Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T20:58:25.872Z Has data issue: false hasContentIssue false
  • English
  • Français

Topology of helical fluid flow

Published online by Cambridge University Press:  17 March 2014

MORTEN ANDERSEN  and
MORTEN BRØNS
MORTEN ANDERSEN
Affiliation:
Department of Applied Mathematics and Computer Science and Fluid.DTU email: post.til.morten.a@gmail.com
MORTEN BRØNS
Affiliation:
Technical University of Denmark, Building 303B, DK-2800 Kongens Lyngby, Denmark email: mobr@dtu.dk
Get access Rights & Permissions [Opens in a new window]

Abstract

Considering a coordinate-free formulation of helical symmetry rather than more traditional definitions based on coordinates, we discuss basic properties of helical vector fields and compare results from the literature obtained with other approaches. In particular, we discuss the role of the stream function for the topology of the streamline pattern in incompressible flows. On this basis, we perform a comprehensive study of the topology of the flow field generated by a helical vortex filament in an ideal fluid. The classical expression for the stream function obtained by Hardin (Hardin, J. C. 1982 Phys. Fluids25, 1949–1952) contains an infinite sum of modified Bessel functions. Using the approach by Okulov (Okulov, V. L. 1995 Russ. J.Eng. Thermophys.5, 63–75) we obtain a closed-form approximation which is considerably easier to analyse. Critical points of the stream function can be found from the zeroes of a single real function of one variable, and we show that three different flow topologies can occur, depending on a single dimensionless parameter. By including the self-induced velocity on the vortex filament by a localised induction approximation, the stream function is slightly modified and an extra parameter is introduced. In this setting two new flow topologies arise, but not more than two critical points occur for any combination of parameters.

Keywords

Helical flowFlow topologyVortex filamentsAsymptotic expansions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 25 , Issue 3 , June 2014 , pp. 375 - 396
Copyright
Copyright © Cambridge University Press 2014 

