Global stability analysis of axisymmetric boundary layer on a slender circular cone with the streamwise adverse pressure gradient

This paper presents a global stability analysis of the boundary layer developed on a slender circular cone. The base flow direction is towards the apex of a cone, and the pressure gradient is positive (adverse) in the flow direction. The decelerating base flow is non-parallel and non-similar. The increased semi-cone angle increases the adverse pressure gradient in the flow direction. For a given semi-cone angle, transverse curvature increases in the flow direction. However, increased semi-cone angle reduces transverse curvature at any streamwise location. The Reynolds number of the flow is defined based on the displacement thickness at the computational domain's inlet. The radius of the cone reduces in the flow direction, which results in the increased transverse curvature. The governing stability equations are derived in the spherical coordinates and discretized using the Chebyshev Spectral collocation method. The discretized equations, along with homogeneous boundary conditions, form a general eigenvalue problem, and it is solved using Arnoldi's iterative algorithm. The global temporal modes have been computed for small semi-cone angles α=2°, 4°, and 6°, azimuthal wave-numbers N=0, 1, 2, and 3 and Re=283, 416, and 610. Thus, the state of base flow is laminar at the inlet (line pq) of the domain pqrs. All the global modes computed within the range of parameters are found temporally stable. Further, the global modes are found less stable at higher semi-cone angles (α) due to the streamwise adverse pressure gradient. The effect of transverse curvature has been found significant at higher semi-cone angles. For a given Re and α, the global modes are found least stable for the helical mode N=1 and most stable for N=3. The eigenmodes’ two-dimensional spatial structure suggests that flow is spatially unstable as the disturbances grow in the streamwise direction.