I have a new toy: it’s called HyperCFD and it calculates a number of interesting quantities for bodies of rotation that are moving faster than the speed of sound. I bought it because I was interested in the stability of the “reentry body” associated with the DF-2 which Jeffrey used to discuss the design (and weight!) of the DF-2’s nuclear warhead but I imagine I will find other uses for it. As it is, I was very surprised at what I found out.
But first, I needed to check its accuracy against some actual data. (After all, it says it’s for amateur rocket designers, which didn’t instill the faith that perhaps it should have in me.) It turns out to be pretty good! (This plot shows the pressure distribution along the reentry vehicle at a fixed speed (Mach 5.8 in this case. I wont bore you with such checks, but you can look at it if you like. I should note that this fairly blunt shape is one of the most stressing for Newtonian approximations in computational fluid dynamics and in the study I’ve quoted, the other shapes are even better simulated by HyperCFD.)
I picked this particular plot to check HyperCFD with because it’s closely related to the sum of all the pressure on the reentry body. It’s important because, when you add it all up, you can calculate the point on the hypersonic body where all the pressure effectively acts. The resulting point on the body is called the center of pressure and it had better be behind the center of mass of a body if the shape is going to be aerodynamically stable.
Image a weathervane. It, of course, pivots around its central pole, which can be thought of as the center of mass of a free rocket or reentry vehicle. To a very good approximation, all the torque or turning force of wind can be thought of as acting at a single point: the “center of pressure,” or Cp. If the Cp is behind the center of mass, the rocket is stable and turns into the “wind.” If the Cp is in front of the center of mass, then the rocket or reentry body will flip. Funny thing is, that might not be stable either since the aerodynamics will change completely if the reentry vehicle is going in “butt end first,” if you will.
Which brings us to the DF-2 warhead. Several readers of Jeffrey’s post commented on the position of the center of mass of the warhead, as indicated by the balance point of the jig that is lifting the warhead. It turns out that the center of mass is too far back for a nose-first reentry and too far forward for a butt-end-first reentry! (And, in fact, the butt-end-first becomes even less stable as it slows down. Oh, yes, before I forget, the calculated Cp’s are definitely on the symmetry axis of the body and I assume that the CG is also, though it doesn’t really have to be.) So what does that mean? It’s obvious, actually.
It doesn’t separate from its rocket body, or if it does, it separates much farther back from the nose. Perhaps an even more interesting question is why so many of us thought it must separate. My guess is that when you see a separated warhead, you assume it separates during flight.
What’s next for HyperCFD? Jeffrey got interested in this warhead because it was reported to be the same design sold to the Libyans. This program provides a little more information for trying to constrain the nuclear devices potential proliferators might try. So I think I want to “design” a nuclear warhead to fit in the Shahab-3 and Shahab-3B warheads. But that is going to be a little while from now.
John Field | January 27, 2009
I remember commenting that the drag center didn’t seem like it was too far off from the balance point – e.g. that just perhaps the CG and CD were one and the same. But, now I believe you that the CG is a little too far back for stable flight.
What does this mean? Is it necessary for the rocket body skin to be much tougher to survive reentry perhaps? Maybe it says something about the reentry speed? What else?
George William Herbert | January 27, 2009
The earlier assumption that the hoisted warhead picture was the RV shell may not be correct. It could be the actual physics package instead, with the RV being slightly larger, and having some ballast at its nose. Several US warheads did that – the physics package inside looks vaguely RV-like, conical with blunt nose, but fits inside an actual RV.
I wonder about this, because a real RV would be unlikely to have open “bolt on here” holes through the thermal protective system on the sides… Much more likely that those would be in that position on a separate inner physics package than the RV body itself.
The RV appears to be close to that same shape (no big suprise there), but could be longer and/or ballasted at the nose…
If say the photo was a 1.0 meter base physics package, with a 1.2 meter base RV, then even without RV ballasting the Cp moves back enough relative to the physics package CG to reach rough stability, and ballasting would significantly improve the situation.
That still leaves enough volume for a … 70 cm or so physics package, which is not particularly challenging to design for once one understands all the initiation and lens tricks used to shrink spherically symmetric primaries/fission weapons. The Iraqis did better than 70 cm, and appear not to have known/used some of the tricks.
