. "Velocity and Turbulence in the Near-Field Region of Tip Vortices from Elliptical Wings: Its Impact on Cavitation." Twenty-First Symposium on Naval Hydrodynamics. Washington, DC: The National Academies Press, 1997.
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Twenty-First Symposium on NAVAL HYDRODYNAMICS
A simple and more appropriate method is applied by computing separately the deviations associated to the velocities of identical signs, in order to make a more realistic estimation of the standard deviation, inferior by a factor four to the classical computation. By this mean, the bias induced by the finite dimension of the measuring volume with respect to size of the vortex core is eliminated. In the case of superimposed fluctuation of positions, the method is not efficient but an other method is proposed.
In conclusion, the effect on the critical cavitation conditions of both the axial flow and the turbulence in the vortex core are found to be negligible.
NOMENCLATURE
a
tip vortex core radius
Cl
lift coefficient
cpmin
minimum pressure coefficient on the vortex axis
cmax
maximum foil chord
k*
non dimensional velocity fluctuations ((u'2+v'2)/U∞2)0.5
p∞
static reference pressure
r*
non dimensional distance to the vortex centre (r/cmax)
U∞
free stream velocity
U*
non dimensional mean axial velocity (U/U∞)
Uo*
non dimensional mean axial velocity at the tip vortex centre (Uo/U∞)
u*
non dimensional instantaneous axial velocity (u/U∞)
u'*
non dimensional axial velocity fluctuation
V*
non dimensional mean tangential velocity (V/U∞)
v*
non dimensional tangential instantaneous velocity (v/U∞)
v'*
non dimensional tangential velocity fluctuation
x*
non dimensional distance to the wing tip in the free stream direction (x/cmax)
δ
turbulent boundary layer thickness for a flat plate of length cmax and U∞ without pressure gradient
ΔV*
non dimensional maximum tangential velocity magnitude (Vmax–Vmin)/U∞
δV*
non dimensional difference between the maximum tangential velocity and the absolute value of the minimum tangential velocity (Vmax–|Vmin|)/U∞
Γ0
foil bound circulation at mid span
Γ*
non dimensional local tip vortex intensity Γ/Γ0
σd
desinent cavitation number
INTRODUCTION
Because of the possible interaction of the tip vortices of large aircrafts on following smaller planes, considerable attention has been given to the evolution of the tangential and axial velocities and the turbulence intensity in the far field region (more than 10 foil chords) (Spreiter and Sacks (1951), Staufenbiel and Vitting, (1990), Chow et al. (1991) Chigier and Cosiglia (1972), Orloff and Grant (1973), Baker and Saffman (1974), Cliffone and Orloff (1974), Singh and Uberoi (1976), McAlister and Takahashi (1991)). In this region, the roll-up of the vortices is fully achieved and the predominant effect is the diffusion of the tip vortex due to viscosity. Because tip vortex cavitation occurs at short distances from the wing tip, most recent works have been concerned on the very near region (less than a chord), Stinebring et al. (1991), Fruman et al. (1992a, 1992b, 1993, 1994, 1995a, b, c, 1996) and Pauchet et al. (1994, 1996), and the intermediate region (comprised between one and ten chords), Arndt and Keller (1991), Arndt et al (1991), Green and Acosta (1991).
In the very near region, the roll-up of the vortex is initiated and a rapid change of its local intensity (circulation) occurs. Moreover, it is in this same region that the boundary layer over the surface of the foil develops into the vortex viscous core. In this region the vortex is not axisymmetric as shown by flow visualization (Francis et Katz (1988), Chow et al. (1991), Pascal (1993), Liang et al. (1991)), Particle Image Velocimetry (PIV) (Green and Acosta (1991), Pogozelski et al. (1993), Shekarriz et al. (1993)) and Laser Doppler Velocimetry measurements (Baker et al. (1974), Accardo et al. (1984), Higuchi et al. (1987), Stinebring et al. (1991), Arndt and Keller (1991), Falçao de Campos (1989, 1992), Fruman, et al. (1992b)). Because of this lack of symmetry and uncompletion of the roll-up it is difficult to obtain, from a characteristic tangential velocity profile, informations on the tip vortex characteristics: vortex intensity and vortex core radius.
It should be pointed out that defining the vortex intensity in the near region is a subject of much debate. Indeed, let us assume that the velocity field is fully known in a plane normal to the vortex axis. The circulation of the velocity vector can be computed then over any closed path around the vortex. If the path encloses the whole wake of the wing, the tip vortex and a reasonable outboard surface, the circulation will be equal to the mid span bound circulation of the foil. If the path is now reduced in size, part of the circulation carried out by the wake will be ignored and the circulation around this path will decrease accordingly. If the viscous core region is reached by reducing the
