Stability Modes in Vortex Structure Formation : Canonical Examples for Rotating Components

Three rather different physical cases have been studied. All represent very practical geometries for which the modal behavior of vortex structures is not completely understood. The work on these problems is ongoing with the objective of obtaining physical confirmation, enhanced understanding and predictive capability for the vortex structures encountered in rotating machines. 1. Surface Flows over Rotating Cones. Experimental studies have shown that the boundarylayer flow over a rotating cone is susceptible to crossflow and centrifugal instability modes of a spiral nature, depending on the cone sharpness. For apex 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 ◦ ), counterrotating vortices are observed. A mathematical description is provided 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 using numerical and asymptotic methods. The stability of inviscid cross-flow modes (type I) is investigated, as well as modes arising from a viscous–Coriolis force balance (type II). The influence of ψ on the number and orientation of the spiral vortices is examined, with comparisons drawn between the two distinct methods as well as with previous experimental studies. The results indicate that increasing ψ has a stabilizing effect on both the type I and type II modes. Favorable agreement is obtained between the numerical and asymptotic methods presented here and existing experimental results for ψ >40 ◦ . Below this half-angle it appears that an alternative instability mechanism is at work, which is not amenable to investigation using the broad-cone formulation. An alternative formulation has been developed and properties of the counter-rotating vortices successfully predicted. 2. Turbine Blades and Swept Cylinders. Suction surface flow visualization on turbine blades at subsonic and transonic speeds showed robust streamwise streaks on a lengthy time-average basis [1] (Fig. 1). The flow on the suction surface, under the influence of a strong favorable pressure gradient, was initially laminar. Further downstream laminar separation and transition to turbulence were encountered. The turbulent boundary layer region then persisted to the trailing edge. The streamwise streaks were unaffected throughout the whole surface regardless of the boundary layer state. Similar behavior was observed by Halstead [2], who had surface film confirmation of the boundary layer state throughout. The streamwise vortex structures, whilst not particularly strong, are persistent and would seem to exert a stabilizing influence on the flow. The lateral spacing between streaks on convex surfaces had been predicted by Kestin and Wood [3]. Observations of streaks on turbine blades and unswept cylinders were to provide a firm basis for referencing the influence of sweep. High speed testing was undertaken on a 38 mm diameter cylinder and lowspeed testing on an unswept 152 mm diameter cylinder. The results are in excellent agreement with the Kestin and Wood theory, providing confirmation of the theory for circular cylinders. Although the Kestin and Wood work is related to unswept circular cylinders it also provides an excellent benchmark for sweep effects on turbomachinery blading. The introduction of sweep brings consideration of a wide range of instabilities. Crossflow instability results from the inflectional behavior of a three-dimensional boundary layer and is thought to be prominent and aggressive. Experimental work on a circular cylinder has been undertaken by the authors over a range of sweep angles from zero to 61 o , giving results for lateral spacing and angular orientation of the vortical streaks. Figure 2 demonstrates that at high-sweep angles the results are consistent with those of Poll [4] and of Takagi et al. [5]. It is shown that STABILITY MODES IN VORTEX STRUCTURE FORMATION: CANONICAL EXAMPLES FOR ROTATING COMPONENTS