Geoff Forden | January 27, 2009
Hi John, I think the nosecone just remains attached to the rocket body with its fins etc that keep it stable as it reenters. With empty fuel tanks it is clearly going to have its total center of mass well forward of its center of pressure.
Conical nosecones represent a real problem for reentry. That is why the Shahab-3B “baby bottle nosecone” was designed so that it could separate from the rocket body. The extra little flange at the back of the reentry vehicle makes all the difference. Nevertheless, my guess is that it still puts some significant constraints on the physics package design. But that, of course, is exactly what I’d be interested in finding out.
Its interesting to note that when Iraq tried to develop a separating nosecone for the Al Hussein, it was with the reduced mass explosive charge. This helped move the CG forward.
Jochen Schischka | January 29, 2009
I think that there can be no doubt about a separating warhead on the DF-2 (unless one assumes that this particular missile was only a non-working blunder).
Since the reentry-speed is roughly proportional to the range, the ram-pressure-induced forces grow with v² and the structural weight of single-staged missiles has to shrink according to the natural logarithm to achieve this speed/range-capability, missiles with a range of over ~500km will generally be too fragile to survive reentry (see Al-Hussain H2). A break-up of the missile would not only grossly reduce the accuracy (because of the unpredictability of the movement of a randomly-shaped “reentry-vehicle” – see Al-Hussain H2 or early experiments with Al-Abbas or Aggregat-4/V-2), but would also endanger the proper function of a radar-fuse (which is essential in combination with a nuclear warhead) because of said unpredictable tumbling.
The solution to this problem could be, as George William Herbert suggested: what we see on the photo with the DF-2-warhead on the crane hook (see Jeffrey’s article) is only the “physics package” with a base diameter of about 0.9-1m (corresponding to the red-painted part of the missile on the photo accompaning this article), while the complete reentry vehicle is about 0.8-0.9m longer and has a base diameter of 1.2-1.3m; The real separation plane is obviously about where the forward bracket of the MEL-erector-arm grasps the missile (more or less similar to that on the Aggregat-4/V-2-Meillerwagen from which this MEL clearly is a descendant), while the mostly empty portion inbetween acts as a flare (aka draws the center of pressure back) on the separated RV and maybe also has a positive influence on the mass distribution of the entire missile.
(BTW, this also explains the dangling wiring at the back end of the hoisted warhead…)
Geoff Forden | January 29, 2009
Jochen, while in my original post, I mentioned this possibility, I am far from convinced; especially by the Al-Hussein analogy. The Al Hussein broke up because, with the reduced warhead mass, its center of gravity was too far back and never “righted” itself and hit the dense part of the atmosphere side on. With a proper weighting, I believe that this missile could reenter safely until I see more of an analysis to show it couldnt. On a related point, my hunch is that the Nodong (and Shahab-3 with a simple conical nosecone) does not separate and uses the rocket body for stability during reentry.
Jochen Schischka | January 29, 2009
Geoff:
As i tried to explain, single-staged missiles with ranges in excess of ~500km are per se doomed to break up on reentry by physical laws: the high reentry speeds at these ranges create too strong aerodynamic forces for a too nimble lightweight-structure (which, according to K.E. Tsiolkovsky’s/H.J. Oberth’s rocket equation, has to be that lightweight to attain these speeds) which is, to make matters worse, also burdened with heavy masses at both ends (the engine and the warhead)!
ALL missiles that leave the denser parts of the atmosphere reenter “side on” (unless they were pre-aligned by attitude thrusters). Not aerodynamic stability is the decisive factor in this context – structural integrity and excessive aerodynamic force-levels are.
If you emphasize a working (nuclear) warhead on your missile with a basically acceptable accuracy without too much expensive hi-tech, a separable, aerodynamically stable reentry vehicle is unavoidable at ranges of ~500+km, period.
And the Al-Hussain H2 with five of the six (or maybe eight?) air bottles in the empty section of the 450kg-warhead was, according to my own estimations, not as aerodynamically instable as you seem to think (as a side-effect, the lengthened airframe also provided a bigger lever arm)…but that is a different story. Of course, Al-Hussain H4 with the separable warhead of the Al-Abbas would have been a preferable solution.
