Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
EnglishFrançais
Journal of Fluid Mechanics Journal of Fluid Mechanics

Article contents

Topological structure of shock induced vortex breakdown

Published online by Cambridge University Press:  19 October 2009

SHUHAI ZHANG ,
HANXIN ZHANG  and
CHI-WANG SHU
SHUHAI ZHANG
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Centre, Mianyang, Sichuan 621000, China
HANXIN ZHANG
Affiliation:
China Aerodynamics Research and Development Centre, Mianyang, Sichuan 621000, China
CHI-WANG SHU
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
Corresponding
E-mail address:
shu@dam.brown.edu
Get access
Rights & Permissions[Opens in a new window]

Abstract

Using a combination of critical point theory of ordinary differential equations and numerical simulation for the three-dimensional unsteady Navier–Stokes equations, we study possible flow structures of the vortical flow, especially the unsteady vortex breakdown in the interaction between a normal shock wave and a longitudinal vortex. The topological structure contains two parts. One is the sectional streamline pattern in the cross-section perpendicular to the vortex axis. The other is the sectional streamline pattern in the symmetrical plane. In the cross-section perpendicular to the vortex axis, the sectional streamlines have spiral or centre patterns depending on a function λ (x, t) = 1/ρ(∂ρ/∂t+∂ρu/∂x), where x is the coordinate corresponding to the vortex axis. If λ > 0, the sectional streamlines spiral inwards in the near region of the centre. If λ < 0, the sectional streamlines spiral outwards in the same region. If λ = 0, the sectional streamlines form a nonlinear centre. If λ changes its sign along the vortex axis, one or more limit cycles appear in the sectional streamlines in the cross-section perpendicular to the vortex axis. Numerical simulation for two typical cases of shock induced vortex breakdown (Erlebacher, Hussaini & Shu, J. Fluid Mech., vol. 337, 1997, p. 129) is performed. The onset and time evolution of the vortex breakdown are studied. It is found that there are more limit cycles for the sectional streamlines in the cross-section perpendicular to the vortex axis. In addition, we find that there are quadru-helix structures in the tail of the vortex breakdown.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 639 , 25 November 2009 , pp. 343 - 372
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below.
If you should have access and can't see this content please contact technical support.

