*Post edited 9:18 am – December 15, 2009 by Luke Maurits*

In order to be able to get some approximate hard numbers up on the Wiki for OHKLA in order to make it look more like we know what we are doing, I have done some basic flight simulations today. There is ample work for improved accuracy, but at the very least these figures should be not terrible.

Here's the gist of how these work:

- A rocket is completely specified by its engine burn time, engine thrust, empty body mass, full fuel mass, payload mass, radius and drag coefficient.
- Simplifying assumptions: engine thrust is constant for duration of burn, fuel mass decreases at a uniform rate throughout burn.
- Air density (for drag calculations) is computed from altitude according to the formula here.
- Because I know very little about aeronautical engineering, my way of handling supersonic drag is as follows: At all times during flight, the rocket's coefficient of drag is D*f(M), where D is a fixed "base drag coefficient" and f(M) is a "drag scaling function" which is a function of Mach number M (yes, Mach number calculations take into account change in air density with altitude). The shape of f(M) is such that f(M) = 1 if M < 0.6 or M > 1.8. In between these values, the drag is increased. Between Mach 1.0 and Mach 1.2, f(M) = 5. Between Mach 0.6 and Mach 1.0, it increases linearly from 1 to 5 and between Mach 1.2 and Mach 1.8 it decreases linearly from 5 to 1. These numbers were "inspired" by this article. With drag coefficients calculated thusly, drag forces are computed using the standard drag equation. This is probably the biggest fudge in the entire simulation, if a real aeronautical engineer can advise me on how to improve this, by all means do so.
- Equations of motion are integrated using RK4.

My main aim here was to get a feeling for what the engine thrust and burn time need to be. For the purpose of some initial investigation, I have used an empty rocket mass of 30kg, a fuel mass of 100kg, a payload mass of 5kg and a radius of 15cm. These values I got by looking at the values for RocketLab's Atea-1 rocket and adding "pessimism" factors.

I had no idea what a realistic base drag coefficient for a rocket should be, so I experimented with D = 0.1, 0.2, 0.3, 0.4.

The graph below shows (for these various drag coefficients) the required burn time (s) for a given thrust amount (N) in order for the apogee of the flight to be 100 km (or, reading it sideways, the required thrust amount for a given burn time).

Now, it's important to realise that these are probably underestimates. This assumes that all of the rocket thrust is directed perfectly upward. In real life there will be wind, which will cause the rocket to tilt somewhat, causing some of the thrust to be wasted on horizontal translation. This is a little tricky to model because, as the rocket tilts, the angle of attack for the stabilising fins will change, causing asymmetric generation of lift to counter the tilting. We're gonna have to model this eventually, but since this a first stab I have left it out.

Note that if my crappy supersonic drag approximation results in more drag than we can actually expect, this will counteract the understiamtion due to the above point about wind, and these estimates may be okay or even *overestimates*. For now I will assume that they are underestimates.

Now, taking a pessimistic drag coefficient estimate of 0.3, a thrust of about 5000 N for 50 seconds is adequate according to the above graph. Since these are underestimates for the rest of this thread I will use 5000 N for 60 seconds (a rather unprincipled correction). For these values:

- 100km of altitude is reached after about 106 seconds
- Maximum altitude is 175 km after 230 seconds
- Maximum velocity is about 1715 m/s or 6174 km/h
- Maximum acceleration experienced is about 120 m/s/s, or a bit more than 12 g
- Max Q is about 46000 Pa

These figures seem relatively reasonable: RocketLabs claim a max launch acceleration of 15 g for their rocket (which is lighter than ours, so the slighter higher result makes sense) and claim a maximum velocity of "> mach 5″. Now, this is kind of technically incorrect – the speed "mach 5″ is air pressure, and hence altitude, dependent. When a rocket is at its fastest, the atmosphere is very thin, so the speed of sound is very fast and the mach number is actually quite low. Assuming that by "mach 5″ they mean 5 times the speed of sound at sea level, that's about 6180 km/h which is extremely close to the maximum speed from my simulation. CSXT claim that one of their rockets reached space in one minute and thirty seconds, only 16 seconds faster than my simulation. So we are certainly seeing figures which match those for real world OHKLA-esque rockets.

So, for rough planning and design purposes, we can work with the rough figures:

- Engine thrust = 5000 N
- Engine burn time = 60 s
- Time to zenith = 4 mins
- Max acceleration = 12 g
- Max velocity = 6175 km/h ("Mach 5″)

I probably won't get much more done today, but in the near future I'll make a Wiki page dedicated to the results of simulations (which we can refine over time), which will include links to the GPLed code I wrote for this and more graphs of various things (altitude and velocity vs time, etc). EDIT: The simulation results will go here, the code and instructions will go here. Keep an eye out!

Input on everything here is extremely welcome, particularly from people who know more about supersonic aerodynamics than I do.