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These are Tim B.'s notes, which I am copying verbatim without his permission. Hopefully he won't slap me too hard.

-- [[BartMassey]] - 07 Apr 2005

# <a name="Rocket Simulator Physics"></a> Rocket Simulator Physics

These are very preliminary notes from Tim B. (writing in the first person below) on how the rocket simulator might deal with physics.

## <a name="Rocket modeling techniques"></a> Rocket modeling techniques

### <a name="Barrowman equations"></a> Barrowman equations

Linear, Slender Body, Subsonic, Steady, Asymptotic to zero angle of attack. Fail to predict variation of Cp with angle of attack. Already know what they are, understand them fairly well, ought to be able to derive them using potential theory as a jumping off point.

### <a name="Barrowman equations w/ correctio"></a> Barrowman equations w/ corrections

Same but retain some 2nd order terms or semi-empirically add in some terms from wing theory depending on how you look at it. Definitely more accurate at moderate and large angles of attack. At this point, don't understand why the corrections are valid in the form given, therefore don't have a good grip on the accuracy and range of validity. Figuring this out definitely means research and so order 2 weeks of time at the current load. Merely adding the corrections only addresses angle of attack weakness of Barrowman, doesn't address other weaknesses.

### <a name="Slender body theory"></a> Slender body theory

Classic technique (so plenty of references). Might be amenable to modern computation al 'la Matlab etc. This may have already been done and written up. _Way_ more complicated then the Barrowman approach, definitely nothing fully closed form, but maybe easy-ish to express for numerical solution? Should be reducible to fully automated solution. Axisymmetric body portion could be given as a 2D list of radius vs. z-axis points. These would effectively be linearly interpolated so the number of points required would be quite small. Fins are a considerable complication, see 2D theory below. Haven't read anything on this published after 1960, so nothing i've seen considers computer techniques. Either need to find that or derive it. Either way, will need to validate the result. (Of course that's true for any of these methods.) It will take at least several weeks to get the theory in shape.

### <a name="Axisymmetric theory"></a> Axisymmetric theory

Just pointing out that if there is an opportunity to make this simplification i'm inclined to take it since our vehicles should be axisymmetric for the foreseeable future.

### <a name="2D theory"></a> 2D theory

Fins aren't axially symmetric in the r[theta]=constant sense. Instead apply wing theory. This isn't too awful. Since we stick with very thin fins the theory is fairly simple, but it is a whole 'nother thing, which takes time. The Barrowman approach makes about the same order of error in Cp variation for the fins as for the body, but for fins that are short compared to body length (our case) the error is probably acceptable. It's not unreasonable to work out a better body theory and glue a Barrowman style fin theory on until something better comes along. Also tons &amp; tons of stuff to read on thin wings. Unfortunately nothing simultaneously very introductory and complete AFAIK, probably does exist somewhere.

### <a name="Vortex theory"></a> Vortex theory

Basic wing theory doesn't explicitly deal with strong vortices such as might form on the upper body tube at large-ish angles of attack. There are fairly tractable classic approaches for this though, and many texts that address them. The vortices become important when modeling behavior above ~10deg. angle of attack. Again it's a whole other thing, which will take time to work out.

There might be something simplistic out there that would handle the un-steady aspects too, which would be relatively exciting.

### <a name="Wing Body Interference"></a> Wing Body Interference

All practical analytic theories that i'm aware of solve the body separate from the fins then stitch the results together using an interference factor something like

         C_total = C_body + C_fin + C_interference

Usually the interference is chiefly due to the transverse flow from the wing against the body. With 2D theory in hand interference is a small complication, but it is also a possible source of error because fully analytic methods are restricted to guesstimate-style approximations. Then again, the error is approaching second order. Still it's yet another thing to get right.

### <a name="Theory for base drag"></a> Theory for base drag

The base drag is the pressure on the blunt base of the body. It actually depends on several independent factors but the big one that is unlike other aspects is its relation to the details of the boundary layer in the vicinity of the base. This is fundamentally different from all previously discussed flow patterns which were treated as potential flow (inviscid). This means that the solution will be quasi-empirical assuming a turbulent boundary layer (a reasonable assumption). I looked into this just a little, and found some seemingly decent stuff, so it looks like we can get some sort of reasonable result... but, if for some reason the heat from the motor, say, was thickening the boundary layer at the base, which seems at least possible, then anything we did initially would probably be fairly wrong. This is just one of those areas where we'll ultimately need to check the simulation against a real flight and see if there is decent agreement. Even being able to check this would be a fine accomplishment because it would mean everything else was worked out sufficiently that we could say with reasonable confidence that unexplained variations in drag were due to un-modeled base drag variation. We could also check against pre-existing wind tunnel data for similar configurations.

### <a name="Transonic theory"></a> Transonic theory

I also looked into this just a little. There might be some acceptable things to do in the transonic regime to get results with accuracy roughly equivalent to the slender body approximations in the sub- and super- sonic regimes. The transonic techniques rely to an extent on either being sufficiently far from M=1 or at Mach one for short enough periods of time that highly unstable shocks do not develop. Fortunately our recent motors seem to be moving us away from the danger zone. (Which LV1b was basically in.) As a first cut we can definitely do something questionable here and argue that the error is of short duration.

### <a name="{Sub,Super}sonic Slender body th"></a> \{Sub,Super\}sonic Slender body theory for small angles

I haven't seen anyone work out a unified framework for both the subsonic and supersonic regimes for slender bodies, so either research---probably ordering papers since PSU doesn't seem to have them---or some in-house development is required. Again this is at least a few weeks work.

