On the instantaneous cutting of a columnar vortex with non-zero axial flow

Abstract

A study of the response of a columnar vortex with non-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.

Transient lift force on a blade during cutting of a vortex with non-zero axial flow

“…The shear force similarly results in an increase in vortex effective core radius σ immediately upstream of the blade leading edge and a corresponding decrease in σ downstream of the blade. The increase (decrease) in effective vortex core radius on the upstream (downstream) side of the vortex from the cutting blade is consistent with the experimental observations and inviscid computations of Marshall & Krishnamoorthy (1997) and the full Navier-Stokes simulations of Liu & Marshall (2004). From figure 3, we observe that both the external pressure gradient force and the circulation gradient force have much smaller magnitudes than the shear force, so the vortex core response to the blade is governed primarily by the shear force.…”

“…This review summarizes the results of a series of experimental studies of orthogonal vortex-blade interaction at high Reynolds number, which were performed during the late 1990s and early 2000s in the wind tunnel at Glasgow University (Doolan, Coton & Galbraith 1999Green, Doolan & Cannon 2000;Early, Green & Coton 2002;Wang et al 2002;Green, Coton & Early 2006). Visualization of vortex-body interaction was also conducted during this time period using low Reynolds number experiments in water, many of which made use of laser-induced fluorescence and particle image velocimetry for vortex imaging and measurement (Johnston and Sullivan 1992;Krishnamoorthy & Marshall 1994Marshall & Krishnamoorthy 1997).…”

“…This steady-state vortex cutting force is imposed on the vortex as the vortex is cut and persists throughout the time 260 D. C. Saunders and J. S. Marshall period following vortex cutting (Marshall 1994). Full Euler equation simulations of the inviscid response of a vortex core with axial flow following instantaneous cutting were examined by Marshall & Krishnamoorthy (1997) and Lee, Burggraf & Conlisk (1998) for incompressible flow and by Yildirim & Hillier (2013) for compressible flow, which show good agreement with the plug-flow model predictions for change in core radius and vortex wave speed.…”

Helicopter tail rotor orthogonal blade vortex interaction

“…The series of experiments conducted at Iowa by Krishnamoorthy and Marshall [8,49,50] particularly examine the effect of the impact parameter on OBVI for a subcritical vortex. The experiments were performed by forming an intake vortex in a cylindrical tank, with tangential jets at the tank top and an orifice through which liquid was removed at the bottom (Fig.…”

“…16 (from Marshall and Krishnamoorthy, [50]). The dye injected into the vortex core is cut off prior to cutting, so that the dye marks only the lateral boundary of the vortex core.…”