This article is going to tick off a fair number of people, but it can't be helped. There is just too much misinformation out there about mid-power rockets and Mach 1 for me to feel comfortable, so I'm going to try and clean some of it up. First, I'm going to talk about what (and, incidentally, who) has actually been supersonic, but only on my way to showing you how to get there yourself.
Let's start with the poor schmucks at the H.A.R.T. Rocket Club in Europe; they think they're the first to have gone supersonic there, but I'm afraid I've got bad news for them. First, European amateurs have gone supersonic as early as 1992. Second, H.A.R.T. hasn't. There should be TWO distinct noises, one on the pad, the other anywhere from about 200 to 500 feet in altitude. Here's why.
In order for a rocket to reach any particular velocity, v, the exhaust gasses from its engine must have a velocity greater than v. This means that a supersonic rocket must have a supersonic exhaust. (Black powder can't supply that, but ammonium perclorate can.) Therefore, the first noise you hear, a monstrous "BANG", is NOT the rocket going supersonic, but its exhaust! The second noise, created when the rocket actually breaks the sound barrier and the shock wave attaches to the nose cone, is much different; a very audible but softer "pop". If you don't hear two distinct and different reports, your bird hasn't been to Mach 1.
Look, don't be discouraged, this isn't something that's easy to do. It took the aircraft industry years to recognize that the barrier existed, years more to believe it could be broken, and years more to actually accomplish it. We're starting with the advantage of all their hard-won knowledge, but knowing the barrier exists and can be broken is only two-thirds of the battle. If you're going to "bust Mach", you need the other third of their data. The supersonic exhaust requirement I talk about above is one of their datum, and in this paper I'll try and give you the rest.
Rocket Vision Hasn't, Either
Another common misconception is that Rocket Vision's Mach Buster rocket is supersonic. The manufacturer (as well as at least one user) claims that this bird goes Mach 1.2 on a G55. The term "supersonic" refers to speeds in excess of Mach 1.2, not Mach 1. Speeds between Mach 0.8 and Mach 1.2 are called "transonic". If the Mach Buster were capable of Mach 1.2 it would be supersonic just barely, but it ain't; here's why.
I've tried to think of a nice way to say this, but I can't; the claim that the Mach Buster gets to Mach 1.2 is just plain wrong. Calculating the Mach Buster's performance with DigiTrak (or wRASP in DigiTrak mode) indicates that she will only reach Mach 1.1. Still transonic, mind you, but less than claimed by the manufacturer, and definitely notsupersonic. (BTW, the Mach Buster does a respectable Mach 1.1 on an F72, no worse than the G55.) (You used to be able to download Rocket Vision's own Mach Buster wRASP file and try it yourself.) Before you hit Launch, select Options, Calculations and choose DigiTrak, and then Data, Rocket and turn on the correct for mach effects (DigiTrak) checkbox. (It's also informative to revise the drag coefficient to a more realistic .6 and rerun the simulations.)
Does the Mach Buster "pop"? You betcha. Is it supersonic? Technically, no, but it flies over Mach 1, which is what people really mean to say.
Mach Is a Real Drag
Why the disparity? Because the effects transonic speeds have on a model's coefficient of drag (CD) were not taken into account. As a body approaches Mach 1, it's effective drag coefficient increases markedly as it begins to generate a shock wave. This wave is initially out in front of the rocket a good distance, but as speed increases, it gets closer and closer, until at Mach 1, it is actually touching the nose cone. This shock wave creates additional drag, effectively increasing the coefficient of drag of the rocket. This is called "compressibility drag". DigiTrak takes this increase into account by adjusting the CD of the rocket based on its speed.
Mach Buster Numbers
Another simulator, the Model Rocket Altitude Predictor, also takes compressibility drag into acount, though only as a rough approximation, and comes up with slightly better numbers for the Mach Buster; Mach 1.13 on a G55 and Mach 1.18 on an F72 with a .4 CD, and 1.03 and 1.08, respectively with a .6 CD. You'll need the values out of the conversion table at right when you try this yourself. I don't own RockSim, Apogee's performance prediction software, but a demo copy of version 4.0, which also takes compressibility into account, comes up with similar numbers; Mach 1.11 on a G55 and Mach 1.2 on an F72.
Shock waves begin to form at different speeds due to a number of factors, most of which can't be efficiently modeled. For pointed bodies (like rockets) these speeds almost always lie between Mach 0.75 and 0.95, though, and DigiTrak allows you to select either 0.8 or 0.9 for the onset of Mach effects. As the speed at which the shock wave begins to form only has a small effect on overall performance, these two choices are generally enough to get your model close to the mark.
