Skip to main content

Exact theory of material spike formation in flow separation

  • Mattia Serra (a1), Jérôme Vétel (a2) and George Haller (a3)

We develop a frame-invariant theory of material spike formation during flow separation over a no-slip boundary in two-dimensional flows with arbitrary time dependence. Based on the exact curvature evolution of near-wall material lines, our theory identifies both fixed and moving flow separation, is effective also over short time intervals, and admits a rigorous instantaneous limit. As a byproduct, we derive explicit formulae for the evolution of material line curvature and the curvature rate for general compressible flows. The material backbone that we identify acts first as the precursor and later as the centrepiece of unsteady Lagrangian flow separation. We also discover a previously undetected spiking point where the backbone of separation connects to the boundary, and derive wall-based analytical formulae for its location. Finally, our theory explains the perception of off-wall separation in unsteady flows and provides conditions under which such a perception is justified. We illustrate our results on several analytical and experimental flows.

Corresponding author
Email address for correspondence:
Hide All
Arnold, V. 1973 Ordinary Differential Equations. MIT Press.
Benczik, I. J., Toroczkai, Z. & Tél, T. 2002 Selective sensitivity of open chaotic flows on inertial tracer advection: catching particles with a stick. Phys. Rev. Lett. 89 (16), 164501.
Bewley, T. R. & Protas, B. 2004 Skin friction and pressure: the ‘footprints’ of turbulence. Physica D 196 (12), 2844.
Cassel, K. W. & Conlisk, A. T. 2014 Unsteady separation in vortex-induced boundary layers. Phil. Trans. R. Soc. Lond. A 372 (2020), 20130348.
Farazmand, M. & Haller, G. 2012 Computing Lagrangian coherent structures from their variational theory. Chaos 22, 013128.
Garth, C., Li, G.-S., Tricoche, X., Hansen, C. D. & Hagen, H. 2009 Visualization of coherent structures in transient 2d flows. In Topology-Based Methods in Visualization II, pp. 113. Springer.
Gurtin, M. E. 1982 An Introduction to Continuum Mechanics, vol. 158. Academic.
Haller, G. 2000 Finding finite-time invariant manifolds in two-dimensional velocity fields. Chaos 10 (1), 99108.
Haller, G. 2004 Exact theory of unsteady separation for two-dimensional flows. J. Fluid Mech. 512, 257311.
Haller, G. 2015 Lagrangian coherent structures. Annu. Rev. Fluid. Mech. 47, 137162.
Jung, C., Tél, T. & Ziemniak, E. 1993 Application of scattering chaos to particle transport in a hydrodynamical flow. Chaos 3 (4), 555568.
Kilic, M. S., Haller, G. & Neishtadt, A. 2005 Unsteady fluid flow separation by the method of averaging. Phys. Fluids 17 (6), 067104.
Klonowska-Prosnak, M. E. & Prosnak, W. J. 2001 An exact solution to the problem of creeping flow around circular cylinder rotating in presence of translating plane boundary. Acta Mechanica 146 (1), 115126.
Liu, C. S. & Wan, Y.-H. 1985 A simple exact solution of the Prandtl boundary layer equations containing a point of separation. Arch. Rat. Mech. Anal. 89 (2), 177185.
Miron, P. & Vétel, J. 2015 Towards the detection of moving separation in unsteady flows. J. Fluid Mech. 779, 819841.
Moore, F. K. 1958 On the separation of the unsteady laminar boundary layer. In Grenzschichtforschung/Boundary Layer Research, pp. 296311. Springer.
Nelson, D. A. & Jacobs, G. B. 2015 DG-FTLE: Lagrangian coherent structures with high-order discontinuous-Galerkin methods. J. Comput. Phys. 295, 6586.
Nelson, D. A. & Jacobs, G. B. 2016 High-order visualiszation of three-dimensional Lagrangian coherent structures with DG-FTLE. Comput. Fluids 139, 197215.
Prandtl, L. 1904 Über Flüssigkeitsbewegung bei sehr kleiner Reibung. Verhandl III, Intern. Math. Kongr. Heidelberg, Auch: Gesammelte Abhandlungen 2, 484491.
Rott, N. 1956 Unsteady viscous flow in the vicinity of a stagnation point. Q. Appl. Maths 13 (4), 444451.
Ruban, A. I., Araki, D., Yapalparvi, R. & Gajjar, J. S. B. 2011 On unsteady boundary-layer separation in supersonic flow. Part 1. Upstream moving separation point. J. Fluid Mech. 678, 124155.
Sears, W. R. & Telionis, D. P. 1971 Unsteady boundary-layer separation. In Recent Research on Unsteady Boundary Layers, vol. 1, pp. 404447. Laval University Press.
Sears, W. R. & Telionis, D. P. 1975 Boundary-layer separation in unsteady flow. SIAM J. Appl. Maths 28 (1), 215235.
Shariff, K., Pulliam, T. H. & Ottino, J. M. 1991 A dynamical systems analysis of kinematics in the time-periodic wake of a circular cylinder. Lect. Appl. Math 28, 613646.
Surana, A., Grunberg, O. & Haller, G. 2006 Exact theory of three-dimensional flow separation. Part 1. Steady separation. J. Fluid Mech. 564, 57103.
Surana, A. & Haller, G. 2008 Ghost manifolds in slow–fast systems, with applications to unsteady fluid flow separation. Physica D 237 (10), 15071529.
Surana, A., Jacobs, G. B., Grunberg, O. & Haller, G. 2008 An exact theory of three-dimensional fixed separation in unsteady flows. Phys. Fluids 20 (10), 107101.
Sychev, V. V. & Sychev, V. V. 1998 Asymptotic Theory of Separated Flows. Cambridge University Press.
Truesdell, C. & Noll, W. 2004 The Non-Linear Field Theories of Mechanics. Springer.
Van Dommelen, L. L. 1981 Unsteady Boundary Layer Separation. Cornell University.
Van Dommelen, L. L. & Shen, S. F. 1982 The genesis of separation. In Numerical and Physical Aspects of Aerodynamic Flows, pp. 293311. Springer.
Weldon, M., Peacock, T., Jacobs, G. B., Helu, M. & Haller, G. 2008 Experimental and numerical investigation of the kinematic theory of unsteady separation. J. Fluid Mech. 611, 111.
Williams, J. C. 1977 Incompressible boundary-layer separation. Annu. Rev. Fluid Mech. 9 (1), 113144.
Yuster, T. & Hackborn, W. W. 1997 On invariant manifolds attached to oscillating boundaries in Stokes flows. Chaos 7 (4), 769776.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification

