Application of the reduced Navier–Stokes methodology to flow stability of Falkner–Skan class flows

Abstract

This investigation ascertains the ability of the reduced Navier–Stokes (RNS) methodology to model linear flow stability. This is accomplished through development and investigation of two reduced forms of the Orr–Sommerfeld equation and of a second order RNS direct numerical simulation (DNS). The stability of five Falkner–Skan flows ( β=1.0, 0.2, 0.0, −0.1, and −0.1988) is investigated for these modified forms of the Orr–Sommerfeld equation (OSE). Neutral stability curves are numerically generated and compared for three forms of the OSE, viz. full Navier–Stokes equations, two-dimensional thin-layer Navier–Stokes equations which exclude only axial diffusion, and two-dimensional reduced Navier–Stokes equations which exclude all axial diffusion, as well as all diffusion in the normal momentum equation. Effects of a deferred corrector to include these terms are also investigated. Results of the computations demonstrate that the reduced forms of the OSE are consistent with the full OSE. With confirmation that the reduced Navier–Stokes equations contain the information required to properly model flow stability, development of a new class of asymptotic theories, stability methods, and approaches to direct numerical simulations, based on the RNS methodology, becomes feasible. Results from full DNS calculations using the RNS equations demonstrate the proper characteristics for disturbance growth and decay of the velocity disturbances. Velocity disturbance profiles are also of the required shape and magnitude.