Modal stability analysis of toroidal pipe flow approaching zero curvature

The present study investigates the modal stability of the steady incompressible flow inside a toroidal pipe for values of the curvature δ (ratio between pipe and torus radii) approaching zero, i.e. the limit of a straight pipe. The global neutral stability curve for 10⁻⁷ ≤ δ ≤ ~10⁻² is traced using a continuation algorithm. Two different families of unstable eigenmodes are identified. For curvatures below 1.5 × 10⁻⁶ , the critical Reynolds number Recr is proportional to δ -1/2 . Hence, the critical Dean number is constant, Decr = 2\,Recr\,√δ ≈ 113 . This behaviour confirms that the Hagen–Poiseuille flow is stable to infinitesimal perturbations for any Reynolds number and suggests that a continuous transition from the curved to the straight pipe takes place as far as it regards the stability properties. For low values of the curvature, an approximate self-similar solution for the steady base flow can be obtained at a fixed Dean number. Exploiting the proposed semi-analytic scaling in the stability analysis provides satisfactory results.