Abstract
The
aim of this paper is to obtain streamline patterns of axisymmetric flow and
their bifurcations for 2-D incompressible flows close to non-simple singular
point. The streamlines of a Hamiltonian vector field system are simplified by
using the homotopy invariance of the index theory. Using a homotopy invariance
of the index, we develop a theory for the sufficient and necessary conditions
for structural bifurcation of axisymmetric flow near non-simple degenerate
critical points. The variation of parameters in the flow field can cause
structural bifurcations. The bifurcation of the degenerate flow structure is
obtained when it is perturbed slightly.
Keywords
Structural stability divergence-free vector field Lagrangian dynamics structural bifurcation singularity classification
References
- [1]. Brøns, M., Voigt, L.K. and Sørensen, J.N., Topology of vortex breakdown bubbles in a cylinder with a rotating bottom and a free surface, J. Fluid Mech. 428 (2001) 133-148.
- [2]. Brøns, M., Voigt, L.K. and Sørensen, J.N., Streamline topology of steady axisymmetric vortex breakdown in a cylinder with co- and counter-rotating end-covers, J. Fluid Mech., 401 (1999) 275-292.
- [3]. Bisgaard, A.V., Brøns, M. and Sørensen, J.N., Vortex breakdown generated by off-axis bifurcation in a cylinder with rotating covers, Acta Mech. 187 (2006) 75-83.
- [4]. Brøns M., Topologic al fluid mechanics of axisymmetric flows, In simulation and identification of organized structures in flows, (ed. J. N. Sørensen et al.), Kluwer Academic Publishers, Dordrecht, (1999) 213-222.
- [5]. Brøns M., Streamline patterns and their bifurcations using methods from dynamical systems, In: Ricca R.L. (eds), An introduction to the geometry and topology of fluid flows, NATO Science Series (Series II: Mathematics, Physics and Chemistry), Springer, Dordrecht, 47 (2001) 167-182.
- [6]. Bisgaard, A. V., Structures and bifurcations in fluid flows with applications to vortex breakdown and wakes, PhD thesis, Department of Mathematics, Technical University of Denmark, (2005).
- [7]. Bakker, P.G., Bifurcation in flow patterns, vol. 2: Nonlinear topics in the mathematical sciences, Dordrecht: Klüver, (1991).
- [8]. Brøns, M. and Hartnack, J.N., Streamline topologies near simple-degenerate critical points in two-dimensional flow away from boundaries, Phys. Fluids, 11(1999) 314-324.
- [9]. Hartnack, J.N., Streamlines topologies near a fixed wall using normal form, Acta Mech., 136 (1999) 55-75.
- [10]. Deliceouğlu, A. and Gürcan, F., Streamline topology near non-simple degenerate critical points in two-dimensional flow with symmetry about an axis, J. Fluid Mech., 606 (2008) 417-432.
- [11]. Gürcan F. and Deliceoğlu, A., Streamline topologies near non-simple degenerate points in two dimensional flows with double symmetry away from boundaries and an application, Phys. Fluids, 17 (2005) 093116.
- [12]. Deliceoğlu, A., Dinamik sistemler ve basit olmayan dejenere nokta civarındaki sıkıştırılamaz akışların topolojik çatallanmaları, Ph.D Thesis, University of Erciyes, (2004).
- [13]. Ma, T. and Wang, S., Interior structural bifurcation and seperation of 2D incompressible flows, J. Math. Phys., 45 (2004) 1762-1776.
- [14]. Ma, T. and Wang, S., Geometric theory of incompressible flows with applications to fluid dynamics (Mathematical Surveys and Monographs, American Mathematical Society), (2005).
- [15]. Ma, T. and Wang, S., Structural classification and stability of divergence-free vector fields, Physica D, 171 (2002) 107-126.