How bad is this effect? Bad. Really bad. A rocket's CD is approximately doubled (e.g., from .6 to 1.2) at Mach 1.1. It is imperative that a rocket's subsonic CD be as small as possible. Major contributors to drag are launch lugs, unpainted paper or wood surfaces, blunt fin edges, fin flutter, etc. All of these things must be avoided at almost any cost, and I'll tell you how later.
Wider Is Not Better
Besides velocity, the amount of drag on a rocket is a combination of only two other things; its CD multiplied by its frontal area. To combat the effects of shock wave induced drag, it is also extremely important to have the rocket present the smallest cross section possible, and that means the smaller the diameter, the better. The Mach Buster has a 1.2" diameter, which is very small for mid and high powered rockets, but not the smallest possible. A minimum diameter rocket would be 24 mm, no bigger than the diameter of the motor, but attaching fins to this bird is problematic. The battle soon becomes one between structural integrity and diameter. The Mach Buster represents a good trade off in this area, giving away some frontal area which it attempts to regain with an exceptionally smooth outer surface, reducing its subsonic CD.
And It Gets Worse
Mr. Barrowmann was real specific that his equations for determining a model rocket's center of pressure (CP) are only to be relied upon under Mach .5. As a rocket approaches Mach .5, its CP shifts rearward slightly, making the bird more stable, but beyond Mach .6 or so, it begins to shift forward, a very bad thing. How much these shifts are depend on the shape of the rocket, and if you have a bad one and it doesn't have enough margin to start with, it'll go unstable at the worst possible time. No matter how well built, sideways flight near the speed of sound generally isn't survivable.
So which shapes are bad and how much additional margin should be designed in? A lovely piece of software, AeroLab, put out by DARK, the Dansk Amatør Raket Klub (Danish Amateur Rocket Club), handles all the complex math for the change in CP .vs. Mach number. It still has a lot of bugs, but it does a great job of displaying both CP and CD changes versus speed. After playing around with it a bit, I pinpointed something that really affects CP shift: fin trailing edge taper. Take a look at the graph below; it plots CP location versus Mach number for various trailing edge tapers - 1% of full fin cord through 40%:
The dashed red line represents the calculated Barrowmann CP location, which helps explain why some model rockets aren't stable when tested with the "spin it on a string" method, but are in flight. As you can see, CP location above Mach 1 isn't affected much, but in the subsonic and lower transonic ranges, it can make a big difference, in one case over 5% of this 1700 mm model's length. There are lots of other things that affect the location of CP versus speed, download AeroLab and check your design before you fly.
So let's talk about ways to hold down CD on a rocket. The rec.models.rockets FAQ only has a few brief words to say on the subject:
The best thing you can do is to NOT use launch lugs. Use a launch tower instead. A polished, smooth finish makes a big difference too. If the design allows, use a boattail and make sure all transitions are smooth (from nosecones/payload sections, etc.). Fin shape is a minor effect if they are relatively thin, otherwise make sure the edges are at least rounded.
For supporting numbers for these facts (and others), check out Model Rocket Drag Analysis. All good advice, but little in the way of achieving these things.
First, let's address finish. Stay away from paper or wood surfaces as these are nearly impossible to get as smooth as plastic. For the body tube consider thin walled PVC or phenolic resin tubes, and for the fins think about fiberglass, fiberglass coated plywood or graphite/epoxy composite. For the nose cone, ABS or polystyrene works well. Put alot of time into painting your model; start with a primer, use a high-quality enamel, and finish with clear coat. Use multiple, light coats of each, sand between each coat with progressively lighter paper, use a tack cloth, and hand rub down the finished model with a piece of cotton or dacron cloth. Eschew decals as these make bumps in the surface.
Transonic mid-power rockets don't need a launch rod at all; the rod is there to keep the rocket stable until its traveling fast enough for the fins to bite in air. A transonic rocket gets up to that speed in a heartbeat, usually in about one foot of travel, or less than .1 seconds. Still, the Model Rocket Safety Code says you gotta have something, so I'd suggest a short launch tower of only a foot or so.
Boat tails are usually a bad idea; most transonics are nearly unstable anyway, and adding a boat tail usually increases fin size so much that flutter soon becomes a problem. Plus, they add an additional transition, not a good thing. Speaking of transitions, body tubes should be one piece, and you should sand the joint between the nose cone and body tube to flush before finishing the rocket. Finish the rocket with the nose cone in place.
Fin flutter is often the most overlooked cause of drag. Most fin materials are stiff enough to remain motionless at speeds less than Mach 0.5, but above this speed, the turbulence over the fin surface can cause it to flutter wildly, sometimes to the point of destruction. The smoother the finish on a rocket, the less flutter, but it still must be accounted for in your design. Getting back to H.A.R.T., you'll see that the fins on their rockets are quite long and thin. As you read their launch reports, you'll see what happens to fins like that; they flutter so badly that they are ripped right off (or out of) the body tube.
