Numerical Analysis of Spatial Hydrodynamic Stability of Shear Flows in Ducts of Constant Cross Section

Abstract

A technique for analyzing the spatial stability of viscous incompressible shear flows in ducts of constant cross section, i.e., a technique for the numerical analysis of the stability of such flows with respect to small time-harmonic disturbances propagating downstream is described and justified. According to this technique, the linearized equations for the disturbance amplitudes are approximated in space in the plane of the duct cross section and are reduced to a system of first-order ordinary differential equations in the streamwise variable in a way independent of the approximation method. This system is further reduced to a lower dimension one satisfied only by physically significant solutions of the original system. Most of the computations are based on standard matrix algorithms. This technique makes it possible to efficiently compute various characteristics of spatial stability, including finding optimal disturbances that play a crucial role in the subcritical laminar–turbulent transition scenario. The performance of the technique is illustrated as applied to the Poiseuille flow in a duct of square cross section.