In reviewing predictions of the Denver 03 data, it was noticed that the initial vortex descent rate was often a bit too shallow (V0 too small). Since V0 is effectively an aircraft-specific input parameter, this discrepancy can not be resolved directly by making adjustments to the model. What had been done up till now is to adjust the model to produce a descent rate that increased with time in an effort to 'catch up' to the more rapidly descending vortex position measurements. The required adjustment was a non-physical negative entrainment coefficient for vertical momentum. In essence this negative entrainment implies that the wake oval is entraining fluid with downward directed momentum from the outside. Using this approach, the computed vortex trajectory would tend to catch up to the data at intermediate times, but then drop below it at later times.
As shown below, initial vortex descent rates that match the data can be obtained by specifying an appropriate value for the initial vortex spacing, b0. Although an attempt to do this was made earlier, the b0 values thus obtained do not seem to work well for the Denver 03 landing data. Oddly enough, several of these previously-determined b0 values were larger than the elliptic loading estimates. As discussed below, this is almost certainly not the case when the aircraft is in a landing configuration with flaps and slats deployed.
The difficulties with the prior b0 estimates, prompted the need to recalculate these using the Denver data.
The initial vortex positions, y0p and
y0s, that give rise to
b0 = y0s - y0p
are assumed to be the centroids of the spanwise distribution of circulation
shed into the wake
, where
is
the spanwise load distribution, and
is the total lift. Most aircraft are designed to have an
elliptic load distribution in cruise flight. The load distribution is
altered significantly by flaps and slats in a landing configuration, however,
and thus calculating b0 from an elliptic load distribution would
be inaccurate for landing studies. Both flaps and slats serve to increase
the loading on just the inboard sections of the wing. The sudden drop in
load at the outboard edge of the flaps and slats results in significant
quantities of circulation being shed into the wake at these locations.
This situation necessarily moves the circulation centroid inboard from the
position that it takes in cruise flight. Thus we should conclude without
doubt that b0 for a landing configuration should be reduced from
the elliptical load value. As mentioned above, the fact the the
previously-determined b0 values were often larger than the
corresponding elliptic loading estimtes make them highly suspect.
It would be an easy matter to compute b0 for a landing
configuration if the loading distribution were known for this phase of
flight. Since we do not have easy access to this information, an
alternative method is to measure the initial vortex descent rate from
the vortex trajectory data and use this information to infer an effective
(non-elliptical) value of b0. To do this,
we simply combine the equations for the initial vortex descent rate
![]() |
(1) |
![]() |
(2) |
![]() |
(3) |
In order to compute V0 from the landing data, we simply apply a linear fit to the vertical vortex position data over the first 25 seconds of descent, as illustrated in the following image
The initial vortex descent rate is then just the slope of the linear fit. Note that we get two independent measures of V0 for each landing - one for the port and another for the starboard vortex.
Data from all trajectories of like aircraft are combined in order to get an average estimate of V0 (and hence b0). The plot below shows all the A319 trajectories from the Denver 2003 dataset
Although there is considerable scatter, the average descent rate and its uncertainty are well defined by the data.
While the method just described allows us to infer b0 indirectly via knowledge of V0, it is also possible to measure b0 directly from the lateral vortex position data. Consider the following plot that shows the lateral vortex position measurements over the first 25 seconds of the test
By applying independent linear fits to the data for the port and starboard vortices, it is possible to determine and effective y0 for each and then deduce b0 via b0 = y0s - y0p. The collection of these measurements for the A319 aircraft are shown in the following plot
Using both the direct and indirect measurements of b0, for the Denver 2003 landing date, it is possible construct the following table
Aircraft | Sample | b0direct | b0indirect | b0prior | b0indirect/span |
---|---|---|---|---|---|
A318 | 10 | 31.18 | 23.60 | 31.60 | 0.692 |
A319 | 112 | 23.19 | 26.22 | 29.40 | 0.769 |
A320 | 84 | 23.34 | 23.86 | 27.30 | 0.704 |
B733 | 93 | 19.73 | 22.42 | 24.00 | 0.776 |
B735 | 31 | 23.26 | 22.24 | 23.80 | 0.770 |
B738 | 18 | 20.84 | 23.78 | 25.00 | 0.693 |
B752 | 65 | 26.96 | 29.26 | 33.10 | 0.770 |
B763 | 13 | 29.43 | 34.31 | 39.60 | 0.721 |
B772 | 6 | 29.31 | 38.27 | 39.50 | 0.628 |
MD82 | 15 | 20.66 | 25.18 | 27.70 | 0.765 |
MD83 | 6 | 23.66 | 28.07 | 27.70 | 0.853 |