References

Benjamin, T. B. 1962 Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14, 593–629.CrossRefGoogle Scholar
Bisset, D., Antonia, R. & Browne, L. 1990 Spatial organization of large structures in the turbulent far wake of a cylinder. J. Fluid Mech. 218, 439–461.CrossRefGoogle Scholar
Blackmore, D. 1994 Simple dynamical models for vortex breakdown of B-type. Acta Mech. 102, 91–101.CrossRefGoogle Scholar
Blackmore, D., Brøns, M. & Goullet, A. 2008 A coaxial vortex ring model for vortex breakdown. Physica D. 237, 2817–2844.CrossRefGoogle Scholar
Bossel, H. H. 1969 Vortex breakdown flow field. Phys. Fluids 12, 498–508.CrossRefGoogle Scholar
Brøns, M., Shen, W. Z., Sørensen, J. N. & Zhu, W. J. 2007 The influence of imperfections on the flow structure of steady vortex breakdown bubbles. J. Fluid Mech. 578, 453–466.CrossRefGoogle 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.CrossRefGoogle Scholar
Brücker, C. & Althaus, W. 1995 Study of vortex breakdown by particle tracking velocimetry (PTV). Part 3. Time-dependent structure and development of breakdown-modes. Exp. Fluids 18, 174–186.CrossRefGoogle Scholar
Chong, M. S., Perry, A. E. & Cantwell, B. J. 1990 A general classification of three-dimensional flow field. Phys. Fluids A 2, 765.CrossRefGoogle Scholar
Crank, J., Martin, H. G. & Melluish, D. M. 1977 Nonlinear Ordinary Differential Equations. Oxford University Press.Google Scholar
Delery, J. M. 1994 Aspects of vortex breakdown. Prog. Aerosp. Sci. 30, 1–59.CrossRefGoogle Scholar
Erlebacher, G., Hussaini, M. Y. & Shu, C.-W. 1997 Interaction of a shock with a longitudinal vortex. J. Fluid Mech. 337, 129–153.CrossRefGoogle Scholar
Escudier, M. P. 1984 Observations of the flow produced in a cylindrical container by a rotating endwall. Exp. Fluids 2, 189–196.CrossRefGoogle Scholar
Faler, J. H. & Leibovich, S. 1977 Disrupted states of vortex flow and vortex breakdown. Phys. Fluids 20, 1385–1400.CrossRefGoogle Scholar
Hall, M. G. 1961 A theory for the core of a leading-edge vortex. J. Fluid Mech. 11, 209–228.CrossRefGoogle Scholar
Hall, M. G. 1967 A new approach to vortex breakdown. Proceedings of the 1967 heat transfer and fluid mechanics institute, University of California, San Diego, La Jolla.Google Scholar
Haller, G. 2004 Exact theory of unsteady separation for two-dimensional flows. J. Fluid Mech. 512, 257–311.CrossRefGoogle Scholar
Howard, L. N. & Gupta, A. S. 1962 On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech. 14, 463–476.CrossRefGoogle Scholar
Hunt, J. C. R., Abell, C. J., Peterka, J. A. & Woo, H. 1978 Kinematical studies of the flow around free or surface-mounted obstacles; applying topology to flow visualization. J. Fluid Mech. 86, 179–200.CrossRefGoogle Scholar
Hussain, A. & Hayakawa, M. 1987 Education of large-scale organized structures in a turbulent plane wake. J. Fluid Mech. 180, 193–229.CrossRefGoogle Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.CrossRefGoogle Scholar
Jiang, G.-S. & Shu, C.-W. 1996 Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228.CrossRefGoogle Scholar
Kalkhoran, I. & Smart, M. 2000 Aspects of shock wave-induced vortex breakdown. Prog. Aerosp. Sci. 36, 63–95.CrossRefGoogle Scholar
Kandil, O., Kandil, H. & Liu, C. H. 1992 Shock/vortex interaction and vortex-breakdown modes. In IUTAM Symposium of Fluid Dynamics of High Angle of Attack (ed. Kawamura, R. & Aihara, Y.), pp. 192–212. Springer.Google Scholar
Klass, M., Schröder, W. & Thomer, O. 2005 Experimental investigation and numerical simulation of oblique shock/vortex interaction. In Vortex Dominated Flows: A Volume Celebrating Lu Ting's 80th Birthday (ed. Blackmore, D., Krause, E. & Tung, C.), pp. 119–134. World Scientific.CrossRefGoogle Scholar
Krause, E., Thomer, O. & Schröder, W. 2003 Strong shock-vortex interaction – a numerical study. Comput. Fluid Dyn. 12, 266–278.Google Scholar
Kurosaka, M., Cain, C. B., Srigrarom, S., Wimer, J. D., Dabiri, D., Johnson III, W. F., Hatcher, J. C., Thompson, B. R., Kikuchi, M., Hirano, K., Yuge, T. & Honda, T. 2006 Azimuthal vorticity gradient in the formative stages of vortex breakdown. J. Fluid Mech. 10, 221–246.Google Scholar
Leibovich, S. 1978 The structure of vortex breakdown. Annu. Rev. Fluid Mech. 10, 221–246.CrossRefGoogle Scholar
Lessen, M., Singh, P. J. & Paillet, F. 1974 The stability of a trailing line vortex. Part 1. Inviscid theory. J. Fluid Mech. 65, 753–763.CrossRefGoogle Scholar
Marques, F., Gelfgat, A. Y. & Lopez, J. M. 2003 Tangent double Hopf bifurcation in a differentially rotating cylinder flow. Phys. Rev. E 68, 016310.Google Scholar
Perry, A. E. & Chong, M. S. 1986 A series-expansion study of the Navier–Stokes equation with applications to three-dimensional separation patterns. J. Fluid Mech. 173, 207–223.CrossRefGoogle Scholar
Ruith, M. R., Chen, P., Meiburg, E. & Maxworth, T. 2003 Three-dimensional vortex breakdown in swirling jets and wakes: direct numerical simulation. J. Fluid Mech. 486, 331–378.CrossRefGoogle Scholar
Sanchez, J., Marques, F. & Lopez, J. M. 2002 A continuation and bifurcation technique for Navier–Stokes flows. J. Comput. Phys. 180, 78–98.CrossRefGoogle Scholar
Sarpkaya, T. 1971 On stationary and travelling vortex breakdown. J. Fluid Mech. 45, 545–559.CrossRefGoogle Scholar
Spall, R. E. 1996 Transition from spiral- to bubble-type vortex breakdown. Phys. Fluids 8, 1330–1332.CrossRefGoogle Scholar
Surana, A., Grunberg, O. & Haller, G. 2006 Exact theory of three-dimensional flow separation. Part 1. Steady separation. J. Fluid Mech. 564, 57–103.CrossRefGoogle Scholar
Tobak, M. & Peake, D. J. 1982 Topology of three-dimensional separated flows. Annu. Rev. Fluid Mech. 14, 61–85.CrossRefGoogle Scholar
Werle, H. 1954 Quelques Résultats expérimentaux sur les ailes en Fleches, aux faibles vitesses. Obtenus en tunnel hydrodynamique. La Recherche Aéronautiale. 41, 15–21.Google Scholar
Zhang, H. 1995 Characteristic analysis of subsonic and supersonic vortical flow. ACTA Aerodyn. Sin. 13, 259–264 (in Chinese).Google Scholar
Zhang, S. & Shu, C.-W. 2007 A new smooth indicator for the WENO schemes and its effect on the convergence to steady state solutions. J. Sci. Comput. 31, 273–305.CrossRefGoogle Scholar
Zhang, S., Zhang, Y.-T. & Shu, C.-W. 2005 Multistage interaction of a shock wave and a strong vortex. Phys. Fluids 17, 116101.CrossRefGoogle Scholar
Zhang, S., Zhang, Y.-T. & Shu, C.-W. 2006 Interaction of an oblique shock wave with a pair of parallel vortices: shock dynamics and mechanism of sound generation. Phys. Fluids 18, 126101.CrossRefGoogle Scholar
Zhang, S., Zhang, H. & Zhu, G. 1996 Numerical simulation of vortex structure in the leeside of supersonic delta wing. ACTA Aerodyn. Sin. 14, 430–435 (in Chinese).Google Scholar
Zhang, S., Zhang, H. & Zhu, G. 1997 Numerical simulation of vortical flows over delta wings and analysis of vortex motion. ACTA Aerodyn. Sin. 15, 121–129 (in Chinese).Google Scholar
Zhang, S. & Zhu, G. 1997 Numerical visualization for three-dimensional vortex. In Fifth triennial international symposium on fluid control, measurement and visualization, Hayama, Japan.Google Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 114 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 21st January 2021. This data will be updated every 24 hours.

11 Cited by
Hostname: page-component-76cb886bbf-tvlwp Total loading time: 0.369 Render date: 2021-01-21T22:01:27.072Z Query parameters: { "hasAccess": "0", "openAccess": "0", "isLogged": "0", "lang": "en" } Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false }

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Topological structure of shock induced vortex breakdown
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Topological structure of shock induced vortex breakdown
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Topological structure of shock induced vortex breakdown
Available formats
×
×

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *