Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-19T23:25:49.041Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal wavepackets in streamwise corner flow

Published online by Cambridge University Press:  04 February 2015

Oliver T. Schmidt ,
Seyed M. Hosseini ,
Ulrich Rist ,
Ardeshir Hanifi  and
Dan S. Henningson
Oliver T. Schmidt*
Affiliation:
Institut für Aerodynamik und Gasdynamik, Universität Stuttgart, D-70569 Stuttgart, Germany
Seyed M. Hosseini
Affiliation:
Linné Flow Center and Swedish e-Science Research Centre (SeRC), KTH Mechanics, SE-100 44 Stockholm, Sweden
Ulrich Rist
Affiliation:
Institut für Aerodynamik und Gasdynamik, Universität Stuttgart, D-70569 Stuttgart, Germany
Ardeshir Hanifi
Affiliation:
Linné Flow Center and Swedish e-Science Research Centre (SeRC), KTH Mechanics, SE-100 44 Stockholm, Sweden Swedish Defence Research Agency, FOI, SE-164 90 Stockholm, Sweden
Dan S. Henningson
Affiliation:
Linné Flow Center and Swedish e-Science Research Centre (SeRC), KTH Mechanics, SE-100 44 Stockholm, Sweden
*
†Email address for correspondence: o.schmidt@iag.uni-stuttgart.de
Get access Rights & Permissions [Opens in a new window]

Abstract

The global non-modal stability of the flow in a right-angled streamwise corner is investigated. Spatially confined linear optimal initial conditions and responses are obtained by use of direct-adjoint looping. Two base states are considered, the classical self-similar solution for a zero streamwise pressure gradient, and a modified solution that mimics leading-edge effects commonly observed in experimental studies. The latter solution is obtained in a reverse engineering fashion from published measurement data. Prior to the global analysis, a classical local linear stability and sensitivity analysis of both base states is conducted. It is found that the base-flow modification drastically reduces the critical Reynolds number through an inviscid mechanism, the so-called corner mode. A survey of the geometry of the two base states confirms that the modification greatly aggravates the inflectional nature of the flow. Global optimals are calculated for subcritical and supercritical Reynolds numbers, and for two finite optimization times. The optimal initial conditions are found to be self-confined in the spanwise directions, and symmetric with respect to the corner bisector. They evolve into streaks or streamwise modulated wavepackets, depending on the base state. Substantial transient growth caused by the Orr mechanism and the lift-up effect is observed.

JFM classification

Instability: Absolute/convective instability Boundary Layers: Boundary layer stability Boundary Layers: Boundary Layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 766 , 10 March 2015 , pp. 405 - 435
Check for updates
Copyright
© 2015 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

