Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-03T01:17:16.948Z Has data issue: false hasContentIssue false
  • English
  • Français

Streamline topology near non-simple degenerate critical points in two-dimensional flow with symmetry about an axis

Published online by Cambridge University Press:  10 July 2008

A. DELİCEOĞLU  and
F. GÜRCAN
A. DELİCEOĞLU
Affiliation:
Department of Mathematics, Erciyes University, Kayseri, Turkey38039
F. GÜRCAN*
Affiliation:
Department of Mathematics, Erciyes University, Kayseri, Turkey38039
*
†Author to whom correspondence should be addressed: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The local flow patterns and their bifurcations associated with non-simple degenerate critical points appearing away from boundaries are investigated under the symmetric condition about a straight line in two-dimensional incompressible flow. These flow patterns are determined via a bifurcation analysis of polynomial expansions of the streamfunction in the proximity of the degenerate critical points. The normal form transformation is used in order to construct a simple streamfunction family, which classifies all possible local streamline topologies for given order of degeneracy (degeneracies of order three and four are considered). The relation between local and global flow patterns is exemplified by a cavity flow.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 606 , 10 July 2008 , pp. 417 - 432
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Brøns, M. 1994 Topological fluid dynamics of interfacial flows. Phys. Fluids 6, 2730–2737.Google Scholar
Brøns, M. 2007 Streamline topology-Patterns in fluid flows and their bifurcations. Adv. Appl. Mech. 41, 1–43.Google Scholar
Brøns, M. & Bisgaard, A. V. 2006 Bifurcation of vortex breakdown patterns in a circular cylinder with two rotating covers. J. Fluid Mech. 568, 329–349.Google Scholar
Brøns, M. & Hartnack, J. N. 1999 Streamline topologies near simple degenerate critical points in two-dimensional flow away from boundaries. Phys. Fluids 11, 314–324.Google Scholar
Brøns, M., Jakobsen, B., Niss, K., Bisgaard, A. V. & Voigt, L. K. 2007 Streamline topology in the near-wake of a circular cylinder at moderate Reynolds numbers. J. Fluid Mech. 584, 23–43.Google Scholar
Brøns, M., Voigt, L. K. & Sørensen, J. N. 1999 Streamline topology of steady axisymmetric vortex breakdown in a cylinder with co- and counter-rotating end-covers. J. Fluid Mech. 401, 275–292.Google Scholar
Brøns, M., Voigt, L. K. & Sørensen, J. N. 2001 Topology of vortex breakdown bubbles in a cylinder with a rotating bottom and a free surface. J. Fluid Mech. 428, 133–148.Google Scholar
Bakker, P. G. 1991 Bifurcation in Flow Patterns. D. Klüwer.Google Scholar
Bisgaard, A. V., Brøns, M. & Sørensen, J. N. 2006 Vortex breakdown generated by off-axis bifurcation in circular cylinder with rotating covers. Acta Mechanica 187, 75–83.Google Scholar
Chien, W. L., Rising, H. & Ottino, J. M. 1986 Laminar mixing and chaotic mixing in several cavity flows. J. Fluid Mech. 170, 355–377.Google Scholar
Dallmann, U. 1983 Topological structures of three-dimensional flow separations. IB 221–82 A 07. DFVLR.CrossRefGoogle Scholar
Dallmann, U. & Gebing, H. 1994 Flow attachment at flow separation lines. Acta Mechanica 4, 47.Google Scholar
Gaskell, P. H., Gürcan, F., Savage, M. D. & Thompson, H. M. 1998 Stokes flow in a double-lid-driven cavity with free surface side-walls. Proc. Inst. Mech. Engrs C 212, 387–403.Google Scholar
Gürcan, F. 1997 Flow bifurcations in rectangular, lid-driven, cavity flows. PhD thesis, University of Leeds.Google Scholar
Gürcan, F. 2003 Streamline topologies in Stokes flow within lid-driven cavities. Theor. Comput. Fluid Dyn. 17, 19–30.Google Scholar
Gürcan, F. & Deliceoğlu, A. 2005 Streamline topologies near non-simple degenerate points in two-dimensinonal flows with double symmetry away from boundaries and an application. Phys. Fluids 17, 093106.Google Scholar
Gürcan, F., Gaskell, P. H., Savage, M. D. & Wilson, M. C. T. 2003 Eddy genesis and transformation of Stokes flow in a double-lid driven cavity. Proc. Inst. Mech. Engrs C 217, 353–364.Google Scholar
Gürcan, F. & Deliceoğlu, A. 2006 Saddle connections near degenerate critical points in Stokes flow within cavities. Appl. Math. Comput. 172, 1133–1144.Google Scholar
Gürcan, F., Wilson, M. C. T. & Savage, M. D. 2006 Eddy genesis and transformation of Stokes flow in a double-lid driven cavity Part 2: deep cavity. Proc. Inst. Mech. Engrs. C 220, 1765–1774.Google Scholar
Hartnack, J. N. 1999 Structural changes in incompressible flow patterns. PhD thesis, Department of Mathematics, Technical University of Denmark.Google Scholar
Hartnack, J. N. 1999 Streamlines topologies near a fixed wall using normal forms. Acta Mechanica 136, 55–75.Google Scholar
Joseph, D. D. & Sturges, L. 1978 The convergence of biorthogonal series for biharmonic and Stokes flow edge problems: Part II. SIAM. J. Appl. Maths 34, 7–27.Google Scholar
Legendre, R. 1956 Séparation de I‘écoulement laminaire tridimensionel. Rech. Aerosp. 54, 3–8.Google Scholar
Meleshko, V. V. 1996 Steady Stokes flow in a rectangular cavity Proc. R. Soc. Lond. A 452, 1999–2022.Google Scholar
Milne-Thomson, L. M. 1957 A general solution of the equations of hydrodynamics. J. Fluid Mech. 2, 88.Google Scholar
Milne-Thomson, L. M. 1968 Theoretical Hydrodynamics. Dover.Google Scholar
Shankar, P. N. & Deshpande, M. D. 2000 Fluid mechanics in the driven cavity. Annu. Rev. Fluid Mech. 32, 93–136.Google Scholar
Smith, R. C. T. 1952 The bending of a semi-infinite strip. Austral. J. Sci. Res. 5, 227–237.Google Scholar
Tobak, M. & Peake, D. J. 1982 Topology of three-dimensional separated flows. Annu. Rev. Fluid Mech. 14, 61–85.Google Scholar
Tophøj, L., Møller, S. & Brøns, M. 2006 Streamline patterns and their bifurcations near a wall with Navier slip boundary conditions. Phys. Fluids 18, 083102.Google Scholar