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Optimal Spaceplane Entry AoA


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To clarify up front, I'm looking for math help - not 'rules of thumb' or best practices.

As a spaceplane enters the atmosphere, it's got three forces acting on it: gravity, lift, and drag. For the purposes of this discussion, I've got a well-developed model that describes drag as a function of lift, and lift as a function of angle of attack (AoA) (accounting for mach number in both cases). This theoretically means that the spaceplane's downrange distance is a function solely of AoA (assuming AoA is constant throughout the entry if need be). It's not going to be a clean function, but it is a function of one variable. I'm only interested in the regime where the flight path is pseudo-ballistic (i.e. the vessel is not 'flying' per se). This question centers around the effects before the atmosphere is significant enough to treat the vessel as a simple glider (say, h > atm_height / 2).

Increasing AoA increases both lift and drag. More lift means the vessel's descent angle shallows and it postpones hitting thicker atmosphere, increasing downrange distance. More lift means more drag, though, which slows the vessel and decreases downrange distance. The question is, what is/are the critical AoA(s) where these two effects cancel? I believe there will be two roots, one where drag is minimized (low AoA) and one where lift is maximized (higher AoA). From my experiments, L/D(max) is not ideal for best range in early stages of entry - and this makes sense since it doesn't account for the delayed effect of prolonging the rarified atmosphere.

I've tried doing the calculus of looking at a snapshot of a purely ballistic trajectory and treating lift and drag as changing the launch angle and speed, but this seems to be a dead end (at least to my sleepy brain). Before I start brute forcing this through a differential equation solver, are there any suggestions to how to find the optimum AoA(s) for downrange distance in a rarified atmosphere?

Potential reading list:

https://ntrs.nasa.gov/api/citations/19720022195/downloads/19720022195.pdf

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35 minutes ago, Booots said:

To clarify up front, I'm looking for math help - not 'rules of thumb' or best practices.

As you're not looking for discussion directly about the game, we'll move this over to the Science and Spaceflight Sub. 

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18 minutes ago, Gargamel said:

As you're not looking for discussion directly about the game, we'll move this over to the Science and Spaceflight Sub. 

Sure, I guess. This is all feeding back into flying a spaceplane in-game, though I suppose like any KSP physics is generalizable to the real world as well. 

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4 hours ago, Booots said:

To clarify up front, I'm looking for math help - not 'rules of thumb' or best practices.

As a spaceplane enters the atmosphere, it's got three forces acting on it: gravity, lift, and drag. For the purposes of this discussion, I've got a well-developed model that describes drag as a function of lift, and lift as a function of angle of attack (AoA) (accounting for mach number in both cases). This theoretically means that the spaceplane's downrange distance is a function solely of AoA (assuming AoA is constant throughout the entry if need be). It's not going to be a clean function, but it is a function of one variable. I'm only interested in the regime where the flight path is pseudo-ballistic (i.e. the vessel is not 'flying' per se). This question centers around the effects before the atmosphere is significant enough to treat the vessel as a simple glider (say, h > atm_height / 2).

Increasing AoA increases both lift and drag. More lift means the vessel's descent angle shallows and it postpones hitting thicker atmosphere, increasing downrange distance. More lift means more drag, though, which slows the vessel and decreases downrange distance. The question is, what is/are the critical AoA(s) where these two effects cancel? I believe there will be two roots, one where drag is minimized (low AoA) and one where lift is maximized (higher AoA). From my experiments, L/D(max) is not ideal for best range in early stages of entry - and this makes sense since it doesn't account for the delayed effect of prolonging the rarified atmosphere.

I've tried doing the calculus of looking at a snapshot of a purely ballistic trajectory and treating lift and drag as changing the launch angle and speed, but this seems to be a dead end (at least to my sleepy brain). Before I start brute forcing this through a differential equation solver, are there any suggestions to how to find the optimum AoA(s) for downrange distance in a rarified atmosphere?

Potential reading list:

https://ntrs.nasa.gov/api/citations/19720022195/downloads/19720022195.pdf

The answer is "it depends".

Even for a regular plane in conventional flight, the optimum AoA for maximum L/D depends heavily on the design of the plane. It also varies with speed because drag is highly nonlinear with speed. Re-entering spaceplanes necessarily have dramatically varying speeds.

Spaceplanes also then have to contend with heating. They don't generally solve for "max downrange" as a longer flight time results in a greater total absorbed energy at a lower temperature, which both gives the heat more time to penetrate through insulation and minimises the proportion that is re-radiated.

