In reviewing predictions of the Denver 03 data, it was noticed that the initial
vortex descent rate was often a bit too shallow (V_{0} too small).
Since V_{0} 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,
b_{0}. Although an attempt to do this was made earlier, the
b_{0} values thus obtained do not seem to work well for the Denver
03 landing data. Oddly enough, several of these previously-determined
b_{0} 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 b_{0} estimates, prompted the need
to recalculate these using the Denver data.

The initial vortex positions, y_{0}_{p} and
y_{0}_{s}, that give rise to
b_{0} = y_{0}_{s} - y_{0}_{p}
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 b_{0} 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 b_{0} for a landing configuration should be reduced from
the elliptical load value. As mentioned above, the fact the the
previously-determined b_{0} values were often larger than the
corresponding elliptic loading estimtes make them highly suspect.

It would be an easy matter to compute b_{0} 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 b_{0}. To do this,
we simply combine the equations for the initial vortex descent rate

(1) |

(2) |

(3) |

In order to compute V_{0} 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 V_{0} 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 V_{0} (and hence b_{0}). 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 b_{0} indirectly
via knowledge of V_{0}, it is also possible to measure b_{0}
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 y_{0} for each
and then deduce b_{0} via b_{0} = y_{0}_{s} -
y_{0}_{p}. The collection of these measurements for the
A319 aircraft are shown in the following plot

Using both the direct and indirect measurements of b_{0}, for the
Denver 2003 landing date, it is possible construct the following table

Aircraft | Sample | b_{0}_{direct} |
b_{0}_{indirect} |
b_{0}_{prior} |
b_{0}_{indirect}/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 |

Note that there is general agreement between b_{0direct} and
b_{0indirect} and that these are almost always smaller than
the b_{0} values computed previously. Also note that the ratio
b_{0indirect}/span is almost always less than the elliptical
loading equivalent, π/4 = 0.7854.
Since the objective here is to provide the model with more accurate values of
V_{0}, the b_{0indirect} values are used in the
modeling efforts. **Thus the b**_{0direct} values only
serve as a consistency check.