Airloads and Moments on an Aircraft Flying Over a Pair of Inclined Trailing Vortices

  • W. P. Jones
  • B. M. Rao


When an aircraft flies across the wake of a preceding one, it is subjected to changing airloads and moments induced by the trailing vortices of the first aircraft. The purpose of this paper is to investigate the magnitude and characteristics of the time dependent aerodynamic forces so produced. Both aircraft are assumed to be in horizontal flight but the direction of flight of the second aircraft is assumed to be inclined at a small angle to the trailing vortices of the first aircraft. The airloads then will change relatively slowly with time and may be estimated with reasonable accuracy by quasi-steady aerodynamic theory without taking Wagner growth of lift effects into account. In the present study, the pilot of the second aircraft is assumed to have sufficient control power to maintain his aircraft in level flight. However, in practice, this condition is likely to be violated when the aircraft is close to one of the trailing vortices of the leading aircraft, particularly when the latter happens to be a large transport plane. In such circumstances the following aircraft could stall and would, in any case, be subjected to big changes of lift and rolling moment as indicated by the results presented. In the development of the theory it is assumed that the trailing vortices are a chord length or more below the following aircraft and that the vorticity distribution over its wings will have negligible effect on the trailing vortices themselves. The pair of vortices will give rise to an upwash distribution over the wings of an approaching aircraft which must be balanced by an equal and opposite velocity distribution induced by the vorticity distribution created over the wing. The effects of the induced velocity components along the span and in the direction of flight are neglected. The problem then is one of finding the appropriate vorticity distribution at each stage of the aircraft’s flight over the trailing vortices. This can be done approximately by using a modified lifting line theory or, more accurately, by lifting surface theory. To illustrate the methods of analysis employed, calculations were done for an aircraft with rectangular wings, but only the airloads on the wings were determined. Values of the lift, rolling moment and pitching moment coefficients, corresponding to vortex inclinations of 0, l0, 20, 30, and 60° were obtained for a rectangular wing of aspect ratio 6 at different times during its passage over the trailing vortices.


Vorticity Distribution Rolling Moment SPANWISE Distribution Spanwise Position Pitching Moment Coefficient 
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.



axes of coordinates

ξ(= x/ℓ or −cosθ)

chordwise coordinates

ŋ(= y/s or −cosφ)

spanwise coordinates

τ(= Ut/ℓ)

distance moved in half chords

semi chord of following aircraft


aspect ratio


distance of trailing vortices from plane of wing

Uo, U

velocities of leading and following aircraft

so, s

semi span of leading and following aircraft

\[\overset{\mathop{\gamma }_{v}}{\mathop \lambda }\,\](=2πUℓτv

trailing vortex strength of leading aircraft angle of inclination of vortex to direction of flight


distance between trailing vortices

Y01(= yo/ℓ or An01)

spanwise coordinate determining location of trailing vortex on left in Fig. 1


downwash distribution over wing at time t


general bound vorticity distribution on wing


corresponding generalized circulation or doublet distribution



nose up pitching moment


rolling moment

CL(=L/½ ρU2S)

lift coefficient

C(= L/ρU2Ss)

rolling moment coefficient


pitching moment coefficient


wing area


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  1. 1.
    Jones, W. P. and Rao, B. M., “Wing-Vortex Interaction,” Texas A&M University, Project Themis Aero Report, Jan. 1970.Google Scholar
  2. 2.
    McGowan, W. A., “Calculated Normal Load Factors on Light Airplanes Traversing the Trailing Vortices of Heavy Transport Airplanes,” NASA TN D-829, March 1969.Google Scholar
  3. 3.
    McGowan, W. A., “Trailing Vortex Hazard,” SAE Transactions, Report No. 68–220, Vol. 77, 1968, pp. 740–753.Google Scholar
  4. 4.
    Wetmore, J. W. and Reeder, J. P., “Aircraft Vortex Wakes in Relation to Terminal Operations,” NASA TN D-1777, April 1963.Google Scholar
  5. 5.
    Spreiter, J. R. and Sacks, A. H., “The Rolling Up of the Trailing Vortex Sheet and Its Effect on the Downwash Behind Wings,” Journal of Aeronautical Sciences, Vol. 18, Jan. 1951.Google Scholar
  6. 6.
    Jahnke, E. and Emde, F. Funktionentafeln. 1933.Google Scholar

Copyright information

© Plenum Press, New York 1971

Authors and Affiliations

  • W. P. Jones
    • 1
  • B. M. Rao
    • 1
  1. 1.Texas A&M UniversityUSA

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