Series Representation of the Eigenvalues of the OrrSommerfeld Equation
Abstract
A series representation of the relation that links the eigenvalues of the OrrSommerfeld equation is developed. This enables the complex frequency parameter to be expressed as a double series in terms of the Reynolds number and wavenumber, both of which are treated as complex variables. The complex coefficients arising in this series are determined by contour integration for the case of the eigenfunctions for a Blasius boundary layer profile. A nonlinear transformation is applied to the partial summations formed from the series in order to improve the convergence, and so to enable predictions of high accuracy to be made from only a few terms. Eigenvalues calculated by this technique are compared with those obtained directly from the OrrSommerfeld equation. The power of the technique is demonstrated by various graphical displays of the amplification contours for both temporal and spatial modes.
 Publication:

Journal of Computational Physics
 Pub Date:
 November 1978
 DOI:
 10.1016/00219991(78)901481
 Bibcode:
 1978JCoPh..29..147G
 Keywords:

 Complex Variables;
 Convergence;
 Eigenvalues;
 Laminar Boundary Layer;
 OrrSommerfeld Equations;
 Power Series;
 Reynolds Number;
 Blasius Flow;
 Boundary Layer Flow;
 Contours;
 Numerical Integration;
 Fluid Mechanics and Heat Transfer