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Stability and sensitivity of a cross-flow-dominated Falkner–Skan–Cooke boundary layer with discrete surface roughness

Published online by Cambridge University Press:  15 August 2017

Mattias Brynjell-Rahkola Open the ORCID record for Mattias Brynjell-Rahkola [Opens in a new window] ,
Nima Shahriari ,
Philipp Schlatter Open the ORCID record for Philipp Schlatter [Opens in a new window] ,
Ardeshir Hanifi Open the ORCID record for Ardeshir Hanifi [Opens in a new window]  and
Dan S. Henningson
Mattias Brynjell-Rahkola*
Affiliation:
KTH Royal Institute of Technology, Linné FLOW Centre and Swedish e-Science Research Centre (SeRC), Department of Mechanics, SE-100 44 Stockholm, Sweden
Nima Shahriari
Affiliation:
KTH Royal Institute of Technology, Linné FLOW Centre and Swedish e-Science Research Centre (SeRC), Department of Mechanics, SE-100 44 Stockholm, Sweden
Philipp Schlatter
Affiliation:
KTH Royal Institute of Technology, Linné FLOW Centre and Swedish e-Science Research Centre (SeRC), Department of Mechanics, SE-100 44 Stockholm, Sweden
Ardeshir Hanifi
Affiliation:
KTH Royal Institute of Technology, Linné FLOW Centre and Swedish e-Science Research Centre (SeRC), Department of Mechanics, SE-100 44 Stockholm, Sweden
Dan S. Henningson
Affiliation:
KTH Royal Institute of Technology, Linné FLOW Centre and Swedish e-Science Research Centre (SeRC), Department of Mechanics, SE-100 44 Stockholm, Sweden
*
†Email address for correspondence: mattiasbr@mech.kth.se
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Abstract

With the motivation of determining the critical roughness size, a global stability and sensitivity analysis of a three-dimensional Falkner–Skan–Cooke (FSC) boundary layer with a cylindrical surface roughness is performed. The roughness size is chosen such that breakdown to turbulence is initiated by a global version of traditional secondary instabilities of the cross-flow (CF) vortices instead of an immediate flow tripping at the roughness. The resulting global eigenvalue spectra of the systems are found to be very sensitive to numerical parameters and domain size. This sensitivity to numerical parameters is quantified using the $\unicode[STIX]{x1D700}$-pseudospectrum, and the dependency on the domain is analysed through an impulse response, structural sensitivity analysis and an energy budget. It is shown that while the frequencies remain relatively unchanged, the growth rates increase with domain size, which originates from the inclusion of stronger CF vortices in the baseflow. This is reflected in a change in the rate of advective energy transport by the baseflow. It is concluded that the onset of global instability in a FSC boundary layer as the roughness height is increased does not correspond to an immediate flow tripping behind the roughness, but occurs for lower roughness heights if sufficiently long domains are considered. However, the great sensitivity results in an inability to accurately pinpoint the exact parameter values for the bifurcation, and the large spatial growth of the disturbances in the long domains eventually becomes larger than can be resolved using finite-precision arithmetic.

JFM classification

Instability: Absolute/convective instability Boundary Layers: Boundary layer stability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 826 , 10 September 2017 , pp. 830 - 850
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Copyright
© 2017 Cambridge University Press 

