Test of sidenotes: this is it.
Although there’s a lot of “science” to be done by going around and “finding” problems, it often ends up being a bit like Don Quixote—we imagine demons where demons are not, and ignore the elephants stepping on our toes. So when we pondered what prevents certain hard engineering problems from being solved, it became clear that it was often not from lack of quality analysis, but instead because not all aspects of the proposed solution worked in conjunction. Instead of thinking “what can’t we do?” and trying to fix that, we so often think, “what can we do right now?” and try to start from there. It’s obvious why this limits thinking.
In this particular case, thinking about why things don’t work and why they do made a lot of things clear. For example, when we wanted to simulate an entire airplane from tip-to-tail, it is much more illustrative to think about what makes this problem hard than about what can be done easily.
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For example, lack of resolution in computational fluid dynamics (CFD), can’t possibly be the problem, since we actually have pretty good CFD of whole vehicles at this point. For the same reason, it can’t really be lack of good finite element methods (FEM). These points could be argued, of course, but it’s pretty clear that the tools exist and yet haven’t changed the design of airplanes much since they became available.
Proof.
Let’s assume that they had changed things. Then either the 707 was an almost perfect design in the first place, or there is something else keeping us from advancing to new airplane designs. Although I think the 707 - 787 airplanes are fantastic, I somehow doubt that the decisions that a small number of airplane designers made over 2 years in the 1950s were magically correct and untouchable by the thousands of future Boeing employees with years more time on their hands.
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Instead of beginning with what we can do and building from there, we began with what was needed and tried to figure out how to supply that. The problem of simulating an entire airplane wasn’t really being held back because of lack of simulation tools or because the tools didn’t have enough resolution. In fact, there have been a number of reasonably successful aerodynamic simulations of whole airplanes. The technology has arrived–we have the right numerical techniques and the computers to do the calculations in a reasonable amount of time. The one thing that is still missing is a holistic approach, one that connects the wires from all the different analyses that are available and creates a useful whole from them. Just to be clear, not everyone agrees that simulations of whole airplanes can be done accurately. Some people claim we’re not even close, and some claim that it’s already being done. But this amount of disagreement itself indicates that the problem is not a “solved” one.
Let’s think about what would be required from a solver we wish to use to do initial design of an airplane.
- It has to give solutions everywhere in the physical domain.
- The results have the predict the same trends as higher fidelity analyses.
- It should be “multi-fidelity” so that models can be exchanged for better ones as required.
- It has to have enough accuracy that the overall solutions are credible, and the pieces have to all have comparable accuracy.
- It has to be fast enough so that a large number of solutions can be computed in order to survey the design space.
That’s about it. There are a number of ways to accomplish each one of these objectives. For example, different types of curve-fitting or surface-fitting methods are very fast and have exactly the same trends as the higher-fidelity analyses they are based on. They don’t, however, give solutions for parts of the domain outside of the range of the higher-fidelity simulations. On the other extreme are “engineering estimate” type methods, such as using drag coefficients and thrust curves. The problem with these is that they aren’t always physical outside of a certain range of conditions.
Instead of doing any of these conventional approaches, our method has been mainly to use the same full, physics-based conservation equations used to develop high-fidelity codes, but to simplify them as much as was required to get acceptable simulation time. This is essentially the classic computing problem, where you have essentially fixed resources (allowed solution time and speed of computer) and you do the best you can with the resources you have available. This is fun because it hearkens back to the early days of scientific computing (and if you want to read a really interesting article about what effect limited resources can have on creativity, among other things, read You and Your Research.). In our case we chose to design the code for a single computer so that the code could be run on a desktop, and we decided that the code had to run in about a second. The code is entirely written in MATLAB, so it has been a challenge to get it to run that fast. We eventually had to relax the requirement so that it now runs in about 3 to 30 seconds depending on the exact conditions.
The inspiration for making the code run fast was basically that we were trying to design a system that hasn’t really be built before, a hypersonic airplane. There aren’t any really successful hypersonic planes flying around (although there have been a few moderately successful tests), so it’s not a problem that’s already solved. Our research sponsors (Air Force Research Lab) correctly realized that we needed to get stability and control, and well as optimized, full-vehicle design earlier into the design process. In order for the control and optimization studies to be worthwhile, they also have to be consistent with the higher-fidelity analyses that will be done later in the design process. Our physics-based approach satisfies this condition.
Our model is different from a true reduced-order model like proper orthogonal decomposition because conservation equations are applied. However, applying a full computational fluid dynamics (CFD) simulation would be prohibitive, so it was still necessary to restrict the size of the problem at each step in order to keep run times small. For example, in many places we eliminated spatial dimensions, so that only 1 or 2 spatial dimensions are used, as appropriate. One interesting thing about writing a code this way is that in a mass averaged sense, the lower-dimensionality solutions can be just as good as 3-D ones. It is difficult or impossible to predict local properties such as maximum heat transfer or locally extinguished parts of the flame, but it turns out that macro-scale properties such as pressure, Mach number and velocity are predicted to good accuracy. I attribute this to the use of full conservation equations throughout. Although solutions with different resolution and different numbers of spatial dimensions probably will not agree in the particulars, they must agree on mass, momentum and energy conservation.
One area in which high resolution and 3-D effects are very important is in combustion. Our model does use a reduced-order model (ROM) for to pre-compute combustion with a high-fidelity simulation. However, we have found that although there is some loss of resolution due to the fact that we used a reynolds-averaged navier-stokes (RANS) approach, this method gives acceptable performance for macro-scale properties. Since our principal goal is to compute forces, which depend only on thermodynamic properties, this approach was sufficient. In fact, in our comparisons the predictions are not any worse than those from commercially-available codes, and they took seconds, rather than days, to produce.
Another major benefit of using a low-order model is that it can be easier for a human to recognize and understand the cause-and-effect relationships between design choices and vehicle performance. Of course, this can be done with high-fidelity methods too, but it is more difficult since the results generally must be manipulated and this seems generally not to happen. Even if a CFD solution is accurate, if it does not yield greater understanding it can be useless. With a limited set of bulk parameters we have been able to, for example, see why certain combustor geometries are stable, predict conditions under which the flame will go out, predict conditions under which inlet unstart will occur, etc., and trace these performance characteristics back to design choices.
This is not to claim that our code is a replacement for the good work being done in CFD by numerous groups. We need more accurate CFD for combustion and we need faster CFD for internal flows, and these are great goals. Our approach has been to develop model reduction methods to get the maximum fidelity allowed by our running time constraints in order to create a unified, full-vehicle solver that could produce credible results for performance of a hypersonic airplane. Along the way, we have found that properly construction low-order models can still be of use in understanding the physics observed in experiments, in creating optimal parametrized designs, and in considering control and operability early in the design process.