No More Convincing Story
Partly so I can remember how I did it, partly to share with people who are interested, this is a rough breakdown of how I did a
Whitcomb Area Rule analyses of the airplane I recently designed in 3D, which I'm calling the Sapphire Edge.
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Fig. 1 - Early render of Sapphire Edge
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But, immediate disclaimer: I am an artist. I am not an aeronautics engineer. I'm not any kind of engineer. Technically, I failed out of my engineering program in college. Ended up becoming an English major, so that I could write science fiction and not have to worry if the math works.
So, all of my designs are art and story first, practicality second. However, I seek to make them look as convincing as possible by putting as many real-world details and principles into the designs as is reasonable. To make the design feel real. To give the illusion of living in a real place. Once you start designing things for the entertainment industry, you learn how important doing this is. You must avoid the trap of designing something in a certain way simply "because it looks cool."
"The five most important factors in animation are story, story, story, storytelling...and story." - Brad Bird, director of The Incredibles.
Everything must be story-driven, and there is no more convincing story behind design than solid engineering fundamentals. That is why I sought to find a way to analyze the cross-sectional area distribution for my aircraft designs. I had learned about this principle at some point, probably when I was doing deeper research while designing aircraft in Kerbal Space Program. I literally spent hundreds of hours in that game designing and testing aircraft to be capable of reaching orbit (Steam says I've spent 740 hours in the game).
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| Fig 2. Schematic of The Elmonica (the spaceplane I designed and used most in KSP) |
Kerbal Space Program lets you cheat a lot, allowing airplanes work even when they don't follow all the rules. Still, it taught me some of the basics.
Eventually I wanted to use the Area Rule on a craft I was designing in 3D (the Apodidae), but didn't know how. I thought maybe I could use a script inside Fusion 360, but remember when I talked about flunking engineering classes? Yeah, that included programming courses. Coding is not my favorite.
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Fig. 3 - Render of The Apodidae
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I finally decided I would just do a very rough estimate and do it long-hand. By chopping a craft up into slices.
The Rule
But, I'm getting ahead of myself. I should talk about what the Whitcomb Area Rule actually is. A little anyway, because the engineering and math it involves is way deeper than my understanding.
Primarily, it has to do with cutting through the air with the least amount of trouble as possible. At transonic speeds (which starts from speeds right before hitting the speed of sound to speeds right after you've hit it), going through air is like dragging a fork through pudding. The biggest problem getting through isn't the friction of hitting the air, forcing it to change direction to move around you, all that business. No, it's the pressure you build up by compressing the air as it's forced to move around you.
Imagine your airplane could be twisted into a ring, its nose kissing its tail. This is a very silly analogy, but it will help make things make sense. Now, you move the craft through a bucket of water, rotating it through from front to back just enough to submerge that lowest part at a time (Fig. 4).
Watch the water rise. If the craft had good cross-sectional area distribution (well, before you bent it into a doughnut), the water will rise and fall with a smooth, even rate. If the area distribution is not smooth, the water will rise and fall in jerks. However, if you think about this, the actual shape of any cross section DOES NOT MATTER. The water will fill around the section submerged at any time without consideration of shape. It is only the smooth transitions between the surface areas of those cross sections that matters.
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Fig. 5 - Simple breakdown of how the Area Rule works. Image credit unknown
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For a craft to operate at trans-sonic speeds, it has to have a shape that displaces air around it at a smooth rate. Because this displacement is an issue of the pressure of air all around the craft, the shape of the cross-section doesn't matter.
When the Area Rule was realized, aircraft designers started tacking on bulbous bodies to the backs and fronts of wings and indenting the fuselages where the wings were longest. They discovered after these alterations that the planes were now significantly faster and more efficient.
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| Fig. 6 - YF-102 and YF-102A...one on left, without body indent, could not go supersonic. Image credit: NASA |
New aircraft were then designed with engines and cockpits and landing gear all strategically placed to achieve the smooth transitions.
Curvy Math
So, as I'd mentioned, I had entertained the idea of using the Area Rule on my Apodidae, but wasn't sure how to do it. I'd even asked on an Autodesk forum how I would go about writing a script in Fusion 360 for generating a cross-sectional analysis of a model. But, it was all overwhelming. I decided to not worry about it for the moment.
Months after finishing the Apodidae, I started working on the Sapphire. And I had the idea to forgo fancy coding and just do it rough and dirty. So, as an approximation analysis, I sliced up a craft into 1-meter chunks.
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Fig. 7 - Sapphire Mk 1 sliced up into 1m sections
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I selected each section's properties in Fusion and wrote the volume of that section down in Excel. (Instead of turning it into a doughnut and running it through water, I'm
basically cutting it into bits and putting each into the water one at a
time). I ended up cutting the craft into 23 sections, so this gave me a pretty useful curve of the data for Excel to turn into a data chart for me.
