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• ), counter-rotating vortices are observed. In this paper we provide a mathematical description of the onset of co-rotating vortices for a family of cones rotating in quiescent fluid, with a view towards explaining the effect of ψ on the underlying transition of dominant instability. We investigate the stability of inviscid cross-flow modes (type I) as well as modes which arise from a viscous-Coriolis force balance (type II), using numerical and asymptotic methods. The influence of ψ on the number and orientation of the spiral vortices is examined, with comparisons drawn between our two distinct methods as well as with previous experimental studies. Our results indicate that increasing ψ has a stabilizing effect on both the type I and type II modes. Favourable agreement is obtained between the numerical and asymptotic methods presented here and existing experimental results for 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.

A study of similarity solutions for laminar swirling axisymmetric flows with both buoyancy and initial momentum flux Phys. Fluids 23, 113601 (2011) On particle spin in two-way coupled turbulent channel flow simulations Phys. Fluids 23, 093302 (2011) First-order virial expansion of short-time diffusion and sedimentation coefficients of permeable particles suspensions Phys. Fluids 23, 083303 (2011) The stochastic Burgers equation with vorticity: Semiclassical asymptotic series solutions with applications J. Math. Phys. 52, 083512 (2011) Roles of particle-wall and particle-particle interactions in highly confined suspensions of spherical particles being sheared at low Reynolds numbers Phys. 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.

Existing experimental and theoretical studies are discussed which lead to the clear hypothesis of a hitherto unidentified convective instability mode that dominates within the boundary-layer flow over slender rotating cones. The mode manifests as Görtler-type counter-rotating spiral vortices, indicative of a centrifugal mechanism. Although a formulation consistent with the classic rotating-disk problem has been successful in predicting the stability characteristics over broad cones, it is unable to identify such a centrifugal mode as the half-angle is reduced. An alternative formulation is developed and the governing equations solved using both short-wavelength asymptotic and numerical approaches to independently identify the centrifugal mode.

In this study, a new centrifugal instability mode, which dominates within the boundary-layer flow over a slender rotating cone in still fluid, is used for the first time to model the problem within an enforced oncoming axial flow. The resulting problem necessitates an updated similarity solution to represent the basic flow more accurately than previous studies in the literature. The new mean flow field is subsequently perturbed leading to disturbance equations that are solved via numerical and short-wavelength asymptotic approaches, importantly yielding favourable comparison with existing experiments. Essentially, the boundary-layer flow undergoes competition between the streamwise flow component, due to the oncoming flow, and the rotational flow component, due to effect of the spinning cone surface, which can be described mathematically in terms of a control parameter, namely the ratio of streamwise to axial flow. For a slender cone rotating in sufficiently strong axial flow rates, the instability mode breaks down to Görtler-type counter-rotating spiral vortices, governed by an underlying centrifugal mechanism, which is consistent with experimental and theoretical studies for a slender rotating cone in otherwise-still fluid.

We present stability analyses for the boundary-layer flow over broad cones (half-angle ψ > 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.

We present stability analyses for the boundary-layer flow over broad cones (half-angle ψ > 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.

In this study, we provide a mathematical description of the onset of counterrotating 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.

The paper presents a linear stability analysis of the temperature-dependent boundary-layer flow over a rotating disk. Gas-and liquid-type responses of the viscosity to temperature are considered, and the disk rotates in both a quiescent and an incident axial flow. Temperature-dependent-viscosity flows are typically found to be less stable than the temperature independent cases, with temperature dependences that produce high wall viscosities yielding the least stable flows. Conversely, increasing the incident axial flow strength produces greater flow stability. Transitional Reynolds numbers for these flows are then approximated through an e N-type analysis, and are found to vary in approximate concordance with the critical Reynolds number. Examination of the component energy contributions shows that flow stability is affected exclusively through changes to the mean flow. The results are discussed in the context of Chemical Vapour Deposition reactors. * The Rayleigh number is the product of the Grashof and Prandtl numbers, i.e. the ratio of buoyancy and viscous forces multiplied by the ratio of momentum and thermal diffusivities.