Spatial Analysis of Hydrodynamic Instabilities With Spectral‐Collocation Methods Using Chebyshev Polynomials

ABSTRACT An open and fully reproducible numerical procedure to solve the Rayleigh stability equation is presented in this work for the spatial analysis of hydrodynamic instabilities in an inviscid shear flow using the Chebyshev spectral‐collocation method. The resulting cubic polynomial eigenvalue problem is linearized via a companion matrix and solved efficiently, from which the most unstable spatial growth rate and its frequency are identified. Verification against the classical reference [1] at dimensionless temporal frequency demonstrates spectral (exponential) convergence of the dispersion relation ; in practice, a resolution of yields grid‐independent predictions. Across the unstable band, the maximum relative difference is for the real component of the wavenumber and for the dimensionless imaginary component of the complex streamwise wavenumber , with the peak growth rate and its frequency matching the reference. The computed real and imaginary eigenfunctions closely reproduce the reported mode shapes. An additional comparison with a shooting method implementation (method used by Michalke) indicated that the spectral formulation attains the same accuracy with superior conditioning and reduced parameter tuning. The proposed workflow thus offers a computationally practical and fully reproducible route to assess shear‐flow stability — complementary to large eddy and direct numeric simulation LES/DNS — while enabling straightforward verification against canonical benchmarks.