Eigenvalue problem for the Navier-Stokes operator in cylindrical coordinates

Abstract

A fast, accurate, and robust numerical algorithm is proposed, suitable for parametric studies of incompressible fluid flow in a pipe. The new algorithm (or its fragments) can have a wider applicability, including cases when the computational domain contains a coordinate singularity along the polar axis r = 0 and when the dependence on the azimuth angle can be represented as a Fourier series, due to the physical symmetry of the problem. The constructed method enables the efficient solution of the eigenvalue problem for the linearized Navier-Stokes operator in cylindrical coordinates. The algorithm is based on a new change in the dependent variables, which makes it possible to circumvent the difficulties associated with coordinate singularities by taking into account the special behavior of analytic functions in the vicinity of the point r = 0. Despite the presence of coordinate singularities, the new algorithm ensures the spectral accuracy. The numerical solution of the linear problem of hydrodynamic stability involves the spatial discretization of the Navier-Stokes operator, its linearization about the stationary solution, and the reduction to the canonical eigenvalue problem of the type λx = Tx. Eigenvalues λ can then be calculated by the QR algorithm. An original method is proposed here for the reduction of the eigenvalue problem to its canonical form, employing the influence matrix technique. This method is economical and is characterized by its low sensitivity to round-off errors.