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Material spike formation in highly unsteady separated flows

  • Mattia Serra (a1), Seán Crouzat (a2), Gaël Simon (a2), Jérôme Vétel (a2) and George Haller (a3)...


We apply the recent frame-invariant theory of separation spike formation to complex unsteady flows, including a turbulent separation bubble, an impinging jet, and flows around a freely moving cylinder and a freely rotating ellipse. We show how the theory captures the onset of material spike formation, without any assumption on the flow type (steady, periodic, unsteady) or separation type (on- or off-wall, fixed or moving boundaries). We uncover new phenomena, such as the transition from on-wall to off-wall separation, the merger of initially distinct spikes, and the presence of severe material spikes that remain hidden to previous approaches. Remarkably, even in steady flows around curved boundaries, we detect material spikes in the absence of flow reversal, the main ingredient to existing separation criteria. Together, our results unveil how an involved network of spikes arises, interacts and merges dynamically, leading to the final ejection of particles from the wall in highly transient flow separation processes.


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Azad, R. S. 1996 Turbulent flow in a conical diffuser: a review. Exp. Therm. Fluid Sci. 13 (4), 318337.
Cassel, K. W. & Conlisk, A. T. 2014 Unsteady separation in vortex-induced boundary layers. Phil. Trans. R. Soc. Lond. A 372, 20130348.
Cermak, J. E. 1976 Aerodynamics of buildings. Annu. Rev. Fluid Mech. 8, 75106.
Cloos, F.-J., Stapp, D. & Pelz, P. F. 2017 Swirl boundary layer and flow separation at the inlet of a rotating pipe. J. Fluid Mech. 811, 350371.
Corke, T. C. & Thomas, F. O. 2015 Dynamic stall in pitching airfoils: aerodynamic damping and compressibility effects. Annu. Rev. Fluid Mech. 47, 479505.
Dandois, J., Mary, I. & Brion, V. 2018 Large-eddy simulation of laminar transonic buffet. J. Fluid Mech. 850, 156178.
Didden, N. & Ho, C.-M. 1985 Unsteady separation in a boundary layer produced by an impinging jet. J. Fluid Mech. 160, 235256.
van Dommelen, L. L. & Shen, S. F. 1982 The genesis of separation. In Numerical and Physical Aspects of Aerodynamic Flows (ed. Cebeci, T.), pp. 293311. Springer.
Duquesne, P., Maciel, Y. & Deschênes, C. 2015 Unsteady flow separation in a turbine diffuser. Exp. Fluids 56, 156.
Elliott, J. W., Smith, F. T. & Cowley, S. J. 1983 Breakdown of boundary layers: (i) on moving surfaces; (ii) in semi-similar unsteady flow; (iii) in fully unsteady flow. Geophys. Astrophys. Fluid Dyn. 25, 77138.
Fang, X. & Tachie, M. F. 2019 On the unsteady characteristics of turbulent separations over a forward–backward-facing step. J. Fluid Mech. 863, 9941030.
Farazmand, M. & Haller, G. 2012 Computing Lagrangian coherent structures from their variational theory. Chaos 22, 013128.
Gresh, T. 2018 Compressor Performance: Aerodynamics for the User, 3rd edn. Butterworth-Heinemann.
Gsell, S., Bourguet, R. & Braza, M. 2016 Two-degree-of-freedom vortex-induced vibrations of a circular cylinder at Re = 3900. J. Fluids Struct. 67, 156172.
Haller, G. 2004 Exact theory of unsteady separation for two-dimensional flows. J. Fluid Mech. 512, 257311.
Haller, G. 2011 A variational theory of hyperbolic Lagrangian coherent structures. Physica D 240 (7), 574598.
Hecht, F. 2012 New development in FreeFem++. J. Numer. Math. 20 (3–4), 251265.
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 Mech. 146, 115126.
Klose, B. F., Serra, M. & Jacobs, G. B.2019 The kinematics of Lagrangian flow separation in external aerodynamics. AIAA J. (submitted) arXiv:1909.04129.
Laizet, S. & Lamballais, E. 2009 High-order compact schemes for incompressible flows: a simple and efficient method with quasi-spectral accuracy. J. Comput. Phys. 228 (16), 59896015.
Laizet, S. & Li, N. 2011 Incompact3d: a powerful tool to tackle turbulence problems with up to O (105) computational cores. Intl J. Numer. Meth. Fluids 67, 17351757.
Lamarche-Gagnon, M.-É. & Vétel, J. 2018 Experimental investigation of unsteady separation in the rotor-oscillator flow. J. Fluid Mech. 844, 546566.
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.
Miron, P., Vétel, J. & Garon, A. 2015 On the flow separation in the wake of a fixed and a rotating cylinder. Chaos 25 (8), 087402.
Mohammed-Taifour, A. & Weiss, J.2016 Unsteadiness in a large turbulent separation bubble 799 383–412.
Moore, F. K. 1958 On the separation of unsteady boundary layer. In Boundary-layer Research (ed. Görtler, H.), pp. 296311. Springer.
Na, Y. & Moin, P. 1998 Direct numerical simulation of a separated turbulent boundary layer. J. Fluid Mech. 374, 379405.
Prandtl, L. 1904 Über Flüssigkeitsbewegung bei sehr kleiner Reibung. Verh. III, Intern. Math. Kongr. Heidelberg 2, 484491.
Rott, N. 1956 Unsteady viscous flows in the vicinity of a separation point. Q. Appl. Maths 13, 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. 1956 Some recent developments in airfoil theory. J. Aero. Sci. 23, 490499.
Sears, W. R. & Telionis, D. P. 1975 Boundary-layer separation in unsteady flow. SIAM J. Appl. Maths 28, 215235.
Serra, M., Vétel, J. & Haller, G. 2018 Exact theory of material spike formation in flow separation. J. Fluid Mech. 845, 5192.
Shah, R. K. & Sekulić, D. P. 2003 Fundamentals of Heat Exchanger Design. John Wiley & Sons.
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.
Sun, M., Liu, Y. & Hu, Z. 2019 Turbulence decay in a supersonic boundary layer subjected to a transverse sonic jet. J. Fluid Mech. 867, 216249.
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., Jacobs, G. B., Grunberg, O. & Haller, G. 2008 An exact theory of three-dimensional fixed separation in unsteady flows. Phys. Fluids 20 (10), 107101.
Tianyun, G., Jianhan, L. & Mingbo, S. 2017 Symmetric/asymmetric separation transition in a supersonic combustor with single-side expansion. Phys. Fluids 29 (12), 126102.
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.
Weymouth, G. D. 2014 Chaotic rotation of a towed elliptical cylinder. J. Fluid Mech. 743, 385398.
Williams, J. C. 1977 Incompressible boundary-layer separation. Annu. Rev. Fluid Mech. 9 (1), 113144.
Wu, W. & Piomelli, U. 2018 Effects of surface roughness on a separating turbulent boundary layer. J. Fluid Mech. 841, 552580.
Yapalparvi, R. & van Dommelen, L. L. 2012 Numerical solution of unsteady boundary-layer separation in supersonic flow: upstream moving wall. J. Fluid Mech. 706, 413430.
Yuster, T. & Hackborn, W. W. 1997 On invariant manifolds attached to oscillating boundaries in Stokes flows. Chaos 7 (4), 769776.
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Material spike formation in highly unsteady separated flows

  • Mattia Serra (a1), Seán Crouzat (a2), Gaël Simon (a2), Jérôme Vétel (a2) and George Haller (a3)...


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