Hostname: page-component-7dd5485656-2pp2p Total loading time: 0 Render date: 2025-10-31T22:58:15.603Z Has data issue: false hasContentIssue false
  • English
  • Français

Self-consistent model for the saturation mechanism of the response to harmonic forcing in the backward-facing step flow

Published online by Cambridge University Press:  22 March 2016

V. Mantič-Lugo Open the ORCID record for V. Mantič-Lugo [Opens in a new window]  and
F. Gallaire
V. Mantič-Lugo*
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, EPFL-STI-IGM-LFMI, CH-1015, Switzerland
F. Gallaire
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, EPFL-STI-IGM-LFMI, CH-1015, Switzerland
*
†Email address for correspondence: vladislav.manticlugo@epfl.ch
Get access Rights & Permissions [Opens in a new window]

Abstract

Certain flows denominated as amplifiers are characterized by their global linear stability while showing large linear amplifications to sustained perturbations. As the forcing amplitude increases, a strong saturation of the response appears when compared to the linear prediction. However, a predictive model that describes the saturation of the response to higher amplitudes of forcing in stable laminar flows is still missing. While an asymptotic analysis based on the weakly nonlinear theory shows qualitative agreement only for very small forcing amplitudes, the linear response to harmonic forcing around the mean flow computed by direct numerical simulations presents a good prediction of the saturation also at higher forcing amplitudes. These results suggest that the saturation process is governed by the Reynolds stress and thus motivate the introduction of a simple self-consistent model. The model consists of a decomposition of the full nonlinear Navier–Stokes equations in a mean flow equation together with a linear perturbation equation around the mean flow, which are coupled through the Reynolds stress. The full fluctuating response and the resulting Reynolds stress are approximated by the first harmonic calculated from the linear response to the forcing around the aforementioned mean flow. This closed set of coupled equations is solved in an iterative manner as partial nonlinearity is still preserved in the mean flow equation despite the assumed simplifications. The results show an accurate prediction of the response energy when compared to direct numerical simulations. The approximated coupling is strong enough to retain the main nonlinear effects of the saturation process. Hence, a simple physical picture is formalized, wherein the response modifies the mean flow through the Reynolds stress in such a way that the correct response energy is attained.

JFM classification

Instability: Nonlinear instability Wakes/Jets: Separated flows Wakes/Jets: Shear layers

Information

Type
Papers
Information
Journal of Fluid Mechanics , Volume 793 , 25 April 2016 , pp. 777 - 797
Check for updates
Copyright
© 2016 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.)

