Procedure for computing b₀

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, y_0p and y_0s, that give rise to b₀ = y_0s - y_0p are assumed to be the centroids of the spanwise distribution of circulation shed into the wake $\Gamma(y) = 1/(\rho U_{ac})dL(y)/dy = 1/(\rho U_{ac})l(y)$, where $l(y)$ is the spanwise load distribution, and $L$ 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

$$V_0 = \frac{ \Gamma_0 }{ 2\pi b_0 }$$ (1)

and the initial vortex circulation

$$\Gamma_0 = \frac{ mg }{ \rho U_{ac} b_0 }$$ (2)

to get

$$b_0 = \sqrt{ \frac{ mg }{ 2\pi \rho U_{ac} V_0 } }$$ (3)

where $m$ is the aircraft mass, $g$ is the gravitational acceleration, $\rho$ is the air density, $U_{ac}$ is the aircraft speed, and V₀ is the measured initial vortex descent rate.

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 = y_0s - y_0p. 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

Note that there is general agreement between b_0direct and b_0indirect and that these are almost always smaller than the b₀ 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 V₀, the b_0indirect values are used in the modeling efforts. Thus the b_0direct values only serve as a consistency check.