Stephen J. Garrett

The stability and transition of the boundary layer on a rotating sphere

This paper is concerned with convective and absolute instabilities in the boundarylayer flow over the outer surface of a sphere rotating in an otherwise still fluid. Viscous and streamline-curvature eects are included and the analysis is conducted between latitudes of 10 and 80 from the axis of rotation. Both convective and absolute instabilities are found at each latitude within specic parameter spaces. The results of the convective instability analysis show that a crossflow instability mode is the most dangerous below =6 6. Above this latitude a streamline-curvature mode is found to be the most dangerous, which coincides with the appearance of reverse flow in the radial component of the mean flow. At low latitudes the disturbances are considered to be stationary, but at higher latitudes they are taken to rotate at 76% of the sphere surface speed, as observed in experimental studies. Our predictions of the Reynolds number and vortex angle at the onset of convective instability are consistent with existing experimental measurements. Results are also presented that suggest that the occurrence of the slowly rotating vortices is associated with the dominance of the streamline-curvature mode at =6 6. The local Reynolds number at the predicted onset of absolute instability matches experimental data well for the onset of turbulence at =3 0; beyond this latitude the discrepancy increases but remains relatively small below =7 0. It is suggested that this absolute instability may cause the onset of transition below =7 0. Close to the pole the predictions of each stability analysis are seen to approach those of existing rotating disk investigations.

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 effect of anisotropic and isotropic roughness on the convective stability of the rotating disk boundary layer

A theoretical study investigating the effects of both anisotropic and isotropic surface roughness on the convective stability of the boundary-layer flow over a rotating disk is described. Surface roughness is modelled using a partial-slip approach, which yields steady-flow profiles for the relevant velocity components of the boundary-layer flow which are a departure from the classic von Karman solution for a smooth disk. These are then subjected to a linear stability analysis to reveal how roughness affects the stability characteristics of the inviscid Type I (or cross-flow) instability and the viscous Type II instability that arise in the rotating disk boundary layer. Stationary modes are studied and both anisotropic (concentric grooves and radial grooves) and isotropic (general) roughness are shown to have a stabilizing effect on the Type I instability. For the viscous Type II instability, it was found that a disk with concentric grooves has a strongly destabilizing effect, whereas a disk with radial grooves or general isotropic roughness has a stabilizing effect on this mode. In order to extract possible underlying physical mechanisms behind the effects of roughness, and in order to reconfirm the results of the linear stability analysis, an integral energy equation for three-dimensional disturbances to the undisturbed three-dimensional boundary-layer flow is used. For anisotropic roughness, the stabilizing effect on the Type I mode is brought about by reductions in energy production in the boundary layer, whilst the destabilizing effect of concentric grooves on the Type II mode results from a reduction in energy dissipation. For isotropic roughness, both modes are stabilized by combinations of reduced energy production and increased dissipation.

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.

Initial condition effects on large scale structure in numerical simulations of plane mixing layers

In this paper, Large Eddy Simulations are performed on the spatially developing plane turbulent mixing layer. The simulated mixing layers originate from initially laminar conditions. The focus of this research is on the effect of the nature of the imposed fluctuations on the large-scale spanwise and streamwise structures in the flow. Two simulations are performed; one with low-level three-dimensional inflow fluctuations obtained from pseudo-random numbers, the other with physically correlated fluctuations of the same magnitude obtained from an inflow generation technique. Where white-noise fluctuations provide the inflow disturbances, no spatially stationary streamwise vortex structure is observed, and the large-scale spanwise turbulent vortical structures grow continuously and linearly. These structures are observed to have a three-dimensional internal geometry with branches and dislocations. Where physically correlated provide the inflow disturbances a “streaky” streamwise structure that is spatially stationary is observed, with the large-scale turbulent vortical structures growing with the square-root of time. These large-scale structures are quasi-two-dimensional, on top of which the secondary structure rides. The simulation results are discussed in the context of the varying interpretations of mixing layer growth that have been postulated. Recommendations are made concerning the data required from experiments in order to produce accurate numerical simulation recreations of real flows.

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.

On the stability of von Kármán rotating-disk boundary layers with radial anisotropic surface roughness

We summarise results of a theoretical study investigating the distinct convective instability properties of steady boundary-layer flow over rough rotating disks. A generic roughness pattern of concentric circles with sinusoidal surface undulations in the radial direction is considered. The goal is to compare predictions obtained by means of two alternative, and fundamentally different, modelling approaches for surface roughness for the first time. The motivating rationale is to identify commonalities and isolate results that might potentially represent artefacts associated with the particular methodologies underlying one of the two modelling approaches. The most significant result of practical relevance obtained is that both approaches predict overall stabilising effects on type I instability mode of rotating disk flow. This mode leads to transition of the rotating-disk boundary layer and, more generally, the transition of boundary-layers with a cross-flow profile. Stabilisation of the type 1 mode means that it may be possible to exploit surface roughness for laminar-flow control in boundary layers with a cross-flow component. However, we also find differences between the two sets of model predictions, some subtle and some substantial. These will represent criteria for establishing which of the two alternative approaches is more suitable to correctly describe experimental data when these become available.

Investigation of Streamwise and Transverse Instabilities on Swept Cylinders and Implications for Turbine Blading

The starting point for this investigation was the observation of robust streamwise streaks in flow visualization on the suction surfaces of blades in a turbine cascade at subsonic and transonic speeds. The spanwise wavelength of an array of streamwise vortices had been predicted and is here confirmed experimentally. Observations of streaks on unswept turbine blades and on circular cylinders confirmed these earlier predictions, providing a firm basis for referencing the new measurements of vortical behavior. The observations made it clear that the boundary layers are highly three dimensional. In this paper observations of streamwise and transverse instabilities on swept circular cylinders, over a range of inclinations, are presented. The circular cylinder is a canonical case and observations relate the streamwise vorticity of the unswept case to the more aggressive crossflow instability at high sweep angles. Introducing sweep brings consideration of a wide range of instabilities. Prominent is crossflow instability resulting from the inflectional behavior of a three-dimensional boundary layer.

On the global linear stability of the boundary layer on rotating bodies

This paper was published as Advances in Turbulence XI: Proceedings of the 11th EUROMECH European Turbulence Conference, June 25-28, 2007, Porto, Portugal; Palma, J. M. L. M.; Silva Lopes, A. (Eds.), pp. 550-552. It is available from http://www.springer.com/materials/mechanics/book/978-3-540-72603-6