Global stability analysis of axisymmetric boundary layers

TL;DR

This study conducts a linear global stability analysis of axisymmetric boundary layers on a cylinder, revealing their convective instability and stability characteristics across different Reynolds numbers.

Contribution

It introduces a comprehensive stability analysis of axisymmetric boundary layers using spectral methods and eigenvalue computations, highlighting curvature effects on stability.

Findings

Flow is temporally stable at all tested Reynolds numbers.

Flow exhibits convective instability with disturbance growth downstream.

Stability characteristics vary with Reynolds number and azimuthal wave number.

Abstract

This paper presents the 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 inlet. The pressure gradient is zero in the streamwise direction. The base flow velocity profile is fully non-parallel and non-similar in nature. The boundary layer grows continuously in the spatial directions. Linearized Navier-Stokes(LNS) equations are derived for the disturbance flow quantities in the cylindrical polar coordinates. The LNS equations along with homogeneous boundary conditions forms a generalized eigenvalues problem. Since the base flow is axisymmetric, the disturbances are periodic in azimuthal direction. Chebyshev spectral collocation method and Arnoldi's iterative algorithm is used for the solution of the general eigenvalues problem. The global temporal modes are computed for the range of Reynolds numbers and different azimuthal wave numbers. The largest imaginary part of the computed eigenmodes are negative and hence the flow is temporally stable. The spatial structure of the eigenmodes shows that the flow is convectively unstable because the disturbance amplitudes grow in size and magnitude when moving towards downstream. The global modes of axisymmetric boundary layer are more stable than that of 2D flat plate boundary layer at low Reynolds number. However, at high Reynolds number they approaches to 2D flat plate boundary layer. Thus, the damping effect of transverse curvature is significant at low Reynolds number. The wave-like nature of the disturbances is found for most unstable eigenmodes.