Jochen Schischka | January 29, 2009
A little afterthought, Geoff:
I’m not sure if i understand your last comment on a missile “righting” itself up outside of the atmosphere if aerodynamically stable (no air – no aerodynamic force, isn’t it?)…
Geoff Forden | January 30, 2009
Jochen—there are aerodynamic forces on the missile before it hits the dense part of the atmosphere, that, for instance, “right” a SCUD.
I think the question is whether or not a DF-2 has reached the sorts of velocities that would insure the rocket body’s destruction. I’d be interested in some sort of discussion of this point, but as of now, 1200 km doesn’t strike me as being sufficiently large to guarantee that, though I must admit I have not done any calculations to estimate it.
Jochen Schischka | January 30, 2009
Geoff:
Obviously, you haven’t ever seen a COESA atmospheric model (ISA extended to a height of 1.000km) – otherwise you’d know that the air density drops below 0.001kg/m³at a height of only 50km (compared to 1.225kg/m³ at sea level or 0.013kg/m³ at 32km where the ISA ends) – and rapidly keeps dropping even lower.
Missiles with ranges of over ~500km and optimum (non-depressed) trajectories will spend most of their flight time above 50km. The generated aerodynamic forces at these heights are, due to the vanishingly small air density, almost ZERO even at high speeds – and they are clearly NOT sufficient to tip over the missile during the “exoatmospheric” part of the flight.
What is more, the burnout-height of liquid-fueled missiles with ranges in excess of ~500km will be above 30km – and thus, a non-post-boosted projectile will (almost) keep the orientation it had at burnout, which means that all these missiles will even end up hitting the denser parts of the atmosphere at about 30km (simulation tells me that there is characteristically an “explosion” of aerodynamic force below that height!) with their rear end ahead.
Not ideal circumstances for something with the consistency of an empty can of beer (but a much higher mass inertia!) traveling at speeds in excess of Mach 6 if you ask me.
And i haven’t even mentioned thermal loading…
If you’re still inclined to not believe me – simply do some math and computer simulation by yourself. I’m highly confident that you will end up with the same (or rather similar) conclusions as i.
Geoff Forden | January 30, 2009
Let’s look at some numbers about the DF-2 trajectory, in particular, lets look at the compressive loads a missile feels just above the engine compartment at maximum aerodynamic pressure during launch (max Q) and the maximum aerodynamic pressure during reentry. First, according to my simulation, Max Q occurs at 49 seconds, while there is still 13,260 kg of propellant in the fuel tanks. That means that the compressive load on the missile body just above the engine is equal to 3.4 × 10^5 N. (Most of this is inertial forces due to the acceleration of the missile body but there is some contribution due to aerodynamic drag. I’m ignoring the force of gravity which would also increase this compressive force but doing so is very conservative in the sense that it would tend to work in the favor of breaking up the missile during reentry. I am also ignoring the body mass of the missile; again, this favors break up.) Now, maximum aerodynamic acceleration during reentry occurs at 576 s—an altitude of roughly 1.5 km. At that point, the compressive force on a point just “behind” the warhead is 3.8 × 10^5 N, or just over 10% more than the missile experienced during Max Q. If China wanted to leave it attached the rocket body, they certainly could even if they had to “beef up” the body strength by 10%.
As for the Al Hussein, there was a very great lack of uniformity. The Iraqis’ themselves say the Al Hussein come down in a stable, nose-down attitude during their tests. The Israelis say that the Al Husseins they observed broke up at about 30 km and Postol and Lewis saw Al Husseins break up at between 10 and 12 km. So there is nothing magical about the 500 km range missile.
Let’s keep these discussions on a professional level.
John Field | January 31, 2009
As insistent as he is, I can’t see what Jochen is talking about. Not at all actually.
Looks all OK to me provided they’re sensible enough to design the missile with this in mind.
On the other hand, it does sort of seem like separation with a tail shroud would mean fewer things for the designers to worry about . . . Maybe the electronics all goes in the back by the wires where it’ll all stay cooler.
Jochen Schischka | January 31, 2009
Geoff, John:
The problem i’m talking about is: the missile will reenter the atmosphere at a high angle of attack, because OUTSIDE OF THE ATMOSPHERE (by my definition a height of over 50km -> see atmospheric density per COESA-model: almost zero times anything still is almost ZERO!) there will be NO SIGNIFICANT AERODYNAMIC FORCES to tip the missile into the direction of the momentary flight vector (this is exactly the reason why the X-15 had additional attitude thrusters installed – although the attained maximum height of the X-15-program was only about 108km, not up to ~210km like i’d expect of the DF-2).
