Numerical solution of the Orr-Sommerfeld equation using the viscous Green function and split-Gaussian quadrature

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:224

Article

McBain, G. D.¹  Chubb, T. H.¹  Armfield, S. W.¹ 

[1] Univ Sydney, Sch Aerosp Mech & Mechatron Engn, Darlington, NSW 2006, Australia

关键词: Two-point boundary value problem;    Eigenvalue problem;    Orr-Sommerfeld equation;    Hydrodynamic stability;    Green function;    Gaussian quadrature;    Product integration;   

DOI  :  10.1016/j.cam.2008.05.040

来源: Elsevier

【 摘 要 】

We continue our study of the construction of numerical methods for solving two-point boundary value problems using Green functions, building oil the Successful use of split-Gauss-type quadrature schemes. Here we adapt the method for eigenvalue problems, in particular the Orr-Sommerfeld equation of hydrodynamic stability theory. Use of the Green function for the Viscous part of the problem reduces the fourth-order ordinary differential equation to an integro-differential equation which we then discretize using the split-Gaussian quadrature and product integration approach of our earlier work along with pseudospectral differentiation matrices for the remaining differential operators. As the latter are only second-order the resulting discrete equations are much more stable than those obtained from the original differential equation. This permits us to obtain results for the standard test problem (plane Poiseuille flow at unit streamwise wavenumber and Reynolds number 10000) that we believe are the most accurate to date. (c) 2008 Elsevier B.V. All rights reserved.

【 授权许可】

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