Fins for transonic rockets should be made from fiberglass, fiberglass coated plywood or graphite/epoxy composite for stiffness to resist flutter. 1/8" G-10 fiberglass or 1/16" graphite/epoxy (e.g., M55J) are good places to start. The design of the fin should place the entire tip cord inside the root cord; in other words, the sweep plus the tip cord length should be less than or equal to the root cord length. The best design is a triangular fin, i.e., one without a tip cord at all. Fins should be as thin as possible (again to help limit cross section), but should have a leading and trailing edges as sharp as possible.
Just like balsa or plywood, multi-material plastics have different bending moments along different axes. Cut your fins from the raw material in such a way that the stiffest axis lies perpendicular to the line of flight. Flutter makes a fin move from left to right and back, so the stiffer it is in this direction, the better.
For reasons I don't completely understand, as rockets approach the speed of sound, their angle of attack oscillates unpredictably. None of the performance predicition programs supporting transonic speeds I know of can model this effect accurately, primarily because there is no totally adequate drag divergence model that can predict these angle of attack variations. If the sound barrier is traversed quickly, the "flutter" encountered is tolerable, but mid-power transonic birds usually spend a fair amount of time there. Just about all you can do is build your bird sturdy enough to handle a little of this and pray. The oscillations typically range between 5 and 15 degrees, so she's not going to turn sideways or anything.
Holding down on diameter is a lot tougher than getting a good starting CD. The worst part is attaching the fins, which are going to be doing their best to get off the bus in a hurry. Generally, integral fins, i.e., fins that are made from the same piece as the body tube, are good, but not as good as through-the-wall fin mountings. Needless to say, the smaller the diameter of the rocket, the thinner the tube wall, and the weaker the through-the-wall attachment becomes. Some interesting fin mounting systems have been developed, but at some point they all rely on the structural integrity of the body tube, and thin tubes don't have much of that. Its a lot easier to give a little away on cross section and attempt to make it up in finish.
Small diameter tubes are just about impossible to find, until you start looking in non-rocketry places. If you have a good lathe, you can bore ¾" PVC pipe or 1" fiberglass antenna spreader rod out to 24 mm. Lexan pipe can be had in thin walled configurations, though it, too, will have to be bored. 24 mm motor mount tube is not a good idea; the surface finish will be worse than plastics, and you'd have to get a custom nose cone made (or turn one up yourself).
Nose Cone Shape
Speaking of nose cones, what shape should they be? In an unpublished paper by Gary A. Crowell, Sr. called The Descriptive Geometry of Nose Cones, he explains that, for the transonic region, conical and ogive shapes are the worst, while rounded tip shapes, such as eliptical and parabolic, are better. He says that this runs counter to "conventional wisdom", probably because we've all seen (pictures of) supersonic sounding rockets with conical nose cones. These are for high supersonic and hypersonic work, though, and these rockets blow through the transonic region in fractions of seconds, much the way our transonic birds hustle through the subsonic region. The best are nose cones specifically designed for supersonic work, the Haack series (of which the Von Karman, the best, is a special case), but there aren't any of these commercially available to the hobbyist.
He also explains something else that is a bit counter-intuitive, that slightly blunted tips on any of these shapes are better:
"While most of the nosecone shapes ideally come to a sharp tip, they are often blunted to some degree as a practical matter for ease of manufacturing, resistance to handling and flight damage, and safety. This blunting is most often specified as a hemispherical ‘tip diameter’ of the nosecone. The term ‘Bluffness Ratio’ is often used to describe a blunted tip, and is equal to the tip diameter divided by the base diameter. Fortunately, there is little or no drag increase for slight blunting of a sharp nose shape. In fact, for constant overall lengths, there is a decrease in drag for bluffness ratios of up to 0.2, with an optimum of 0.15. Whether by design or coincidence, most commercially-made tangent ogive hobby nosecones are blunted to a bluffness ratio of about .1."
Rounded Nose Shock Wave
Pointed Nose Shock Wave
Most Estes ogives have a slight blunting of the tip, some of which, like the one found on the Banshee, have bluffness ratios of .15. Their tips usually take the form of an added hemisphere (as opposed to the flat tip found on bullets and artilery shells, called a Me'plat diameter) which is one of the best for reducing transonic drag. Surprisingly enough, Estes also makes a nose cone that is nearly Haack-shaped, their PNC-55AC found on the AIM-120 AMRAAM and Bull Pup 12D semi-scale models. This part is 1.325" in diameter, though, and requires a G80 motor and some exotic materials to get over Mach 1. While nose cone shape is important, it is secondary to a small CD and diameter.