They also need to contend with structural loads and keeping vital elements out of the airstream.

If you want an answer along the lines of "the best AoA is 43°" then I'm sorry to disappoint.

This is a highly complex computational challenge.

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Posted (edited)
6 hours ago, RCgothic said:

The answer is "it depends".

Even for a regular plane in conventional flight, the optimum AoA for maximum L/D depends heavily on the design of the plane. It also varies with speed because drag is highly nonlinear with speed. Re-entering spaceplanes necessarily have dramatically varying speeds.

Spaceplanes also then have to contend with heating. They don't generally solve for "max downrange" as a longer flight time results in a greater total absorbed energy at a lower temperature, which both gives the heat more time to penetrate through insulation and minimises the proportion that is re-radiated.

They also need to contend with structural loads and keeping vital elements out of the airstream.

If you want an answer along the lines of "the best AoA is 43°" then I'm sorry to disappoint.

This is a highly complex computational challenge.

You're not disappointing because I'm not looking for a simple answer. Like I said, I'm trying to simplify the maths (where possible).

The first part is taken care of - I've already got a method for characterizing Lift and Drag at any (reasonable) AoA through the range of speeds that will be encountered.

I'm also ignoring the heating constraint and structural constraints for now because KSP's parts have such high heat tolerances and rigidity. I can adjust afterwards if vessels start exploding. I'm also not necessarily looking to maximize downrange distance - rather, I want to know what AoA range is productive to use in a control loop to bring the impact point to the desired landing location (though this would lead to maximizing downrange distance at its limit).

Since it's not looking like there will be an analytical answer, it's time to start setting up a MATLAB script! Any suggestions for simplifications and assumptions that might help make the computation simpler/faster?

Edited by Booots
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I'm not sure if you are looking for an analytical solution but people way smarter than I have convinced me that is just not feasible in KSP for a lot of reasons.  The feedback I got was that only an iterative solution with ongoing correction would work.  The KSP or Ferram aero models have round-off error, every craft is different, etc.  Are you scripting this in kOS or similar?  The Trajectories mod can help inform the error signal quite a bit for aero descents

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this is usually determined experimentally during the initial wind tunnel testing and later in the test flight stage. computational models can aid in initial design efforts. but you still need to prove the design experimentally. once the envelop is established then you can specify what is optimal (and safe) for the operation of the craft. 

for a space plane a high aoa is preferable as it exposes the biggest cross section at re-entry. this spreads out the heating over more of the surface and aids in deceleration. however you don't want to put the wings into a stall condition either, as that limits the controllability of the craft.  you might also limit the aoa to reduce g loads or to save energy for a later phase of flight, like subsonic approach and landing.

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In case anyone doesn't know: Booots wrote Kerbal Wind Tunnel.

Booots, speaking as someone who finds the thermal system the most interesting and challenging aspect of KSP, and has done a LOT of atmo-diving in planes, you can only handwave it in cases like Duna and Sarnus.

That said:

  • I would definitely limit your calcs to between 5 and 25 degrees AOA, for reasons I'm sure you're aware.
  • Disregard the subsonic regime, since it's a discontinuity, and will be pretty much a rounding error on range.
  • Inclination/course is going to be your second-most influential variable, due to the effects of centripetal acceleration. You will get wildly different ranges flying east vs. west.
  • I don't know that this project has an operational utility (other than fun) given that one can stay in orbit for "free", and just re-enter later. If you're looking to use it to make big changes in landing latitude (as opposed to longitude), see my prior point.

This is really similar to a project I shelved last year, seeing if I could calculate re-entry flight paths that minimize total heating, and/or max radiated wattage. I suspect there are some fascinating windows in there. I have the atmo/temp models, I just don't have the drive to get it done.

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Actually for  real spacecrafts the AoA is defined by thermal protection. That's why the space shuttle did S shaped approach: AoA stays constant but rolling the lift vector sideways let you control range. 

For kerbal parts without too much realism added, I think 2 different AoA should be considered during reentry:

1. Low Drag phase: 60-90 degree AoA inspired by starship. Lift is completely neglectable and reduced AoA results in slightly increased range for the control loop.

