Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-21T05:51:48.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Algebraic disturbance growth by interaction of Orr and lift-up mechanisms

Published online by Cambridge University Press:  14 September 2017

M. J. Philipp Hack Open the ORCID record for M. J. Philipp Hack [Opens in a new window]  and
Parviz Moin
M. J. Philipp Hack*
Affiliation:
Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA
Parviz Moin
Affiliation:
Center for Turbulence Research, Stanford University, Stanford, CA 94305, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Algebraic disturbance growth in spatially developing boundary-layer flows is investigated using an optimization approach. The methodology builds on the framework of the parabolized stability equations and avoids some of the limitations associated with adjoint-based schemes. In the Blasius boundary layer, non-parallel effects are shown to significantly enhance the energy gain due to algebraic growth mechanisms. In contrast to parallel flow, the most energetic perturbations have finite frequency and are generated by the simultaneous activity of the Orr and lift-up mechanisms. The highest amplification occurs in a limited region of the parameter space that is characterized by a linear relation between the wavenumber and frequency of the disturbances. The frequency of the most highly amplified perturbations decreases with Reynolds number. Adverse streamwise pressure gradient further enhances the amplification of disturbances while preserving the linear trend between the wavenumber and frequency of the most energetic perturbations.

JFM classification

Boundary Layers: Boundary Layers Boundary Layers: Boundary layer stability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 829 , 25 October 2017 , pp. 112 - 126
Check for updates
Copyright
© 2017 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

Åkervik, 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, 501–513.Google Scholar
Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11 (1), 134–150.Google Scholar
Bertolotti, F. P., Herbert, T. & Spalart, P. R. 1992 Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech. 242, 441–474.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4 (8), 1637–1650.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
Farrell, B. F. 1987 Developing disturbances in shear. J. Atmos. Sci. 44 (16), 2191–2199.Google Scholar
Hack, M. J. P. & Zaki, T. A. 2014 Streak instabilities in boundary layers beneath free-stream turbulence. J. Fluid Mech. 741, 280–315.Google Scholar
Hack, M. J. P. & Zaki, T. A. 2015 Modal and nonmodal stability of boundary layers forced by spanwise wall oscillations. J. Fluid Mech. 778, 389–427.Google Scholar
Hack, M. J. P. & Zaki, T. A. 2016 Data-enabled prediction of streak breakdown in pressure-gradient boundary layers. J. Fluid Mech. 801, 43–64.Google Scholar
Haj-Hariri, H. 1994 Characteristics analysis of the parabolized stability equations. Stud. Appl. Maths 92, 41–53.Google Scholar
Herbert, T. 1994 Parabolized stability equations. In Special Course in Transition Modelling. AGARD Rep. 793, pp. 4(1)–4(34).; https://www.sto.nato.int/publications/AGARD/ AGARD-R-793/AGARDR793.pdf.Google Scholar
Herbert, T. 1997 Parabolized stability equations. Annu. Rev. Fluid Mech. 29, 245–283.Google Scholar
Jimenéz, J. 2013 How linear is wall-bounded turbulence? Phys. Fluids 25, 110814.Google Scholar
Landahl, M. T. 1975 Wave breakdown and turbulence. SIAM J. Appl. Maths 28 (4), 735–756.Google Scholar
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.CrossRefGoogle Scholar
Li, F. & Malik, M. R. 1996 On the nature of the PSE approximation. Theor. Comput. Fluid Dyn. 8, 253–273.Google Scholar
Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289–309.Google Scholar
Monokrousos, A., Åkervik, 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
Orr, W. M. F. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part I: a perfect liquid. Part II: a viscous liquid. Proc. R. Irish Acad. 27, 9–138.Google Scholar
Reddy, S. & Henningson, D. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.Google Scholar
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1993 Pseudospectra of the Orr–Sommerfeld operator. SIAM J. Appl. Maths 53 (1), 15–47.Google Scholar
Schlichting, H. & Gersten, K. 2006 Boundary Layer Theory, 10th edn. Springer.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Tempelmann, D., Hanifi, A. & Henningson, D. S. 2010 Spatial optimal growth in three-dimensional boundary layers. J. Fluid Mech. 646, 5–37.Google Scholar
Towne, A.2016 Advancements in jet turbulence and noise modeling: accurate one-way solutions and empirical evaluation of the nonlinear forcing of wavepackets. PhD thesis, California Institute of Technology.Google Scholar
Trefethen, L. N., Trefethen, A. N., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578–584.Google Scholar
Zaki, T. A. & Durbin, P. A. 2006 Continuous mode transition and the effects of pressure gradients. J. Fluid Mech. 563, 357–358.Google Scholar