Abstract
We consider the linear stability of incompressible attachment-line flow within the spatial framework. No similarity or symmetry assumptions for the instability modes are introduced and the full two-dimensional representation of the modes is used. The perturbation equations are discretized on a two-dimensional staggered grid. A high order finite difference scheme has been developed which gives rise to a large, sparse, quadratic, eigenvalue problem for the instability modes. The benefits of the Jacobi–Davidson method for the solution of this eigenvalue system are demonstrated and the approach is validated in some detail. Spatial stability results are presented subsequently. In particular, instability predictions at very high Reynolds numbers are obtained which show almost equally strong instabilities for symmetric and antisymmetric modes in this regime.
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References
Juillen, J. and Arnal, D., Experimental study of boundary layer suction effects. In: Kobayashi, R. (ed.), Laminar Turbulent Transition. Springer-Verlag, Berlin (1995) pp. 174–179.
Fokkema, D. R., Sleijpen, G.L.G. and Van der Vorst, H.A., Jacobi-Davidson style QR and QZ algorithms for the partial reduction of matrix pencils. Report No. 941, University of Utrecht (1996).
Hall, P., Malik, M.R. and Poll, D.I.A., On the stability of an infinitely swept attachment line boundary layer. Proc. Roy. Soc. Lond. A 395 (1984) 229–345.
Hall, P. and Malik, M.R., On the instability of a three-dimensional attachment-line boundary layer: Weakly nonlinear theory and a numerical approach. J. Fluid Mech. 163 (1986) 257–282.
Heeg, R.S. and Geurts, B.J., Spatial stability of the compressible attachment-line boundary layer and generalized similarity properties. (1997) Available at: http://www.ub.utwente.nl/webdocs/tw/0000000f.pdf.
Joslin, R.D., Simulation of three-dimensional symmetric and asymmetric instabilities in attachment-line boundary layers. AIAA J. 34 (1995) 2432–2434.
Lin, R.S. and Malik, M.R., On the stability of attachment-line boundary layers. Part 1. The incompressible swept Hiemenz flow. J. Fluid Mech. 311 (1996) 239–255.
Van der Ploeg, A., Preconditioning techniques for non-symmetric matrices with application to temperature calculations of cooled concrete. Internat. J. Numer. Methods Engrg. 35 (1992) 1311–1328.
Poll, D.I.A., Transition in the infinite swept attachment line boundary layer. The Aeronautical Quarterly 30 (1979) 607–629
Sorensen, D.C., Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13 (1992) 357–385.
Sleijpen, G.L.G. and Fokkema, D.R., BiCGstab(l) for linear equations involving matrices with complex spectrum. Electronic Transactions on Numerical Analysis (ETNA) 1 (1993) 11–32.
Sleijpen, G.L.G., Booten, A.G.L., Fokkema, D.R. and Van der Vorst, H.A., Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT 36 (1996) 595–633.
Theofilis, V., The discrete temporal eigenvalue spectrum of the generalised Hiemenz flow as solution of the Orr-Sommerfeld equation. J. Engrg. Math. 28 (1994) 241–259.
Theofilis, V., Spatial stability of incompressible attachment-line flow Theoret. Comput. Fluid Dynamics 7 (1994) 159–171.
Theofilis, V., On the linear and nonlinear instability of the incompressible swept attachment-line boundary layer. Spatial stability of incompressible attachment-line flow J. Fluid Mech. 355 (1998) 193–227.
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Heeg, R.S., Geurts, B.J. Spatial Instabilities of the Incompressible Attachment-Line Flow Using Sparse Matrix Jacobi–Davidson Techniques. Flow, Turbulence and Combustion 59, 315–329 (1997). https://doi.org/10.1023/A:1001181628630
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DOI: https://doi.org/10.1023/A:1001181628630