As the cutoff of the DF-2 will be at a height of about 60km and the guided part of the flight will end with cutoff (no air vanes!), it will reenter the atmosphere UNGUIDED at an angle of approximately 70° (cutoff angle ~34° + angle of ballistic flight vector at reentry ~36°).
Drag and lift will now grow significantly with the angle of attack (this is why aircraft typically start and land with the nose pointing upward – to generate more lift at low velocities).
It might be right that the level of forces on the DF-2 body are roughly about the same (i’m getting ~500kN and ~700kN in my simulation…) during ascent and descent IF the missile reenters “aligned”, BUT THAT WILL SIMPLY NOT BE THE CASE.
Thus the aerodynamic forces will, at least for several seconds, be MUCH higher (as in 2, 3, 5 or 10 times higher!) – so the missile structure will absolutely certain be overburdened and break up if it is only built strong enough to survive the ascent loads (mostly generated by the thrust).
And, due to the rocket equation, if a single-stage missile shall be capable of reaching higher velocities (and thus ranges), the mass ratio of the fueled missile to the burnt-out missile has to be lower if i can’t enhance the Isp anymore (and there are physical limits to that!).
Lower mass ratio now means automatically a less sturdy missile, because i’ll have to economize on material thickness and other strength-enhancing design features – thus the “magical 500+km” (assuming reasonable propellant combinations – LF/LH clearly is not an option in this context).
On the other hand, if i separate the warhead, i don’t have to worry anymore about designing the missile to survive the worst-case reentry scenario, but can reduce the structural integrity enough to only endure the ascent loads, thus gain a crucial lot of “dead-weight” to enhance the missile’s performance.
I hope i did satisfyingly explain my point of view this time?
(BTW Geoff: in my simulation i get max-Q at a height of 8-9km during ascent at around fight second 50 with about 13,4t of unused propellant and at 4-5km at descent around flight second 535.)
Geoff Forden | January 31, 2009
Sorry Jochen. If the missile is properly designed to reenter the atmosphere, it will right itself before it breaks apart. I get significant aerodynamic forces at about 35 km. That, or perhaps a little higher, is where it rights itself.
As far as I’m concerned, this calculation proves that China _could_ design the DF-2 to reenter with its warhead attached. Whether or not it does is not determinable from first principles but is left up to the engineering of the missile. I suspect we have different ideas as to where that might lead, but, as I say, the real answer cannot be determined by calculations alone.
Jochen Schischka | January 31, 2009
Geoff:
Let’s have a closer look at the general design characteristics of liquid-fueled missiles.
In essence, we’re talking about an empty, thin-walled tube (the tanks) with two heavy weights attached to both ends (the warhead and the engine block).
As long as the involved forces (thrust, drag) affect this structure longitudinally, it can withstand considerable loadings without problems.
But this kind of structure can only take a lot less stress in lateral direction (as during flight with high angle of attack), and be (due to the high moment of inertia: about 60% of the mass is concentrated at both ends in case of the DF-2) especially prone to torsional moments.
Especially if the structure has to be lightweight enough to reach the magical range of over ~500km (BTW this refers to REALISTIC range including propellant reserves/residuals corresponding to a theoretical range without residuals of about 600km). You simply can’t allow yourself to build the missile sturdy enough to withstand the worst-case lateral reentry – otherwise your missile will fall too short and/or your payload will be nonexistent (the latter would be a particularly counterproductive idea: aerodynamic indifference or instability will for certain make matters even worse)!
Also, let’s not forget that the increase in aerodynamic force during reentry is relatively sharp (while burnt-out liquid-fueled missiles tend to be not “very” aerodynamically stable, if at all, and will respond rather sluggish to turning moments because of most of the mass being concentrated at both ends): if my simulation delivers realistic data in that respect, then the aerodynamic forces increase by a factor of 10 about every 10 seconds from a height of approximately 30-35km down to 4-5km (thus my comment about a “force explosion” at these heights)!
And i haven’t even talked about other kinematical/physical effects to consider like precession, nutation or thermal loading that will additionally strain and/or weaken the structure…