Analysis of boundary layer flow over a broad rotating cone in still fluid with non-stationary modes

Abstract

This study focuses on analyzing the effects of traveling modes on the boundary layer flow over a rotating cone in a still fluid system. Non-stationary modes are known to manifest in the boundary layer of rotating cones with highly polished, very smooth surfaces. In this paper, only the broad rotating cone (defined as a cone with the half angle ψ ≥ 40°) system is considered. An asymptotic analytical method is used to solve the governing equations and output the waveangle and wavenumber of the system. This is then compared to a numerical formulation that uses a Chebyshev spectral method. The resulting solutions show that increasing the wave frequency destabilizes the system with a much stronger destabilization for the viscous wall type II modes than the inviscid cross-flow type I modes, where the type I mode is the dominant mode seen in experiments. This result suggests that a slower frequency wave should be selected in order to increase the stability of the system. It was also observed that the negative frequency values have a minimum of the critical Reynolds number values for each cone half angle. It also shows that there is a comparison limit for high frequency positive values. After this, an energy balance analysis is conducted to see the effect on the total mechanical energy transferred between the basic flow and the perturbation quantities. This showed that as the frequency of the traveling modes increases, the energy transferred decreases.