Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T21:40:57.608Z Has data issue: false hasContentIssue false
  • English
  • Français

Viscid–inviscid pseudo-resonance in streamwise corner flow

Published online by Cambridge University Press:  04 March 2014

Oliver T. Schmidt  and
Ulrich Rist
Oliver T. Schmidt*
Affiliation:
Institut für Aerodynamik und Gasdynamik, Universität Stuttgart, Pfaffenwaldring 21, 70569 Stuttgart, Germany
Ulrich Rist
Affiliation:
Institut für Aerodynamik und Gasdynamik, Universität Stuttgart, Pfaffenwaldring 21, 70569 Stuttgart, Germany
*
†Email address for correspondence: o.schmidt@iag.uni-stuttgart.de
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of streamwise corner flow is investigated by means of direct numerical simulation at subcritical Reynolds numbers. The flow is harmonically forced, and global modes are extracted through a spectral decomposition. Spatial amplification in the near-corner region is observed even though the flow is shown to be subcritical in terms of spatial linear theory. This apparent discrepancy is resolved by extending the local analysis to include non-modal effects. It is demonstrated that the amplification is a result of the interaction between two coexistent spatial transient growth processes that can be associated with different parts of the linear stability spectrum. A detailed investigation of the underlying mechanisms shows that the transient amplification behaviour is caused by pseudo-resonance between the inviscid corner mode, and different sets of viscous modes. By comparison with studies of other locally inflectional flows, it is found that viscid–inviscid pseudo-resonance might be a general phenomenon leading to selective noise amplification.

