Zahir Hussain

The cross-flow instability of the boundary layer on a rotating cone

Experimental studies have shown that the boundary-layer flow over a rotating cone is susceptible to cross-flow and centrifugal instability modes of spiral nature, depending on the cone sharpness. For half-angles (ψ) ranging from propeller nose cones to rotating disks (ψ ≥ 40°), the instability triggers co-rotating vortices, whereas for sharp spinning missiles (ψ 40°. Below this half-angle we suggest that an alternative instability mechanism is at work, which is not amenable to investigation using the formulation presented here.

Boundary-Layer Transition on Broad Cones rotating in an Imposed Axial Flow

We present stability analyses for the boundary-layer flow over broad cones (half-angle > 40 ◦ ) rotating in imposed axial flows. Preliminary convective instability analyses are presented that are based on the Orr–Sommerfeld equation for a variety of axial-flow speeds. The results are discussed in terms of the limited existing experimental data and previous stability analyses on related bodies. The results of an absolute instability analysis are also presented which are intended to further those by Garrett & Peake 21 through the use of a more rigorous steady-flow formulation. Axial flow is seen to delay the onset of both convective and absolute instabilities.

The instability of the boundary layer over a disk rotating in an enforced axial flow

We consider the convective instability of stationary and traveling modes within the boundary layer over a disk rotating in a uniform axial flow. Complementary numerical and high Reynolds number asymptotic analyses are presented. Stationary and traveling modes of type I (crossflow) and type II (streamline curvature) are found to exist within the boundary layer at all axial flow rates considered. For low to moderate axial flows, slowly traveling type I modes are found to be the most amplified, and quickly traveling type II modes are found to have the lower critical Reynolds numbers. However, near-stationary type I modes are expected to be selected due to a balance being struck between onset and amplification. Axial flow is seen to stabilize the boundary layer by increasing the critical Reynolds numbers and reducing amplification rates of both modes. However, the relative importance of type II modes increases with axial flow and they are, therefore, expected to dominate for sufficiently high rates. The application to chemical vapour deposition(CVD) reactors is considered.

The centrifugal instability of a slender rotating cone.

In this study, we provide a mathematical description of the onset of counter-rotating circular vortices observed for a family of slender rotating cones (of half-angles 15° or less) in quiescent fluid. In particular, we apply appropriate scalings in order to simplify the basic-flow profiles, which are subsequently perturbed, accounting for the effects of streamline curvature. A combined large Reynolds number and large vortex wavenumber analysis is used to obtain an estimate for the asymptotic right-hand branch of neutral stability for a slender rotating cone. Our results confirm our earlier predictions pertaining to the existence of the new Görtler mode and capture the effects of the governing centrifugal instability mechanism. Meanwhile, favourable comparisons are drawn with existing numerical neutral stability curve results.