Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-17T11:26:03.310Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact theory of material spike formation in flow separation

Published online by Cambridge University Press:  20 April 2018

Mattia Serra Open the ORCID record for Mattia Serra [Opens in a new window] ,
Jérôme Vétel Open the ORCID record for Jérôme Vétel [Opens in a new window]  and
George Haller Open the ORCID record for George Haller [Opens in a new window]
Mattia Serra*
Affiliation:
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
Jérôme Vétel
Affiliation:
Department of Mechanical Engineering, Polytechnique Montréal, Montréal, QC, H3C 3A7, Canada
George Haller
Affiliation:
Institute for Mechanical Systems, ETH Zürich, 8092 Zürich, Switzerland
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

JFM classification

Nonlinear Dynamical Systems: Pattern formation Wakes/Jets: Separated flows Mathematical Foundations: Topological fluid dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 845 , 25 June 2018 , pp. 51 - 92
Check for updates
Copyright
© 2018 Cambridge University Press 

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

Arnold, V. 1973 Ordinary Differential Equations. MIT Press.Google Scholar
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.Google Scholar
Bewley, T. R. & Protas, B. 2004 Skin friction and pressure: the ‘footprints’ of turbulence. Physica D 196 (12), 28–44.Google Scholar
Cassel, K. W. & Conlisk, A. T. 2014 Unsteady separation in vortex-induced boundary layers. Phil. Trans. R. Soc. Lond. A 372 (2020), 20130348.Google Scholar
Farazmand, M. & Haller, G. 2012 Computing Lagrangian coherent structures from their variational theory. Chaos 22, 013128.CrossRefGoogle ScholarPubMed
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. 1–13. Springer.Google Scholar
Gurtin, M. E. 1982 An Introduction to Continuum Mechanics, vol. 158. Academic.Google Scholar
Haller, G. 2000 Finding finite-time invariant manifolds in two-dimensional velocity fields. Chaos 10 (1), 99–108.Google Scholar
Haller, G. 2004 Exact theory of unsteady separation for two-dimensional flows. J. Fluid Mech. 512, 257–311.Google Scholar
Haller, G. 2015 Lagrangian coherent structures. Annu. Rev. Fluid. Mech. 47, 137–162.Google Scholar
Jung, C., Tél, T. & Ziemniak, E. 1993 Application of scattering chaos to particle transport in a hydrodynamical flow. Chaos 3 (4), 555–568.CrossRefGoogle Scholar
Kilic, M. S., Haller, G. & Neishtadt, A. 2005 Unsteady fluid flow separation by the method of averaging. Phys. Fluids 17 (6), 067104.Google Scholar
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), 115–126.Google Scholar
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), 177–185.Google Scholar
Miron, P. & Vétel, J. 2015 Towards the detection of moving separation in unsteady flows. J. Fluid Mech. 779, 819–841.Google Scholar
Moore, F. K. 1958 On the separation of the unsteady laminar boundary layer. In Grenzschichtforschung/Boundary Layer Research, pp. 296–311. Springer.Google Scholar
Nelson, D. A. & Jacobs, G. B. 2015 DG-FTLE: Lagrangian coherent structures with high-order discontinuous-Galerkin methods. J. Comput. Phys. 295, 65–86.Google Scholar
Nelson, D. A. & Jacobs, G. B. 2016 High-order visualiszation of three-dimensional Lagrangian coherent structures with DG-FTLE. Comput. Fluids 139, 197–215.Google Scholar
Prandtl, L. 1904 Über Flüssigkeitsbewegung bei sehr kleiner Reibung. Verhandl III, Intern. Math. Kongr. Heidelberg, Auch: Gesammelte Abhandlungen 2, 484–491.Google Scholar
Rott, N. 1956 Unsteady viscous flow in the vicinity of a stagnation point. Q. Appl. Maths 13 (4), 444–451.Google Scholar
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, 124–155.CrossRefGoogle Scholar
Sears, W. R. & Telionis, D. P. 1971 Unsteady boundary-layer separation. In Recent Research on Unsteady Boundary Layers, vol. 1, pp. 404–447. Laval University Press.Google Scholar
Sears, W. R. & Telionis, D. P. 1975 Boundary-layer separation in unsteady flow. SIAM J. Appl. Maths 28 (1), 215–235.Google Scholar
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, 613–646.Google 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.Google Scholar
Surana, A. & Haller, G. 2008 Ghost manifolds in slow–fast systems, with applications to unsteady fluid flow separation. Physica D 237 (10), 1507–1529.Google Scholar
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.Google Scholar
Sychev, V. V. & Sychev, V. V. 1998 Asymptotic Theory of Separated Flows. Cambridge University Press.Google Scholar
Truesdell, C. & Noll, W. 2004 The Non-Linear Field Theories of Mechanics. Springer.Google Scholar
Van Dommelen, L. L. 1981 Unsteady Boundary Layer Separation. Cornell University.Google Scholar
Van Dommelen, L. L. & Shen, S. F. 1982 The genesis of separation. In Numerical and Physical Aspects of Aerodynamic Flows, pp. 293–311. Springer.Google Scholar
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, 1–11.Google Scholar
Williams, J. C. 1977 Incompressible boundary-layer separation. Annu. Rev. Fluid Mech. 9 (1), 113–144.CrossRefGoogle Scholar
Yuster, T. & Hackborn, W. W. 1997 On invariant manifolds attached to oscillating boundaries in Stokes flows. Chaos 7 (4), 769–776.Google Scholar

Serra et al. supplementary movie 1

Movie 1: Caption included in Figure 1

Download Serra et al. supplementary movie 1(Video)
Video 965.5 KB

Serra et al. supplementary movie 10

Movie 10: Caption included in Figure 23

Download Serra et al. supplementary movie 10(Video)
Video 1.6 MB

Serra et al. supplementary movie 2

Movie 2: Caption included in Figure 2

Download Serra et al. supplementary movie 2(Video)
Video 1.8 MB

Serra et al. supplementary movie 3

Movie 3: Caption included in Figure 8

Download Serra et al. supplementary movie 3(Video)
Video 1.5 MB

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.

Download Serra et al. supplementary movie 4(Video)
Video 1.6 MB

Serra et al. supplementary movie 5

Movie 5: Caption included in Figure 11

Download Serra et al. supplementary movie 5(Video)
Video 21 MB

Serra et al. supplementary movie 6

Movie 6: Caption included in Figure 13

Download Serra et al. supplementary movie 6(Video)
Video 1.6 MB

Serra et al. supplementary movie 7

Movie 7: Caption included in Figure 15

Download Serra et al. supplementary movie 7(Video)
Video 2.2 MB

Serra et al. supplementary movie 8

Movie 8: Caption included in Figure 18

Download Serra et al. supplementary movie 8(Video)
Video 1 MB

Serra et al. supplementary movie 9

Movie 9: Caption included in Figure 20

Download Serra et al. supplementary movie 9(Video)
Video 3.3 MB