I spent quite an amount of time developing and using the exact same DNS method (Finite Volume, upwind), for my own research (the same paper I presented at AIAA - linked above), as this is bread and butter for supersonic flows. But for my supersonic and hypersonic flows I used the whole

*Navier Stokes Equations* (Conservation of Mass, Momentum, Energy and Entropy balance).

I say this because these three engineers chose to use

*Euler Equations* which basically are an abbreviated form of the Navier Stokes Equations, which assume no heat transfer occurs in the flow field and no viscous effects are considered (hence no effects related to friction).

I was scratching my head for a while, wondering why the engineers would neglect friction when talking about hypersonic flows and over simplify, using ideal gas law, all the while talking about temperature gradients in the order of thousands of degrees and temperatures on the skin. It made little sense to have used the Euler Equations in my mind.

But then I remembered that you can still treat the air flow outside of the boundary layers as inviscid, and shock waves can be mathematically treated as adiabatic (no heat transfer in radiation and conduction) affairs, even though it makes little sense at first glance (since shock waves in fact dissipate an enormous amount of heat).

But theoretical shock waves (Rankine Hugoniot jump relations) can be derived from the Euler equations, actually. Shocks are like black holes, that is, they are a type of mathematical singularity. They operate on an infinitesimally thin line called the "shock front" and are like pushing the "reset" button for pressure and temperature.

So the word of caution here is that the engineers really are modelling only the flow of gases outside of the boundary layer (the layer with friction and maybe turbulence right next to the skin), and any temperatures from rocket combustion are treated as "boundary conditions," in laymen terms, mathematically imposed.

Any shock compression follows the Rankine Hugoniot jump relations, just a consequence of the Euler equations, and gas expansions are basically an isentropic consequence of ideal gas law. No radiation allowed, no conduction and no friction allowed. No chemical reactions either. Note hypersonic flows involve all of the above especially next to the skin.

So they are just following how the gas transports energy and changes pressure and temperature as it is "thrashed around" the fuselage, so to speak (this gives you an idea of how much energy you are really talking about when you speed up to Mach 15). This is, for lack of a better term, a "supersonic simulation," not a hypersonic one.

In the software, you set the engine temperature, ambient pressures and expected free-stream fluid velocities to set an initial condition for shocks to develop naturally. The simulations are good for "surrounding flows" and calculating drag, but otherwise, thermal effects on the skin and skin temperatures will be significantly off in the hypersonic range.

Their temperature assessments will be off, actually, as much as 100% (double) error by the time you get to Mach 15, compared to using the non-ideal gas corrections. Mostly because real air molecules absorb energy, and because you allow heat to move around otherwise. Also, they ignore chemical adsorption and radiative heat transfer to the skin (infra-red from the combustion process), which is really a huge amount of energy (but in all honesty, I would also would have ignored radiation in my calculations at least until I was a graduate student studying radiative heat transfer in participating media

This is a necessary correction that NASA had to take care of for rocket engines on the Lunar Module, for example.

Anyhow, in simple terms (and in English) Their Cart3D temperatures are more than double what you might actually encounter if you bother to use the full Navier Stokes Equations and consider a non-ideal gas correction. When you correct the math you get much lower temperatures.

Correcting for non-ideal gas laws by assuming molecules can absorb energy, produces an effect known in Hypersonics as a "thermal relaxation effect" where Mother Nature "gives you a break," so to speak. Without this "break" from Mother Nature, designing a Moon mission, would have been absolutely impossible and the Space Shuttle would simply not exist. I first learned about this thermodynamic correction in my first few classes in Intro to Hypersonics, back in 1997-8.

This is their "small letter disclaimer" on Page 15.

At M_{∞}=16.969, approximately 20 percent of the aft portion of the fuselage, including the empennage, is surrounded by fluid at very high static temperatures. As the rocket mode is used to raise M_{∞} beyond 17, the percentage of fuselage engulfed with nacelle plumes will further increase and the thermal environment will become increasingly severe.

