Ultraspherical Spectral Method for Incompressible Boundary Layer Stability Calculations

Linear stability theory constitutes an important approach to predicting the amplification of disturbances in fluid boundary layers and eventual transition to turbulence. In this work, we implement the ultraspherical polynomial spectral method (i.e. US method) developed by Olver & Townsend (siam REVIEW 55, no. 3 (2013), pp. 462-489) to the eigenvalue problems associated with incompressible laminar boundary layer stability. The US method is a spectral coefficient method developed for linear, higher-order differential equations involving the representation of derivatives using ultraspherical (or Gegenbauer) polynomials. The method is favourable as it results in sparse, almost-banded, well-conditioned matrices. First, the feasibility of the numerical scheme is demonstrated by solving for the temporal eigenvalue spectrum of the Orr-Sommerfeld equation for plane channel (Poiseuille) flow at Re=10,000 (classic Orszag (J. Fluid Mech., 50.4 (1971), pp. 689-703) test case). The ease of implementation, accuracy of solution, and computational complexity of the US spectral method are compared against existing schemes for this problem including historic asymptotic formulae for the eigenvalues, finite difference, and Chebyshev collocation and coefficient based methods. The US method is then extended to capture the frequencies and wavelengths of boundary layer disturbances leading to transition, for Blasius and other self-similar incompressible boundary layer profiles. Finally, we highlight the favourable aspects of extending the present numerical method to compressible, and more complex boundary layer flows of interest.