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Decay of a Vortex Pair behind an Aircraft

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Aircraft Wake Turbulence and Its Detection

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.

The work reported herein was sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under Contract F44620-70-C-0052.

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Abbreviations

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

τrθ :

shear in θ direction lying in plane of r and 8

(τ0)ℓ:

value of τrθ at edge of vortex for laminar flow

(τ0)t :

value of τrθ 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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Author information

Authors and Affiliations

  1. Nielsen Engineering & Research, Inc., The Netherlands

    J. N. Nielsen & Richard G. Schwind

Authors
  1. J. N. Nielsen
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  2. Richard G. Schwind
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Editor information

Editors and Affiliations

  1. Boeing Scientific Research Laboratories, Post Office Box 3981, Seattle, Washington, USA

    John H. Olsen  & Arnold Goldburg  & 

  2. Air Force Office of Scientific Research, Arlington, Virginia, USA

    Milton Rogers

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© 1971 Plenum Press, New York

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Nielsen, J.N., Schwind, R.G. (1971). Decay of a Vortex Pair behind an Aircraft. In: Olsen, J.H., Goldburg, A., Rogers, M. (eds) Aircraft Wake Turbulence and Its Detection. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-8346-8_23

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  • DOI: https://doi.org/10.1007/978-1-4684-8346-8_23

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4684-8348-2

  • Online ISBN: 978-1-4684-8346-8

  • eBook Packages: Springer Book Archive

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