Optimization of Propagation at Terminal Velocity

In modeling and simulation, a problem that has often occurred for me is the problem of low ballistic coefficient particles through the atmosphere. The oddness here is that the least energetic of debris is the most difficult to model. The debris which is considered below the kinetic energy threshold for injury is the worst of all. There is a very good reason for this.

Bounce.

When we model an object propagating through the atmosphere, the particle’s path is modeled as undergoing the influence of the following acceleration components:

1) geocentric gravity
2) centripetal acceleration coaxial with Earth’s rotation
3) Coriolis acceleration due to Earth’s rotation
4) aerodynamic drag
5) aerodynamic lift
6) thrust

Any control can, of course, be modeled simply applying drag and lift to a body of known moment of inertia in three vehicle axes in a 6 degree-of-freedom (6DOF) simulation. Most, however, can be modeled more simply in a three degree-of-freedom (3DOF) simulation.

Let’s assume no commanded directional or thrust. We will also consider lift similar to how we consider drag. Then our model should be that the acceleration due to drag (per Prandtl’s interpretation of Bernoulli) is the atmospheric density divided by twice the ballistic coefficient. Here, ballistic coefficient is defined as the mass divided by the empirically derived drag coefficient multiplied by the presented area. The drag coefficient at high velocities will be a function of the velocity, of course, but may be averaged over a trajectory to find a mean Cd.

Anyway, this ballistic coefficient or beta can be used in aerodynamic analysis.

The topic is optimization of propagation of particles at terminal velocity. What we do here is to find terminal velocity as a function of the atmospheric density. We iterate using a fourth order Runge-Kutta algorithm based on the Prantdl model of aerodynamic drag, the Coriolis model of acceleration in a rotating frame of reference, and the Newtonian model of acceleration due to gravity along with Galilean relativity and the Newtonian laws of motion.

With these things in mind, we can find the terminal velocity of an item as the square root of the velocity whose downward component is is such that the deceleration due to drag in the up direction (that is to say, the descent rate) exactly counterbalances the acceleration due to gravity. This is simply done within the propagation routine.

Next, we find the point at which the item achieves terminal velocity. This is when we change the algorithm. Now, what I suggest, is to alter the model.

From the terminal velocity time on, we simply calculate the amount of time it will take a particle of a given ballistic coefficient to transverse an atmospheric layer. We consider that the particle now takes the wind velocity over the entire altitude layer and calculate the time it will take such a particle to make the distance. In this way, the only real difficulty is in accurately modeling the fall time of the particle. I will see how this works next week.

By marc

I am a scientist interested in laser safety, modeling and simulation, and statistics. I speak some Spanish, German, and Chinese.