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References

Åkervik, E., Brandt, L., Henningson, D. S., Hœpffner, J., Marxen, O. & Schlatter, P. 2006 Steady solutions of the Navier–Stokes equations by selective frequency damping. Phys. Fluids 18 (6), 068102.Google Scholar
Alizard, F. & Robinet, J.-C. 2007 Spatially convective global modes in a boundary layer. Phys. Fluids 19 (11), 114105.CrossRefGoogle Scholar
Bagheri, S, Åkervik, E., Brandt, L. & Henningson, D. S. 2009 Matrix-free methods for the stability and control of boundary layers. AIAA J. 47 (5), 1057–1068.Google Scholar
Bech, K. H., Henningson, D. S. & Henkes, R. A. W. M. 1998 Linear and nonlinear development of localized disturbances in zero and adverse pressure gradient boundary-layers. Phys. Fluids 10 (6), 1405–1418.CrossRefGoogle Scholar
Brandt, L., Cossu, C., Chomaz, J.-M., Huerre, P. & Henningson, D. S. 2003 On the convectively unstable nature of optimal streaks in boundary layers. J. Fluid Mech. 485, 221–242.CrossRefGoogle Scholar
Canton, J., Schlatter, P. & Örlü, R. 2016 Modal instability of the flow in a toroidal pipe. J. Fluid Mech. 792, 894–909.CrossRefGoogle Scholar
Cerqueira, S. & Sipp, D. 2014 Eigenvalue sensitivity, singular values and discrete frequency selection mechanism in noise amplifiers: the case of flow induced by radial wall injection. J. Fluid Mech. 757, 770–799.Google Scholar
Chevalier, M., Schlatter, P., Lundbladh, A. & Henningson, D. S.2007 SIMSON – a pseudo-spectral solver for incompressible boundary layer flows. Tech. Rep. TRITA-MEK 2007:07. Department of Mechanics, Royal Institute of Technology, KTH.Google Scholar
Citro, V., Giannetti, F., Luchini, P. & Auteri, F. 2015 Global stability and sensitivity analysis of boundary-layer flows past a hemispherical roughness element. Phys. Fluids 27 (8), 084110.Google Scholar
Cooke, J. C. 1950 The boundary layer of a class of infinite yawed cylinders. Math. Proc. Camb. Phil. Soc. 46 (04), 645–648.Google Scholar
Deville, O., Fischer, P. F. & Mund, E. H. 2002 High-Order Methods for Incompressible Fluid Flow. Cambridge University Press.Google Scholar
von Doenhoff, A. E. & Braslow, A. L. 1961 The effect of distributed surface roughness on laminar flow. In Boundary Layer and Flow Control: its Principles and Application (ed. Lachmann, G. V.), pp. 657–681. Pergamon.CrossRefGoogle Scholar
Ehrenstein, U. & Gallaire, F. 2005 On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer. J. Fluid Mech. 536, 209–218.CrossRefGoogle Scholar
Falkner, V. M. & Skan, S. W. 1931 Some approximate solutions of the boundary layer equations. Phil. Mag. 12 (80), 865–896.Google Scholar
Fischer, P. F., Lottes, J. W. & Kerkemeier, S. G.2008 Nek5000. Available at: http://nek5000.mcs.anl.gov.Google Scholar
Garnaud, X., Lesshafft, L., Schmid, P. J. & Huerre, P. 2013 Modal and transient dynamics of jet flows. Phys. Fluids 25 (4), 044103.Google Scholar
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.Google Scholar
Gregory, N., Stuart, J. T. & Walker, W. S. 1955 On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk. Phil. Trans. R. Soc. Lond. A 248 (943), 155–199.Google Scholar
Hill, D. C. 1995 Adjoint systems and their role in the receptivity problem for boundary layers. J. Fluid Mech. 292, 183–204.Google Scholar
Högberg, M. & Henningson, D. 1998 Secondary instability of cross-flow vortices in Falkner–Skan–Cooke boundary layers. J. Fluid Mech. 368, 339–357.Google Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.Google Scholar
Kurz, H. B. E. & Kloker, M. J. 2016 Mechanisms of flow tripping by discrete roughness elements in a swept-wing boundary layer. J. Fluid Mech. 796, 158–194.Google Scholar
Lehoucq, R. B., Sorensen, D. C. & Yang, C. 1997 ARPACK User’s Guide: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM.Google Scholar
Lesshafft, L.2017 Artificial eigenmodes in truncated flow domains. arXiv:1704.08450v1.Google Scholar
Loiseau, J.-C., Robinet, J.-C., Cherubini, S. & Leriche, E. 2014 Investigation of the roughness-induced transition: global stability analyses and direct numerical simulations. J. Fluid Mech. 760, 175–211.Google Scholar
Mack, C. J., Schmid, P. J. & Sesterhenn, J. L. 2008 Global stability of swept flow around a parabolic body: connecting attachment-line and crossflow modes. J. Fluid Mech. 611, 205–214.CrossRefGoogle Scholar
Malik, M. R., Li, F., Choudhari, M. M. & Chang, C.-L. 1999 Secondary instability of crossflow vortices and swept-wing boundary-layer transition. J. Fluid Mech. 399, 85–115.Google Scholar
Peplinski, A., Schlatter, P., Fischer, P. F. & Henningson, D. S. 2014 Stability tools for the spectral-element code Nek5000; application to jet-in-crossflow. In ICOSAHOM’12: International Conference on Spectral and High Order Methods for Partial Differential Equations (ed. Azaïez, M., El Fekih, H. & Hesthaven, J. S.), pp. 349–359. Springer.Google Scholar
Peplinski, A., Schlatter, P. & Henningson, D. S. 2015 Global stability and optimal perturbation for a jet in cross-flow. Eur. J. Mech. (B/Fluids) 49, 438–447.CrossRefGoogle Scholar
Saric, W. S., Reed, H. L. & White, E. B. 2003 Stability and transition of three-dimensional boundary layers. Annu. Rev. Fluid Mech. 35, 413–440.Google Scholar
Schlichting, H. 1979 Boundary-Layer Theory, 7th edn. McGraw-Hill.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows, Applied Mathematical Sciences, vol. 142. Springer.Google Scholar
Toh, K. C. & Trefethen, L. N. 1996 Calculation of pseudospectra by the Arnoldi iteration. SIAM J. Sci. Stat. Comput. 17 (1), 1–15.Google Scholar
Trefethen, L. N. 1992 Pseudospectra of matrices. Numer. Anal. 91, 234–266.Google Scholar
Wassermann, P. & Kloker, M. 2002 Mechanisms and passive control of crossflow-vortex-induced transition in a three-dimensional boundary layer. J. Fluid Mech. 456, 49–84.CrossRefGoogle Scholar