Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-28T02:19:10.237Z Has data issue: false hasContentIssue false
  • English
  • Français

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:[email protected]
Get access

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

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

[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