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.


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.

Corporate Sponsorship Pays

Pretty soon, the useless U.S. yellow journalism/infotainment-lobotomy casserole will inform us that the Supreme Court has “made a decision” that Hobby Lobby (which is a corporation, and corporations ARE people, my friend) is within its rights to refuse to allow those filthy whores to get their whore pills on Hobby Lobby’s dime, nosireee Bob. They will then act as if it had been anything but a fait accompli long before any of the interested parties brought it to this level. A few years back, interested parties provided the ridiculous Citizens United case as an absurdly over-the-top flouting of the Constitution that demanded immediate action. The Supreme Court had, at that point, been appointed largely by purchased politicians of the past few decades. The strategy of purchasing politicians paid massive dividends to the original investors. With the foregone conclusion of the Citizens United case, effectively legalizing bribery as an branch of political campaigning, the path has been forged for the long fight to completely unravel the ideals presented in the Constitution.

Establish the primacy of corporations over people.

Put “those people” back in their place.

Overrule Separation of Church and State.

The sentences above appear on the manifestos of a disquietingly high percentage of “conservative” power brokers.

By the way…

Why didn’t Hobby Lobby ever raise objections to health plans offering Viagra coverage?


Not So Much on Expertise as Science

Ok, so I am easily distracted. I seem to have gone off the claimed topic a bit in my previous post. This time, I will actually attempt to stay on point.

Expertise is an important indicator of the soundness of a science. There are a number of metrics by which we can measure degrees of expertise. The difficulty is in ascertaining the consistency of the parts. That is to say that if a discipline is real, its practitioners should be able to readily identify experts in the field while just as easily dismissing charlatans out of hand. Another feature which we generally overlook is that the tendency is for real science to arrive at solutions independently from different directions and in agreement. This is the case, that is, for mature sciences. In newer topics, we expect merely that disparate groups or disciplines are converging towards a solution or model.

We generally find that new publications do not disprove old theorems. Rather, they propose refinements. Adjustments of models over decades of rigorous science is expected. Overturning of well-established scientific principles, on the other hand, would require an enormous amount of incontrovertible evidence inconsistent with theory.

The Existence of Expertise

On easy way to see if your pursuit is real or bullshit is the existence of experts. People in science often see it as a battlefield wherein truth is approached through bitter struggle between competing factions supporting incompatible models of reality.

On this, the fools and leaders of fools capitalize.

Unlike silly detective shows, the world does not really lend itself well to discovery. Large effects have been known since ancient times. Think of these as a first order approximation. Less obvious effects took more time, but are still old by most standards. These are second order approximations. Then, through numerous iterations, we achieve finer and finer approximations of reality. This, now, is where we find ourselves. Scientific argument has reached the point where, in most fields, academic disagreement is on scales which are not accessible to those outside narrow focal areas. My late thermodynamics professor (if you’ve used dive tables, you have used his work) said with some pride that an important constant had been changed within his lifetime. He had been born in the 1920s, and I think that it may have been Avogadro’s number that had changed in the third decimal place.

I wasn’t taking notes, sorry.

Simulations I create of movement of objects through the atmosphere rely on work done from the 17th Century to the early 20th Century. From Galilean relativity and Newtonian mechanics and gravitation, through a rotating frame of reference (Coriolis), to simple models of atmospheric resistance originating with Bournouilli and standardized by Prandtl. More complex models exist to account for geographic variation in gravity, and to more closely approximate acceleration due to drag, and to incorporate gravitational effects of the sun and moon, and include relativistic effects. For real-world simulations, however, difference in estimate impacts between the two are small.

That is where science is. As far as modeling the world, it is being done deep in the decimal places. Discoveries and inventions still appear and do improve our understanding of the world, but not very much of it noticeably alters our lives. A 10% improvement in energy storage per unit of mass (say in batteries) would be dramatic from an engineering standpoint. It wouldn’t really make much difference to an individual, though. Your iPad would be slightly lighter or last slightly longer.

We have reached a point where any person whose model would overturn major established theory can be dismissed as a crackpot out of hand. Absent overwhelming supporting evidence, we can safely dismiss any such theory out of hand.

I Understand the Obsession

Long, long ago, a little known spot on the body journalistic happened upon a new World War in the making. Powered by such luminaries as Peter Arnett and Christiane Amanpour, the world populace was informed of the goings-on of a then little-known patch of the world known as Iraq. Fast-forward what seems like a hundred years and you will find a smoldering shell of tear-inducing toxins presented by the comically dim (and pornstar-named) Wolf Blitzer. This enterprise has reached the point where any group of reasonably intelligent, college educated people could outperform the entire cast (i.e. non-technical personnel) of CNN in presenting a greatly superior product. You could limit their instructions to 1) require strong evidence for reporting, and 2) discard ridiculous bullshit.

… but I digress…

We all want to know what happened to, say, Malaysian Airlines Flight 370. But this craving for news won’t be sated with the fluff and bullshit proffered by CNN’s clown show. While they do not intend to mislead (see propaganda organization Fox News), they must be aware that any “reporting” done on this subject is as substantial as cotton candy (and equally nourishing). If they were interested in real reporting, they would add minor updates to a piece covering all known details. They would not elevate each crackpot theory that comes along to headline “BREAKING NEWS”.