5. Axisymmetric Boundary Layer on a Cylinder

This chapter presents a linear global stability analysis of the incompressible axisymmetric boundary layer on a circular cylinder. The base flow is parallel to the axis of the cylinder at the inflow boundary, fully non-parallel and non-similar. Linearized Navier-Stokes (LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations and homogeneous boundary conditions form a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in the azimuthal direction. The Chebyshev spectral collocation method and Arnoldi’s iterative algorithm are used for the numerical solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wavenumbers. The largest imaginary part of the computed eigenmodes is negative; hence, the flow is temporally stable. The spatial structure of the eigenmodes shows that the disturbance amplitudes grow in size and magnitude while moving downstream. The global modes of the axisymmetric boundary layer are more stable than that of the 2D flat-plate boundary layer at low Reynolds number. However, at a higher Reynolds number, they approach to 2D flat-plate boundary layer. Thus, the damping effect of transverse curvature is significant at a low Reynolds number. The wave-like nature of the disturbance amplitudes is found in the streamwise direction for the least stable eigenmodes.