Green Function and Integral Equation Method for the Orr-Sommerfeld Equation

Li Jia-chun

Volume 3 Issue 5

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Article Navigation > Applied Mathematics and Mechanics > 1982 > 3(5): 597-604

Li Jia-chun. Green Function and Integral Equation Method for the Orr-Sommerfeld Equation[J]. Applied Mathematics and Mechanics, 1982, 3(5): 597-604.

Citation:

Li Jia-chun. Green Function and Integral Equation Method for the Orr-Sommerfeld Equation[J]. Applied Mathematics and Mechanics, 1982, 3(5): 597-604.

Citation:

Li Jia-chun. Green Function and Integral Equation Method for the Orr-Sommerfeld Equation[J]. Applied Mathematics and Mechanics, 1982, 3(5): 597-604.

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Green Function and Integral Equation Method for the Orr-Sommerfeld Equation

Li Jia-chun

Institute of Mechanics, Academia Sinica, Beijing

Received Date: 1981-11-02
Publish Date: 1982-10-15

Abstract

Abstract

In this paper, an alternative method has been presented to solve the Orr-Sommerfeld equation in the linear theory of stability. To begin with, we define a Green function which is expressed in terms of matrix. Subsequently, its reciprocity has been shown. Finally, a linear integral equation equivalent to the original Orr-Sommerfeld equation is derived. The method is applied to the cases with two solid walls and any velocity distribution of the main flow at any Reynolds number.

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References(8)

References

[1]	Lin, C. C., The Theorg of Hydrodynamic Stability, Cam. Uni. Press, (1955).
[2]	Lakin, W. D., Ng, B. S. and Reid, W. H., Approximation to the eigenvalue relation for the Orr-Sommerfeld problem. Phi. Trans. Roy. Soc. Lon. 289,No. 1360, (1978).
[3]	李家春,流体运动稳定性和渐近方法,力学与实践,3,4. (1981).
[4]	Thomas, L. H., The stability of plane Poiseuille flow. Phys. Rev. 91,(1953), 780.
[5]	Dolph, C. L. and Lewis, D. C., On the application of finite systems of ordinary differential equations to perturbations of plane Poiseuille flow,Quar. Appl. Math. 16, (1958),97.
[6]	Gallagher, A. P. and Mercer, A. M., On the behaviour of small disturtances in plane Couette flow. Jour. Fluid Mech. 13, (1962),91.
[7]	Grosch, C. E. and Salwen, H., The stability of steadg and time-dependent plane Poiseuille flow. Jour. Fluid Mech. 34, (1968), 177.
[8]	Orszag, S. A., Accurate solution of the Orr-Sommerfeld stability equation,Jour. Fluid Mech. 50, (1971), 689.

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