The thermal environment will also depend on how the SABRE nozzles are gimbaled. However, the plumes are so under-expanded that it is unlikely this will substantially alleviate the impingement effects. At M_{∞}= 12.189 for a perfect gas (γ= 1.4), the freestream total temperature is approximately 30 times T_{∞}= 231° K, and the freestream recovery temperature is roughly 26 times T_{∞}. However, based on real gas chemistry and the edge pressure = atmospheric pressure, the freestream total and recovery temperatures are, respectively, nearly 13 and 11 times the freestream temperature (as shown in Table 3). Here, the assumed recovery factor is 0.85.

Because Cart3D simulated temperatures are based on Euler equations for a perfect gas, they do not translate directly into fuselage skin temperatures. The surface equilibrium radiation thermal environment will differ when simulations are conducted with air and hydrogen/oxygen chemistry and account for viscous, plume radiation, and real gas (γ~ 1.3) effects. This level of physics will provide information such as surface temperatures, flow separation, and realistic effects of shock-shock/boundary interactions and vertical tail bow shock/boundary layer interactions. Nevertheless, the fundamental fluid phenomena will remain the same. High-temperature gas has the ability to emit significant radiation in the UV, visible, and IR regions of the electromagnetic spectrum, leading to potentially substantial heating of the aft fuselage surface. Radiative processes augment the convective heating.

By the way, above, "Total Temperature" (and Total Pressure) is synonymous to Stagnation temperature (T

_{o}) (and Stagnation Pressure, P

_{o}), the temperature (pressure) of a stream of air that comes to a stop at the nose or leading edge of a wing, and is usually the highest anywhere on the fuselage.

So take these NASA engineer's "exact" results with a big grain of salt. They are missing a lot of detail.

*But* the bad news is that, even cut in half to correct for non ideal gas effects (never mind heat transfer), the temperatures around the places where the plumes hit the fuselage would be in the order of at least 10 times higher than ambient temperature. At T

_{o} = 13*231 K = 3003 K = 2729 C on the leading edge of the vertical stabilizer, the conditions are already harsher than anything the Space Shuttle skin ever endured at 1533 K = 1,260 °C = 2,300 °F (The maximum operating temperature if RCC panels is 1922 K)...

http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20100016285.pdfThese workers are MOSTLY correct in that the plumes would expand greatly and basically cook the tail. The transport of energy and pressures around the fuselage skin would be very similar with or without heat transfer, with or without viscosity, because the plumes are just expanding and compressing gases in free-stream. The geometric dilation of the plumes with altitude and speed would be the same. As you go higher and faster and the static pressure is lower, but your nozzle pressure remains the same, higher, relative to the ambient, and thus you create shock waves trailing the nozzles, and expansion compression "diamonds."

http://www.aerospaceweb.org/question/propulsion/q0220.shtmlhttp://www.allstar.fiu.edu/aero/rocket3.htmThe plume edges expand outwardly the higher up (and the faster) you go to get into orbit. The nozzles become very "under-expanded" (over-pressured) at high altitude, engulfing the tail, with nozzle shocks intersecting the skin in front of the vertical stabilizer. That whole tail end of Skylon needs a very different thermal rating.

Their Lift to Drag ratio discussion, I found much less interesting. Basically they're saying that at Mach 3, REL's lift and drag figures are too "optimistic," but at hight speeds (M=15) their figures are "pessimistic," because the plumes actually help you equalize the front-to back pressure distribution, thus reduce drag and increase lift - never mind that you just melted/ablated your tail off.

So their use of "favourable" and unfavourable" terminology in the paper depends on what you're talking about. Favourable for L/D. VERY unfavourable for heat transfer purposes. They flip flop on the terms (the term is more commonly used for pressure gradients in engineering for discussions). Talking about plumes and L/D ratios is like talking about apples and bananas; they are related because both are fruits and grow on trees, but otherwise, makes little sense to equate lift to drag ratios to plume-widths. These are related only by way of the under-expansion of rocket nozzles which applies to this specific project (and rockets) but it's not obvious for airplanes in general.

And their "computations being superior to REL engineering practices" comments fully deserve to be in the "WTF" thread. It seems they they hired a used car salesman to sell their "Cart3d" method

~ ~ ~

Just because we're engineers and geeks doesn't mean we're good writers or presenters. You should see some of the presentations in meetings. Professional engineers from industry are notoriously bad at conveying their message - only a couple of steps away from pointing and grunting at the screen.

That is part of the reason that industry started demanding that college students take courses in technical communication. So you don't end up confusing your audience.