Numerical method for distribution of eigenvalues in laminar ripe flows

In order to understand in more detail the dynamic structure of the linearized Navier-Stokes equation, a distribution of eigenvalues and classification of modes have been investigated especially for Poiseuille pipe flow. Generally, the accuracy of the eigenvalue gradually diminishes as its order increases. It is therefore difficult to obtain a correct eigenvalue distribution including higher order ones. This report aims to develop an improved numerical method for determining a distribution of eigenvalues with sufficient accuracy in general pipe flows. The method of expansion in orthogonal functions due to H. Salwen is reformulated in a Hilbert space. Then a finite dimensional approximate system is derived by the Galerkin method. In the Galerkin approximation, it is difficult to determine the dimension of approximate subspace a priori. A criterion for this truncation problem is presented based on a property of operator invariance of the subspace. The numerical method is applied to Poiseuille pipe flow, and the result is compared with asymptotic and other numerical methods.