Vortex Wake Inverse Modeling Results

In version 1.0, the kinematic relations for the vortex lateral and vertical positions were removed from the viper equation system and replaced with linear interpolations between the measured data values. The measured data values were also used to produce a proxy crosswind. No smoothing was applied to the measured data when computing the trajectories or the proxy crosswind. In the crosswind calculation, The effect of vortex induction (for a tilted vortex system) was taken into account by making use of the computed vortex circulations. Iteration was used to allow the crosswind and computed circulations to come into equilibrium. The resulting proxy crosswind data were then curve fit with a 2nd order polynomial.

The lateral position kinematic relation was approximately satisfied since the vortex velocities were estimated from the computed circulations, and because the proxy crosswind was fit with a smooth function. No attempt was made to satisfy the vertical kinematic relation and thus it has an uncontrolled error.

After performing these runs, it was determined that the viper equations did not treat the buoyancy terms correctly and that the local atmospheric density was not computed correctly. Although these problems have now been corrected, they may adversely affect the version 1.0 results.

The main conclusion from the version 1.0 model is that the proxy crosswind estimates are too noisy to produce sensible results in most cases. Basically measurement error in the vortex trajectory data is amplified when taking the time derivative necessary to compute the proxy crosswind. The large degree of scatter often leads to unreasonable proxy crosswind profiles, which produce large second derivatives. The large second derivatives lead to unreasonable circulation time histories.

Click the link below to view the version 1.0 results.

Version 1.0 Results

Several improvements were made in version 1.1. These include

Noise in the measured vortex trajectory data was reduced at the outset through use of a third order smoothing spline.
Both lateral and vertical kinematic relations are retained within the viper equation system.
The actual computed vortex velocity was used in the proxy crosswind calculation instead of an estimated based on the vortex circulation.
The buoyancy term was corrected in the viper equation set.
The density profile calculation was corrected.

In version 1.1, the proxy crosswind calculation effectively inverts the lateral vortex trajectory kinematic relation. An exact inversion would be achieved if we allowed for separate proxy crosswind profiles for the port and starboard vortices. In this implementation, a least-squares approach is used to generate a single proxy crosswind profile. It was determined that this approach leads to good satisfaction of the lateral kinematic relations in most cases.

The vertical kinematic relation is solved in its original form and thus is not constrained by the measured values. In version 2.0 we will constrain it by computing a second proxy environmental profile. For now, errors will be present between the computed and measured vertical vortex trajectories.

Since we use iteration to bring the proxy crosswind calculation in equilibrium with the computed vortex velocities, there is no difference between a forward and inverse run. That is, exactly the same solutions can be generate with a standard forward run if the proxy crosswind profile is utilized.

The version 1.1 results are much improved over version 1.0. The proxy crosswind profiles are much more reasonable in most cases, as are the vortex circulation time histories. There are still a few isolated cases where unreasonable proxy crosswind profiles lead to unreasonable vortex circulations. These cases can be improved by constraining the proxy crosswind above and below with the measured crosswind values.

In general the circulation time histories are not all that different between the standard and inverse runs. This is true even in cases where there is significant in the computed lateral vortex trajectories. Most of the notable differences in the circulation time histories appear in cases where the proxy crosswind profile (perhaps incorrectly) produces large second derivatives. The vertical vortex trajectory is rather insensitive to the details of the proxy crosswind, and generally there is almost no visible difference between the results computed with the standard and inverse runs.

Click the link below to view the version 1.1 results.

Version 1.1 Results