Linear and Secondary Instabilities in Incompressible Axisymmetric Boundary Layers: Effect of Transverse Curvature

The incompressible boundary layer in the axial flow past a cylinder has been shown Tutty et. al.(\cite{tutty}) to be stabler than a two-dimensional boundary layer, with the helical mode being the least stable. In this paper the secondary instability of this flow is studied. The laminar flow is shown here to be always stable at high transverse curvatures to secondary disturbances, which, together with a similar observation for the linear modes implies that the flow past a thin cylinder is likely to remain laminar. The azimuthal wavenumber of the pair of least stable secondary modes ($m_+$ and $m_-$) are related to that of the primary ($n$) by $m_+=2n$ and $m_-=-n$. The base flow is shown to be inviscidly stable at any curvature.