Advanced Question Formula for reentry heating and G-forces

[BruceJohnJennerLawso]

Hello,

Does anyone know the formula for calculating heatshield temperature during reentry?

 

[N_Molson]

That's an old question. Almost every addon developper, myself included has been asking one day. No clear (usable) response so far. It seems to be one of the most well kept secrets of the Universe.

 

[boogabooga]

While you seek a simple "equation," what you are asking for would take several graduate level engineering courses to understand well.


I'll give you a starting point
View attachment Gas Dynamics Relations.zip

In the worksheet "Shockwave Flow", Column A gives the Mach number, and Column D gives you how much you need to multiply ambient temperature by to find the temperature behind a normal shock. This does not take into account skin friction or hypersonic effects, such as ionization.

So, if you assume a craft is flying at low-to-moderate supersonic speed and is blunt enough to have a bow shock, you can find approximately the air temperature around the craft. There is still the matter of heat transfer by convection, conduction, and radiation to find the actual hull temperature.

 

[Urwumpe]

The heat flux in Watts per square meter, is calculated by

Q = 0.5 * rho * v³

With rho being the density of the air at that altitude. v is of course, velocity.

This can also be written as

Q = q * v

with q being the dynamic pressure.

There is also a much more sophisticated formula somewhere, which also includes the conductive heat exchange. I have it at home somewhere.

(Yes, the actual accurate heating requires serious calculations - but simplified approximations are not that hard to implement)

 

[boogabooga]

Compressible flow is dominated by the Mach number. I refuse to accept any calculation that does not take that into account.

Dynamic pressure based on "1/2 rho v squared" is for incompressible flow. It quickly becomes meaningless above Mach 0.3.

Then again, the equations for compressible flow will become meaningless for hypersonic flow.

 

[Urwumpe]

Compressible flow is dominated by the Mach number. I refuse to accept any calculation that does not take that into account.

Dynamic pressure based on "1/2 rho v squared" is for incompressible flow. It quickly becomes meaningless above Mach 0.3.

Then again, the equations for compressible flow will become meaningless for hypersonic flow.

Well, your choice to be wrong. From Mach 10 upwards, all effects are no longer depending on the Mach number at all, but on the actual velocity. It would be still wrong calling the flow incompressible, but practically, it shares many properties of it.

From Mach 5 on, you are already closer to the real measurements by assuming incompressible flow, instead of compressible flow (Because viscosity starts to dominate the aerodynamics). Yes, in that region, both compressible and incompressible flow are not getting you accurate simulations. It is simpler to calculate for Mach 15 than for Mach 8.

 

[boogabooga]

Respect to those involved,
but universal "1/2 rho v squared" based heating is what gives us shuttle re-entries with no plasma streaks, and Mach 3 flight with vigorous plasma streaks.

 

[N_Molson]

While you seek a simple "equation," what you are asking for would take several graduate level engineering courses to understand well.

This is exactly what I was referring to when I was talking about "not usable answers". It is obvious that I, for example, 29 years-old, am not going to take several graduate engineering level courses to developp an add-on :dry:

In simulations, we have sometimes to take shortcuts, even at the cost of precision. The idea is not to have a formula that "exactly describes the complex thermodynamical phenomenons that take place during reentry" but a formula that simulates the thermal behavior of the heat shield in a credible and usable way.

The relevant data we can ask to Orbiter through the C++ functions are velocity (horizontal, vertical), heat shield area (we have to admit it is a flat circle for simplification), vessel mass, Mach number, vessel AoA, vessel drag, air density, dynamic pressure, and I probably miss other parameters...

So I think it would be very convienient to have a "generic" formula that allows to simulate credible reentry profiles for existing or fictional capsules, allowing the addon to track the shield's temperature and then detect a breakup situation.

Subtilities of hypersonic behavior of lifting bodies might be interesting, but here only if there is a real impact in term of gameplay, IMHO. I don't see the point of having the .dll calculating realistic parameters (and we know that Orbiter has serious limitations on atmosphere modeling anyway) if there is no use to it. We can also add variables to tell if the first crewmember has a cat or a dog, or if it is black or white. :2cents:

So yeah I definitively would be interested by a "ready to use" C++ translated formula, even basic (one may always refine it later) that describes with a fair amount of realism that kind of phenomenon.

As said George Sand (aka Amantine Aurore Lucile Dupin, 1804-1876),, "Art for art is an empty thing. Art for the sake of the good, art for the sake of the beautiful, this is what I am searching for".

 

[Urwumpe]

Respect to those involved,
but universal "1/2 rho v squared" based heating is what gives us shuttle re-entries with no plasma streaks, and Mach 3 flight with vigorous plasma streaks.

Yes, if you assume a spherical space shuttle. :lol:

In reality, the aerodynamic heatflux function works right, but you need to include local velocity and local density. Behind the nose shock for example, you have much higher density and lower velocity than free-stream. Like in reality, you have no uniform generation of plasma around the Shuttle, but rather it starts at local hotspots.

From Mach 5 up, you are better served with the incompressible solution, since the air effectively is incompressible at that speed. Moving faster does no longer compress the air and increase pressure, but instead increases temperature. Thus the formation of plasma at relatively low heat flux already.

 

[BruceJohnJennerLawso]

Yes, if you assume a spherical space shuttle. :lol:

In reality, the aerodynamic heatflux function works right, but you need to include local velocity and local density. Behind the nose shock for example, you have much higher density and lower velocity than free-stream. Like in reality, you have no uniform generation of plasma around the Shuttle, but rather it starts at local hotspots.

From Mach 5 up, you are better served with the incompressible solution, since the air effectively is incompressible at that speed. Moving faster does no longer compress the air and increase pressure, but instead increases temperature. Thus the formation of plasma at relatively low heat flux already.

That appears to be a reasonable argument from what I can understand. I don't recall the specifics, but think the vesselAPI doesn't have a ATM density function, so I can just use the Q= q * v function for the purpose.

Heat flux is probably the energy into the heat shield (in watts?). What units should I use for velocity?

 

[Bibi Uncle]

The formula that Urwumpe wrote is probably the best looking in Orbiter, because it is the one used by Orbiter to calculate the size and the opacity of the reentry flames. I would just add this:
s = min(1, (P-P0)/(5*P0)), where P is the the heat flux (P = 0.5 * rho * v^3), P0 is a user defined limit (8e6 by default in Orbiter, I found it by trial and error) and s is the opacity. This is from Orbiter Programmer's Guide, p.12.

I don't recall the specifics, but think the vesselAPI doesn't have a ATM density function, so I can just use the Q= q * v function for the purpose.

Heat flux is probably the energy into the heat shield (in watts?). What units should I use for velocity?

There is VESSEL::GetAtmDensity() and VESSEL::GetAirspeed().

In fact, the units is in W/(m^2). Correct me if I'm wrong, but here is my demonstration.
Since you have the atmospheric density (in kg/(m^3)) and airspeed (in m/s) ^3, you finally have units of kg/(s^3) and since W = (kg * m^2)/(s^3), you have units of W/(m^2). Also, a heat flux is, by definition, a rate of heat energy transfer through a given surface.