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

[1]Abramowitz, M. & Stegun, I. A. (1964) Handbook of Mathematical Functions, Superintendent of Documents, US Government Printing Office, Washington, DC.Google Scholar
[2]Alekseenko, S. V., Okulov, P. A. & Shtork, S. I. (1999) Helical vortices in swirl flow. J. Fluid Mech. 382, 195–243.Google Scholar
[3]Alekseenko, S. V., Kuibin, P. A. & Okulov, V. L. (2007) Theory of Concentrated Vortices, Springer-Verlag, Berlin, Germany.Google Scholar
[4]Batchelor, G. K. (1967) An Introduction to Fluids Dynamics, Cambridge University Press, Cambridge, UK.Google Scholar
[5]Brøns, M. (1994) Topological fluid dynamics of interfacial flows. Phys. Fluids 6 (8), 2730–2737.Google Scholar
[6]Brøns, M. (2007) Streamline topology: Patterns in fluid flows and their bifurcations. Adv. Appl. Mech. 41, 1–42.Google Scholar
[7]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
[8]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
[9]Childress, S., Landman, M. & Strauss, H. (1990) Steady motion with helical symmetry at large Reynolds number. In: Moffatt, H. K. and Tsinober, A. (editors), Proceedings of the IUTAM Symposium on Topological Fluid Mechanics, Cambridge University Press, Cambridge, UK.Google Scholar
[10]DaAAAARios, L. S. (1906) Sul moto d'un liquido indefinito con un filetto vorticoso di forma qualunque. Rendiconti del Circolo Matematico di Palermo 22, 117–135.CrossRefGoogle Scholar
[11]Dean, W. R. (1927) Note on the notion of fluid in a curved pipe. Phil. Mag. 4, 208–223.Google Scholar
[12]Delbende, I., Rossi, M. & Daube, O. (2012) DNS of flows with helical symmetry. Theor. Comput. Fluid Dyn. 26, 141–160.Google Scholar
[13]Deliceoğlu, A. (2013) Topology of two-dimensional flow associated with degenerate dividing streamline on a free surface. Euro. J. Appl. Math. 24 (1), 77–101.Google Scholar
[14]Dritschel, D. G. (1991) Generalised helical Beltrami flows in hydrodynamics and magnetohydrodynamics. J. Fluid Mech. 222, 525–541.Google Scholar
[15]Ettinger, B. & Titi, E. S. (2009) Global existence and uniqueness of weak solutions of three-dimensional Euler equations with helical symmetry in the absence of vorticity stretching. SIAM J. Math. Anal. 41, 269–296.Google Scholar
[16]Fukumoto, Y. & Okulov, V. L. (2005) The velocity field induced by a helical vortex tube. Phys. Fluids 17, 107101.Google Scholar
[17]Germano, M. (2009) The Dean equations extended to a helical pipe flow. J. Fluid Mech. 203, 289–305.Google Scholar
[18]Gupta, B. P. & Loewy, R. G. (1974) Theoretical analysis of the aerodynamic stability of multiple, interdigitated helical vortices. AIAA Journal 12, 1381–1387.Google Scholar
[19]Hardin, J. C. (1982) The velocity field induced by a helical vortex filament. Phys. Fluids 25, 1949–1952.Google Scholar
[20]Ijzermans, R. H. A., Hajmeijer, R. & vanAAAALangen, P. J. (2007) Accumulation of heavy particles around a helical vortex filament. Phys. Fluids 19, 107102.Google Scholar
[21]Ito, H. (2007) Flow in curved pipes. Jpn Soc. Mech. Eng. 30, 543–552.Google Scholar
[22]Landman, M. J. (1990) On the generation of helical waves in circular pipe flow. Phys. Fluids A 2, 738–747.CrossRefGoogle Scholar
[23]Landman, M. J. (1990) Time-dependent helical waves in rotating pipe flow. J. Fluid Mech. 221, 289–310.Google Scholar
[24]Liu, S. & Masliiyah, J. H. (1993) Axially invariant laminar flow in helical pipes with a finite pitch. J. Fluid Mech. 251, 315–353.Google Scholar
[25]Lo Jacono, D., Nazarinia, M. & Brøns, M. (2009) Experimental vortex breakdown topology in a cylinder with a free surface. Phys. Fluids 11, 111704.Google Scholar
[26]Lucas, D. & Dritschel, D. G. (2009) A family of helically symmetric vortex equilibria. J. Fluid Mech. 634, 245–268.Google Scholar
[27]Mezic, I., Leonard, A. & Wiggins, S. (1998) Regular and chaotic particle motion near a helical vortex filament. Physica D 111, 179–201.Google Scholar
[28]Okulov, V. L. (1995) The velocity field induced by helical vortex filaments with cylindrical or conic supporting surface. Russ. J. Eng. Thermophys. 5, 63–75.Google Scholar
[29]Okulov, V. L. (2004) On the stability of multiple helical vortices. J. Fluid Mech. 521, 319–342.Google Scholar
[30]Okulov, V. L. & Sørensen, J. N. (2007) Stability of helical tip vortices in a rotor far wake. J. Fluid Mech. 576, 1–25.Google Scholar
[31]Olver, F. W. J. (1954) The asymptotic expansion of Bessel functions of large order. Philos. Trans. R. Soc. A 247, 328–368.Google Scholar
[32]Olver, F. W. J. (1954) The asymptotic solution of linear differential equations of the second order for large values of a parameter. Philos. Trans. R. Soc. A 247, 307–327.Google Scholar
[33]Ricca, R. L. (1994) The effect of torsion on the motion of a helical vortex filament. J. Fluid Mech. 273, 241–259.CrossRefGoogle Scholar
[34]Ricca, R. L. (1996) The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filaments dynamics. Fluid Dyn. Res. 18, 245–268.Google Scholar
[35]Sarasua, L. G., Sicardi Schifino, A. C. & Gonzalez, R. (1998) The stability of steady, helical vortex filaments in a tube. Phys. Fluids 11, 1096–1103.CrossRefGoogle Scholar
[36]Sørensen, J. N.. (2011) Instability of helical tip vortices in rotor far wakes. J. Fluid Mech. 682, 1–4.CrossRefGoogle Scholar
[37]Throumoulopoulos, G. N. & Tasso, H. (1999) Ideal magnetohydrodynamics equilibria with helical symmetry and incompressible flows. J. Plasma Phys. 62, 449–459.Google Scholar
[38]Tuttle, E. R. (1990) Laminar flow in twisted pipes. J. Fluid Mech. 219, 545–570.Google Scholar
[39]Zabielski, L. & Mestel, A. J. (1998) Steady flow in a helically symmetric pipe. J. Fluid Mech. 370, 297–320.Google Scholar
[40]Zabielski, L. & Mestel, A. J. (1998) Unsteady flow in a helically symmetric pipe. J. Fluid Mech. 370, 321–345.Google Scholar