Linear stability of curved pipe flow approaching zero curvature

This study investigates the linear stability of the flow in a torus for very low curvatures, approaching the limit of a straight pipe, i.e. zero curvature. An in-house developed method is employed for both computing the base flow and the stability analysis. The equations are solved on a 2D domain with three components, employing a spectral collocation method based on Chebychev and Fourier nodes in the radial and azimuthal direction, respectively. The linear stability analysis is carried out for curvatures ?? < 10^-2 approaching zero. Inhomogeneous disturbances are considered in both the radial and azimuthal directions, whereas normal modes are assumed in the streamwise direction. The Navier-Stokes equations are linearised with respect to both the computed base flow and the parabolic Poiseuille velocity profile. In the latter case, perturbations are allowed to have non-zero curvature. In this way, one can assess for which parameters (??, Re) the base flow is indistinguishable from the parabolic velocity profile regarding the stability properties. For the curvatures here investigated, the critical Reynolds number increases monotonically as the curvature decreases, going asymptotically to infinity as ?? → 0, i.e. approaching the straight pipe which is known to be linearly stable.