On the Reynolds stress and inviscid solution to the hydrodynamic stability problem

Abstract

The linear instability of inviscid, incompressible annular flows, a problem commonly known as the Circular Rayleigh Problem in hydrodynamic stability, is analyzed. Tollmien’s series solutions are presented for the problem under consideration. A neutrally stable eigensolution is demonstrated to exist for wave numbers \(k_s > 0\), and it is established that instability arises exclusively for wave numbers within the interval \(0< k < k_s\). Furthermore, the behavior of the imaginary part of the complex wave speed \( c \), namely \( c_i \), as \( k \) approaches \( k_s^{-} \) in the context of the Circular Rayleigh problem, is studied using Tollmien–Lin’s perturbation formula. For singular neutral modes, the Reynolds stress (\(\tau \)) decreases proportionally to the inverse square of the radial distance, unlike the parallel flow case where it is constant.