Applied Mathematics and Mechanics (English Edition) ›› 2019, Vol. 40 ›› Issue (6): 877-888.doi: https://doi.org/10.1007/s10483-019-2484-9

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A trigonometric series expansion method for the Orr-Sommerfeld equation

Ying TAN1,2, Weidong SU1,2   

  1. 1. State Key Laboratory for Turbulence and Complex Systems, Peking University, Beijing 100871, China;
    2. Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, China
  • Received:2018-03-07 Revised:2018-11-25 Online:2019-06-01 Published:2019-06-01
  • Contact: Weidong SU E-mail:swd@pku.edu.cn
  • Supported by:
    Project supported by the National Natural Science Foundation of China (Nos. 11221062, 11521091, and 91752203)

Abstract: A trigonometric series expansion method and two similar modified methods for the Orr-Sommerfeld equation are presented. These methods use the trigonometric series expansion with an auxiliary function added to the highest order derivative of the unknown function and generate the lower order derivatives through successive integrations. The proposed methods are easy to implement because of the simplicity of the chosen basis functions. By solving the plane Poiseuille flow (PPF), plane Couette flow (PCF), and Blasius boundary layer flow with several homogeneous boundary conditions, it is shown that these methods yield results with the same accuracy as that given by the conventional Chebyshev collocation method but with better robustness, and that obtained by the finite difference method but with fewer modal number.

Key words: generalized variational principle, dynamical analysis of structure, elastoplastic analysis, blast-resistant underground structure, hydrodynamical stability, spectral method, trigonometric function, parallel shear flow, eigenvalue problem, Orr-Sommerfeld equation

2010 MSC Number: 

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