It was clear that I did not have smooth distribution. You can see in Fig. 8 that significant deficiency in surface area around the 12 meter mark, plus a bunch of other troubled spots.
But, I wanted a standard to compare my curve to, so I could better plan how to improve things. And that's where I made a mistake.
On one website, someone said that, to have an ideal area distribution, this curve should look like a normal (or "bell") curve. This was based on an
out-dated belief that ideal cross sectional distribution should follow something called the Sears-Haak body. So, I used the NORMAL function Excel has and carefully generated a normal curve that closely matched my bad curve.
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Fig. 8 - Sapphire's distribution vs. normal distribution (with differences listed as percents)
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Then I went through a long process of trying to add more surface area to the segments in my design that were lacking or to remove surface area to areas that were excessive. I did a major reshaping of my original rough mesh in Blender (I almost always start my designs in Blender with very simple geometry, then I import them into Fusion). I made the cockpit higher. I stretched out the leading corner of the inner wings, making them almost into strakes (a common design feature of supersonic aircraft).
Then I sliced it up in Fusion and did the analysis all over again (Fig. 9).
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| Fig. 9 - Mk1 is blue, Mk2 (second analysis) is green. |
But I didn't fix things as much as I had hoped. I still had that same dip at the 12 meter mark. SO, I added antishock bodies. Those little bulbs between the fuselage and the engine nacelle (seen in Fig. 10) served that purpose, but I put nozzles all over them to turn them into RCS modules (the things that shoot little jets of gas to let you change orientation in weightlessness).
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Fig. 10 - Mk2, after aerodynamic fixes, including antishock bodies added on inner wings and pylons addded to the ends of the outer wings and the vertical stabilizers.
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Those fixes were all warranted. But some of the other fixes I tried were possibly unnecessary or even misguided.
Apparently the study of what the "ideal" area distribution body looks like is a deep topic (I'm not even sure what to reference for this, because I don't understand the two main documents that talk about this...
such as this article and this
report from NASA).
First of all, I technically should have been studying what kind of shape a craft should have for hyper-sonic flight. Not trans-sonic flight. Because to make it to orbit, an airplane would need to be able to break Mach 5 while in atmosphere. That's a big problem, because at that speed air isn't just hard to cut through but completely crumbles apart.
Also, you just have to look at a few super-sonic craft to realize that they do not have a normal distribution of their cross-sectional area. They are all biased toward the rear. Think of the SR-71. Or the Concord. Or the space shuttle!
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| Fig. 11 - SR-71, with clear bias in cross-sectional area to the rear. Image credit: NASA |
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| Fig. 12 - Space shuttle, which has bias to the rear, but you can ALSO notice how the external tank is positioned forward of the orbiter and boosters, to make sure there's a smoother transition of cross-section. Image credit: NASA |
In fact, it seems that it's standard practice to have the back end of the distribution curve end abruptly. Which is what mine has. Because I was designing my craft to FEEL like a super-sonic craft, I instinctively ended up matching this practice. What was most important all a long was smoothing out the transitions between cross-sectional area values.
But, because I was confused by that website showing a normal curve, I tried to nudging my design's distribution to be closer to normal. So I added extra-long pylons to the ends of the wings and the vertical stabilizers (that you can see clearly in Fig. 13).
But...now I kind of like them And, my airplane isn't real. It's just a piece of 3D art. So...I can cheat. I can decide to just leave it. I WILL BREAK THE GREAT RULE OF DESIGN AND DO SOMETHING BECAUSE IT LOOKS COOL.
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Fig. 13 - Sapphire Mk 2 from side.
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Come at me, design police. You'll never take me alive. Because I'll be flying up to SPACE.
More Guidelines Than Actual Rules
Also, those pylons on the ends of the wings actually serve an important design purpose, so it's fine. Virgin's SpaceShipTwo has these
fancy rotating wings that help stabilize reentry, and I sort of mimicked a little of their design. However, SpaceShipTwo doesn't reach actual orbit, so it doesn't deal with full-on reentry velocities or temperatures, so it may not be the best design to steal.
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Fig. 14 - Virgin Galactic's SpaceShipTwo, with wings in glide configuration. Image credit: Virgin Galactic.
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But, again, we run into the reality that my plane is all pretend. My design would never work anyway, because it's too skinny to carry enough liquid hydrogen to burn into full orbit.
I'll work on fixing many of these issues when I design my next pretend space plane. That one will probably be more of a cargo-sized craft, instead of being fighter-sized, like the Apodidae and the Sapphire Edge. It will probably end up looking more like my clunky Elmonica from KSP.
Which I'm actually really looking forward to. I may have to write another piece on what I learn during that process.
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Elmonica, taking off in Kerbal Space Program
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