Decay of a Vortex Pair behind an Aircraft

  • J. N. Nielsen
  • Richard G. Schwind


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.


Ring Vortex Outer Region Eddy Viscosity Vortex Pair Vortex Sheet 
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lateral spacing between a pair of trailing vortices


wing chord


constants of integration


nondimensional parameters, equation (59)


dx + i dy


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


exponential function


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


Г/2π, circulation parameter


value of K corresponding to vorticity rolled up in vortex core


value of K at r = ri




value of K for v = vl


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


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


static pressure


static pressure of free stream


vx+i vy


cylindrical coordinates with positive z along downstream axis of vortex


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


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


outer radius of vortex where Г = 0.99 Г0


value of r where K = K1 and v = v1


wing semispan


semispan of trailing vortices


time measured behind wing trailing edge


velocity components along r, θ, z directions, respectively


turbulent fluctuating values of u, v, and w


components of velocity along x,y axes


maximum value of v


free-stream velocity


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


circulation around any contour enclosing entire trailing vortex of radius ro


initial value of Г near wing for no decay

axial velocity defect at vortex centerline

constant in equation (3)


absolute viscosity of air


laminar kinematic viscosity of air


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)


vorticity in crossflow plane of a turbulent vortex


mass density of air


shear in z direction lying in plane of r and τ


shear in θ direction lying in plane of r and 8


value of τ at edge of vortex for laminar flow


value of τ at edge of vortex for turbulent flow


stream function


stream function for pair of trailing vortices and their induced crossflow


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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