Experiments have been performed on the tip vortex trailing from a rectangular NACA 0012 half-wing. Preliminary studies showed the vortex to be insensitive to the introduction of a probe and subject only to small wandering motions. Meaningful velocity measurements could therefore be made using hot-wire probes.Detailed analysis of the effects of wandering was performed to properly reveal the flow structure in the core region and to give confidence in measurements made outside the core. A theory has been developed to correct mean-velocity profiles for the effects of wandering and to provide complete quantitative estimates of its amplitude and contributions to Reynolds stress fields. Spectral decomposition was found to be the most effective method of separating these contributions from velocity fluctuations due to turbulence.Outside the core the flow structure is dominated by the remainder of the wing wake which winds into an ever-increasing spiral. There is no large region of axisymmetric turbulence surrounding the core and little sign of turbulence generated by the rotational motion of the vortex. Turbulence stress levels vary along the wake spiral in response to the varying rates of strain imposed by the vortex. Despite this complexity, the shape of the wake spiral and its turbulent structure reach an approximately self-similar form.On moving from the spiral wake to the core the overall level of velocity fluctuations greatly increases, but none of this increase is directly produced by turbulence. Velocity spectra measured at the vortex centre scale in a manner that implies that the core is laminar and that velocity fluctuations here are a consequence of inactive motion produced as the core is buffeted by turbulence in the surrounding spiral wake. Mean-velocity profiles through the core show evidence of a two-layered structure that dies away with distance downstream.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112096001929

The structure and development of a wing-tip vortex

In this paper, we study the interaction of two co-rotating trailing vortices. It is well-known that vortices of like-sign ultimately merge to form a single vortex, and there has been much work on measuring and predicting the initial conditions for the onset of merger, especially concerning the critical vortex core radius. However, the physical mechanism causing this merger has received little attention. In this work, we directly measure the structure of the antisymmetric vorticity field that causes the co-rotating vortices to be pushed towards each other during merger. We discover that the form of the antisymmetric vorticity comprises two counter-rotating vortex pairs, whose induced velocity field readily pushes the two centroids together. The merging velocity computed from the antisymmetric vorticity field agrees closely with the merging velocity measured directly from the total (original) flow field.The co-rotating vortex pair evolves through four distinct phases. The initial stage comprises a diffusive growth, which can be either viscous or turbulent. In either case, the number of turns that they rotate around one another until the vortices start to merge increases with Reynolds number (Re). If one observes the streamlines in a rotating reference frame (moving with the vortices), then one finds an inner and outer recirculating region of the flow bounded by a separatrix streamline. When the vortices grow large enough in the first stage, diffusion across the separatrix places vorticity into the outer recirculating region of the flow, and this leads to the generation of the antisymmetric vorticity, causing convective merger. This second (convective) stage corresponds to the motion of the vortex centroids towards each other, and is a process which is almost independent of viscosity. During the late part of this stage, the antisymmetric vorticity is diminished by a symmetrization process, and the final merging into one vorticity structure is achieved by a second diffusive stage. The fourth and ultimate phase is one where the merged vortex core grows by diffusion.

The physical mechanism for vortex merging

Implicit finite-difference procedures for the primitive form of the incompressible Navier-Stokes and the compressible Euler equations are used to compute vortex wake flows. The partial differential equations in strong conservation-law form are transformed to cluster grid points in regions with large changes in vorticity. In addition to clustering, fourth-order accurate, spatial difference operators are used to help resolve the flowfield gradients. The use of implicit time-differencing permits large time steps to be taken since temporal variations are typically small. Computational efficiency is achieved by approximate factorization. Both two-dimensional and preliminary three-dimensional calculations are described. I. Introduction T HE concentrated vorticity in the near wake of a large aircraft can pose a destructive threat to smaller aircraft within the same airspace. Consequently, experimental and theoretical efforts have been under way to understand, predict (for use in avoidance systems), and possibly reduce the vortex wake hazard. Most theoretical models developed to treat this problem rely on tracing discrete vortices,1"3 but an alternate and potentially more powerful approach is to use finitedifference procedures.4'5 Computer programs based on such methods can ultimately account for flowfield nonlinear effects with few ad hoc assumptions. Implicit finite-difference procedures are developed here to solve the incompressible Navier-Stokes equations and compressible Euler equations for simplified two- and threedimensional, unsteady vortex wake flows. This paper is divided into five interdependent sections. The flowfield and its numerical implications are discussed in Sec. II. The incompressible equations are developed for simulation as a system of first-order partial-differential equations in Sec. Ill, and the finite-difference algorithms are described in Sec. IV. Simulation based on the compressible flow equations is discussed in Sec. V. and, finally, simple wake-vortex flow calculations using both kinds of modeling are presented in Sec. VI.

