6. Axisymmetric Boundary Layer on a Circular Cone

This paper presents the linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cone. The base flow is considered parallel to the axis of the cone at the inlet. The angle of attack is zero, and hence, the base flow is axisymmetric. The favourable pressure gradient develops in the streamwise direction due to the cone angle. The Reynolds number is calculated based on the cone radius (a) at the inlet and free-stream velocity (\(U_{\infty }\)). The base flow velocity profile is fully non-parallel and non-similar. Therefore, linearized Navier-Stokes equations (LNS) are derived for the disturbance flow quantities in the spherical coordinates. The LNS is discretized using Chebyshev spectral collocation method. The discretized LNS and the homogeneous boundary conditions form a general eigenvalues problem. Arnoldi’s iterative algorithm is used for the numerical solution of the general eigenvalues problem. The Global temporal modes are computed for the range of Reynolds number from 174 to 1046, semi-cone angles \(2^\circ \), \(4^\circ \), \(6^\circ \) and azimuthal wavenumbers from 0 to 5. It is found that the Global modes are more stable at a higher semi-cone angle \(\alpha \) due to the development of a favourable pressure gradient. The effect of transverse curvature is reduced at higher semi-cone angles (\(\alpha \)). The spatial structure of the eigenmodes shows that the flow is convectively unstable. The spatial growth rate (\(A_x\)) increases with the increase in semi-cone angle (\(\alpha \)) from \(2^\circ \) to \(6^\circ \). Thus, the effect of an increase in semi-cone angle (\(\alpha \)) is to reduce the temporal growth rate (\(\omega _i\)) and to increase the spatial growth rate (\(A_x\)) of the Global modes at a given Reynolds number.