Accelerations of one airplane encountering a vortex or pair of vortices in the wake of another airplane were computed in a parametric study. The approach taken in the study was to systematically investigate the effects of progressively more nearly complete descriptions of the interaction of an airplane with a wake-vortex system, in order to compare the theoretical effects of some of the major simplifying assumptions that are commonly made in formulating mathematical models to represent this interaction. Despite their theoretical nature, this study and other related studies have practical significance: For example, mathematical models of airplane/vortex interaction are needed for pilot-training flight simulators. The models must represent the dominant vortex-encounter effects, yet must be simple enough to be adaptable to a variety of airplanes and simulators.

In this study, the axis of the vortices was parallel to the *x* axis of a Cartesian coordinate system fixed with respect to the Earth. The *y *and *z* axes were the lateral horizontal and vertical axes, respectively. In the case of a pair of vortices, the origin of the coordinate system was located between the vortices. Using a previously developed mathematical model of wake vortices, the *y* and *z* components of the vortex velocity at a given *y*,*z* location were taken to be (1) proportional to the weight and inversely proportional to the velocity and wing span of the wake-generating airplane and (2) proportional to functions of *y* and *z* that decrease with distance from the vortex cores and that include the radii and the *y *and *z* positions of the vortex cores as parameters.

In addition to the vortex coordinate system described above, there was a coordinate system fixed to the airplane body axes and five aerodynamic-surface axis systems, to model the right and left wing panels, the right and left horizontal tail panels, and the vertical tail panel. The body-axis system could be located at an arbitrary position and orientation relative to the vortex axis system. The orientation was specified by the standard airplane yaw, pitch, and roll Euler angles. The five aerodynamic-surface axis systems were placed at arbitrary positions and orientations relative to the airplane body-axis system. The equations for the wing and tail surfaces were written in general terms for planar aerodynamic surfaces with sweep, taper, and dihedral. No fuselage was modeled, and the aerodynamic surfaces were treated as originating at the center line of the airplane.

Each surface was broken up into *N *spanwise incremental areas. The angle of attack and sideslip at the three-quarter-chord point of each incremental area was calculated independently of those of its neighbors. The vortex and encountering-airplane velocities were transformed into the aerodynamic-surface axis systems and used to compute aerodynamic forces, which were assumed to act at the quarter-chord point of each incremental area. The aerodynamic forces were then transformed back into the body axis system and used to compute the accelerations of the encountering airplane.

As an example case, the vortex flow field modeled in the study had the nominal characteristics of the wake of a Boeing 767 airplane, while the encountering airplane had the nominal characteristics of a Boeing 757 airplane. The majority of cases considered in this study involved roll-dominant encounters, in which the longitudinal axis of the encountering airplane was nearly parallel to the vortex *x *axis. The first case considered was that of a drag-less rectangular wing in the flow field of a single vortex. Then in a sequence of increasingly complex cases, the study progressed to the case of a complete airplane with (1) aerodynamic surfaces characterized by taper, sweep, and dihedral; (2) aerodynamic behavior that included stalling; and (3) the vortex pair in ground effect. The effects of the pitch, roll, and yaw attitudes of the airplane on the calculated accelerations were also investigated.

The numerical results of the calculations were plotted as contours of constant acceleration in a 300-by-300-ft (91-by-91-m) area centered on vortex pairs. The following conclusions were drawn from the results:

- The effects of the single vortex field extended to larger distances from the vortex core than did the effects of a counter-rotating vortex pair. However, near the cores, the effects of the vortex pair were greater and covered a larger area. In addition, the acceleration contours for the vortex pair were more complicated with more sign reversals. Thus, it appears that a vortex pair poses a potential hazard greater than that of a single vortex.
- The dominant accelerations were in roll and the
*z*body axis, although significant yawing, pitching, and lateral accelerations were calculated when the vertical and horizontal tail surfaces were added to the mathematical model. Longitudinal acceleration was not a major factor. - A nonzero lift coefficient, drag, wing dihedral, and localized stalling had negligible effects on the accelerations relative the effects of taper ratio and sweep.
- Significant distortion of all the acceleration contours occurred when the attitude of the encountering airplane was changed by 20° about any axis.
- In the case of an encounter with vortex perpendicular to the longitudinal airplane axis, the
*z*and pitch accelerations were generally comparable to those for a parallel encounter, except for small areas of much larger accelerations around the vortex cores. However, the rolling, yawing, and lateral accelerations were zero because of the symmetry of the airplane. - The effect of a ground plane at 150 ft (45.7 m) below the vortex system was minimal except when the encountering airplane was near the ground.

*This work was done by Eric C. Stewart of *Langley Research Center*. **L-17831 *

##### NASA Tech Briefs Magazine

This article first appeared in the September, 1999 issue of *NASA Tech Briefs* Magazine.

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