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Galerkin-Laguerre Spectral Solution of Self-Similar Boundary Layer Problems

Published online by Cambridge University Press:  20 August 2015

F. Auteri  and
L. Quartapelle
F. Auteri*
Affiliation:
Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale, Via La Masa 34, 20156 Milano, Italy
L. Quartapelle
Affiliation:
Politecnico di Milano, Dipartimento di Ingegneria Aerospaziale, Via La Masa 34, 20156 Milano, Italy
*
Corresponding author.Email address:auteri@aero.polimi.it
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Abstract

In this work the Laguerre basis for the biharmonic equation introduced by Jie Shen is employed in the spectral solution of self-similar problems of the boundary layer theory. An original Petrov-Galerkin formulation of the Falkner-Skan equation is presented which is based on a judiciously chosen special basis function to capture the asymptotic behaviour of the unknown. A spectral method of remarkable simplicity is obtained for computing Falkner-Skan-Cooke boundary layer flows. The accuracy and efficiency of the Laguerre spectral approximation is illustrated by determining the linear stability of nonseparated and separated flows according to the Orr-Sommerfeld equation. The pentadiagonal matrices representing the derivative operators are explicitly provided in an Appendix to aid an immediate implementation of the spectral solution algorithms.

Keywords

34L1634L3065L6076E05Laguerre polynomialssemi-infinite intervalboundary layer theoryFalkner-Skan equationCooke equationOrr-Sommerfeld equationlinear stability of parallel flows
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 5 , November 2012 , pp. 1329 - 1358
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Asaithambi, A., A second order finite-difference method for the Falkner-Skan equation, Appl. Math. Comput., 156 (2004), 779–786.Google Scholar
[2]Asaithambi, A., Numerical solution of the Falkner-Skan equation using piecewise linear functions, Appl. Math. Comput., 159 (2004), 267–273.Google Scholar
[3]Asaithambi, A., Solution of the Falkner-Skan equations by recursive evaluation of Taylor coefficients, J. Comput. Appl. Math., 176 (2005), 203–214.Google Scholar
[4]Boyd, J. P., Orthogonal rational functions on a semi-infinite interval, J. Comput. Phys., 70 (1987), 63–88.CrossRefGoogle Scholar
[5]Cebeci, T. and Keller, H. B., Shooting and parallel shooting methods for solving the Falkner-Skan boundary-layer equation, J. Comput. Phys., 7 (1971), 289–300.CrossRefGoogle Scholar
[6]Cooke, J. C., The boundary layer of a class of infinite yawed cylinders, Math. Proc. Cambridge Philos. Soc., 46 (1950), 645–648.Google Scholar
[7]Dobrinsky, A., Adjoint Analysis for Receptivity Prediction, Ph.D. Thesis, Rice University, Houston, 2002.Google Scholar
[8]Drazin, P. G. and Reid, W. H., Hydrodynamics Stability, Cambridge University Press, New York, 1981.Google Scholar
[9]Falkner, V. M. and Skan, S. W., Some approximate solutions of the boundary layer equations, Philos. Mag., 12 (1931), 865–896.Google Scholar
[10]Fazio, R., The Blasius problem formulated as a free boundary value problem, Acta Mech., 95 (1992), 1–7.Google Scholar
[11]Fischer, T. M., A Galerkin approximation for linear eigenvalue problems in two and three-dimensional boundary layer flows, Lect. Notes Math., 1431 (1990), 100–108.Google Scholar
[12]Hager, W. H., Blasius: a life in research and education, Exp. Fluids, 34 (2003), 566–571.Google Scholar
[13]Hartree, D. R., On an equation occurring in Falkner and Skan’s approximate treatment of the equations of the boundary layer, Proc. Cambridge Philos. Soc., 33 (1937), 223–239.Google Scholar
[14]Heeg, R. S., Dijkstra, D. and Zandbergen, P. J., The stability of Falkner-Skan flows with several inflection points, ZAMP, 50 (1999), 82–93.Google Scholar
[15]Mack, L. M., Boundary layer linear stability theory, in Special course on stability and transition of laminar flow, AGARD Report No. 709, 1984.Google Scholar
[16]Meyer, R. E., Introduction to Mathematical Fluid Dynamics, Wiley, New York, 1971.Google Scholar
[17]Parand, K., Shahini, M. and Dehghan, M., Solution of a laminar boundary layer via a numerical method, Commun. Nonlinear Sci. Numer. Simulat., 15 (2010), 360–367.CrossRefGoogle Scholar
[18]Parand, K. and Taghavi, A., Rational scaled generalized Laguerre function collocation method for solving the Blasius equation, J. Comput. Appl. Math., 233 (2009), 980–989.Google Scholar
[19]Salama, A. A., Higher order method for solving free boundary problems, Numer. Heat Trans. B Fundament., 45 (2004), 385–394.CrossRefGoogle Scholar
[20]Schlichting, H. and Gersten, K., Boundary Layer Theory, Springer-Verlag, Berlin, 2000.Google Scholar
[21]Schmid, P. J. and Henningson, D. S., Stability and Transition in Shear Flows, Springer-Verlag, New York, 2001.Google Scholar
[22]Shen, J., Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM J. Numer. Anal., 38 (2000), 1113–1133.Google Scholar
[23]Shen, J. and Wang, L. L., Some recent advances on spectral methods for unbounded domains, Commun. Comput. Phys., 5 (2009), 195–241.Google Scholar
[24]Zhang, J. and Chen, B., An iterative method for solving the Falkner-Skan equation, Appl. Numer. Math., 210 (2009), 215–222.Google Scholar