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Decay of a Vortex Pair behind an Aircraft

  • J. N. Nielsen
  • Richard G. Schwind

Abstract

A model of a trailing vortex pair behind an aircraft is presented which is thought to represent a case of extreme vortex persistency and which therefore is relevant from the safety point of view. Three stages are considered in the analysis: a rolling-up stage directly behind the aircraft, a second stage in which the vortices act independently as constant strength equilibrium turbulent vortices, and a third stage where the vortices physically interact and decay in strength. An overall theory is presented encompassing all three stages and aimed at obtaining equilibrium solutions. Calculative examples are presented for all stages.

Keywords

Ring Vortex Outer Region Eddy Viscosity Vortex Pair Vortex Sheet 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Symbols

a

lateral spacing between a pair of trailing vortices

c

wing chord

C1,C2

constants of integration

Cθ,CZ

nondimensional parameters, equation (59)

$d\overrightarrow{r}$

dx + i dy

D

drag of single vortex of a trailing pair or drag of generator forming vortex

Ei

exponential function

f,g,h

characteristic functions depending only on rl/r0, equation (49)

K

Г/2π, circulation parameter

Kc

value of K corresponding to vorticity rolled up in vortex core

Ki

value of K at r = ri

K0

Гo/2π

K1

value of K for v = vl

L

torque on outer edge of cross section of vortex of unit thickness

m2

mass flow per unit time across area S2 of control volume enclosing vortex

p

static pressure

P

static pressure of free stream

$\overrightarrow{q}$

vx+i vy

r,θ,z

cylindrical coordinates with positive z along downstream axis of vortex

ri

value of r where the eye of the vortex joins the logarithmic region

rj

value of r where the logarithmic region of the vortex joins the outer region

r0

outer radius of vortex where Г = 0.99 Г0

rl

value of r where K = K1 and v = v1

s

wing semispan

sv

semispan of trailing vortices

t

time measured behind wing trailing edge

u,v,w

velocity components along r, θ, z directions, respectively

u′,v′,w′

turbulent fluctuating values of u, v, and w

vx,vy

components of velocity along x,y axes

vl

maximum value of v

W

free-stream velocity

x,y

Cartesian axes in crossflow plane; x = r sin θ, y = r cos θ

Г

circulation around any contour enclosing entire trailing vortex of radius ro

Г0

initial value of Г near wing for no decay

axial velocity defect at vortex centerline

constant in equation (3)

μ

absolute viscosity of air

υ

laminar kinematic viscosity of air

υt

turbulent kinematic viscosity, eddy viscosity

\({{\upsilon }_{t}}_{_{0}}\)

eddy viscosity for turbulent shear at edge of vortex, equation (30)

\({{\upsilon }_{t}}_{_{1}}\)

eddy viscosity for turbulent shear near axis of vortex, equation (45)

ξ

vorticity in crossflow plane of a turbulent vortex

ρ

mass density of air

τrz

shear in z direction lying in plane of r and τ

τ

shear in θ direction lying in plane of r and 8

0)ℓ

value of τ at edge of vortex for laminar flow

0)t

value of τ at edge of vortex for turbulent flow

ψ

stream function

ψT

stream function for pair of trailing vortices and their induced crossflow

ω

angular velocity of eye of vortex

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Copyright information

© Plenum Press, New York 1971

Authors and Affiliations

  • J. N. Nielsen
    • 1
  • Richard G. Schwind
    • 1
  1. 1.Nielsen Engineering & Research, Inc.The Netherlands

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