Article purchase

Temporarily unavailable

References

Akervik, E., Ehrenstein, U., Gallaire, F. & Henningson, D. S. 2008 Global two-dimensional stability measures of the flat plate boundary-layer flow. Eur. J. Mech. (B/Fluids) 27 (5), 501–513.CrossRefGoogle Scholar
Alizard, F., Cherubini, S. & Robinet, J.-C. 2009 Sensitivity and optimal forcing response in separated boundary layer flows. Phys. Fluids 21 (6), 064108.Google Scholar
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75, 750–756.CrossRefGoogle Scholar
Barkley, D., Gomes, M. G. M. & Henderson, R. D. 2002 Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech. 473, 167–190.CrossRefGoogle Scholar
Beaume, C., Chini, G. P., Julien, K. & Knobloch, E. 2015 Reduced description of exact coherent states in parallel shear flows. Phys. Rev. E 91, 043010.CrossRefGoogle ScholarPubMed
Benitez, M. & Bermudez, A. 2011 A second order characteristics finite element scheme for natural convection problems. J. Comput. Appl. Maths 235, 3270–3284.Google Scholar
Blackburn, H. M., Barkley, D. & Sherwin, S. J. 2008 Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603, 271–304.Google Scholar
Boujo, E. & Gallaire, F. 2015 Sensitivity and open-loop control of stochastic response in a noise amplifier flow: the backward-facing step. J. Fluid Mech. 762, 361–392.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.CrossRefGoogle Scholar
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.Google Scholar
Corbett, P. & Bottaro, A. 2000 Optimal perturbations for boundary layers subject to stream-wise pressure gradient. Phys. Fluids 12 (1), 120–130.Google Scholar
Dergham, G., Sipp, D. & Robinet, J.-Ch. 2013 Stochastic dynamics and model reduction of amplifier flows: the backward facing step flow. J. Fluid Mech. 719, 406–430.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn. Cambridge University Press.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1993 Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids A 5, 2600–2609.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1996 Generalized stability theory. Part 1. Autonomous operators. J. Atmos. Sci. 53, 2025–2040.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2003 Structural stability of turbulent jets. J. Atmos. Sci. 60 (17), 2101–2118.Google Scholar
Farrell, B. F. & Ioannou, P. J. 2012 Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow. J. Fluid Mech. 708, 149–196.Google Scholar
Garnaud, X., Lesshafft, L., Schmid, P. J. & Huerre, P. 2013 The preferred mode of incompressible jets: linear frequency response analysis. J. Fluid Mech. 716, 189–202.Google Scholar
Gayme, D. F., McKEON, B. J., Papachristodoulou, A., Bamieh, B. & Doyle, J. C. 2010 A streamwise constant model of turbulence in plane Couette flow. J. Fluid Mech. 665, 99–119.Google Scholar
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.Google Scholar
Lanzerstorfer, D. & Kuhlmann, H. C. 2012 Three-dimensional instability of the flow over a forward-facing step. J. Fluid Mech. 695, 390–404.CrossRefGoogle Scholar
Lu, L. & Papadakis, G. 2011 Investigation of the effect of external periodic flow pulsation on a cylinder wake using linear stability analysis. Phys. Fluids 23 (9), 094105.Google Scholar
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2014 Self-consistent mean flow description of the nonlinear saturation of the vortex shedding in the cylinder wake. Phys. Rev. Lett. 113, 084501.CrossRefGoogle ScholarPubMed
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2015 A self-consistent model for the saturation dynamics of the vortex shedding around the mean flow in the unstable cylinder wake. Phys. Fluids 27, 074103.Google Scholar
Marquet, O. & Sipp, D. 2010 Global Sustained Perturbations in a Backward-Facing Step Flow. (IUTAM Bookseries) , vol. 18. Springer.Google Scholar
Marquet, O., Sipp, D. & Lesshafft, L.2010 Global stability analysis of open shear flows without global modes, unpublished.Google Scholar
Mittal, S. 2008 Global linear stability analysis of time-averaged flows. Intl J. Numer. Meth. Fluids 58, 111–118.Google Scholar
Monokrousos, A., Akervik, E., Brandt, L. & Henningson, D. S. 2010 Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers. J. Fluid Mech. 650, 181–214.Google Scholar
Noiray, N., Durox, D., Schuller, T. & Candel, S. 2008 A unified framework for nonlinear combustion instability analysis based on the flame describing function. J. Fluid Mech. 615, 139–167.CrossRefGoogle Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358.Google Scholar
Sipp, D. & Marquet, O. 2012 Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theor. Comput. Fluid Dyn. 27, 1–19.Google Scholar
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic stability. J. Fluid Mech. 4, 1–21.Google Scholar
Stuart, J. T. 1960 On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. The basic behaviour in plane Poiseuille flow. J. Fluid Mech. 9, 353–370.Google Scholar
Thomas, V. L., Lieu, B. K., Jovanović, M. R., Farrell, B. F., Ioannou, P. J. & Gayme, D. F. 2014 Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow. Phys. Fluids 26, 105112.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.Google Scholar
Turton, S. E., Tuckerman, L. S. & Barkley, D. 2015 Prediction of frequencies in thermosolutal convection from mean flows. Phys. Rev. E 91, 043009.Google Scholar
Zielinska, B., Goujon-Durand, S., Dusek, J. & Wesfreid, J. 1997 Strongly nonlinear effect in unstable wakes. Phys. Rev. Lett. 79, 3893–3896.CrossRefGoogle Scholar