iнтерполяцiї в задачi про адвекцiю пасивної домiшки у вiдомому полi швидкостi / О.А.Гуржiй, В.В.Мелешко, Г.Я.Ф.ван Хейст // Доповеди АН України. – 1996. – N8. – c.48-54.
Meleshko V.V. On Chaplygin's investigations of two-dimensional vortex structures in an inviscid fluid / V.V.Meleshko, van Heijst G.J.F.// Journal of Fluid Mechanics. – 1994. – V.272. – p.157-182.
Trieling R.R. Monopolar vortices in an irrotational annular shear flow / R.R. Trieling, A.H. Linssen, G.J.F. van Heijst // Journal of Fluid Mechanics. – 1998. – Vol.360. – p.273-294.
Published
2016-10-31
How to Cite
Gourjii, A. A., van Heijst, G. J. F., & Zannetti, L. (2016). Two-dimensional vortex pair interaction with the wedge. Bulletin of V.N. Karazin Kharkiv National University, Series «Mathematical Modeling. Information Technology. Automated Control Systems», 31(1), 16-37. Retrieved from https://periodicals.karazin.ua/mia/article/view/6805
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