Calculation of the Orr-Sommerfeld stability equation for the plane Poiseuille flow

Abstract

The stability of plane Poiseuille flow depends on eigenvalues and solutions which are generated by solving Orr-Sommerfeld equation with input parameters including real wavenumber and Reynolds number . In the reseach of this paper, the Orr-Sommerfeld equation for the plane Poiseuille flow was solved numerically by improving the Chebyshev collocation method so that the solution of the Orr-Sommerfeld equation could be approximated even and odd polynomial by relying on results of proposition 3.1 that is proved in detail in section 2. The results obtained by this method were more economical than the modified Chebyshev collocation if the comparison could be done in the same accuracy, the same collocation points to find the most unstable eigenvalue. Specifically, the present method needs 49 nodes and only takes 0.0011s to create eigenvalue while the modified Chebyshev collocation also uses 49 nodes but takes 0.0045s to generate eigenvalue with the same accuracy to eight digits after the decimal point in the comparison with , see [4], exact to eleven digits after the decimal point.