### <a name="{Sub,Super}sonic Slender body th"></a> \{Sub,Super\}sonic Slender body theory to first order

Barrowman is a first order theory with zealous truncation of higher order terms. There are theoretical considerations which i somewhat understand that suggest that some of the resulting approximations are actually invalid, at least for certain aspects of the flow. The first order slender body theory referred to here, in addition to retaining more terms relating to angle of attack, also retains terms of higher order in certain cases. There is no example i'm aware of that does this in a complete fashion, so that is something else that needs to either be found or worked out. The references to the Barrowman equations with corrections i found retain or add higher order terms, but don't really explain what's going on. The corrections appear to be ad hoc attempts to fix the problems introduced by the zealous truncations. My gut feeling is that the corrections are a considerable improvement, but fall short of the accuracy of a rigorous first order theory. I'd be happier if i could trace the corrections to some analytical result.

### <a name="{Sub,Super}sonic Slender body th"></a> \{Sub,Super\}sonic Slender body theory to 2nd order

There is a published method of computation known as "Second order" theory which dates to ~1949, due to Van Dyke. I even found computer sheets that purport to reduce the method to one of routine calculation, but i confess not to fully understand them since the computers referred to were actually human beings. It's not unlikely that a little research would turn up extensions to this method that would handle our typical airframes (what i have is axisymmetric only). This might be a good way to go, i'm not sure. It may be that carrying out the full slender body computation makes more sense.

My impression is that first order theory is considerably simpler than 2nd order, but that while the full slender body computations are only possible numerically, the extra complication of second order theory is of about the same degree as the full computation _if_ we are allowed to treat Numeric\_Integration[] as a simple function call that returns in a reasonable amount of time. For this reason 2nd order theory might be the wrong way to go.

### <a name="Method of Characteristics"></a> Method of Characteristics

The same basic thoughts against 2nd order theory apply to the spiffy, but deeply annoying method of characteristics. This is a low-computation numerical method that solves the non-linear supersonic flow problem. My thought is that with cheaper computation a reality, it would be better to use a more bone-headed numerical method that comparatively burned cycles yet was easier to set up and use.

## <a name="Other strategies"></a> Other strategies

### <a name="Find someone else&#39;s open-source"></a><a name="Find someone else&#39;s open-source "></a> Find someone else's open-source simulator

One thing that someone who was not me could possibly do is check to see what's around. There are definitely a few projects. Maybe some other colleges have code lying around? Those Dutch are sometimes into this sort of thing. I looked a year or two ago, but there may be other things now, or something i missed. Also that guy out past Beaverton we visited (completely forgot his name, totally remember the sandwiches) might have some thoughts.

### <a name="Buy a simulator"></a> Buy a simulator

Brian already has one. There are several available for model rocketry. I presume the commercial ones are prohibitive. Possible NASA code as we said before. The big difficulty with this is that it's so hard to tell what the blasted thing is doing. On the other hand for plugging into a dynamic simulator phenomenologically the force coefficients can come from anywhere.

Another fine use for a 3rd party simulator is validating our own version.

One difficulty with, for example, Brian's simulator is i don't recall it gives normal force coefficients but maybe i'm just forgetting.

### <a name="Build a numerical simulator"></a> Build a numerical simulator

If i had my 'druthers, and possibly ethical and reliable cloning technology, this is what i'd do. I'm not really the one to lead such a project. Ideally someone great would step up. If by some miracle i even tried to assume a leadership role, success would hinge on dumb luck in attracting some decent talent to provide the momentum. It does seem that net-wide the pool of interest, and possibly talent, should exist, so perhaps it's either, like the truth, already out there, or the right seed crystal could be dropped?

### <a name="CFD"></a> CFD

I'd really like to find out that CFD is the way to go, because that's the most fun, but it's also what i understand the least, which is pretty scary considering how well i understand the other stuff.

On the other hand there does seem to be fewer layers of theory present in CFD approaches so maybe there are potential implementation advantages even for us?

## <a name="Related issues"></a> Related issues

### <a name="Consistent units, nomenclature,"></a><a name="Consistent units, nomenclature, "></a> Consistent units, nomenclature, representations

This is just my attempt at emulating the anal-retentive, or more positively, it's really a lot of work translating everyone's notion of notation as i read the literature. This is extra true in aerodynamics where there are often many unstated assumptions. My utopian dream therefore is to first do no self-mutilation by keeping the project internals consistent, and, with wild optimism, hope to spread a well thought out standard by example. (Forgive the digression, but this optimism thing is slightly less stupid than it seems. We've actually got feedback at least once from someone working on standards concerning our setups. Admittedly they were mostly telling us we were doing it wrong, but we were having some impact.)

### <a name="A scheme for solving the equatio"></a> A scheme for solving the equations of motion including transformation

of reference frames

There's a fair amount of work involved in just setting up something that uses force coefficients from whatever source to simulate the flight path. If i'm otherwise holding you up this is something that could be done independently. Possibly it's also an independent branch for me to hold you up on.

### <a name="Find an aerodynamicist"></a> Find an aerodynamicist

I'm not really sure ultimately how we should proceed. It would be nice to borrow a smarter aerodynamicist for a little bit to give us a sense of where it's reasonable to go. Does anyone know of an idle one hanging about?