Type Description Title

Serra et al. supplementary movie 4
Movie 4: Same as Movie 3, but for a longer integration time (T=3). Movie 4 is described in the sentence right above Figure 9.

 Video (1.6 MB)
1.6 MB

Serra et al. supplementary movie 1
Movie 1: Caption included in Figure 1

 Video (965 KB)
965 KB

Serra et al. supplementary movie 5
Movie 5: Caption included in Figure 11

 Video (21.0 MB)
21.0 MB

Serra et al. supplementary movie 10
Movie 10: Caption included in Figure 23

 Video (1.6 MB)
1.6 MB

Serra et al. supplementary movie 6
Movie 6: Caption included in Figure 13

 Video (1.6 MB)
1.6 MB

Serra et al. supplementary movie 2
Movie 2: Caption included in Figure 2

 Video (1.8 MB)
1.8 MB

Serra et al. supplementary movie 7
Movie 7: Caption included in Figure 15

 Video (2.2 MB)
2.2 MB

Serra et al. supplementary movie 8
Movie 8: Caption included in Figure 18

 Video (1.0 MB)
1.0 MB

Serra et al. supplementary movie 9
Movie 9: Caption included in Figure 20

 Video (3.3 MB)
3.3 MB

Serra et al. supplementary movie 3
Movie 3: Caption included in Figure 8

 Video (1.5 MB)
1.5 MB


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed