Laguerre collocation solutions to boundary layer type problems

Abstract

In this paper a Laguerre collocation type method based on usual Laguerre functions is designed in order to solve high order nonlinear boundary value problems as well as eigenvalue problems, on semi-infinite domain. The method is first applied to Falkner–Skan boundary value problem. The solution along with its first two derivatives are computed inside the boundary layer on a fine grid which cluster towards the fixed boundary. Then the method is used to solve a generalized eigenvalue problem which arise in the study of the stability of the Ekman boundary layer. The method provides reliable numerical approximations, is robust and easy implementable. It introduces the boundary condition at infinity without any truncation of the domain. A particular attention is payed to the treatment of boundary conditions at origin. The dependence of the set of solutions to Falkner–Skan problem on the parameter embedded in the system is reproduced correctly. For Ekman eigenvalue problem, the critical Reynolds number which assure the linear stability is computed and compared with existing results. The leftmost part of the spectrum is validated using QZ as well as some Jacobi–Davidson type methods.

Authors

Călin-Ioan Gheorghiu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Keywords

Collocation; usual Laguerre functions; semi-infinite domain; Falkner-Skan equation; Ekman eigenvalue problem; leftmost eigenpair.

References

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Cite this paper as

C.I. Gheorghiu, Laguerre collocation solutions to boundary layer type problems. Numer. Algor., 64 (2013) 385-401
doi: 10.1007/s11075-012-9670-y

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About this paper

Journal

Numerical Algorithms

Publisher Name

Springer

Print ISSN

1017-1398

Online ISSN

1572-9265

Google Scholar Profile

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References

References

1. Acheson, D.J.: Elementary Fluid Dynamics. Clarendon Press, Oxford (1992)
2. Allen, L., Bridges, T.J.: Hydrodynamic stability of the Ekman boundary layer including interaction with a compliant surface: a numerical framework. Eur. J. Mech. B Fluids 22, 239–258 (2003)
3. Ascher, U., Mattheij, R.M.M., Russell, R.D.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs (1988)
4. Bernardy, C., Maday, Y.: Spectral methods. In: Ciarlet, P., Lions, L. (eds.) Handbook of Numerical Analysis, V.5 (Part 2). North-Holland (1997)
5. Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd ed. Dover Publications, New York (2000)
6. Boyd, J.P., Rangan, C., Bucksbaum, P.H.: Pseudospectral methods on a semi-infinite interval with application to the hydrogen atom: a comparison of the mapped Fourier sine method with Laguerre series and rational Chebyshev expansions. J. Comput. Phys. 188, 56–74 (2003)
7. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New-York (1987)
8. Cebeci, T., Cousteix, J.: Modeling and Computation of Boundary Layer Flows. Horizons Publishing, Long Beach, CA and Springer, Heidelberg (1998)
9. Cebeci, T., Shao, J.P.: A non-iterative method for boundary-layer equations—Part II: two-dimensional laminar and turbulent flows. Int. J. Numer. Methods Fluids (2003).
10. Fang, T., Zhang, J.: An exact analytical solution of the Falkner–Skan equation with mass transfer and wall stretching. Int. J. Nonlinear Mech. 43, 1000–1006 (2008)
11. Fang, T., Yao, S., Zhang, J., Zhong, Y., Tao, H.: Momentum and heat transfer of the Falkner–Skan flow with algebraic decay: an analytical solution. Commun. Nonlinear Sci. 17, 2476–2488 (2012)
12. Finch, S.: Prandtl–Blasius Flow. http://algo.inria.fr/csolve/bla.pdf (2008). Accessed 12 June 2012
13. Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1998)
14. Funaro, D.: Polynomial Approximation for Differential Problems. Springer, Berlin (1992)
15. Gheorghiu, C.I.: SpectralMethods for Differential Problems. Casa Cartii de Stiinta Publishing House, Cluj-Napoca (2007)
16. Gheorghiu, C.I., Dragomirescu, I.F.: Spectral methods in linear stability. Application to thermal convection with variable gravity field. Appl. Numer. Math. (2009).
17. Greenberg, L., Marletta, M.: The Ekman flow and related problems: spectral theory and numerical analysis. Math. Proc. Camb. Philos. Soc. 136, 719–764 (2004)
18. Hochstenbach, M.E., Plestenjak, B.: Backward errors, condition numbers, and pseudospectra for the multiparameter eigenvalue problems. Linear Algebra Appl. 375, 63–81 (2003)
19. Hoepffner, J.: Implementation of boundary conditions. http://www.lmm.jussieu.fr/~hoepffner/boundarycondition.pdf (2010). Accessed 25 Aug 2011
20. Ioss, G., Bruun, H.: True, Bifurcation of the stationary Ekman flow into a stable periodic flow. Arch. Ration. Mech. Anal. 68, 227–256 (1978)
21. Keller, H.B.: Numerical Methods for Two-Point Boundary-Value Problems. Blaisdell, New York (1968)
22. Liao, S.-J.: A uniformly valid analytic solution of two dimensional viscous flow over a semi-infinite plat plate. J. Fluid Mech. 385, 101–128 (1999)
23. Lilly, D.K.: On the instability of Ekman boundary flow. J. Atmos. Sci. 23, 481–494 (1966)
24. Magyari, E.: Falkner–Skan flows past moving boundaries: an exactly solvable case. Acta Mech. 203, 13–21 (2009)
25. Melander, M.V.: An algorithmic approach to the linear stability of the Ekman layer. J. Fluid Mech. 132, 283–293 (1983)
26. Motsa, S.S., Sibanda, P.: An efficient numerical method for solving Falkner–Skan boundary layer flows. Int. J. Numer. Methods Fluids (2011). doi: 10.1002/fld.2570
27. van Noorden, T., Rommes, J.: Computing a partial generalized real Schur form using the Jacobi–Davidson method. Numer. Linear Algebra Appl. (2007). doi: 10.1002/nla.523
28. Ockendon, H., Ockendon, J.R.: Viscous Flow. Cambridge University Press, Cambridge (1995)
29. Parand, K., Dehghan, M., Pirkhedri, A.: Sinc-collocation methods for solving the Blasius equation. Phys. Lett. A 373, 1237–1244 (2009)
30. Parand, K., Dehghan, M., Pirkhedri, A.: The use of sinc-collocation method for solving Falkner–Skan boundary-layer equation. Int. J. Numer. Methods Fluids (2010). doi: 10.1002/fld.2493
31. Riley, N., Weidman, P.D.: Multiple solutions of the Falkner–Skan equation past a stretching boundary. SIAM J. Appl. Math. 49, 1350–1358 (1989)
32. Rosales-Vera, M., Valencia, A.: Solutions of Falkner–Skan equation with heat transfer by Fourier series. Int. Commun. Heat Mass Transf. 37, 761–765 (2010)
33. Rosenhead, L. (ed.): Laminar Boundary Layers. Clarendon Press, Oxford (1963). Paperback edition: Dover, New York (1988)
34. Sachdev, P.L., Kudenatti, R.B., Bujurke, N.M.: Exact analytic solution of a boundary value problem for the Falkner–Skan equation. Stud. Appl. Math. 120, 1–16 (2008)
35. Schlichting, H.: Boundary Layer Theory, 4th ed. McGraw-Hill, New York (1960)
36. Schmid, P.J., Henningson, D.S.: Stability and Transition in Shear Flows. Springer, New York (2001)
37. Shen, J.: Stable and efficient spectral methods in unbounded domains using Laguerre functions. SIAM J. Numer. Anal. 38, 1113–1133 (2000)
38. Shen, J., Wang, L.-L.: Some recent advances on spectral methods for unbounded domains. Commun. Comput. Phys. 5, 195–241 (2009)
39. Shen, J., Tang, T., Wang, L.-L.: Spectral Methods. Algorithms, Analysis and Applications. Springer, Berlin (2011)
41. Trefethen, L.N., Trefethen, A.E., Reddy, S.C., Discroll, T.A.: Hydrodynamic stability without eigenvalues. Science 261, 578–584 (1993)
42. Trefethen, L.N.: Computation of pseudospectra. Acta Numer. 8, 247–295 (1999)
43. Weideman, J.A.C., Reddy, S.C.: A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26, 465–519 (2000)

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