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.
Ying TAN, Weidong SU
. A trigonometric series expansion method for the Orr-Sommerfeld equation[J]. Applied Mathematics and Mechanics, 2019
, 40(6)
: 877
-888
.
DOI: 10.1007/s10483-019-2484-9
[1] THOMAS, L. H. The stability of plane Poiseuille flow. Physical Review, 91(4), 780-783(1953)
[2] DOLPH, C. L. and LEWIS, D. C. On the application of infinite systems of ordinary differential equations to perturbations of plane Poiseuille flow. Quarterly of Applied Mathematics, 16(2), 97-110(1958)
[3] GALLAGHER, A. P. and MERCER, A. M. On the behaviour of small disturbances in plane Couette flow. Journal of Fluid Mechanics, 13(1), 91-100(1962)
[4] GROSCH, C. E. and SALWEN, H. The stability of steady and time-dependent plane Poiseuille flow. Journal of Fluid Mechanics, 34(1), 177-205(1968)
[5] ORSZAG, S. A. Accurate solution of the Orr-Sommerfeld stability equation. Journal of Fluid Mechanics, 50(4), 689-703(1971)
[6] HERBERT, T. Analysis of the subharmonic route to transition in boundary layers. 22nd Aerospace Sciences Meeting, American Institute of Aeronautics and Astronautics, New York (1984)
[7] ZHOU, H. and ZHAO, G. F. Hydrodynamic Stability (in Chinese), National Defence Industrial Press, Beijing (2004)
[8] LANCZOS, C. Applied Analysis, Prentice Hall, Englewood Cliffs (1956)
[9] GAD-EL-HAK, M. The fluid mechanics of microdevices-the freeman scholar lecture. Journal of Fluids Engineering, 121(1), 5-33(1999)
[10] QIAN, T., WANG, X. P., and SHENG, P. A variational approach to the moving contact line hydrodynamics. Journal of Fluid Mechanics, 564, 333-360(2006)
[11] HE, Q. and WANG, X. P. The effect of the boundary slip on the stability of shear flow. Zeitschrift für Angewandte Mathematik und Mechanik, 88(9), 729-734(2008)
[12] SCHMID, P. J. and HENNINGSON, D. S. Stability and Transition in Shear Flows, Springer, New York (2001)
[13] MALIK, M. R. Numerical method for hypersonic boundary layer stability. Journal of Computational Physics, 86, 376-413(1990)
[14] ANTAR, B. N. The eigenvalue spectrum of the Orr-Sommerfeld problem. NASA STI/Recon Technical Report N, 77N11344, NASA, Washington, D. C. (1976)
[15] JONDINSON, R. The flat plate boundary layer, part 1, numerical integration of the OrrSommerfeld equation. Journal of Fluid Mechanics, 43(4), 801-811(1970)
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