Abstract
Bu
makalenin amacı basit olmayan tekil nokta civarındaki 2-Boyutlu sıkıştırılamaz
akışlar için eksenel simetrik akışların akış çizgi modellerini ve onların
çatallanmalarını elde etmektir. Hamiltoniyen vektör alan sisteminin akış
çizgileri, indeks teorisinin homotopi değişmezliği kullanılarak
basitleştirildi. İndeksin homotopi değişmezliği kullanılarak, basit olmayan
dejenere nokta civarında, eksenel simetrik akışın yapısal çatallanması için
yeterli ve gerekli koşullar için bir teori geliştirildi. Akış alanındaki parametrelerin değişimi
yapısal çatallanmalara neden olabilir. Dejenere akış yapısının çatallanması, bu
parametrelerin hafifçe değiştirilmesiyle elde edildi.
Keywords
Yapısal kararlılık divergense-serbest vektör alanı Lagrange dinamiği yapısal çatallanma tekillik sınıflandırması
References
- [1]. Brøns, M., Voigt, L.K. and Sørensen, J.N., Topology of vortex breakdown bubbles in a cylinder with a rotating bottom and a free surface, J. Fluid Mech. 428 (2001) 133-148.
- [2]. Brøns, M., Voigt, L.K. and Sørensen, J.N., Streamline topology of steady axisymmetric vortex breakdown in a cylinder with co- and counter-rotating end-covers, J. Fluid Mech., 401 (1999) 275-292.
- [3]. Bisgaard, A.V., Brøns, M. and Sørensen, J.N., Vortex breakdown generated by off-axis bifurcation in a cylinder with rotating covers, Acta Mech. 187 (2006) 75-83.
- [4]. Brøns M., Topologic al fluid mechanics of axisymmetric flows, In simulation and identification of organized structures in flows, (ed. J. N. Sørensen et al.), Kluwer Academic Publishers, Dordrecht, (1999) 213-222.
- [5]. Brøns M., Streamline patterns and their bifurcations using methods from dynamical systems, In: Ricca R.L. (eds), An introduction to the geometry and topology of fluid flows, NATO Science Series (Series II: Mathematics, Physics and Chemistry), Springer, Dordrecht, 47 (2001) 167-182.
- [6]. Bisgaard, A. V., Structures and bifurcations in fluid flows with applications to vortex breakdown and wakes, PhD thesis, Department of Mathematics, Technical University of Denmark, (2005).
- [7]. Bakker, P.G., Bifurcation in flow patterns, vol. 2: Nonlinear topics in the mathematical sciences, Dordrecht: Klüver, (1991).
- [8]. Brøns, M. and Hartnack, J.N., Streamline topologies near simple-degenerate critical points in two-dimensional flow away from boundaries, Phys. Fluids, 11(1999) 314-324.
- [9]. Hartnack, J.N., Streamlines topologies near a fixed wall using normal form, Acta Mech., 136 (1999) 55-75.
- [10]. Deliceouğlu, A. and Gürcan, F., Streamline topology near non-simple degenerate critical points in two-dimensional flow with symmetry about an axis, J. Fluid Mech., 606 (2008) 417-432.
- [11]. Gürcan F. and Deliceoğlu, A., Streamline topologies near non-simple degenerate points in two dimensional flows with double symmetry away from boundaries and an application, Phys. Fluids, 17 (2005) 093116.
- [12]. Deliceoğlu, A., Dinamik sistemler ve basit olmayan dejenere nokta civarındaki sıkıştırılamaz akışların topolojik çatallanmaları, Ph.D Thesis, University of Erciyes, (2004).
- [13]. Ma, T. and Wang, S., Interior structural bifurcation and seperation of 2D incompressible flows, J. Math. Phys., 45 (2004) 1762-1776.
- [14]. Ma, T. and Wang, S., Geometric theory of incompressible flows with applications to fluid dynamics (Mathematical Surveys and Monographs, American Mathematical Society), (2005).
- [15]. Ma, T. and Wang, S., Structural classification and stability of divergence-free vector fields, Physica D, 171 (2002) 107-126.
Details
| Primary Language | English |
|---|---|
| Journal Section | Natural Sciences |
| Authors | |
| Publication Date | September 30, 2018 |
| Submission Date | June 6, 2018 |
| Acceptance Date | July 24, 2018 |
| Published in Issue | Year 2018Volume: 39 Issue: 3 |