A solution of the parabolized Navier–Stokes stability model in discrete space by two-directional differential quadrature and application to swirl intense flows

In this paper we propose a two-directional differential quadrature discretization of the parabolized Navier–Stokes differential system, which models the hydrodynamic instability of non-parallel swirling flows. A specific operational differentiation matrix was derived based on the properties of the orthogonal base. Numerical experiments comparing the new approach with the standard collocation methods were also carried out for two models: the trailing vortex model and a laboratory controlled swirl intense flow model. Numerical simulations are included that show good robustness properties of the proposed method. The method proposed in this paper offers two major advantages: affords an extended analysis of complex swirling flow that captures the streamwise development and the upstream extent of the instability and provides a reduced order model of the Tollmien–Schlichting modes used for the global stability analysis. The computational storage is significantly decreased compared with the related application of linear radial collocation approach.