2. Lift phase: 30 degree AoA is about maximum lift from wings and body. Since I enter with low PE to quickly cross first part, the lift turns the ballistic trajectory enough to not hit the ground before speed is shed off. Increase AoA and range is reduced since L/D is reduced. Decrease AoA and range increases as speed stays higher and trajectory more turned into flight. L/D is pretty constant from 10 to 30 degree if I remember charts correctly.

 

But I never tried to bring this different approaches into a single mathematical optimisation as various effects dominate each other. I used trajectories mod to get downrange predictions and added a mechjeb control loop with above logic: Decrease AoA to get further, increase to shorten.

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I suppose I should have phrased my question more as a function of L/D than AoA. As @CBase mentioned, the Shuttle kept an AoA that would offer some lift and thermal protection and then scaled lift by banking to adjust the lift component - effectively varying L/D.

@FleshJeb, thanks for the free advertising! :) I'm trying to make a kOS script to finely control landing latitude once reentry interface has started. I need to know what range of AoA (read L/D) I can use for control before further input in that direction becomes counterproductive. I used Wind Tunnel to get lift and drag polars at various supersonic speeds. I agree, subsonic regime is likely a rounding error - and it follows normal gliding flight assumptions.

@CBase brings up the 'intuitive' approach of decreasing AoA to go further but increasing to shorten. The Shuttle did the opposite, though, increasing lift to go further with drag staying constant. That contradiction is why I'm so taken with this question. Why is the technique that was used in the real world not working for my control loop in Kerbal? Adding lift through AoA increases drag immediately, but decreases drag later because you stay in thinner atmo longer. Lowering lift through AoA decreases drag in the moment but you hit thicker air sooner. Can we figure out at what L/D these effects cancel out (either local maximum or local minimum), and is that L/D a function of planet radius, atmospheric characteristics, or otherwise? I'm working on a Simulink model of entry/descent to start to characterize this, but it's going to have to be brute force, I think.

I'm using the assumptions from that NASA paper I linked: cylindrical planet, stationary (ground-relative) atmosphere, instantaneous vessel pitch changes, point mass vessel.

@FleshJeb, could I get a copy of your atmo/temp models, please?

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10 hours ago, CBase said:

Actually for  real spacecrafts the AoA is defined by thermal protection. That's why the space shuttle did S shaped approach: AoA stays constant but rolling the lift vector sideways let you control range. 

For kerbal parts without too much realism added, I think 2 different AoA should be considered during reentry:

1. Low Drag phase: 60-90 degree AoA inspired by starship. Lift is completely neglectable and reduced AoA results in slightly increased range for the control loop.

2. Lift phase: 30 degree AoA is about maximum lift from wings and body. Since I enter with low PE to quickly cross first part, the lift turns the ballistic trajectory enough to not hit the ground before speed is shed off. Increase AoA and range is reduced since L/D is reduced. Decrease AoA and range increases as speed stays higher and trajectory more turned into flight. L/D is pretty constant from 10 to 30 degree if I remember charts correctly.

 

But I never tried to bring this different approaches into a single mathematical optimisation as various effects dominate each other. I used trajectories mod to get downrange predictions and added a mechjeb control loop with above logic: Decrease AoA to get further, increase to shorten.

Now 1 makes perfect sense as lift is low here anyway, then switch to 2 then you get into real atmosphere. 
As most spaceplanes in KSP is SSTO you probably want to cut short and use engines to get you to runway, dump lox then entering the atmosphere. 
KSP also give you some cheats like opening cargo bays increase  drag who is obviously unrealistic. 
 

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On 4/2/2022 at 7:46 PM, Booots said:

@FleshJeb, could I get a copy of your atmo/temp models, please?

Invited you to the PM conversation that contains the resources.

I think once you've got your stuff dialed in, I might appropriate some of your methodology. It should be a similar process, just different things to optimize.

Now to find the energy...

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On 4/3/2022 at 4:46 AM, Booots said:

Why is the technique that was used in the real world not working for my control loop in Kerbal?

I do think it depends what your nominal AoA is during the landing phase without corrections on target. If you are using a very low (0-5°) AoA slightly raising it should improve range as L/D improves. Actually I could not find what the AoA for shuttle was after thermal braking was done and heat shield not required.

Choosing the nominal AoA will depend on other mission requirements: longer glide allows for bigger latitude changes (not that important if spaceplane and Kerbal SC is on equator), shorter glide for more spontaneus player decisions to land after doing some stuff in orbit. The shuttle final approach was in manual control, so maybe even visual view might influence the decision.

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