Global stability analysis of axisymmetric boundary layer over a circular cone

Abstract

This paper presents a 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. A favorable pressure gradient develops in the streamwise direction due to cone angle. The Reynolds number is calculated based on the cone radius a at the inlet and freestream velocity U ∞ . The base flow velocity profile is fully nonparallel and nonsimilar. Linearized Navier-Stokes equations (LNSEs) are derived for the disturbance flow quantities in the spherical coordinates. The LNSEs are discretized using the Chebyshev spectral collocation method. The discretized LNSEs along with 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, semicone angles 2 ∘ , 4 ∘ , and 6 ∘ , and azimuthal wave numbers from 0 to 5. It is found that the global modes are more stable at higher semicone angle α , due to the development of favorable pressure gradient. The effect of transverse curvature is reduced at higher α . The spatial structure of the eigenmodes shows that the flow is convectively unstable. The spatial growth rate A x increases with an increase in α from 2 ∘ to 6 ∘ . Thus, the effect of an increase in α is to reduce the temporal growth rate ω i and increase the A x of the global modes at a given Reynolds number.