One last note on nose cones; their shape also affects CP location during flight. Unlike trailing edge taper, nose cone shape affects CP in the upper transonic and supersonic regions, too. The only thing changed on the rocket in the graph below is the shape of the cone, everything else, including cone length, is the same:
As it turns out, weight isn't much of a consideration, either. If you follow these guidelines, you'll wind up with a relatively light rocket anyway, but anything under four or so ounces dry has a chance with 24 mm motors, and you can go up to 5 ounces or so on 29 mm motors. In preparation for my Red Lightning Mach buster, I did some studies with wRASP in Digitrak mode and developed the graph shown below. In it you can see the relative impact on speed weight has versus diameter:
Weight does has one significant consideration. The more a rocket weighs, the harder it is for drag to steal speed when the motor thrust curve starts to drop. Therefore motors with "spikey" power curves (what the high power fellas call "regressive" motors), such as the G55, need a heavier bird to get beyond Mach 1, but engines with flatter curves, like the F72, can fly lighter rockets due to their more constant acceleration. The ideal motor would be "progressive", i.e., its thrust increases during its burn, but none of the motors us lowly mid-power users have available to us exhibit such a curve.
And just what motors are those? I won't bore you with the gory details, but there are many ammonium perchlorate motors with supersonic exhausts. Reloads are generally too low impulse or too heavy to be of much use, so we're limited to the single use motors. There are four that stand out as having the best chance of making it to Mach 1, the F72, F101 and G55 in the 24 mm catagory, and the G80 in the 29 mm range. You might think that the G80 would be best, but it requires a larger diameter bird, say around 1.3", and the wider cross section takes away all you'd gain in impulse.
I Haven't Been, Either
My White Lightning rockets attempted to keep the diameter down to about 1" or 25 mm (0.985" to 1.08", or 25 to 27.5 mm) while using more and more exotic materials to keep the rocket in one piece during flight. To be quite honest, I was just like most "Yeager-meisters" when I started; I didn't know much about transonic considerations, and my designs reflected this. I was overly concerned with diameter, ignored fin flutter and took surface finish lightly. When I wised up and started doing some research, I wound up with a design that looks very much like Rocket Vision's Mach Buster. The latest version's still got some bad idea "hang over", like a paper body tube and exotic materials to make up for a tiny body, but it should work ... "pop" nice and loudly on an F72.
White Lightning II
White Lightning II was a BT-50W with 1/16" plywood fins epoxied to the surface, pointed nose cone from an Alpha, and no paint at all. It shreaded on an F72.
White Lightning III
White Lightning III was a BT-50B from an Alpha III mated to a bored out fin canister from the same bird with a 1" paper tube and epoxy. The nose cone was the same one used on WL2, and the transition was done with balsa filler. It had two sparing coats of Krylon white. It also shreaded on an F72.
White Lightning IV
White Lightning IV is a complete rethink based on my research, and sports M55J graphite/epoxy through-the-wall fins, a thin walled PVC motor mount with integral engine block, and the Banshee's blunted tangent ogive nose cone that comes tipped with a hemisphere for a "bluffness ratio" of .15. Body diameter is still very low, the upper tube being Estes' HBT-1090, the same one used on their Hijax E2X model. She's painted with two coats of White Primer, three coats of Jet White and two coats of Crystal Clear, all LustreKote from Top Flite. This should fly to a conservatively estimated Mach 1.1.
Red Lightning isn't built yet, but will be my attempt at a 29 mm transonic bird since the F72 will probably be discontinued (if it hasn't been already). She has G-10 fiberglass fins of triangular configuration, uses the Estes PNC-55AC, and the same style PVC engine mount as WL4 bored from 1" pipe. I estimate that on a G80 she'll get to Mach 1.04.
What To Expect
Don't expect to get much past Mach 1 on mid-power motors; 1.1 is more than enough to hear the "pop" and prove your point. High impulse 24 mm engines like the F72 and G55 just don't have the burn time needed to get fully into the supersonic region (i.e., beyond Mach 1.2), and 29 mm HO engines like the G80 make for a larger diameter rocket and too much drag to get there, either. High power rocket kits are available that can get into the supersonic region very comfortably; Hawk Mountain's Transonic II, for example, can go Mach 1.4 and even higher!
The Drag Force on a Sphere - an evaluated Mathematica notebook that gives the basics on drag. By H. Edward Donley of the Mathematics Department at the Indiana University of Pennsylvania.
The Causes of Drag in Model Rockets - an excerpt from Topics in Advanced Model Rocketry about those drag components most effecting model rockets. By Gordon K Mandell, George J Caporaso and William P Bengen.