Modal stability analysis of toroidal pipe flow approaching zero curvature
Peer reviewed, Journal article
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https://hdl.handle.net/11250/3131876Utgivelsesdato
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Sammendrag
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,
De cr = 2 Re cr
√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.