ALBERT

Description: A study of the response of a columnar vortex with on-zero axial flow to impulsive cutting has been performed. The flow evolution is computed based on the vorticity-velocity formulation of the axisymmetric Euler equation using a Lagrangian vorticity collocation method. The vortex response is compared to analytical predictions obtained using the plug-flow model of Lundgren & Ashurst (1989). The plug-flow model indicates that axial motion on a vortex core with variable core area behaves in a manner analogous to one-dimensional gas dynamics in a tube, with the vortex core area playing a role analogous to the gas density. The solution for impulsive cutting of a vortex obtained from the plug-flow model thus resembles the classic problem of impulsive motion of a piston in a tube, with formation of an upstream-propagating vortex 'shock' (over which the core radius changes discontinuously) and a downstream-propagating vortex 'expansion wave' on opposite sides of the cutting surface. Direct computations of the vortex response from the Euler equation reveal similar upstream- and downstream-propagating waves following impulsive cutting for cases where the initial vortex flow is subcritical. These waves in core radius are produced by a series of vortex rings, embedded within the columnar vortex core, having azimuthal vorticity of alternating sign. The effect of the compression and expansion waves is to bring the axial and radial velocity components to nearly zero behind the propagating vortex rings, in a region on both sides of the cutting surface with ever-increasing length. The change in vortex core radius and the variation in pressure along the cutting surface agree very well with the predictions of the plug-flow model for subcritical flow after the compression and expansion waves have propagated sufficiently far away. For the case where the ambient vortex flow is supercritical, no upstream-propagating wave is possible on the compression side of the vortex, and the vortex axial flow is observed to impact on the cutting surface in a manner similar to that commonly observed for a non-rotating jet impacting on a wall. The flow appears to approach a steady state near the point of impact after a sufficiently long time. The vortex response on the expansion side of the cutting surface exhibits a downstream-propagating vortex expansion wave for both the subcritical and supercritical conditions. The results of the vortex response study are used to formulate and verify predictions for the net normal force exerted by the vortex on the cutting surface. An experimental study of the cutting of a vortex by a thin blade has also been performed in order to verify and assess the limitations of the instantaneous vortex cutting model for application to actual vortex-body interaction problems.

Description: A study has been performed of the interaction of periodic vortex rings with a central columnar vortex, both for the case of identical vortex rings and the case of rings of alternating sign. Numerical calculations, both based on an adaptation of the Lundgren-Ashurst (1989) model for the columnar vortex dynamics and by numerical solution of the axisymmetric Navier-Stokes and Euler equations in the vorticity-velocity formulation using a viscous vorticity collocation method, are used to investigate the response of the columnar vortex to the ring-induced velocity field. In all cases, waves of variable core radius are observed to build up on the columnar vortex core due to the periodic axial straining and compression exerted by the vortex rings. For sufficiently weak vortex rings, the forcing by the rings serves primarily to set an initial value for the axial velocity, after which the columnar vortex waves oscillate approximately as free standing waves. For the case of identical rings, the columnar vortex waves exhibit a slow upstream propagation due to the nonlinear forcing. The cores of the vortex rings can also become unstable due to the straining flow induced by the other vortex rings when the ring spacing is sufficiently small. This instability causes the ring vorticity to spread out into a sheath surrounding the columnar vortex. For the case of rings of alternating sign, the wave in core radius of the columnar vortex becomes progressively narrower with time as rings of opposite sign approach each other. Strong vortex rings cause the waves on the columnar vortex to grow until they form a sharp cusp at the crest, after which an abrupt ejection of vorticity from the columnar vortex is observed. For inviscid flow with identical rings, the ejected vorticity forms a thin spike, which wraps around the rings. The thickness of this spike increases in a viscous flow as the Reynolds number is decreased. Cases have also been observed, for identical rings, where a critical point forms on the columnar vortex core due to the ring-induced flow, at which the propagation velocity of upstream waves is exactly balanced by the axial flow within the vortex core when measured in a frame translating with the vortex rings. The occurrence of this critical point leads to trapping of wave energy downstream of the critical point, which results in large-amplitude wave growth in both the direct and model simulations. In the case of rings of alternating sign, the ejected vorticity from the columnar vortex is entrained and carried off by pairs of rings of opposite sign, which move toward each other and radially outward under their self- and mutually induced velocity fields, respectively.

Description: Numerical calculations are performed for the problem of penetration into a vortex core of a blade travelling normal to the vortex axis, where the plane formed by the blade span and the direction of blade motion coincides with the normal plane of the vortex axis at the point of penetration. The calculations are based on a computational method, applicable for unsteady three-dimensional flow past immersed bodies, in which a collocation solution of the vorticity transport equation is obtained on a set of Lagrangian control points. Differences between this method and other Lagrangian vorticity-based methods in the literature are discussed. Lagrangian methods of this type are particularly attractive for problems of unsteady vortex-body interaction, since control points need only be placed on the surface of the body and in regions of the flow with non-negligible vorticity magnitude. The computations for normal blade-vortex interaction (BVI) are performed for an inviscid fluid and focus on the relationship between the vortex core deformation due to penetration of the blade into the vortex ambient position and the resulting unsteady pressure field and unsteady force acting on the blade. Computations for cases with different vortex circulations are performed, and the accuracy of an approximate formulation using rapid distortion theory is assessed by comparison with the full computational results for unsteady blade force. The force generated from blade penetration into the vortex ambient position is found to be of a comparable magnitude to various other types of unsteady BVI forces, such as that due to cutting of the vortex axial flow.