JFM classification

Compressible Flows: Compressible boundary layers Instability: Absolute/convective instability Boundary Layers: Boundary layer stability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 743 , 25 March 2014 , pp. 327 - 357
Copyright
© 2014 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., Cherubini, S. & Robinet, J.-C. 2009 Sensitivity and optimal forcing response in separated boundary layer flows. Phys. Fluids 21 (6), 064108.CrossRefGoogle Scholar
Alizard, F., Robinet, J.-C. & Guiho, F. 2013 Transient growth in a right-angled streamwise corner. Eur. J. Mech. Fluids 37, 99–111.CrossRefGoogle Scholar
Alizard, F., Robinet, J.-C. & Rist, U. 2010 Sensitivity analysis of a streamwise corner flow. Phys. Fluids 22 (1), 014103.CrossRefGoogle Scholar
Babucke, A.2009 Direct numerical simulation of noise-generation mechanisms in the mixing layer of a jet. PhD thesis, Universität Stuttgart.Google Scholar
Babucke, A., Kloker, M. J. & Rist, U. 2008 DNS of a plane mixing layer for the investigation of sound generation mechanisms. Comput. Fluids 37 (4), 360–368.CrossRefGoogle Scholar
Bagheri, S. 2013 Koopman-mode decomposition of the cylinder wake. J. Fluid Mech. 726, 596–623.CrossRefGoogle Scholar
Balachandar, S. & Malik, M. R. 1995 Inviscid instability of streamwise corner flow. J. Fluid Mech. 282, 187–201.CrossRefGoogle Scholar
Barclay, W. H. 1973 Experimental investigation of the laminar flow along a straight 135-deg corner. Aeronaut. Q. 24, 147–154.CrossRefGoogle Scholar
Barclay, W. H. & El-Gamal, H. A. 1983 Streamwise corner flow with wall suction. AIAA J. 21, 31–37.CrossRefGoogle Scholar
Barclay, W. H. & El-Gamal, H. A. 1984 Further solutions in streamwise corner flow with wall suction. AIAA J. 22, 1169–1171.CrossRefGoogle Scholar
Barclay, W. H. & Ridha, A. H. 1980 Flow in streamwise corners of arbitrary angle. AIAA J. 18, 1413–1420.CrossRefGoogle Scholar
Carrier, G. 1947 The boundary layer in a corner. Q. Appl. Maths 4, 367–370.CrossRefGoogle Scholar
Chen, K. K., Tu, J. H. & Rowley, C. W. 2011 Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses. J. Nonlinear Sci. 22 (6), 887–915.CrossRefGoogle Scholar
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.CrossRefGoogle Scholar
Chu, B.-T. On the energy transfer to small disturbances in fluid flow (Part I). Acta Mechanica 1 (3), 215–234.CrossRefGoogle Scholar
Dhanak, M. R. 1992 Instability of flow in a streamwise corner. ICASE Report No. 92-70. Institute for Computer Applications in Science and Engineering, NASA Langley Research Center.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.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability. Cambridge University Press.CrossRefGoogle Scholar
Duck, P. W., Stow, S. R. & Dhanak, M. R. 1999 Non-similarity solutions to the corner boundary-layer equations (and the effects of wall transpiration). J. Fluid Mech. 400, 125–162.CrossRefGoogle Scholar
El-Gamal, H. A. & Barclay, W. H. 1978 Experiments on the laminar flow in a rectangular streamwise corner. Aeronaut. Q. 29, 75–97.CrossRefGoogle Scholar
Farrell, B. F. 1988 Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31, 2093–2102.CrossRefGoogle Scholar
Galionis, I. & Hall, P. 2005 Spatial stability of the incompressible corner flow. Theor. Comput. Fluid Dyn. 19, 77–113.CrossRefGoogle 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.CrossRefGoogle Scholar
Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14 (2), 222–224.CrossRefGoogle 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.CrossRefGoogle Scholar
Hanifi, A., Schmid, P. J. & Henningson, D. S. 1996 Transient growth in compressible boundary layer flow. Phys. Fluids 8, 826–837.CrossRefGoogle Scholar
Kloker, M. J. 1997 A robust high-resolution split-type compact FD scheme for spatial direct numerical simulation of boundary-layer transition. Appl. Sci. Res. 59 (4), 353–377.CrossRefGoogle 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.CrossRefGoogle 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
Lin, R. S., Wang, W. P. & Malik, M. R. 1996 Linear stability of incompressible viscous flow along a corner. In Boundary Layer and Free Shear Flows: Proceedings of the ASME Fluids Engineering Division, Summer Meeting vol. 237, pp. 633–638.Google Scholar
Mack, L. M. 1984 Boundary-Layer Linear Stability Theory. (AGARD Special Course on Stability and Transition of Laminar Flow) .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.CrossRefGoogle Scholar
Morkovin, M. V., Reshotko, E. & Herbert, T. 1994 Transition in open flow systems – a reassessment. Bull. Am. Phys. Soc. 39 (9), 1882.Google Scholar
Nomura, Y. 1962 Theoretical and experimental investigations on the incompressible viscous flow around the corner. Memo. no. 2, pp. 115–145. Defence Academy of Japan.Google Scholar
Pal, A. & Rubin, S. G. 1971 Asymptotic features of viscous flow along a corner. Q. Appl. Maths 29, 91–108.CrossRefGoogle Scholar
Parker, S. J. & Balachandar, S. 1999 Viscous and inviscid instabilities of flow along a streamwise corner. Theor. Comput. Fluid Dyn. 13, 231–270.CrossRefGoogle 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.CrossRefGoogle Scholar
Ridha, A. 1992 On the dual solutions associated with boundary-layer equations in a corner. J. Engng Maths 26, 525–537.CrossRefGoogle Scholar
Ridha, A. 2002 Combined free and forced convection in a corner. Intl J. Heat Mass Transfer 45 (10), 2191–2205.CrossRefGoogle Scholar
Rist, U. & Fasel, H. 1995 Direct numerical simulation of controlled transition in a flat-plate boundary layer. J. Fluid Mech. 298, 211–248.CrossRefGoogle Scholar
Rowley, C. W., Mezić, I., Bagheri, S., Schlatter, P. & Henningson, D. S. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 641 (1), 115–127.CrossRefGoogle Scholar
Rubin, S. G. 1966 Incompressible flow along a corner. J. Fluid Mech. 26 (1), 97–110.CrossRefGoogle 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.CrossRefGoogle Scholar
Rubin, S. G. & Tannehill, J. C. 1992 Parabolized/reduced Navier–Stokes computational techniques. Annu. Rev. Fluid Mech. 24, 117–144.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, P. J. 1994 Optimal energy density growth in Hagen–Poiseuille flow. J. Fluid Mech. 277, 197–225.CrossRefGoogle Scholar
Schmidt, O. T. & Rist, U. 2011 Linear stability of compressible flow in a streamwise corner. J. Fluid Mech. 688, 569–590.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.CrossRefGoogle 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.CrossRefGoogle Scholar
Trefethen, L. N. 1991 Pseudospectra of matrices. In Numerical Analysis (ed. Griffiths, D. F. & Watson, G. A.), pp. 234–266. Longman Scientific & Technical.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578–584.CrossRefGoogle ScholarPubMed
Tumin, A. & Reshotko, E. 2001 Spatial theory of optimal disturbances in boundary layers. Phys. Fluids 13 (7), 2097–2104.CrossRefGoogle Scholar
Weinberg, B. C. & Rubin, S. G. 1972 Compressible corner flow. J. Fluid Mech. 56, 753–774.CrossRefGoogle Scholar
Wright, T. G.2002 Eigtool. Available at: 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.CrossRefGoogle Scholar
Zamir, M. & Young, A. D. 1979 Pressure gradient and leading edge effects on the corner boundary layer. Aeronaut. Q. 30, 471–483.CrossRefGoogle Scholar