Abstract
This article deals with the linear dynamics of a transitional boundary layer subject to two-dimensional Tollmien–Schlichting instabilities. We consider a flat plate including the leading edge, characterized by a Reynolds number based on the length of the plate equal to Re = 6 × 105, inducing a displacement thickness-based Reynolds number of 1,332 at the end of the plate. The global linearized Navier–Stokes equations only display stable eigenvalues, and the associated eigen-vectors are known to poorly represent the dynamics of such open flows. Therefore, we resort to an input–output approach by considering the singular value decomposition of the global resolvent. We then obtain a series of singular values, an associated orthonormal basis representing the forcing (the so-called optimal forcings) as well as an orthonormal basis representing the response (the so-called optimal responses). The objective of this paper is to analyze these spatial structures and to closely relate their spatial downstream evolution to the Orr and Tollmien–Schlichting mechanisms. Analysis of the spatio-frequential diagrams shows that the optimal forcings and responses are respectively localized, for all frequencies, near the upstream neutral point (branch I) and the downstream neutral point (branch II). It is also shown that the spatial growth of the dominant optimal response favorably compares with the e N method in regions where the dominant optimal forcing is small. Moreover, thanks to an energetic input–output approach, it is shown that this spatial growth is solely due to intrinsic amplifying mechanisms related to the Orr and Tollmien–Schlichting mechanisms, while the spatial growth due to the externally supplied power by the dominant optimal forcing is negligible even in regions where the dominant optimal forcing is strong. The energetic input–output approach also yields a general method to assess the strength of the instability in amplifier flows. It is based on a ratio comparing two quantities of same physical dimension, the mean-fluctuating kinetic energy flux of the dominant optimal response across some boundary and the supplied mean external power by the dominant optimal forcing. For the present boundary-layer flow, we have computed this amplification parameter for each frequency and discussed the results with respect to the Orr and Tollmien–Schlichting mechanisms.
Similar content being viewed by others
References
Huerre P., Rossi M.: Hydrodynamic instabilities in open flows. In: Godrèche, C., Manneville, P. (eds) Hydrodynamics and Nonlinear Instabilities, pp. 81–294. Cambridge University Press, Cambridge, UK (1998)
Zebib A.: Stability of a viscous flow past a circular cylinder. J. Eng. Math. 21(2), 155–165 (1987)
Jackson C.P.: A finite-element study of the onset of vortex shedding in flow past variously-shaped bodies. J. Fluid Mech. 182, 23–45 (1987)
Provansal M., Mathis C., Boyer L.: Benard-Von Karman instability—transient and forced regimes. J. Fluid Mech. 182, 1–22 (1987)
Sipp D., Lebedev A.: 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 (2007). doi:10.1017/S0022112007008907
Chomaz J.M.: Global instabilities in spatially developing flows: non-normality and nonlinearity. Ann. Rev. Fluid Mech. 37, 357 (2005)
Marquet O., Lombardi M., Chomaz J.M., Sipp D., Jacquin L.: Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities. J. Fluid Mech. 622, 1–21 (2009)
Trefethen L.N., Trefethen A.E., Reddy S.C., Driscoll T.A.: Hydrodynamic stability without eigenvalues. Science 261(5121), 578–584 (1993)
Schmid P.J.: Nonmodal stability theory. Ann. Rev. Fluid Mech. 39, 129–162 (2007)
Marquet O., Sipp D., Chomaz J.M., Jacquin L.: Amplifier and resonator dynamics of a low-Reynolds-number recirculation bubble in a global framework. J. Fluid Mech. 605, 429 (2008)
Blackburn H.M., Barkley D., Sherwin S.J.: Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603, 271 (2008)
Cantwell C.D., Barkley D.: Computational study of subcritical response in flow past a circular cylinder. Phys. Rev. E 82(2, part 2), 026315 (2010)
Cantwell C.D., Barkley D., Blackburn H.M.: Transient growth analysis of flow through a sudden expansion in a circular pipe. Phys. Fluids 22(3), 034101 (2010)
Schmid P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010)
Farrell B., Ioannou P.: Optimal excitation of three-dimensional perturbations perturbations in viscous constant shear-flow. Phys. Fluids A 5(6), 2600 (1993)
Alizard F., Robinet J.C.: Spatially convective global modes in a boundary layer. Phys. Fluids 19(11), 114105 (2007)
Akervik E., Ehrenstein U., Gallaire F., Henningson D.S.: Global two-dimensional stability measures of the flat plate boundary-layer flow. Eur. J. Mech. B-Fluids 27(5), 501 (2008)
Alizard F., Cherubini S., Robinet J.C.: Sensitivity and optimal forcing response in separated boundary layer flows. Phys. Fluids 21(6), 064108 (2009)
Sipp D., Marquet O., Meliga O., Barbagallo A.: Dynamics and control of global instabilities in open flows: a linearized approach. Appl. Mech. Rev. 63, 030801 (2010)
Monokrousos A., Akervik E., Brandt L., Henningson D.: Global three-dimensional optimal disturbances in the blasius boundary-layer flow using time-steppers. J. Fluid Mech. 650, 181 (2010)
Ehrenstein U., Gallaire F.: On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer. J. Fluid Mech. 536, 209 (2005)
Farrell B.: Optimal excitation of perturbations in viscous shear-flow. Phys. Fluids 31(8), 2093–2102 (1988)
Lopez J., Marques F., Rubio A., Avila M.: Crossflow instability of finite bödewadt flows: transients and spiral waves. Phys. Fluids 21, 114107 (2009)
Do Y., Lopez J., Marques F.: Optimal harmonic response in a confined bödewadt boundary layer flow. Phys. Rev. E 82, 036301 (2010)
Bewley T.R.: Flow control: new challenges for a new renaissance. Prog. Aerosp. Sci. 37(1), 21–58 (2001)
Jiménez, J.: Localized amplification of energy in turbulent channels. Center for Turbulence Research, Annual Research Briefs (2009)
Lehoucq R.B., Sorensen D.C.: Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM J. Matrix Anal. Appl. 17(4), 789–821 (1996)
Amestoy P.R., Duff I.S., Koster J., L’Excellent J.Y.: A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23(1), 15–41 (2001)
Gaster M.: On the effects of boundary-layer growth on flow stability. J. Fluid Mech. 66, 465–480 (1974)
Drazin P., Reid W.: Hydrodynamic Stability. Cambridge University Press, Cambridge, UK (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by T. Colonius.
Rights and permissions
About this article
Cite this article
Sipp, D., Marquet, O. Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theor. Comput. Fluid Dyn. 27, 617–635 (2013). https://doi.org/10.1007/s00162-012-0265-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00162-012-0265-y