Implicit Finite-Difference Procedures for the Computation of Vortex Wakes

https://hal.archives-ouvertes.fr/hal-00081677/document

Physics of vortex merging

The viscous evolution of two co-rotating vortices is analysed using direct two-dimensional numerical simulations of the Navier–Stokes equations. The article focuses on vortex interaction regimes before merging. Two parameters are varied: a steepness parameter n which measures the steepness of the initial vorticity profiles in a given family of profiles, and the Reynolds number Re (between 500 and 16 000). Two distinct relaxation processes are identified. The first one is non-viscous and corresponds to a rapid adaptation of each vortex to the external (strain) field generated by the other vortex. This adaptation process, which is profile dependent, is described and explained using the damped Kelvin modes of each vortex. The second relaxation process is a slow diffusion phenomenon. It is similar to the relaxation of any non-Gaussian axisymmetrical vortex towards the Gaussian. The quasi-stationary solution evolves on a viscous-time scale toward a single attractive solution which corresponds to the evolution from two initially Gaussian vortices. The attractive solution is analysed in detail up to the merging threshold a/b ≈ 0.22 where a and b are the vortex radius and the separation distance respectively. The vortex core deformations are quantified and compared to those induced by a single vortex in a rotating strain field. A good agreement with the asymptotic predictions is demonstrated for the eccentricity of vortex core streamlines. A weak anomalous Reynolds number dependence of the solution is also identified. This dependence is attributed to the advection–diffusion of vorticity towards the hyperbolic points of the system and across the separatrix connecting these points. A Re1/3 scaling for the vorticity at the central hyperbolic point is obtained. These findings are discussed in the context of a vortex merging criterion.

/pdf/viscous-interactions-of-two-co-rotating-vortices-before-3eb8c2rst2.pdf

Viscous interactions of two co-rotating vortices before merging

A merging distance criterion for equal-strength corotational vortices is derived from low-turbulence wind tunnel flow visualization data. The vortex separation distance is normalized by defining a vortex core diameter based on circulation defect and angular momentum defect. Merging may take place for larger separation distances than predicted from earlier two-dimensional inviscid calculations, which indicates that viscosity and possibly three-dimensional effects are important factors in the merging phenomenon. Hot-wire velocity distributions and rolling moment measurements show that attenuation of the vortex hazard is associated with vortex merging. Nomenclature bg = generator wingspan bf = follower wingspan CL = lift coefficient cf = rolling-moment coefficient c = wing chord d = vortex separation distance d0 = initial vortex separation distance dc = vortex core diameter (usually based on peak tangential speed) dM = vortex core diameter (based on angular momentum defect equal to that in a Rankine vortex) dr = vortex core diameter (based on circulation defect equal to that in a Rankine vortex) dv —vortex core diameter (based on outer boundary of.the rotational portion of the vortex)

Merging of Aircraft Trailing Vortices

A merging-distance criterion for equal-strength corotational vortices is derived from low-turbulence wind-tunnel flow-visualization data. The vortex separation distance is normalized by defining a vortex core diameter based on circulation defect and angular-momentum defect. Merging may take place for larger separation distances than predicted from earlier two-dimensional inviscid calculations, which indicates that viscosity and possibly three-dimensional effects are important factors in the merging phenomenon. Hot-wire velocity distributions and rolling-moment measurements show that attenuation of the vortex hazard is associated with vortex merging.

Steven A. Brandt

Papers

Merging of Aircraft Trailing Vortices

Merging of aircraft trailing vortices