Alizard, F., Rist, U. & Robinet, J.-C. 2009 Linear instability of streamwise corner flow. Adv. Turbulence 12, 67–70.Google Scholar
Alizard, F., Robinet, J.-C. & Guiho, F. 2012 Transient growth in a right-angled streamwise corner. Eur. J. Mech. (B/Fluids) 37, 99–111.Google Scholar
Alizard, F., Robinet, J.-C. & Rist, U. 2010 Sensitivity analysis of a streamwise corner flow. Phys. Fluids 22 (1), 014103.Google Scholar
Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11, 134.Google Scholar
Bagheri, S., Brandt, L. & Henningson, D. S. 2009 Input–output analysis, model reduction and control of the flat-plate boundary layer. J. Fluid Mech. 620, 263–298.Google Scholar
Balachandar, S. & Malik, M. R. 1995 Inviscid instability of streamwise corner flow. J. Fluid Mech. 282, 187–201.Google Scholar
Barclay, W. H. & Ridha, A. H. 1980 Flow in streamwise corners of arbitrary angle. AIAA J. 18, 1413–1420.Google Scholar
Biau, D., Soueid, H. & Bottaro, A. 2008 Transition to turbulence in duct flow. J. Fluid Mech. 596, 133–142.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637.CrossRefGoogle Scholar
Cherubini, S., De Palma, P., Robinet, J.-C. & Bottaro, A. 2010 Rapid path to transition via nonlinear localized optimal perturbations in a boundary-layer flow. Phys. Rev. E 82 (6), 066302.CrossRefGoogle Scholar
Corbett, P. & Bottaro, A. 2000 Optimal perturbations for boundary layers subject to stream-wise pressure gradient. Phys. Fluids 12, 120.Google Scholar
Dhanak, M. R.1992 Instability of flow in a streamwise corner. NASA Contractor Rep. 191410, ICASE Rep. No. 92-70.Google Scholar
Dhanak, M. R. 1993 On the instability of flow in a streamwise corner. Proc. R. Soc. Lond. A 441, 201–210.Google Scholar
Dhanak, M. R. & Duck, P. W. 1997 The effects of freestream pressure gradient on a corner boundary layer. Proc. R. Soc. Lond. A 453 (1964), 1793–1815.Google Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Edwards, W. S., Tuckerman, L. S., Friesner, R. A. & Sorensen, D. C. 1994 Krylov methods for the incompressible Navier–Stokes equations. J. Comput. Phys. 110, 82–102.CrossRefGoogle Scholar
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18, 487.Google Scholar
Farrell, B. F. 1988 Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31, 2093–2102.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1993 Optimal excitation of three-dimensional perturbations in viscous constant shear flow. Phys. Fluids A 5, 1390.Google Scholar
Fischer, P. F., Lottes, J. W. & Kerkemeier, S. G.2008 nek5000 web page.http://nek5000.mcs.anl.gov.Google Scholar
Galionis, I. & Hall, P. 2005 Spatial stability of the incompressible corner flow. Theor. Comput. Fluid Dyn. 19, 77–113.Google Scholar
Ghia, K. N. 1975 Incompressible streamwise flow along an unbounded corner. AIAA J. 13, 902–907.Google Scholar
Ghia, K. N. & Davis, R. T. 1974 A study of compressible potential and asymptotic viscous flows for corner region. AIAA J. 12, 355–359.Google Scholar
Karp, M. & Cohen, J. 2014 Tracking stages of transition in Couette flow analytically. J. Fluid Mech. 748, 896–931.Google Scholar
Kornilov, V. I. & Kharitonov, A. M. 1982 On the part played by the local pressure gradient in forming the flow in a corner. Fluid Dyn. 17, 242–246.Google Scholar
Lakin, W. D. & Hussaini, M. Y. 1984 Stability of the laminar boundary layer in a streamwise corner. Proc. R. Soc. Lond. A 393, 101–116.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 (02), 243–251.CrossRefGoogle Scholar
Lindzen, R. S. 1988 Instability of plane parallel shear flow (toward a mechanistic picture of how it works). Pure Appl. Geophys. 126 (1), 103–121.Google Scholar
Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404 (1), 289–309.Google Scholar
Mack, L. M.1984 Boundary-layer linear stability theory. AGARD Rep. No. 709.Google Scholar
MATLAB 2013 Version 8.1.0.604 (R2013a). The MathWorks Inc.Google Scholar
Mikhail, A. G. & Ghia, K. N. 1978 Viscous compressible flow in the boundary region of an axial corner. AIAA J. 16, 931–939.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
Monokrousos, A., Bottaro, A., Brandt, L., Di Vita, A. & Henningson, D. S. 2011 Nonequilibrium thermodynamics and the optimal path to turbulence in shear flows. Phys. Rev. Lett. 106 (13), 134502.CrossRefGoogle ScholarPubMed
Nomura, Y. 1962 Theoretical and experimental investigations on the incompressible viscous flow around the corner. Memo Defence Acad. Japan 2, 115–145.Google Scholar
Orr, W. M. F. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. In Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, vol. 27. JSTOR.Google Scholar
Pal, A. & Rubin, S. G. 1971 Asymptotic features of viscous flow along a corner. Q. Appl. Maths 29, 91–108.Google Scholar
Parker, S. J. & Balachandar, S. 1999 Viscous and inviscid instabilities of flow along a streamwise corner. Theor. Comput. Fluid Dyn. 13, 231–270.Google Scholar
Patera, A. T. 1984 A spectral element method for fluid dynamics: laminar flow in a channel expansion. J. Comput. Phys. 54 (3), 468–488.Google Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252 (1), 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
Ridha, A. 1992 On the dual solutions associated with boundary-layer equations in a corner. J. Engng Maths 26, 525–537.Google Scholar
Ridha, A. 2003 Flow along streamwise corners revisited. J. Fluid Mech. 476, 223–265.Google Scholar
Rubin, S. G. 1966 Incompressible flow along a corner. J. Fluid Mech. Digital Arch. 26 (01), 97–110.Google Scholar
Rubin, S. G. & Grossman, B. 1971 Viscous flow along a corner: numerical solution of the corner layer equations. Q. Appl. Maths 29, 169–186.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Schmidt, O. T. & Rist, U. 2011 Linear stability of compressible flow in a streamwise corner. J. Fluid Mech. 688, 569–590.Google Scholar
Schmidt, O. T. & Rist, U. 2014 Viscid–inviscid pseudo-resonance in streamwise corner flow. J. Fluid Mech. 743, 327–357.CrossRefGoogle Scholar
Sipp, D. & Marquet, O. 2013 Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theor. Comput. Fluid Dyn. 27 (5), 617–635.Google Scholar
Tannehill, J., Anderson, D. & Pletcher, R. 1997 Computational Fluid Mechanics and Heat Transfer. Taylor & Francis.Google Scholar
Theofilis, V. 2003 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 39, 249–315.Google Scholar
Trefethen, L. N. 1991 Pseudospectra of matrices. Numer. Anal. 91, 234–266.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578–584.Google Scholar
Weinberg, B. C. & Rubin, S. G. 1972 Compressible corner flow. J. Fluid Mech. 56, 753–774.Google Scholar
Wright, T. G.2002 Eigtool. http://www.comlab.ox.ac.uk/pseudospectra/eigtool/.Google Scholar
Zamir, M. 1981 Similarity and stability of the laminar boundary layer in a streamwise corner. Proc. R. Soc. Lond. A 377, 269–288.Google Scholar
Zamir, M. & Young, A. D. 1970 Experimental investigation of the boundary layer in a streamwise corner. Aeronaut. Q. 21, 313–339.Google Scholar